Research ArticleJoint Inventory Pricing and Advertising Decisions withSurplus and Stockout Loss Aversions

Bing-Bing Cao1 Zhi-Ping Fan12 Hongyan Li3 and Tian-Hui You1

1Department of Information Management and Decision Sciences School of Business Administration Northeastern UniversityShenyang 110167 China2State Key Laboratory of Synthetical Automation for Process Industries Northeastern University Shenyang 110819 China3Department of Economics and Business Economics School of Business and Social Sciences Aarhus University 8210 Aarhus Denmark

Correspondence should be addressed to Zhi-Ping Fan zpfanmailneueducn

Received 29 December 2015 Accepted 26 April 2016

Academic Editor Francisco R Villatoro

Copyright copy 2016 Bing-Bing Cao et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The newsvendor models considering decision-makersrsquo behavioral factors remain a fruitful research area in operation managementfield in past decade In this paper we further extend the current literatures to look into joint inventory pricing and advertisingdecisions considering loss aversion effects under the newsvendor settingThe purpose is to explore how the loss aversions affect theoptimal policy of order quantity price and advertising effort levelWe present an integrated utilitymodel tomeasure both economicpayoff and loss aversion utility of the newsvendor where surplus loss aversion and stockout loss aversion are first separately definedand quantified Then we analyze the optimal solution conditions of the integrated model under exogenous and endogenous pricecases respectively Under exogenous price case we find that the uniquely optimal policy exists and is presented in the closed formUnder endogenous price case the optimal policy is determined under mild conditions we also provide the solutions when orderquantity factor or advertising effort level is fixed in this case In addition the sensitivity analysis shows that the loss aversions affectthe optimal decisions of order quantity price and advertising effort level in a systematic way

1 Introduction

Both business practices and academic research show that thedecisions on operations management (OM) in real worldoften deviate from the optimal solutions of the traditionalanalytical models in operations research (see [1ndash3]) This isbecause the traditional analytical models are usually based ona strong assumption of the newsvendorrsquos perfect rationalityIn reality the newsvendor usually cannot exhibit perfectrationality but can exhibit some psychological behaviorswhich cause the derivation of the decisions from the tradi-tional optimal solutions Hence it is difficult to describe thenewsvendorrsquos real decision-making process using the tradi-tional analytical models In order to bridge the gap betweentraditional models and real world situations some scholarsconduct studies on the integration of behavioral factors intothe traditional analytical models (see [4ndash6])

The behavioral factors which have been identified in theOM area include bounded rationality (see [6 7]) reference

dependence (see [8]) decision bias (see [9]) fairness concern(see [10 11]) overconfidence (see [12]) mental account-ing (see [13]) and loss aversion (see [5 14 15]) We willconcentrate on the behavioral factor of loss aversion Theloss aversion implies that the newsvendor is more sensitiveto the gains than to the absolutely commensurable lossesIt is reasonable in the real business environment becausethe decision-makers may perform different preferences fromrisk-neutrality It can be seen from the existing literatures thatthe loss aversion effect has been identified and addressed inthe OM area (see [16ndash18]) Although the existing studies havemade great contributions to the OM problem consideringbehavioral factors the study on the joint inventory pricingand advertising decisions considering the loss aversion is stilllacking In this paper we look into joint inventory pricingand advertising decisions considering the loss aversion underthe newsvendor setting Given that the retailer may havedifferent sensitivity to the loss caused by the overstock and

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 1907680 14 pageshttpdxdoiorg10115520161907680

2 Discrete Dynamics in Nature and Society

the loss caused by the out-of-stock the loss aversion isdistinguished as the surplus loss aversion and the stockoutloss aversion We focus on discovering the impact of the twoloss aversions on the joint inventory pricing and advertisingeffort level decisions Since price and advertising are themostimportant and direct marketing tools to balance demandand supply when facing a stochastic demand we consider aprice and advertising effort level dependent demand functionwith a stochastic demand factor The loss-averse newsvendorneeds to determine the optimal order quantity retail priceand the advertising effort level before the beginning of theselling season

Given the great complexity of the joint decision-makingproblem we first analyze the economic payoff and loss aver-sion utility of the newsvendor separately and then establishan integrated utility function based on Bellrsquos integratedmodel(see [19]) In integrated function the economic payoff ismea-sured by the profit function the loss aversion utility consistsof two parts a surplus loss aversion utility and a stockout lossaversion utility The loss aversion utilities are measured bya linear function Furthermore the optimal solution condi-tions on the inventory price and advertising effort level arepresented by analyzing the characteristics of the integratedutility function under the exogenous price case and the endo-genous price case

The main contribution of the study is extending theexisting loss-averse newsvendor models and joint inventoryand pricing models to be more realistic settings In detail wefirst identify and quantify two types of loss aversions thatis stockout loss aversion and surplus loss aversion in thenewsvendor environment Next based on the utility maxi-mization theory we integrate the economic payoff and theloss aversion utility and determine a total utility model Fur-thermore we provide the structural properties of the optimalsolutions to the integrated model under the exogenous pricecase and the endogenous price case Under the exogenousprice case the optimal order quantity and advertising effortlevel exist and are given in the closed formUnder the endoge-nous case the optimal order quantity price and advertisingeffort level can be determined simultaneously under mildconditions In addition we also provide the optimal solu-tions to the joint decision model when the order quantityfactor or the advertising effort level is fixed Moreover thesensitivity analysis shows the robustness of research resultsFinally we give a numerical example and show that both thestockout loss aversion and the surplus loss aversion affect theoptimal order quantity price and advertising effort level in asystematic way

The rest of the paper is organized as follows Section 2 out-lines related literatures Section 3 describes the newsvendorrsquosutility framework and constructs the utility model Section 4solves the loss-averse newsvendor problem with advertisingeffect under the exogenous price case Section 5 solves theproblem under the endogenous price case and provides theoptimal solutionswhen the order quantity factor or the adver-tising effort level is fixed Section 6 concludes with a brief dis-cussion of future research directions All proofs are providedin the technical appendix

2 Literature Review

Our study is closely related to three streams of literatures theloss-averse newsvendor models the newsvendor and pricingmodels and the advertising optimization models There is agreat amount of studies in these areas and it is difficult toexhaust the literatures For the sake of brevity we only focuson the latest and most representative studies here

Extensive behavioral experiments show that the psycho-logical behaviors play an important role in newsvendorrsquos deci-sions under uncertainty (see [6 8 19ndash24]) The loss-aversenewsvendor problemhas been a fruitful research topic in pastfew years Lee et al [5] analyze the impact of the loss aversionof the newsvendor on hisher optimal options decisionsTheyfind that a loss-averse newsvendorwill order lesswithout sup-plying options Herweg [14] extends the classical newsvendorproblem with the expectation-based loss aversion They statethe order quantity for the loss-averse newsvendor is less thanthat for the risk-neutral newsvendor Wang and Webster [17]focus on the loss aversion in classic newsvendor settings andfind that loss aversion affects the optimal inventory policyThey also find that optimal order quantity may increase inwholesale price but decrease in retail price in this situationWang [18] extends the standard newsvendor problem intothe game setting where the multiple loss-averse newsvendorsand one risk-neutral supplier are considered and showsthat the newsvendorsrsquo total order quantity increases withthe increase of loss aversion Nagarajan and Shechter [23]address the newsvendor problem based on the prospecttheory through an experimental study They maintain thatthe real order quantity deviates from the theoretical optimalorder quantity and the prospect theory cannot explain thereason of the deviation Ma et al [25] study the loss-aversenewsvendor problem with two ordering opportunities andmarket information updating and build a penalty model forthe loss-averse newsvendor to obtain the target profit Xuet al [26] focus on the optimal decision for the loss-aversenewsvendor problem under conditional value at risk Theyintroduce the legacy loss into the analysis of the loss-aversenewsvendor problem and analyze the effect of the legacy losson the optimal order quantity

In addition the newsvendor problem is also analyzedwith other psychological behaviors such as reference depen-dence decision bias bounded rationality and inequalityaversion Interested readers may please refer to the recentlypublished papers for a thorough review [2 3 6 22 27 28]Although the above studies have made great contributionsto the newsvendor model with loss aversion they seldomconsider the price and the advertising effect simultaneouslyand they do not describe clearly the impacts of the aversionsto the surplus loss and stockout loss on the optimal policy

The newsvendor and pricing problem is the most typicaltopic in the interface between the OM and marketing It isone of the extensions of the classical newsvendor model byconsidering the endogenous price and refers to the determi-nation of the order quantity and price in order to maximizethe newsvendorrsquos expected profit in an uncertain demandframework (see [29 30])The newsvendor and pricingmodelis a fundamental and significant model in OM (see [5]) and

Discrete Dynamics in Nature and Society 3

has attracted continuously extensive attention from both theacademia and the practice We refer the interested readers to[29 31ndash33] for detailed literature review

Advertising effort is another decision variable in ourstudy and works as one of the indispensable marketing toolsto increase demands Recently there are increasing interestsfrom operation researchers about the joint OM and adver-tising decisions For detailed survey of the advertising effectand its extensions we refer the interested readers to [34ndash40]However how to integrate the newsvendor problem pricingand advertising effort level with the loss aversion behaviorremains unresolved But it can be seen that several studiesattempt the integration of OM decisions advertising effortand behavioral factors For example Zhang et al [41] studythe cooperative advertising with reference price effect in avertical supply chain and find that the firm will invest morein national advertising if impact of the reference price on theoptimal policy is larger Yang et al [42] introduce the inequal-ity aversion into the research on the cooperative advertisingin a distribution channel By equivalent analysis they statethat the channel coordination can be achieved under themildconditions

Our study is a realistic extension of the aforementionedpapers but it differs from them significantly in that the psy-chological behaviors that is loss aversions of the newsven-dor and advertising effect are simultaneously taken intoaccount and that solutions for both the exogenous price caseand the endogenous price case are presented

3 The Formulations

We consider joint inventory pricing and advertising deci-sions for a loss-averse newsvendor with newsvendor settingsIn this problem apart from the traditional business objectiveof economic payoff the newsvendor is driven by the eco-nomic payoff and loss aversion Here we apply the classicweighted sum utility model proposed by Bell [19] to integratethe economic payoff and loss aversion utility it is shownbelow

Utility = economic payoff

+ psychological satisfaction(1)

where the economic payoff can be measured by a newsven-dorrsquos profit during the selling season the psychological satis-faction means also loss aversion utility and can be measuredby the psychological differences between the realized profitand the expected profit of the newsvendorWe further presentthe profit and loss aversion utility of the newsvendor in detailin the following sections

31 Profit Economic Payoff of the Newsvendor In the jointinventory pricing and advertising decisions the newsvendorplaces an order of quantity 119876 at a unit purchasing cost 119888and sells at price 119901 The inventory cannot be replenishedduring the selling season In addition the newsvendor alsodoes advertising to promote the products and the advertisingeffort level 119860 depends on the newsvendorrsquos advertisinginvestment The price the advertising effort level and the

market uncertainty can affect the demand Without loss ofgenerality consider that the demand is composed of the twoparts (see [43 44]) One is the deterministic part whichis related to the price and advertising effort level Usuallythis part is nonincreasing in the price (see [29 45 46]) andnondecreasing in the advertising effort level (see [47ndash49])The other is the stochastic part which is denoted by a randomfactor 120576 120576 isin [119872119873] Let 119891(120576) and 119865(120576) denote the probabilitydensity function and the cumulative distribution function ofthe random factor 120576 respectively and 120583 and 120590 denote themean and the standard deviation respectively The demandfunction can be additive or multiplicative (see [29 45])Since the optimal policies for the additive demand functioncan be easily adapted to the ones for the multiplicativedemand function (see [46]) and the model is tractable for theadditive demand function we use the linear additive demandfunction it is given by

119863(120576) = 119910 (119901) + 119896119860 + 120576 (2)

where 119910(119901) = 119886 minus 119887119901 119886 and 119887 denote the market size andthe price sensitivity respectively 119886 gt 0 119887 ge 0 119860 denotes theadvertising effort level 119860 ge 0 and 119896 denotes the advertisingsensitivity 119896 gt 0 We assume that the advertising cost is con-vex in the advertising effort level and the cost function of theadvertising effort level 119860 is 11986022 It is commonly used in lit-eratures (see [42 50ndash52])The parameters should be properlychosen to assure a positive demand for some range of 119901 and119860 In addition if there is unsatisfied customer demand at theend of the selling season a shortage cost 119904 incurs and if thereis excess stock by end of the selling season an salvage value Vincurs where 119901 gt 119888 gt V

Therefore the profit function of the newsvendor can bewritten as

Π =

(119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(3)

32 Utility of Loss Aversion Psychological Satisfaction Lossaversion is first recognized by Kahneman and Tversky [53]in the framework of prospect theory and it is an importantpsychological concept which receives increasing attention intheOM especially in behavioral OM in recent years (see [5 916ndash18 53ndash56]) Loss aversion implies that the newsvendor hasdifferent sensitivity to the perceived losses and the perceivedgains (see [16 17 57]) Specifically if the realized profit of thenewsvendor is less than his expectation then the newsvendormay feel extra loss beyond the actual economic lost sales Infact the newsvendor is often averse to the loss at the decisionmaking phase

Moreover loss aversion is directly related to a referencepoint denoted by Π

0 Generally reference point can be the

expected profit of the newsvendor (see [5 16 17 55]) In factthe selection of reference point is also a subjective choice ofthe newsvendor and it is often relevant to themarket environ-ment newsvendorrsquos business strategies and the competitiveposition Theoretically the reference point may be any arbi-trary value in the profit range [Πmin

Π

max] of the newsvendor

4 Discrete Dynamics in Nature and Society

Utility

ProfitΠ00 Πmax

Figure 1 Loss aversion function

where Πmin denotes the theoretical minimum of the profitof the newsvendor it may be negative and Πmax denotes thetheoretical maximum of the profit of the newsvendorΠmax

=

(119901 minus 119888)119863 minus 119860

22 Although theoretically the expected profit

may be negative the newsvendor usually does not choose anegative reference point since the businesses are always profitdriven Hence the reference point is usually determined inthe range of [0 Πmax

]Based on the above analysis the newsvendorrsquos perception

on the gain and the loss can be described in Figure 1 (see [9 1617]) It is easy to see that the newsvendor perceives loss whenthe profit is less than the reference point Π

0 and the utility

caused by the loss decreases faster than the utility caused bythe gain increases when the profit is greater than the referencepoint

In this paper to analyze the impacts of the surplus lossaversion and the stockout loss aversion without distractionsfrom the perceived gain the theoretic maximum of thenewsvendorrsquos profit is considered as reference point that isΠ0= Π

max The theoretical maximum profit can be achievedwhen the order quantity is equal to the realized demand Inthis case the newsvendor will not perceive gain and Figure 1can be transformed into Figure 2

Furthermore both overstock and out-of-stock cause lossof profit Since the newsvendor may react differently tooverstock and out-of-stock situations we distinguish the losscaused by overstock and by out-of-stock hereThe loss causedby overstock is named as surplus loss and it occurs when theorder quantity of the newsvendor is greater than the realizeddemandThe loss caused by out-of-stock is named as stockoutloss and it occurs when the order quantity of the newsvendoris lower than the realized demandThe newsvendor is usuallyaverse to both the surplus loss and the stockout loss and thedegree of the surplus loss aversion may be different from theone of the stockout loss aversion

Given the separation of the surplus loss aversion and thestockout loss aversion we apply linear loss aversion function

Utility

Profit0 Πmax

Figure 2 Loss aversion function when Π0= Π

max

to integrate the two types of loss aversion utilities (see [9 1658]) In the following we provide the specific illustration

If the realized demand is lower than the order quantitythat is 119863 lt 119876 the newsvendor experiences the surplusloss ΔΠ

119863lt119876 and the surplus loss aversion utility is related

to the difference between the reference point (ie theoreticalmaximum profit Π

0) and the real profit it is denoted by

LA (ΔΠ119863lt119876

) = minus120572ΔΠ119863lt119876

(4)

where 120572 denotes the degree of the surplus loss aversion120572 ge 0 The greater the parameter 120572 is the more averse thenewsvendor is to surplus loss If 120572 = 0 the newsvendor issurplus loss neutral Since ΔΠ

119863lt119876= Π

maxminus Π119863lt119876

= (119888 minus

V)(119876 minus 119863) the surplus loss aversion utility can be written as

LA (ΔΠ119863lt119876

) = minus120572 (119888 minus V) (119876 minus 119863) (5)

Analogously if the realized demand is greater than theorder quantity that is 119863 gt 119876 the newsvendor experiencesthe stockout loss ΔΠ

119863ge119876 The stockout loss aversion utility is

denoted by

LA (ΔΠ119863ge119876

) = minus120573ΔΠ119863ge119876

(6)

where 120573 denotes the degree of the stockout loss aversion120573 ge 0 The greater the parameter 120573 is the more sensitive thenewsvendor is to stockout loss If 120573 = 0 the newsvendor isthe stockout loss neutral Since ΔΠ

119863ge119876= Π

maxminus Π119863ge119876

=

(119901 minus 119888 + 119904)(119863 minus 119876) the stockout loss aversion utility can bewritten as

LA (ΔΠ119863ge119876

) = minus120573 (119901 minus 119888 + 119904) (119863 minus 119876) (7)

If 120572 = 120573 then the newsvendor exhibits the surplus lossaversion and stockout loss aversion with the same degree If120572 gt 120573 (120572 lt 120573) then the newsvendor is more averse to thesurplus (stockout) loss than to the stockout (surplus) lossThe

Discrete Dynamics in Nature and Society 5

Utility

Profit0

L120572

L120573

L

L

Πmax

Figure 3 Surplus and stockout loss aversion function when Π0=

Π

max

loss aversion utility curvesmay show two kinds of relations asdescribed in Figure 3

In Figure 3 119871120572represents the utility curve of the surplus

loss aversion when 120572 lt 120573 and 119871represents the utility curve

when 120572 gt 120573 119871120573and 119871

represent the utility curves of the

stockout loss aversion when 120572 lt 120573 and 120572 gt 120573 respectively

33 The Integrated Utility Model While the economic payoffand the loss aversion utility are two separate decision objec-tives they both are somehow related to and measured by theprofit of the newsvendor On the basis of (1) (3) (5) (7) and(8) a total utility of the newsvendor can be written as

119880 = Π minus LA (ΔΠ119863lt119876

) minus LA (ΔΠ119863ge119876

) (8)

Furthermore if the realized demand is lower than theorder quantity that is 119863 lt 119876 we have the utility functionthat is

119880119863lt119876

= Π119863lt119876

minus LA (ΔΠ119863lt119876

)

= (119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863)

minus 120572 (119888 minus V) (119876 minus 119863) minus119860

2

2

(9)

If the realized demand is greater than or equal to the orderquantity that is119863 ge 119876 we have the utility function that is

119880119863ge119876

= Π119863ge119876

minus LA (ΔΠ119863ge119876

)

= (119901 minus 119888)119876 minus 119904 (119863 minus 119876)

minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

(10)

Therefore the total utility of the newsvendor can bewritten as

119880 =

(119901 minus 119888)119863 minus (1 + 120572) (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(11)

For ease of exposition we induce the order quantityfactor 119911 = 119876 minus 119910(119901) minus 119896119860 into the model Thus 119863 lt

119876 is equivalent to 120576 lt 119911 and 119863 ge 119876 is equivalent

to 120576 ge 119911 Then by substituting the demand function (2)into (11) the integrated utility function can be rewrittenas

119880 =

(119901 minus 119888) [119910 (119901) + 119896119860 + 120576] minus (1 + 120572) (119888 minus V) (119911 minus 120576) minus119860

2

2

120576 lt 119911

(119901 minus 119888) [119910 (119901) + 119896119860 + 119911] minus 119904 (120576 minus 119911) minus 120573 (119901 minus 119888 + 119904) (120576 minus 119911) minus

119860

2

2

120576 ge 119911

(12)

Then our original decision making problem becomesthe expected utility maximization problem with followingobjective function that is

max 119864 [119880]

= (119901 minus 119888) [119910 (119901) + 119896119860 + 120583]

minus (1 + 120572) (119888 minus V) Λ (119911)

minus (1 + 120573) (119901 minus 119888 + 119904) 120579 (119911) minus

119860

2

2

(13)

where Λ(119911) = int119911119872(119911 minus 120576)119891(120576)119889120576 and 120579(119911) = int119873

119911(120576 minus 119911)119891(120576)119889120576

For the convenience of the description 119864[119880] can berewritten as

119864 [119880] = 120593 (119901) minus (1 + 120572) 119871 (119911) minus (1 + 120573) 119878 (119911) (14)

6 Discrete Dynamics in Nature and Society

where 119871(119911) = (119888minusV)Λ(119911) denotes surplus loss when the orderquantity is greater than the realized demand 119878(119901 119911) = (119901 minus119888 + 119904)120579(119911) denotes the stockout loss when the order quantityis lower than the realized demand 120593(119901) = (119901minus119888)[119910(119901)+119896119860+120583] minus 119860

22 denotes the expected profit function for the risk-

neutral newsvendor andΩ(119901 119911) = 120593(119901)minus119871(119911)minus119878(119911) denotesthe expected profit function for the loss-neutral newsvendor

In the following wewill address the optimal solution con-ditions under the exogenous price case and the endogenousprice case respectively

4 Solutions under the Exogenous Price

In some industries the retail price of the product is deter-mined by the competitive market and the newsvendor doesnot have pricing powerThis situation is named as exogenousprice case The exogenous price case is common in thefuriously competitivemarket In this section we look into theorder quantity and advertising effort level solutions when theretail price is exogenous

According to (13) given price 119901 we have the first- andsecond-order partial derivatives of 119864[119880] with respect to theorder quantity factor 119911 and the advertising effort level119860 thatis

120597119864 [119880]

120597119911

= minus (1 + 120572) (119888 minus V) 119865 (119911)

+ (1 + 120573) (119901 minus 119888 + 119904) [1 minus 119865 (119911)]

(15)

120597119864 [119880]

120597119860

= 119896 (119901 minus 119888) minus 119860 (16)

120597

2119864 [119880]

120597119911

2

= minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)

lt 0

(17)

120597

2119864 [119880]

120597119911120597119860

= 0(18)

120597

2119864 [119880]

120597119860

2= minus1 lt 0

(19)

120597

2119864 [119880]

120597119860120597119911

= 0(20)

Then the Hessian matrix is obtained that is

119867119860119911

= [

minus1 0

0 minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)]

(21)

Since119867119860119911

is negative definite the total utility function ofthe newsvendor (as shown in (13)) is jointly concave in orderquantity factor 119911 and advertising effort level 119860 Thereforeaccording to (15) and (16) we have Lemmas 1 and 2 below

Lemma 1 Given price 119901 there exists a unique optimaladvertising effort level 119860lowast that is

119860

lowast= 119896 (119901 minus 119888) (22)

Lemma 2 Given price 119901 there exists a unique optimal orderquantity factor 119911lowast and it satisfies

119865 (119911

lowast) =

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904) (23)

Furthermore the optimal order quantity factor 119911lowast can beobtained that is

119911

lowast= 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] (24)

Then we know that the optimal policy of the advertisingeffort level and order quantity is to order119876lowast units to sell at anexogenous price 119901 with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 and 119876lowast is specified by Lemmas 1and 2 that is

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888)

+ 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]

(25)

Since there is no direct evidence on the values of the lossaversion degree parameters we conduct a sensitivity analysisto analyze the impact of the surplus loss aversion degreeand the stockout loss aversion degree on the optimal orderquantity and the advertising effort level The general findingsare presented as the following propositions

Proposition 3 Given price 119901 the optimal advertising effortlevel119860lowast is independent of the surplus and stockout loss aversionbehaviors

Proof Since 119860lowast = 119896(119901 minus 119888) if the price 119901 is exogenous thatis the price 119901 is not related to the newsvendorrsquos surplus andstockout loss aversion behaviors then we have that the opti-mal advertising effort level 119860lowast is also unrelated to the news-vendorrsquos surplus and stockout loss aversion behaviors

Proposition 4 If 120572 = 120573 then the optimal order quantity 119876lowastis irrelevant to both parameters 120572 and 120573 In this situation theoptimal order quantity equals the loss-neutral order quantity

Proof If 120572 = 120573 by (25) we have

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119865

minus1[

119901 minus 119888 + 119904

119901 minus V + 119904]

(26)

thus the order quantity is irrelevant to the stockout and sur-plus loss aversion behaviors and the optimal order quantity isthe loss-neutral solution

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

the loss caused by the out-of-stock the loss aversion isdistinguished as the surplus loss aversion and the stockoutloss aversion We focus on discovering the impact of the twoloss aversions on the joint inventory pricing and advertisingeffort level decisions Since price and advertising are themostimportant and direct marketing tools to balance demandand supply when facing a stochastic demand we consider aprice and advertising effort level dependent demand functionwith a stochastic demand factor The loss-averse newsvendorneeds to determine the optimal order quantity retail priceand the advertising effort level before the beginning of theselling season

Given the great complexity of the joint decision-makingproblem we first analyze the economic payoff and loss aver-sion utility of the newsvendor separately and then establishan integrated utility function based on Bellrsquos integratedmodel(see [19]) In integrated function the economic payoff ismea-sured by the profit function the loss aversion utility consistsof two parts a surplus loss aversion utility and a stockout lossaversion utility The loss aversion utilities are measured bya linear function Furthermore the optimal solution condi-tions on the inventory price and advertising effort level arepresented by analyzing the characteristics of the integratedutility function under the exogenous price case and the endo-genous price case

The main contribution of the study is extending theexisting loss-averse newsvendor models and joint inventoryand pricing models to be more realistic settings In detail wefirst identify and quantify two types of loss aversions thatis stockout loss aversion and surplus loss aversion in thenewsvendor environment Next based on the utility maxi-mization theory we integrate the economic payoff and theloss aversion utility and determine a total utility model Fur-thermore we provide the structural properties of the optimalsolutions to the integrated model under the exogenous pricecase and the endogenous price case Under the exogenousprice case the optimal order quantity and advertising effortlevel exist and are given in the closed formUnder the endoge-nous case the optimal order quantity price and advertisingeffort level can be determined simultaneously under mildconditions In addition we also provide the optimal solu-tions to the joint decision model when the order quantityfactor or the advertising effort level is fixed Moreover thesensitivity analysis shows the robustness of research resultsFinally we give a numerical example and show that both thestockout loss aversion and the surplus loss aversion affect theoptimal order quantity price and advertising effort level in asystematic way

The rest of the paper is organized as follows Section 2 out-lines related literatures Section 3 describes the newsvendorrsquosutility framework and constructs the utility model Section 4solves the loss-averse newsvendor problem with advertisingeffect under the exogenous price case Section 5 solves theproblem under the endogenous price case and provides theoptimal solutionswhen the order quantity factor or the adver-tising effort level is fixed Section 6 concludes with a brief dis-cussion of future research directions All proofs are providedin the technical appendix

2 Literature Review

Our study is closely related to three streams of literatures theloss-averse newsvendor models the newsvendor and pricingmodels and the advertising optimization models There is agreat amount of studies in these areas and it is difficult toexhaust the literatures For the sake of brevity we only focuson the latest and most representative studies here

Extensive behavioral experiments show that the psycho-logical behaviors play an important role in newsvendorrsquos deci-sions under uncertainty (see [6 8 19ndash24]) The loss-aversenewsvendor problemhas been a fruitful research topic in pastfew years Lee et al [5] analyze the impact of the loss aversionof the newsvendor on hisher optimal options decisionsTheyfind that a loss-averse newsvendorwill order lesswithout sup-plying options Herweg [14] extends the classical newsvendorproblem with the expectation-based loss aversion They statethe order quantity for the loss-averse newsvendor is less thanthat for the risk-neutral newsvendor Wang and Webster [17]focus on the loss aversion in classic newsvendor settings andfind that loss aversion affects the optimal inventory policyThey also find that optimal order quantity may increase inwholesale price but decrease in retail price in this situationWang [18] extends the standard newsvendor problem intothe game setting where the multiple loss-averse newsvendorsand one risk-neutral supplier are considered and showsthat the newsvendorsrsquo total order quantity increases withthe increase of loss aversion Nagarajan and Shechter [23]address the newsvendor problem based on the prospecttheory through an experimental study They maintain thatthe real order quantity deviates from the theoretical optimalorder quantity and the prospect theory cannot explain thereason of the deviation Ma et al [25] study the loss-aversenewsvendor problem with two ordering opportunities andmarket information updating and build a penalty model forthe loss-averse newsvendor to obtain the target profit Xuet al [26] focus on the optimal decision for the loss-aversenewsvendor problem under conditional value at risk Theyintroduce the legacy loss into the analysis of the loss-aversenewsvendor problem and analyze the effect of the legacy losson the optimal order quantity

In addition the newsvendor problem is also analyzedwith other psychological behaviors such as reference depen-dence decision bias bounded rationality and inequalityaversion Interested readers may please refer to the recentlypublished papers for a thorough review [2 3 6 22 27 28]Although the above studies have made great contributionsto the newsvendor model with loss aversion they seldomconsider the price and the advertising effect simultaneouslyand they do not describe clearly the impacts of the aversionsto the surplus loss and stockout loss on the optimal policy

The newsvendor and pricing problem is the most typicaltopic in the interface between the OM and marketing It isone of the extensions of the classical newsvendor model byconsidering the endogenous price and refers to the determi-nation of the order quantity and price in order to maximizethe newsvendorrsquos expected profit in an uncertain demandframework (see [29 30])The newsvendor and pricingmodelis a fundamental and significant model in OM (see [5]) and

Discrete Dynamics in Nature and Society 3

has attracted continuously extensive attention from both theacademia and the practice We refer the interested readers to[29 31ndash33] for detailed literature review

Advertising effort is another decision variable in ourstudy and works as one of the indispensable marketing toolsto increase demands Recently there are increasing interestsfrom operation researchers about the joint OM and adver-tising decisions For detailed survey of the advertising effectand its extensions we refer the interested readers to [34ndash40]However how to integrate the newsvendor problem pricingand advertising effort level with the loss aversion behaviorremains unresolved But it can be seen that several studiesattempt the integration of OM decisions advertising effortand behavioral factors For example Zhang et al [41] studythe cooperative advertising with reference price effect in avertical supply chain and find that the firm will invest morein national advertising if impact of the reference price on theoptimal policy is larger Yang et al [42] introduce the inequal-ity aversion into the research on the cooperative advertisingin a distribution channel By equivalent analysis they statethat the channel coordination can be achieved under themildconditions

Our study is a realistic extension of the aforementionedpapers but it differs from them significantly in that the psy-chological behaviors that is loss aversions of the newsven-dor and advertising effect are simultaneously taken intoaccount and that solutions for both the exogenous price caseand the endogenous price case are presented

3 The Formulations

We consider joint inventory pricing and advertising deci-sions for a loss-averse newsvendor with newsvendor settingsIn this problem apart from the traditional business objectiveof economic payoff the newsvendor is driven by the eco-nomic payoff and loss aversion Here we apply the classicweighted sum utility model proposed by Bell [19] to integratethe economic payoff and loss aversion utility it is shownbelow

Utility = economic payoff

+ psychological satisfaction(1)

where the economic payoff can be measured by a newsven-dorrsquos profit during the selling season the psychological satis-faction means also loss aversion utility and can be measuredby the psychological differences between the realized profitand the expected profit of the newsvendorWe further presentthe profit and loss aversion utility of the newsvendor in detailin the following sections

31 Profit Economic Payoff of the Newsvendor In the jointinventory pricing and advertising decisions the newsvendorplaces an order of quantity 119876 at a unit purchasing cost 119888and sells at price 119901 The inventory cannot be replenishedduring the selling season In addition the newsvendor alsodoes advertising to promote the products and the advertisingeffort level 119860 depends on the newsvendorrsquos advertisinginvestment The price the advertising effort level and the

market uncertainty can affect the demand Without loss ofgenerality consider that the demand is composed of the twoparts (see [43 44]) One is the deterministic part whichis related to the price and advertising effort level Usuallythis part is nonincreasing in the price (see [29 45 46]) andnondecreasing in the advertising effort level (see [47ndash49])The other is the stochastic part which is denoted by a randomfactor 120576 120576 isin [119872119873] Let 119891(120576) and 119865(120576) denote the probabilitydensity function and the cumulative distribution function ofthe random factor 120576 respectively and 120583 and 120590 denote themean and the standard deviation respectively The demandfunction can be additive or multiplicative (see [29 45])Since the optimal policies for the additive demand functioncan be easily adapted to the ones for the multiplicativedemand function (see [46]) and the model is tractable for theadditive demand function we use the linear additive demandfunction it is given by

119863(120576) = 119910 (119901) + 119896119860 + 120576 (2)

where 119910(119901) = 119886 minus 119887119901 119886 and 119887 denote the market size andthe price sensitivity respectively 119886 gt 0 119887 ge 0 119860 denotes theadvertising effort level 119860 ge 0 and 119896 denotes the advertisingsensitivity 119896 gt 0 We assume that the advertising cost is con-vex in the advertising effort level and the cost function of theadvertising effort level 119860 is 11986022 It is commonly used in lit-eratures (see [42 50ndash52])The parameters should be properlychosen to assure a positive demand for some range of 119901 and119860 In addition if there is unsatisfied customer demand at theend of the selling season a shortage cost 119904 incurs and if thereis excess stock by end of the selling season an salvage value Vincurs where 119901 gt 119888 gt V

Therefore the profit function of the newsvendor can bewritten as

Π =

(119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(3)

32 Utility of Loss Aversion Psychological Satisfaction Lossaversion is first recognized by Kahneman and Tversky [53]in the framework of prospect theory and it is an importantpsychological concept which receives increasing attention intheOM especially in behavioral OM in recent years (see [5 916ndash18 53ndash56]) Loss aversion implies that the newsvendor hasdifferent sensitivity to the perceived losses and the perceivedgains (see [16 17 57]) Specifically if the realized profit of thenewsvendor is less than his expectation then the newsvendormay feel extra loss beyond the actual economic lost sales Infact the newsvendor is often averse to the loss at the decisionmaking phase

Moreover loss aversion is directly related to a referencepoint denoted by Π

0 Generally reference point can be the

expected profit of the newsvendor (see [5 16 17 55]) In factthe selection of reference point is also a subjective choice ofthe newsvendor and it is often relevant to themarket environ-ment newsvendorrsquos business strategies and the competitiveposition Theoretically the reference point may be any arbi-trary value in the profit range [Πmin

Π

max] of the newsvendor

4 Discrete Dynamics in Nature and Society

Utility

ProfitΠ00 Πmax

Figure 1 Loss aversion function

where Πmin denotes the theoretical minimum of the profitof the newsvendor it may be negative and Πmax denotes thetheoretical maximum of the profit of the newsvendorΠmax

=

(119901 minus 119888)119863 minus 119860

22 Although theoretically the expected profit

may be negative the newsvendor usually does not choose anegative reference point since the businesses are always profitdriven Hence the reference point is usually determined inthe range of [0 Πmax

]Based on the above analysis the newsvendorrsquos perception

on the gain and the loss can be described in Figure 1 (see [9 1617]) It is easy to see that the newsvendor perceives loss whenthe profit is less than the reference point Π

0 and the utility

caused by the loss decreases faster than the utility caused bythe gain increases when the profit is greater than the referencepoint

In this paper to analyze the impacts of the surplus lossaversion and the stockout loss aversion without distractionsfrom the perceived gain the theoretic maximum of thenewsvendorrsquos profit is considered as reference point that isΠ0= Π

max The theoretical maximum profit can be achievedwhen the order quantity is equal to the realized demand Inthis case the newsvendor will not perceive gain and Figure 1can be transformed into Figure 2

Furthermore both overstock and out-of-stock cause lossof profit Since the newsvendor may react differently tooverstock and out-of-stock situations we distinguish the losscaused by overstock and by out-of-stock hereThe loss causedby overstock is named as surplus loss and it occurs when theorder quantity of the newsvendor is greater than the realizeddemandThe loss caused by out-of-stock is named as stockoutloss and it occurs when the order quantity of the newsvendoris lower than the realized demandThe newsvendor is usuallyaverse to both the surplus loss and the stockout loss and thedegree of the surplus loss aversion may be different from theone of the stockout loss aversion

Given the separation of the surplus loss aversion and thestockout loss aversion we apply linear loss aversion function

Utility

Profit0 Πmax

Figure 2 Loss aversion function when Π0= Π

max

to integrate the two types of loss aversion utilities (see [9 1658]) In the following we provide the specific illustration

If the realized demand is lower than the order quantitythat is 119863 lt 119876 the newsvendor experiences the surplusloss ΔΠ

119863lt119876 and the surplus loss aversion utility is related

to the difference between the reference point (ie theoreticalmaximum profit Π

0) and the real profit it is denoted by

LA (ΔΠ119863lt119876

) = minus120572ΔΠ119863lt119876

(4)

where 120572 denotes the degree of the surplus loss aversion120572 ge 0 The greater the parameter 120572 is the more averse thenewsvendor is to surplus loss If 120572 = 0 the newsvendor issurplus loss neutral Since ΔΠ

119863lt119876= Π

maxminus Π119863lt119876

= (119888 minus

V)(119876 minus 119863) the surplus loss aversion utility can be written as

LA (ΔΠ119863lt119876

) = minus120572 (119888 minus V) (119876 minus 119863) (5)

Analogously if the realized demand is greater than theorder quantity that is 119863 gt 119876 the newsvendor experiencesthe stockout loss ΔΠ

119863ge119876 The stockout loss aversion utility is

denoted by

LA (ΔΠ119863ge119876

) = minus120573ΔΠ119863ge119876

(6)

where 120573 denotes the degree of the stockout loss aversion120573 ge 0 The greater the parameter 120573 is the more sensitive thenewsvendor is to stockout loss If 120573 = 0 the newsvendor isthe stockout loss neutral Since ΔΠ

119863ge119876= Π

maxminus Π119863ge119876

=

(119901 minus 119888 + 119904)(119863 minus 119876) the stockout loss aversion utility can bewritten as

LA (ΔΠ119863ge119876

) = minus120573 (119901 minus 119888 + 119904) (119863 minus 119876) (7)

If 120572 = 120573 then the newsvendor exhibits the surplus lossaversion and stockout loss aversion with the same degree If120572 gt 120573 (120572 lt 120573) then the newsvendor is more averse to thesurplus (stockout) loss than to the stockout (surplus) lossThe

Discrete Dynamics in Nature and Society 5

Utility

Profit0

L120572

L120573

L

L

Πmax

Figure 3 Surplus and stockout loss aversion function when Π0=

Π

max

loss aversion utility curvesmay show two kinds of relations asdescribed in Figure 3

In Figure 3 119871120572represents the utility curve of the surplus

loss aversion when 120572 lt 120573 and 119871represents the utility curve

when 120572 gt 120573 119871120573and 119871

represent the utility curves of the

stockout loss aversion when 120572 lt 120573 and 120572 gt 120573 respectively

33 The Integrated Utility Model While the economic payoffand the loss aversion utility are two separate decision objec-tives they both are somehow related to and measured by theprofit of the newsvendor On the basis of (1) (3) (5) (7) and(8) a total utility of the newsvendor can be written as

119880 = Π minus LA (ΔΠ119863lt119876

) minus LA (ΔΠ119863ge119876

) (8)

Furthermore if the realized demand is lower than theorder quantity that is 119863 lt 119876 we have the utility functionthat is

119880119863lt119876

= Π119863lt119876

minus LA (ΔΠ119863lt119876

)

= (119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863)

minus 120572 (119888 minus V) (119876 minus 119863) minus119860

2

2

(9)

If the realized demand is greater than or equal to the orderquantity that is119863 ge 119876 we have the utility function that is

119880119863ge119876

= Π119863ge119876

minus LA (ΔΠ119863ge119876

)

= (119901 minus 119888)119876 minus 119904 (119863 minus 119876)

minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

(10)

Therefore the total utility of the newsvendor can bewritten as

119880 =

(119901 minus 119888)119863 minus (1 + 120572) (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(11)

For ease of exposition we induce the order quantityfactor 119911 = 119876 minus 119910(119901) minus 119896119860 into the model Thus 119863 lt

119876 is equivalent to 120576 lt 119911 and 119863 ge 119876 is equivalent

to 120576 ge 119911 Then by substituting the demand function (2)into (11) the integrated utility function can be rewrittenas

119880 =

(119901 minus 119888) [119910 (119901) + 119896119860 + 120576] minus (1 + 120572) (119888 minus V) (119911 minus 120576) minus119860

2

2

120576 lt 119911

(119901 minus 119888) [119910 (119901) + 119896119860 + 119911] minus 119904 (120576 minus 119911) minus 120573 (119901 minus 119888 + 119904) (120576 minus 119911) minus

119860

2

2

120576 ge 119911

(12)

Then our original decision making problem becomesthe expected utility maximization problem with followingobjective function that is

max 119864 [119880]

= (119901 minus 119888) [119910 (119901) + 119896119860 + 120583]

minus (1 + 120572) (119888 minus V) Λ (119911)

minus (1 + 120573) (119901 minus 119888 + 119904) 120579 (119911) minus

119860

2

2

(13)

where Λ(119911) = int119911119872(119911 minus 120576)119891(120576)119889120576 and 120579(119911) = int119873

119911(120576 minus 119911)119891(120576)119889120576

For the convenience of the description 119864[119880] can berewritten as

119864 [119880] = 120593 (119901) minus (1 + 120572) 119871 (119911) minus (1 + 120573) 119878 (119911) (14)

6 Discrete Dynamics in Nature and Society

where 119871(119911) = (119888minusV)Λ(119911) denotes surplus loss when the orderquantity is greater than the realized demand 119878(119901 119911) = (119901 minus119888 + 119904)120579(119911) denotes the stockout loss when the order quantityis lower than the realized demand 120593(119901) = (119901minus119888)[119910(119901)+119896119860+120583] minus 119860

22 denotes the expected profit function for the risk-

neutral newsvendor andΩ(119901 119911) = 120593(119901)minus119871(119911)minus119878(119911) denotesthe expected profit function for the loss-neutral newsvendor

In the following wewill address the optimal solution con-ditions under the exogenous price case and the endogenousprice case respectively

4 Solutions under the Exogenous Price

In some industries the retail price of the product is deter-mined by the competitive market and the newsvendor doesnot have pricing powerThis situation is named as exogenousprice case The exogenous price case is common in thefuriously competitivemarket In this section we look into theorder quantity and advertising effort level solutions when theretail price is exogenous

According to (13) given price 119901 we have the first- andsecond-order partial derivatives of 119864[119880] with respect to theorder quantity factor 119911 and the advertising effort level119860 thatis

120597119864 [119880]

120597119911

= minus (1 + 120572) (119888 minus V) 119865 (119911)

+ (1 + 120573) (119901 minus 119888 + 119904) [1 minus 119865 (119911)]

(15)

120597119864 [119880]

120597119860

= 119896 (119901 minus 119888) minus 119860 (16)

120597

2119864 [119880]

120597119911

2

= minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)

lt 0

(17)

120597

2119864 [119880]

120597119911120597119860

= 0(18)

120597

2119864 [119880]

120597119860

2= minus1 lt 0

(19)

120597

2119864 [119880]

120597119860120597119911

= 0(20)

Then the Hessian matrix is obtained that is

119867119860119911

= [

minus1 0

0 minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)]

(21)

Since119867119860119911

is negative definite the total utility function ofthe newsvendor (as shown in (13)) is jointly concave in orderquantity factor 119911 and advertising effort level 119860 Thereforeaccording to (15) and (16) we have Lemmas 1 and 2 below

Lemma 1 Given price 119901 there exists a unique optimaladvertising effort level 119860lowast that is

119860

lowast= 119896 (119901 minus 119888) (22)

Lemma 2 Given price 119901 there exists a unique optimal orderquantity factor 119911lowast and it satisfies

119865 (119911

lowast) =

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904) (23)

Furthermore the optimal order quantity factor 119911lowast can beobtained that is

119911

lowast= 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] (24)

Then we know that the optimal policy of the advertisingeffort level and order quantity is to order119876lowast units to sell at anexogenous price 119901 with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 and 119876lowast is specified by Lemmas 1and 2 that is

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888)

+ 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]

(25)

Since there is no direct evidence on the values of the lossaversion degree parameters we conduct a sensitivity analysisto analyze the impact of the surplus loss aversion degreeand the stockout loss aversion degree on the optimal orderquantity and the advertising effort level The general findingsare presented as the following propositions

Proposition 3 Given price 119901 the optimal advertising effortlevel119860lowast is independent of the surplus and stockout loss aversionbehaviors

Proof Since 119860lowast = 119896(119901 minus 119888) if the price 119901 is exogenous thatis the price 119901 is not related to the newsvendorrsquos surplus andstockout loss aversion behaviors then we have that the opti-mal advertising effort level 119860lowast is also unrelated to the news-vendorrsquos surplus and stockout loss aversion behaviors

Proposition 4 If 120572 = 120573 then the optimal order quantity 119876lowastis irrelevant to both parameters 120572 and 120573 In this situation theoptimal order quantity equals the loss-neutral order quantity

Proof If 120572 = 120573 by (25) we have

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119865

minus1[

119901 minus 119888 + 119904

119901 minus V + 119904]

(26)

thus the order quantity is irrelevant to the stockout and sur-plus loss aversion behaviors and the optimal order quantity isthe loss-neutral solution

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

has attracted continuously extensive attention from both theacademia and the practice We refer the interested readers to[29 31ndash33] for detailed literature review

Advertising effort is another decision variable in ourstudy and works as one of the indispensable marketing toolsto increase demands Recently there are increasing interestsfrom operation researchers about the joint OM and adver-tising decisions For detailed survey of the advertising effectand its extensions we refer the interested readers to [34ndash40]However how to integrate the newsvendor problem pricingand advertising effort level with the loss aversion behaviorremains unresolved But it can be seen that several studiesattempt the integration of OM decisions advertising effortand behavioral factors For example Zhang et al [41] studythe cooperative advertising with reference price effect in avertical supply chain and find that the firm will invest morein national advertising if impact of the reference price on theoptimal policy is larger Yang et al [42] introduce the inequal-ity aversion into the research on the cooperative advertisingin a distribution channel By equivalent analysis they statethat the channel coordination can be achieved under themildconditions

Our study is a realistic extension of the aforementionedpapers but it differs from them significantly in that the psy-chological behaviors that is loss aversions of the newsven-dor and advertising effect are simultaneously taken intoaccount and that solutions for both the exogenous price caseand the endogenous price case are presented

3 The Formulations

We consider joint inventory pricing and advertising deci-sions for a loss-averse newsvendor with newsvendor settingsIn this problem apart from the traditional business objectiveof economic payoff the newsvendor is driven by the eco-nomic payoff and loss aversion Here we apply the classicweighted sum utility model proposed by Bell [19] to integratethe economic payoff and loss aversion utility it is shownbelow

Utility = economic payoff

+ psychological satisfaction(1)

where the economic payoff can be measured by a newsven-dorrsquos profit during the selling season the psychological satis-faction means also loss aversion utility and can be measuredby the psychological differences between the realized profitand the expected profit of the newsvendorWe further presentthe profit and loss aversion utility of the newsvendor in detailin the following sections

31 Profit Economic Payoff of the Newsvendor In the jointinventory pricing and advertising decisions the newsvendorplaces an order of quantity 119876 at a unit purchasing cost 119888and sells at price 119901 The inventory cannot be replenishedduring the selling season In addition the newsvendor alsodoes advertising to promote the products and the advertisingeffort level 119860 depends on the newsvendorrsquos advertisinginvestment The price the advertising effort level and the

market uncertainty can affect the demand Without loss ofgenerality consider that the demand is composed of the twoparts (see [43 44]) One is the deterministic part whichis related to the price and advertising effort level Usuallythis part is nonincreasing in the price (see [29 45 46]) andnondecreasing in the advertising effort level (see [47ndash49])The other is the stochastic part which is denoted by a randomfactor 120576 120576 isin [119872119873] Let 119891(120576) and 119865(120576) denote the probabilitydensity function and the cumulative distribution function ofthe random factor 120576 respectively and 120583 and 120590 denote themean and the standard deviation respectively The demandfunction can be additive or multiplicative (see [29 45])Since the optimal policies for the additive demand functioncan be easily adapted to the ones for the multiplicativedemand function (see [46]) and the model is tractable for theadditive demand function we use the linear additive demandfunction it is given by

119863(120576) = 119910 (119901) + 119896119860 + 120576 (2)

where 119910(119901) = 119886 minus 119887119901 119886 and 119887 denote the market size andthe price sensitivity respectively 119886 gt 0 119887 ge 0 119860 denotes theadvertising effort level 119860 ge 0 and 119896 denotes the advertisingsensitivity 119896 gt 0 We assume that the advertising cost is con-vex in the advertising effort level and the cost function of theadvertising effort level 119860 is 11986022 It is commonly used in lit-eratures (see [42 50ndash52])The parameters should be properlychosen to assure a positive demand for some range of 119901 and119860 In addition if there is unsatisfied customer demand at theend of the selling season a shortage cost 119904 incurs and if thereis excess stock by end of the selling season an salvage value Vincurs where 119901 gt 119888 gt V

Therefore the profit function of the newsvendor can bewritten as

Π =

(119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(3)

32 Utility of Loss Aversion Psychological Satisfaction Lossaversion is first recognized by Kahneman and Tversky [53]in the framework of prospect theory and it is an importantpsychological concept which receives increasing attention intheOM especially in behavioral OM in recent years (see [5 916ndash18 53ndash56]) Loss aversion implies that the newsvendor hasdifferent sensitivity to the perceived losses and the perceivedgains (see [16 17 57]) Specifically if the realized profit of thenewsvendor is less than his expectation then the newsvendormay feel extra loss beyond the actual economic lost sales Infact the newsvendor is often averse to the loss at the decisionmaking phase

Moreover loss aversion is directly related to a referencepoint denoted by Π

0 Generally reference point can be the

expected profit of the newsvendor (see [5 16 17 55]) In factthe selection of reference point is also a subjective choice ofthe newsvendor and it is often relevant to themarket environ-ment newsvendorrsquos business strategies and the competitiveposition Theoretically the reference point may be any arbi-trary value in the profit range [Πmin

Π

max] of the newsvendor

4 Discrete Dynamics in Nature and Society

Utility

ProfitΠ00 Πmax

Figure 1 Loss aversion function

where Πmin denotes the theoretical minimum of the profitof the newsvendor it may be negative and Πmax denotes thetheoretical maximum of the profit of the newsvendorΠmax

=

(119901 minus 119888)119863 minus 119860

22 Although theoretically the expected profit

may be negative the newsvendor usually does not choose anegative reference point since the businesses are always profitdriven Hence the reference point is usually determined inthe range of [0 Πmax

]Based on the above analysis the newsvendorrsquos perception

on the gain and the loss can be described in Figure 1 (see [9 1617]) It is easy to see that the newsvendor perceives loss whenthe profit is less than the reference point Π

0 and the utility

caused by the loss decreases faster than the utility caused bythe gain increases when the profit is greater than the referencepoint

In this paper to analyze the impacts of the surplus lossaversion and the stockout loss aversion without distractionsfrom the perceived gain the theoretic maximum of thenewsvendorrsquos profit is considered as reference point that isΠ0= Π

max The theoretical maximum profit can be achievedwhen the order quantity is equal to the realized demand Inthis case the newsvendor will not perceive gain and Figure 1can be transformed into Figure 2

Furthermore both overstock and out-of-stock cause lossof profit Since the newsvendor may react differently tooverstock and out-of-stock situations we distinguish the losscaused by overstock and by out-of-stock hereThe loss causedby overstock is named as surplus loss and it occurs when theorder quantity of the newsvendor is greater than the realizeddemandThe loss caused by out-of-stock is named as stockoutloss and it occurs when the order quantity of the newsvendoris lower than the realized demandThe newsvendor is usuallyaverse to both the surplus loss and the stockout loss and thedegree of the surplus loss aversion may be different from theone of the stockout loss aversion

Given the separation of the surplus loss aversion and thestockout loss aversion we apply linear loss aversion function

Utility

Profit0 Πmax

Figure 2 Loss aversion function when Π0= Π

max

to integrate the two types of loss aversion utilities (see [9 1658]) In the following we provide the specific illustration

If the realized demand is lower than the order quantitythat is 119863 lt 119876 the newsvendor experiences the surplusloss ΔΠ

119863lt119876 and the surplus loss aversion utility is related

to the difference between the reference point (ie theoreticalmaximum profit Π

0) and the real profit it is denoted by

LA (ΔΠ119863lt119876

) = minus120572ΔΠ119863lt119876

(4)

where 120572 denotes the degree of the surplus loss aversion120572 ge 0 The greater the parameter 120572 is the more averse thenewsvendor is to surplus loss If 120572 = 0 the newsvendor issurplus loss neutral Since ΔΠ

119863lt119876= Π

maxminus Π119863lt119876

= (119888 minus

V)(119876 minus 119863) the surplus loss aversion utility can be written as

LA (ΔΠ119863lt119876

) = minus120572 (119888 minus V) (119876 minus 119863) (5)

Analogously if the realized demand is greater than theorder quantity that is 119863 gt 119876 the newsvendor experiencesthe stockout loss ΔΠ

119863ge119876 The stockout loss aversion utility is

denoted by

LA (ΔΠ119863ge119876

) = minus120573ΔΠ119863ge119876

(6)

where 120573 denotes the degree of the stockout loss aversion120573 ge 0 The greater the parameter 120573 is the more sensitive thenewsvendor is to stockout loss If 120573 = 0 the newsvendor isthe stockout loss neutral Since ΔΠ

119863ge119876= Π

maxminus Π119863ge119876

=

(119901 minus 119888 + 119904)(119863 minus 119876) the stockout loss aversion utility can bewritten as

LA (ΔΠ119863ge119876

) = minus120573 (119901 minus 119888 + 119904) (119863 minus 119876) (7)

If 120572 = 120573 then the newsvendor exhibits the surplus lossaversion and stockout loss aversion with the same degree If120572 gt 120573 (120572 lt 120573) then the newsvendor is more averse to thesurplus (stockout) loss than to the stockout (surplus) lossThe

Discrete Dynamics in Nature and Society 5

Utility

Profit0

L120572

L120573

L

L

Πmax

Figure 3 Surplus and stockout loss aversion function when Π0=

Π

max

loss aversion utility curvesmay show two kinds of relations asdescribed in Figure 3

In Figure 3 119871120572represents the utility curve of the surplus

loss aversion when 120572 lt 120573 and 119871represents the utility curve

when 120572 gt 120573 119871120573and 119871

represent the utility curves of the

stockout loss aversion when 120572 lt 120573 and 120572 gt 120573 respectively

33 The Integrated Utility Model While the economic payoffand the loss aversion utility are two separate decision objec-tives they both are somehow related to and measured by theprofit of the newsvendor On the basis of (1) (3) (5) (7) and(8) a total utility of the newsvendor can be written as

119880 = Π minus LA (ΔΠ119863lt119876

) minus LA (ΔΠ119863ge119876

) (8)

Furthermore if the realized demand is lower than theorder quantity that is 119863 lt 119876 we have the utility functionthat is

119880119863lt119876

= Π119863lt119876

minus LA (ΔΠ119863lt119876

)

= (119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863)

minus 120572 (119888 minus V) (119876 minus 119863) minus119860

2

2

(9)

If the realized demand is greater than or equal to the orderquantity that is119863 ge 119876 we have the utility function that is

119880119863ge119876

= Π119863ge119876

minus LA (ΔΠ119863ge119876

)

= (119901 minus 119888)119876 minus 119904 (119863 minus 119876)

minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

(10)

Therefore the total utility of the newsvendor can bewritten as

119880 =

(119901 minus 119888)119863 minus (1 + 120572) (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(11)

For ease of exposition we induce the order quantityfactor 119911 = 119876 minus 119910(119901) minus 119896119860 into the model Thus 119863 lt

119876 is equivalent to 120576 lt 119911 and 119863 ge 119876 is equivalent

to 120576 ge 119911 Then by substituting the demand function (2)into (11) the integrated utility function can be rewrittenas

119880 =

(119901 minus 119888) [119910 (119901) + 119896119860 + 120576] minus (1 + 120572) (119888 minus V) (119911 minus 120576) minus119860

2

2

120576 lt 119911

(119901 minus 119888) [119910 (119901) + 119896119860 + 119911] minus 119904 (120576 minus 119911) minus 120573 (119901 minus 119888 + 119904) (120576 minus 119911) minus

119860

2

2

120576 ge 119911

(12)

Then our original decision making problem becomesthe expected utility maximization problem with followingobjective function that is

max 119864 [119880]

= (119901 minus 119888) [119910 (119901) + 119896119860 + 120583]

minus (1 + 120572) (119888 minus V) Λ (119911)

minus (1 + 120573) (119901 minus 119888 + 119904) 120579 (119911) minus

119860

2

2

(13)

where Λ(119911) = int119911119872(119911 minus 120576)119891(120576)119889120576 and 120579(119911) = int119873

119911(120576 minus 119911)119891(120576)119889120576

For the convenience of the description 119864[119880] can berewritten as

119864 [119880] = 120593 (119901) minus (1 + 120572) 119871 (119911) minus (1 + 120573) 119878 (119911) (14)

6 Discrete Dynamics in Nature and Society

where 119871(119911) = (119888minusV)Λ(119911) denotes surplus loss when the orderquantity is greater than the realized demand 119878(119901 119911) = (119901 minus119888 + 119904)120579(119911) denotes the stockout loss when the order quantityis lower than the realized demand 120593(119901) = (119901minus119888)[119910(119901)+119896119860+120583] minus 119860

22 denotes the expected profit function for the risk-

neutral newsvendor andΩ(119901 119911) = 120593(119901)minus119871(119911)minus119878(119911) denotesthe expected profit function for the loss-neutral newsvendor

In the following wewill address the optimal solution con-ditions under the exogenous price case and the endogenousprice case respectively

4 Solutions under the Exogenous Price

In some industries the retail price of the product is deter-mined by the competitive market and the newsvendor doesnot have pricing powerThis situation is named as exogenousprice case The exogenous price case is common in thefuriously competitivemarket In this section we look into theorder quantity and advertising effort level solutions when theretail price is exogenous

According to (13) given price 119901 we have the first- andsecond-order partial derivatives of 119864[119880] with respect to theorder quantity factor 119911 and the advertising effort level119860 thatis

120597119864 [119880]

120597119911

= minus (1 + 120572) (119888 minus V) 119865 (119911)

+ (1 + 120573) (119901 minus 119888 + 119904) [1 minus 119865 (119911)]

(15)

120597119864 [119880]

120597119860

= 119896 (119901 minus 119888) minus 119860 (16)

120597

2119864 [119880]

120597119911

2

= minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)

lt 0

(17)

120597

2119864 [119880]

120597119911120597119860

= 0(18)

120597

2119864 [119880]

120597119860

2= minus1 lt 0

(19)

120597

2119864 [119880]

120597119860120597119911

= 0(20)

Then the Hessian matrix is obtained that is

119867119860119911

= [

minus1 0

0 minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)]

(21)

Since119867119860119911

is negative definite the total utility function ofthe newsvendor (as shown in (13)) is jointly concave in orderquantity factor 119911 and advertising effort level 119860 Thereforeaccording to (15) and (16) we have Lemmas 1 and 2 below

Lemma 1 Given price 119901 there exists a unique optimaladvertising effort level 119860lowast that is

119860

lowast= 119896 (119901 minus 119888) (22)

Lemma 2 Given price 119901 there exists a unique optimal orderquantity factor 119911lowast and it satisfies

119865 (119911

lowast) =

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904) (23)

Furthermore the optimal order quantity factor 119911lowast can beobtained that is

119911

lowast= 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] (24)

Then we know that the optimal policy of the advertisingeffort level and order quantity is to order119876lowast units to sell at anexogenous price 119901 with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 and 119876lowast is specified by Lemmas 1and 2 that is

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888)

+ 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]

(25)

Since there is no direct evidence on the values of the lossaversion degree parameters we conduct a sensitivity analysisto analyze the impact of the surplus loss aversion degreeand the stockout loss aversion degree on the optimal orderquantity and the advertising effort level The general findingsare presented as the following propositions

Proposition 3 Given price 119901 the optimal advertising effortlevel119860lowast is independent of the surplus and stockout loss aversionbehaviors

Proof Since 119860lowast = 119896(119901 minus 119888) if the price 119901 is exogenous thatis the price 119901 is not related to the newsvendorrsquos surplus andstockout loss aversion behaviors then we have that the opti-mal advertising effort level 119860lowast is also unrelated to the news-vendorrsquos surplus and stockout loss aversion behaviors

Proposition 4 If 120572 = 120573 then the optimal order quantity 119876lowastis irrelevant to both parameters 120572 and 120573 In this situation theoptimal order quantity equals the loss-neutral order quantity

Proof If 120572 = 120573 by (25) we have

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119865

minus1[

119901 minus 119888 + 119904

119901 minus V + 119904]

(26)

thus the order quantity is irrelevant to the stockout and sur-plus loss aversion behaviors and the optimal order quantity isthe loss-neutral solution

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Utility

ProfitΠ00 Πmax

Figure 1 Loss aversion function

where Πmin denotes the theoretical minimum of the profitof the newsvendor it may be negative and Πmax denotes thetheoretical maximum of the profit of the newsvendorΠmax

=

(119901 minus 119888)119863 minus 119860

22 Although theoretically the expected profit

may be negative the newsvendor usually does not choose anegative reference point since the businesses are always profitdriven Hence the reference point is usually determined inthe range of [0 Πmax

]Based on the above analysis the newsvendorrsquos perception

on the gain and the loss can be described in Figure 1 (see [9 1617]) It is easy to see that the newsvendor perceives loss whenthe profit is less than the reference point Π

0 and the utility

caused by the loss decreases faster than the utility caused bythe gain increases when the profit is greater than the referencepoint

In this paper to analyze the impacts of the surplus lossaversion and the stockout loss aversion without distractionsfrom the perceived gain the theoretic maximum of thenewsvendorrsquos profit is considered as reference point that isΠ0= Π

max The theoretical maximum profit can be achievedwhen the order quantity is equal to the realized demand Inthis case the newsvendor will not perceive gain and Figure 1can be transformed into Figure 2

Furthermore both overstock and out-of-stock cause lossof profit Since the newsvendor may react differently tooverstock and out-of-stock situations we distinguish the losscaused by overstock and by out-of-stock hereThe loss causedby overstock is named as surplus loss and it occurs when theorder quantity of the newsvendor is greater than the realizeddemandThe loss caused by out-of-stock is named as stockoutloss and it occurs when the order quantity of the newsvendoris lower than the realized demandThe newsvendor is usuallyaverse to both the surplus loss and the stockout loss and thedegree of the surplus loss aversion may be different from theone of the stockout loss aversion

Given the separation of the surplus loss aversion and thestockout loss aversion we apply linear loss aversion function

Utility

Profit0 Πmax

Figure 2 Loss aversion function when Π0= Π

max

to integrate the two types of loss aversion utilities (see [9 1658]) In the following we provide the specific illustration

If the realized demand is lower than the order quantitythat is 119863 lt 119876 the newsvendor experiences the surplusloss ΔΠ

119863lt119876 and the surplus loss aversion utility is related

to the difference between the reference point (ie theoreticalmaximum profit Π

0) and the real profit it is denoted by

LA (ΔΠ119863lt119876

) = minus120572ΔΠ119863lt119876

(4)

where 120572 denotes the degree of the surplus loss aversion120572 ge 0 The greater the parameter 120572 is the more averse thenewsvendor is to surplus loss If 120572 = 0 the newsvendor issurplus loss neutral Since ΔΠ

119863lt119876= Π

maxminus Π119863lt119876

= (119888 minus

V)(119876 minus 119863) the surplus loss aversion utility can be written as

LA (ΔΠ119863lt119876

) = minus120572 (119888 minus V) (119876 minus 119863) (5)

Analogously if the realized demand is greater than theorder quantity that is 119863 gt 119876 the newsvendor experiencesthe stockout loss ΔΠ

119863ge119876 The stockout loss aversion utility is

denoted by

LA (ΔΠ119863ge119876

) = minus120573ΔΠ119863ge119876

(6)

where 120573 denotes the degree of the stockout loss aversion120573 ge 0 The greater the parameter 120573 is the more sensitive thenewsvendor is to stockout loss If 120573 = 0 the newsvendor isthe stockout loss neutral Since ΔΠ

119863ge119876= Π

maxminus Π119863ge119876

=

(119901 minus 119888 + 119904)(119863 minus 119876) the stockout loss aversion utility can bewritten as

LA (ΔΠ119863ge119876

) = minus120573 (119901 minus 119888 + 119904) (119863 minus 119876) (7)

If 120572 = 120573 then the newsvendor exhibits the surplus lossaversion and stockout loss aversion with the same degree If120572 gt 120573 (120572 lt 120573) then the newsvendor is more averse to thesurplus (stockout) loss than to the stockout (surplus) lossThe

Discrete Dynamics in Nature and Society 5

Utility

Profit0

L120572

L120573

L

L

Πmax

Figure 3 Surplus and stockout loss aversion function when Π0=

Π

max

loss aversion utility curvesmay show two kinds of relations asdescribed in Figure 3

In Figure 3 119871120572represents the utility curve of the surplus

loss aversion when 120572 lt 120573 and 119871represents the utility curve

when 120572 gt 120573 119871120573and 119871

represent the utility curves of the

stockout loss aversion when 120572 lt 120573 and 120572 gt 120573 respectively

33 The Integrated Utility Model While the economic payoffand the loss aversion utility are two separate decision objec-tives they both are somehow related to and measured by theprofit of the newsvendor On the basis of (1) (3) (5) (7) and(8) a total utility of the newsvendor can be written as

119880 = Π minus LA (ΔΠ119863lt119876

) minus LA (ΔΠ119863ge119876

) (8)

Furthermore if the realized demand is lower than theorder quantity that is 119863 lt 119876 we have the utility functionthat is

119880119863lt119876

= Π119863lt119876

minus LA (ΔΠ119863lt119876

)

= (119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863)

minus 120572 (119888 minus V) (119876 minus 119863) minus119860

2

2

(9)

If the realized demand is greater than or equal to the orderquantity that is119863 ge 119876 we have the utility function that is

119880119863ge119876

= Π119863ge119876

minus LA (ΔΠ119863ge119876

)

= (119901 minus 119888)119876 minus 119904 (119863 minus 119876)

minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

(10)

Therefore the total utility of the newsvendor can bewritten as

119880 =

(119901 minus 119888)119863 minus (1 + 120572) (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(11)

For ease of exposition we induce the order quantityfactor 119911 = 119876 minus 119910(119901) minus 119896119860 into the model Thus 119863 lt

119876 is equivalent to 120576 lt 119911 and 119863 ge 119876 is equivalent

to 120576 ge 119911 Then by substituting the demand function (2)into (11) the integrated utility function can be rewrittenas

119880 =

(119901 minus 119888) [119910 (119901) + 119896119860 + 120576] minus (1 + 120572) (119888 minus V) (119911 minus 120576) minus119860

2

2

120576 lt 119911

(119901 minus 119888) [119910 (119901) + 119896119860 + 119911] minus 119904 (120576 minus 119911) minus 120573 (119901 minus 119888 + 119904) (120576 minus 119911) minus

119860

2

2

120576 ge 119911

(12)

Then our original decision making problem becomesthe expected utility maximization problem with followingobjective function that is

max 119864 [119880]

= (119901 minus 119888) [119910 (119901) + 119896119860 + 120583]

minus (1 + 120572) (119888 minus V) Λ (119911)

minus (1 + 120573) (119901 minus 119888 + 119904) 120579 (119911) minus

119860

2

2

(13)

where Λ(119911) = int119911119872(119911 minus 120576)119891(120576)119889120576 and 120579(119911) = int119873

119911(120576 minus 119911)119891(120576)119889120576

For the convenience of the description 119864[119880] can berewritten as

119864 [119880] = 120593 (119901) minus (1 + 120572) 119871 (119911) minus (1 + 120573) 119878 (119911) (14)

6 Discrete Dynamics in Nature and Society

where 119871(119911) = (119888minusV)Λ(119911) denotes surplus loss when the orderquantity is greater than the realized demand 119878(119901 119911) = (119901 minus119888 + 119904)120579(119911) denotes the stockout loss when the order quantityis lower than the realized demand 120593(119901) = (119901minus119888)[119910(119901)+119896119860+120583] minus 119860

22 denotes the expected profit function for the risk-

neutral newsvendor andΩ(119901 119911) = 120593(119901)minus119871(119911)minus119878(119911) denotesthe expected profit function for the loss-neutral newsvendor

In the following wewill address the optimal solution con-ditions under the exogenous price case and the endogenousprice case respectively

4 Solutions under the Exogenous Price

In some industries the retail price of the product is deter-mined by the competitive market and the newsvendor doesnot have pricing powerThis situation is named as exogenousprice case The exogenous price case is common in thefuriously competitivemarket In this section we look into theorder quantity and advertising effort level solutions when theretail price is exogenous

According to (13) given price 119901 we have the first- andsecond-order partial derivatives of 119864[119880] with respect to theorder quantity factor 119911 and the advertising effort level119860 thatis

120597119864 [119880]

120597119911

= minus (1 + 120572) (119888 minus V) 119865 (119911)

+ (1 + 120573) (119901 minus 119888 + 119904) [1 minus 119865 (119911)]

(15)

120597119864 [119880]

120597119860

= 119896 (119901 minus 119888) minus 119860 (16)

120597

2119864 [119880]

120597119911

2

= minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)

lt 0

(17)

120597

2119864 [119880]

120597119911120597119860

= 0(18)

120597

2119864 [119880]

120597119860

2= minus1 lt 0

(19)

120597

2119864 [119880]

120597119860120597119911

= 0(20)

Then the Hessian matrix is obtained that is

119867119860119911

= [

minus1 0

0 minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)]

(21)

Since119867119860119911

is negative definite the total utility function ofthe newsvendor (as shown in (13)) is jointly concave in orderquantity factor 119911 and advertising effort level 119860 Thereforeaccording to (15) and (16) we have Lemmas 1 and 2 below

Lemma 1 Given price 119901 there exists a unique optimaladvertising effort level 119860lowast that is

119860

lowast= 119896 (119901 minus 119888) (22)

Lemma 2 Given price 119901 there exists a unique optimal orderquantity factor 119911lowast and it satisfies

119865 (119911

lowast) =

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904) (23)

Furthermore the optimal order quantity factor 119911lowast can beobtained that is

119911

lowast= 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] (24)

Then we know that the optimal policy of the advertisingeffort level and order quantity is to order119876lowast units to sell at anexogenous price 119901 with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 and 119876lowast is specified by Lemmas 1and 2 that is

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888)

+ 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]

(25)

Since there is no direct evidence on the values of the lossaversion degree parameters we conduct a sensitivity analysisto analyze the impact of the surplus loss aversion degreeand the stockout loss aversion degree on the optimal orderquantity and the advertising effort level The general findingsare presented as the following propositions

Proposition 3 Given price 119901 the optimal advertising effortlevel119860lowast is independent of the surplus and stockout loss aversionbehaviors

Proof Since 119860lowast = 119896(119901 minus 119888) if the price 119901 is exogenous thatis the price 119901 is not related to the newsvendorrsquos surplus andstockout loss aversion behaviors then we have that the opti-mal advertising effort level 119860lowast is also unrelated to the news-vendorrsquos surplus and stockout loss aversion behaviors

Proposition 4 If 120572 = 120573 then the optimal order quantity 119876lowastis irrelevant to both parameters 120572 and 120573 In this situation theoptimal order quantity equals the loss-neutral order quantity

Proof If 120572 = 120573 by (25) we have

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119865

minus1[

119901 minus 119888 + 119904

119901 minus V + 119904]

(26)

thus the order quantity is irrelevant to the stockout and sur-plus loss aversion behaviors and the optimal order quantity isthe loss-neutral solution

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Utility

Profit0

L120572

L120573

L

L

Πmax

Figure 3 Surplus and stockout loss aversion function when Π0=

Π

max

loss aversion utility curvesmay show two kinds of relations asdescribed in Figure 3

In Figure 3 119871120572represents the utility curve of the surplus

loss aversion when 120572 lt 120573 and 119871represents the utility curve

when 120572 gt 120573 119871120573and 119871

represent the utility curves of the

stockout loss aversion when 120572 lt 120573 and 120572 gt 120573 respectively

33 The Integrated Utility Model While the economic payoffand the loss aversion utility are two separate decision objec-tives they both are somehow related to and measured by theprofit of the newsvendor On the basis of (1) (3) (5) (7) and(8) a total utility of the newsvendor can be written as

119880 = Π minus LA (ΔΠ119863lt119876

) minus LA (ΔΠ119863ge119876

) (8)

Furthermore if the realized demand is lower than theorder quantity that is 119863 lt 119876 we have the utility functionthat is

119880119863lt119876

= Π119863lt119876

minus LA (ΔΠ119863lt119876

)

= (119901 minus 119888)119863 minus (119888 minus V) (119876 minus 119863)

minus 120572 (119888 minus V) (119876 minus 119863) minus119860

2

2

(9)

If the realized demand is greater than or equal to the orderquantity that is119863 ge 119876 we have the utility function that is

119880119863ge119876

= Π119863ge119876

minus LA (ΔΠ119863ge119876

)

= (119901 minus 119888)119876 minus 119904 (119863 minus 119876)

minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

(10)

Therefore the total utility of the newsvendor can bewritten as

119880 =

(119901 minus 119888)119863 minus (1 + 120572) (119888 minus V) (119876 minus 119863) minus119860

2

2

119863 lt 119876

(119901 minus 119888)119876 minus 119904 (119863 minus 119876) minus 120573 (119901 minus 119888 + 119904) (119863 minus 119876) minus

119860

2

2

119863 ge 119876

(11)

For ease of exposition we induce the order quantityfactor 119911 = 119876 minus 119910(119901) minus 119896119860 into the model Thus 119863 lt

119876 is equivalent to 120576 lt 119911 and 119863 ge 119876 is equivalent

to 120576 ge 119911 Then by substituting the demand function (2)into (11) the integrated utility function can be rewrittenas

119880 =

(119901 minus 119888) [119910 (119901) + 119896119860 + 120576] minus (1 + 120572) (119888 minus V) (119911 minus 120576) minus119860

2

2

120576 lt 119911

(119901 minus 119888) [119910 (119901) + 119896119860 + 119911] minus 119904 (120576 minus 119911) minus 120573 (119901 minus 119888 + 119904) (120576 minus 119911) minus

119860

2

2

120576 ge 119911

(12)

Then our original decision making problem becomesthe expected utility maximization problem with followingobjective function that is

max 119864 [119880]

= (119901 minus 119888) [119910 (119901) + 119896119860 + 120583]

minus (1 + 120572) (119888 minus V) Λ (119911)

minus (1 + 120573) (119901 minus 119888 + 119904) 120579 (119911) minus

119860

2

2

(13)

where Λ(119911) = int119911119872(119911 minus 120576)119891(120576)119889120576 and 120579(119911) = int119873

119911(120576 minus 119911)119891(120576)119889120576

For the convenience of the description 119864[119880] can berewritten as

119864 [119880] = 120593 (119901) minus (1 + 120572) 119871 (119911) minus (1 + 120573) 119878 (119911) (14)

6 Discrete Dynamics in Nature and Society

where 119871(119911) = (119888minusV)Λ(119911) denotes surplus loss when the orderquantity is greater than the realized demand 119878(119901 119911) = (119901 minus119888 + 119904)120579(119911) denotes the stockout loss when the order quantityis lower than the realized demand 120593(119901) = (119901minus119888)[119910(119901)+119896119860+120583] minus 119860

22 denotes the expected profit function for the risk-

neutral newsvendor andΩ(119901 119911) = 120593(119901)minus119871(119911)minus119878(119911) denotesthe expected profit function for the loss-neutral newsvendor

In the following wewill address the optimal solution con-ditions under the exogenous price case and the endogenousprice case respectively

4 Solutions under the Exogenous Price

In some industries the retail price of the product is deter-mined by the competitive market and the newsvendor doesnot have pricing powerThis situation is named as exogenousprice case The exogenous price case is common in thefuriously competitivemarket In this section we look into theorder quantity and advertising effort level solutions when theretail price is exogenous

According to (13) given price 119901 we have the first- andsecond-order partial derivatives of 119864[119880] with respect to theorder quantity factor 119911 and the advertising effort level119860 thatis

120597119864 [119880]

120597119911

= minus (1 + 120572) (119888 minus V) 119865 (119911)

+ (1 + 120573) (119901 minus 119888 + 119904) [1 minus 119865 (119911)]

(15)

120597119864 [119880]

120597119860

= 119896 (119901 minus 119888) minus 119860 (16)

120597

2119864 [119880]

120597119911

2

= minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)

lt 0

(17)

120597

2119864 [119880]

120597119911120597119860

= 0(18)

120597

2119864 [119880]

120597119860

2= minus1 lt 0

(19)

120597

2119864 [119880]

120597119860120597119911

= 0(20)

Then the Hessian matrix is obtained that is

119867119860119911

= [

minus1 0

0 minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)]

(21)

Since119867119860119911

is negative definite the total utility function ofthe newsvendor (as shown in (13)) is jointly concave in orderquantity factor 119911 and advertising effort level 119860 Thereforeaccording to (15) and (16) we have Lemmas 1 and 2 below

Lemma 1 Given price 119901 there exists a unique optimaladvertising effort level 119860lowast that is

119860

lowast= 119896 (119901 minus 119888) (22)

Lemma 2 Given price 119901 there exists a unique optimal orderquantity factor 119911lowast and it satisfies

119865 (119911

lowast) =

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904) (23)

Furthermore the optimal order quantity factor 119911lowast can beobtained that is

119911

lowast= 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] (24)

Then we know that the optimal policy of the advertisingeffort level and order quantity is to order119876lowast units to sell at anexogenous price 119901 with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 and 119876lowast is specified by Lemmas 1and 2 that is

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888)

+ 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]

(25)

Since there is no direct evidence on the values of the lossaversion degree parameters we conduct a sensitivity analysisto analyze the impact of the surplus loss aversion degreeand the stockout loss aversion degree on the optimal orderquantity and the advertising effort level The general findingsare presented as the following propositions

Proposition 3 Given price 119901 the optimal advertising effortlevel119860lowast is independent of the surplus and stockout loss aversionbehaviors

Proof Since 119860lowast = 119896(119901 minus 119888) if the price 119901 is exogenous thatis the price 119901 is not related to the newsvendorrsquos surplus andstockout loss aversion behaviors then we have that the opti-mal advertising effort level 119860lowast is also unrelated to the news-vendorrsquos surplus and stockout loss aversion behaviors

Proposition 4 If 120572 = 120573 then the optimal order quantity 119876lowastis irrelevant to both parameters 120572 and 120573 In this situation theoptimal order quantity equals the loss-neutral order quantity

Proof If 120572 = 120573 by (25) we have

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119865

minus1[

119901 minus 119888 + 119904

119901 minus V + 119904]

(26)

thus the order quantity is irrelevant to the stockout and sur-plus loss aversion behaviors and the optimal order quantity isthe loss-neutral solution

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

where 119871(119911) = (119888minusV)Λ(119911) denotes surplus loss when the orderquantity is greater than the realized demand 119878(119901 119911) = (119901 minus119888 + 119904)120579(119911) denotes the stockout loss when the order quantityis lower than the realized demand 120593(119901) = (119901minus119888)[119910(119901)+119896119860+120583] minus 119860

22 denotes the expected profit function for the risk-

neutral newsvendor andΩ(119901 119911) = 120593(119901)minus119871(119911)minus119878(119911) denotesthe expected profit function for the loss-neutral newsvendor

In the following wewill address the optimal solution con-ditions under the exogenous price case and the endogenousprice case respectively

4 Solutions under the Exogenous Price

In some industries the retail price of the product is deter-mined by the competitive market and the newsvendor doesnot have pricing powerThis situation is named as exogenousprice case The exogenous price case is common in thefuriously competitivemarket In this section we look into theorder quantity and advertising effort level solutions when theretail price is exogenous

According to (13) given price 119901 we have the first- andsecond-order partial derivatives of 119864[119880] with respect to theorder quantity factor 119911 and the advertising effort level119860 thatis

120597119864 [119880]

120597119911

= minus (1 + 120572) (119888 minus V) 119865 (119911)

+ (1 + 120573) (119901 minus 119888 + 119904) [1 minus 119865 (119911)]

(15)

120597119864 [119880]

120597119860

= 119896 (119901 minus 119888) minus 119860 (16)

120597

2119864 [119880]

120597119911

2

= minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)

lt 0

(17)

120597

2119864 [119880]

120597119911120597119860

= 0(18)

120597

2119864 [119880]

120597119860

2= minus1 lt 0

(19)

120597

2119864 [119880]

120597119860120597119911

= 0(20)

Then the Hessian matrix is obtained that is

119867119860119911

= [

minus1 0

0 minus [(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] 119891 (119911)]

(21)

Since119867119860119911

is negative definite the total utility function ofthe newsvendor (as shown in (13)) is jointly concave in orderquantity factor 119911 and advertising effort level 119860 Thereforeaccording to (15) and (16) we have Lemmas 1 and 2 below

Lemma 1 Given price 119901 there exists a unique optimaladvertising effort level 119860lowast that is

119860

lowast= 119896 (119901 minus 119888) (22)

Lemma 2 Given price 119901 there exists a unique optimal orderquantity factor 119911lowast and it satisfies

119865 (119911

lowast) =

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904) (23)

Furthermore the optimal order quantity factor 119911lowast can beobtained that is

119911

lowast= 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)] (24)

Then we know that the optimal policy of the advertisingeffort level and order quantity is to order119876lowast units to sell at anexogenous price 119901 with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 and 119876lowast is specified by Lemmas 1and 2 that is

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888)

+ 119865

minus1[

(1 + 120573) (119901 minus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]

(25)

Since there is no direct evidence on the values of the lossaversion degree parameters we conduct a sensitivity analysisto analyze the impact of the surplus loss aversion degreeand the stockout loss aversion degree on the optimal orderquantity and the advertising effort level The general findingsare presented as the following propositions

Proposition 3 Given price 119901 the optimal advertising effortlevel119860lowast is independent of the surplus and stockout loss aversionbehaviors

Proof Since 119860lowast = 119896(119901 minus 119888) if the price 119901 is exogenous thatis the price 119901 is not related to the newsvendorrsquos surplus andstockout loss aversion behaviors then we have that the opti-mal advertising effort level 119860lowast is also unrelated to the news-vendorrsquos surplus and stockout loss aversion behaviors

Proposition 4 If 120572 = 120573 then the optimal order quantity 119876lowastis irrelevant to both parameters 120572 and 120573 In this situation theoptimal order quantity equals the loss-neutral order quantity

Proof If 120572 = 120573 by (25) we have

119876

lowast= 119910 (119901) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119865

minus1[

119901 minus 119888 + 119904

119901 minus V + 119904]

(26)

thus the order quantity is irrelevant to the stockout and sur-plus loss aversion behaviors and the optimal order quantity isthe loss-neutral solution

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Actually because the stockout loss aversion and thesurplus loss aversion require actions on the order quantityin the opposite directions therefore when the newsvendorexhibits equal aversions to the stockout loss and the surplusloss the two loss aversions still affect the decisions but theyoffset each other In the end it shows the optimal optionwhich is the same with the loss-neutral decision

Proposition 5 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast decreases with parameter120572

Proof According to (25) we know

120597119876

lowast

120597120572

=

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(27)

Obviously 120597119876lowast120597120572 lt 0 and the conclusion holds

Proposition 6 If parameter 120572 is not equal to parameter 120573then the optimal order quantity 119876lowast increases with parameter120573

Proof According to (25) we know

120597119876

lowast

120597120573

=

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911lowast)

(28)

Obviously 120597119876lowast120597120573 gt 0 and the conclusion holds

It can be seen from (25) and above propositions thatthe optimal order quantity for the loss-averse newsvendordeviates systematically from the one for the loss-neutralnewsvendor as shown in (26) Specifically if the newsvendoris sensitive to the stockout loss but not sensitive to the surplusloss that is the effect of the surplus loss aversion on the deci-sion can be neglected then the loss-averse newsvendor willorder more than the loss-neutral one and the order quantityincreases with the stockout loss aversion degree Similarlyif the newsvendor is sensitive to the surplus loss but notsensitive to the stockout loss that is the effect of the stockoutloss aversion on the decision can be neglected then the loss-averse newsvendor will order less than the loss-neutral oneand the order quantity decreases with the surplus loss aver-sion degree If the newsvendor is sensitive to both the stock-out loss and surplus loss the order quantity fluctuates aroundthe loss-neutral one for the different degrees of the stockoutloss aversion and the surplus loss aversion

In order to show the scale of the effects of the two lossaversions on the optimal policy (since the advertising effortlevel is not related to the loss aversions here we only analyzethe scale of the effects of the loss aversions on the orderquantity) an illustrative example is shown in Figure 4 Inthe example the parameters are considered to be as follows

005

115

2

005

115

2300

310

320

330

340

350

Inventory Q0

Qlowast

120573120572

Figure 4 The effects of the loss aversions on the optimal orderquantity

119886 = 200 119887 = 15 119888 = 18 V = 5 119904 = 20119872 = 100 119873 = 200and the exogenous price 119901 = 30 and the order quantityfactor 120576 follows the uniform distribution in [119872119873] that is120576 sim 119880[100 200] Since the degree of the loss aversion of thenewsvendor is usually not greater than 2 [17 18] we considerthat the loss aversion parameters are in a reasonable range of0 le 120572 le 2 and 0 le 120573 le 2 and the tendency of the effect is fullyreflected in this range

We can see from Figure 4 that the order quantity of theloss-averse newsvendor 119876lowast deviates from the one of loss-neutral newsvendor 1198760 in systematic way as described inPropositions 3ndash6 When loss aversion is higher (less) thanthe one of the stockout loss aversion that is the newsvendoris more sensitive to the surplus loss aversion (stockout lossaversion) the order quantity 119876lowast is lower (higher) than theloss-neutral one 1198760 When surplus loss aversion parameteris equal to the stockout loss aversion parameter the orderquantity119876lowast is equal to the loss-neutral one1198760 and it is shownby the intersection line Obviously the effect of the surplusloss aversion on the order quantity is opposite to the one ofstockout loss aversion We can also see that compared withthe nonclassified loss aversion the classified loss aversionsthat is surplus loss aversion and stockout loss aversion showthe clearer and more specific effects on the order quantity

5 Solutions under the Endogenous Price

In some situations the newsvendor may have the pricingpower and it is usually called endogenous price case Theendogenous price case is common in the monopoly marketIn this section we investigate the optimal solutions of theorder quantity price and advertising effort level in differentsituations

According to (13) the first- and second-order partialderivatives of 119864[119880] with respect to the price 119901 are obtainedas follows

120597119864 [119880]

120597119901

= 119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) (29)

120597

2119864 [119880]

120597119901

2= minus2119887 lt 0 (30)

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

On the basis of the above analysis we have Lemma 7which follows directly from (29) and (30)

Lemma 7 For fixed advertising effort level 119860 and orderquantity factor 119911 the optimal price is determined uniquely as afunction of 119860 and 119911

119901

lowast=

1

2119887

[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911)] (31)

In (31) since 119901 gt 119888 120573 gt max[(119896119860+119886+120583minus119887119888)120579(119911)minus1 0]It can be seen from Lemmas 2 and 7 that the price is

related directly to the stockout loss aversion parameter andis related indirectly to the surplus loss aversion parameterthrough order quantity factor 119911 It is necessary to pointout that the effect of the price on the order quantity is thesame as the one of the surplus loss aversion on the orderquantity Specifically if the price is higher (lower) then theorder quantity is less (greater) analogously if the surplus lossaversion parameter is higher (lower) then the order quantityis less (greater) too

In the following we consider solving the model forthe joint order quantity price and advertising effort leveldecisions in two cases one is for 2119887 = 119896

2 and the other isfor 2119887 = 1198962 The specific solving processes are given below

If 2119887 = 119896

2 then we substitute 119860lowast = 119860(119901) and 119901lowast = 119901(119911)into (13) and then the optimization problemmax

119860119911119901119864[119880(119860

119911 119901)] is converted into an optimization problemwith a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we have Theorem 8 For the convenience of the descriptionlet119883(sdot) = 119891(sdot)[1 minus 119865(sdot)] and it is a hazard rate function

Theorem 8 When 2119887 = 119896

2 if 119865(120576) satisfies condition (a) 2119887minus119896

2gt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 or (b) 2119887 minus 1198962 lt 0 and

2119883(119911)

2+ 119889119883(119911)119889119911 lt 0 then 119911lowast is the largest 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfiescondition (c) 2119887 minus 1198962 gt 0 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 +120583 minus 119887119888 + (2119887 minus 119896

2)119904 minus (1 + 120573)(120583 minus 119872) gt 0 or condition (d)

2119887 minus 119896

2lt 0 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 and 119886 + 120583 minus 119887119888 + (2119887 minus

119896

2)119904 minus (1+120573)(120583minus119872) gt 0 then 119911lowast is the unique 119911 in the region

[119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore if 2119887 = 119896

2 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast withadvertising effort level119860lowast where 119911lowast is determined accordingtoTheorem 8119860lowast is specified by Lemma 1 and 119901lowast is specifiedby Lemma 7

Analogously if 2119887 = 1198962 we haveTheorem 9

Theorem 9 If 2119887 = 1198962 then the policy is to order 119876lowast units tosell at the price 119901lowast with the advertising effort level 119860lowast where119860

lowast is specified by Lemma 1 119911lowast is specified by Lemma 2 and 119901lowastis bound price

Proof See Appendix

Therefore if 2119887 = 1198962 then the optimal policy is to order119876

lowast (119876lowast = 119910(119901lowast) + 119896119860 + 119911lowast) units to sell at the price 119901lowast with

advertising effort level119860lowast where 119901lowast is determined accordingtoTheorem 9119860lowast is specified by Lemma 1 and 119911lowast is specifiedby Lemma 2

According to Theorems 8 and 9 the optimal solution of119901

lowast 119860lowast and 119911lowast can be determined Then since 119876 = 119910(119901) +

119896119860+119911 and 119910(119901) = 119886minus119887119901 the optimal order quantity119876lowast withendogenous price is

119876

lowast= 119910 (119901

lowast) + 119896119860

lowast+ 119911

lowast

= 119886 minus 119887119901

lowast+ 119896

2(119901

lowastminus 119888)

+ 119865

minus1[

(1 + 120573) (119901

lowastminus 119888 + 119904)

(1 + 120572) (119888 minus V) + (1 + 120573) (119901lowast minus 119888 + 119904)]

(32)

In reality the newsvendormay decide three decision vari-ables simultaneously or may make decisions successively forsome internal or external factors For example the newsven-dor may have a fixed advertising investment for financial rea-sons or the newsvendor may have unchangeable preferenceprice for the competition reason or the newsvendormay havea fixed order quantity factor such as the mean of 120576 If simul-taneously the newsvendorrsquos optimal policy can be obtainedby Theorem 8 or Theorem 9 If successively the news-vendorrsquos optimal policy can be obtained by the followinganalysis

Here we provide the analysis of the optimal solutionswhen one of the three decision variables is fixed for somereasons and conduct it in the following three conditions

(1) If the price 119901 is fixed then the optimization problemmax119860119911119901

119864[119880(119860 119911 119901)] can be converted into max119860119911119864[119880(119860

119911 119901)] and the optimal solution of the advertising effort level119860 and order quantity factor 119911 follows the one when the priceis exogenous discussed in Section 4

(2) If the order quantity factor 119911 is fixed then the opti-mization problem max

119860119911119901119864[119880(119860 119911 119901)] can be converted

into max119860119901119864[119880(119860 119901)] and the optimal solution of the

advertising effort level119860 and retail price 119901 can be determinedby the following discussion

According to (13) the second-order mixed partial deriva-tive of expected utility function is obtained below

120597

2119864 [119880]

120597119901120597119860

= 119896

120597

2119864 [119880]

120597119860120597119901

= 119896

(33)

Then according to (19) (30) and (33) we have theHessianmatrix with respect to the price119901 and the advertisingeffort level 119860 that is

119867119860119901=

[

[

[

[

120597

2119864 [119880]

120597119860

2

120597

2119864 [119880]

120597119860120597119901

120597

2119864 [119880]

120597119901120597119860

120597

2119864 [119880]

120597119901

2

]

]

]

]

= [

minus1 119896

119896 minus2119887

] (34)

For (34) we know that |1205972119864[119880]1205971198602| = minus1 lt 0 and|119867119860119901| = 2119887 minus 119896

2 In the following we conduct the analysis

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

under the scenarios 2119887 gt 119896

2 2119887 = 119896

2 and 2119887 lt 119896

2respectively

(i) If 2119887 gt 1198962 then the Hessian matrix is negative definiteTherefore if 2119887 gt 1198962 the constructed model is concave withrespect to 119860 and 119901 and there exists uniquely joint optimalsolution of 119860 and 119901 to maximize the newsvendorrsquos utility By(16) and (29) the optimal solution of 119860 and 119901 is determinedthat is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)] (35)

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(36)

In (35) and (36) since 119860lowast ge 0 and 119901lowast gt 119888 we have 0 le 120573 lt(119886 + 120583 minus 119887119888)120579(119911) minus 1

Therefore when 2119887 gt 1198962 if 120573 satisfies 0 le 120573 lt (119886 + 120583 minus119887119888)120579(119911) minus 1 then the optimal solution of 119860 and 119901 exists asshown in (35) and (36) if not there is no feasible solution

Remark 10 If the order quantity factor 119911 is fixed and 2119887 gt 1198962then there exists uniquely joint optimal solution of 119860 and 119901

Proposition 11 Given 119911

lowast the advertising effort level 119860lowastdecreases with parameter 120573 but it is not related to parameter120572

Proof According to (35) we know

120597119860

lowast

120597120573

= minus

119896120579 (119911)

2119887 minus 119896

2 (37)

Apparently 120597119860lowast120597120573 le 0 and the advertising effort level is notrelated to parameter 120572

Proposition 12 Given 119911lowast the endogenous price 119901lowast decreaseswith parameter 120573 but it is not related to parameter 120572

Proof According to (36) we know

120597119901

lowast

120597120573

= minus

120579 (119911)

2119887 minus 119896

2 (38)

Apparently 120597119901lowast120597120573 le 0 and the price is not related toparameter 120572

Proposition 13 Given 119911lowast If 119887 ge 1198962 then the order quantity119876

lowast increases with parameter 120573 and if 11989622 lt 119887 lt 1198962 thenthe order quantity 119876lowast decreases with parameter 120573 The orderquantity 119876lowast is not related to parameter 120572

Proof According to (35) (36) and 119876lowast = 119910(119901) + 119896119860lowast + 119911lowast =119886 minus 119887119901 + 119896

2(119901 minus 119888) + 119911 we know

120597119876

lowast

120597120573

=

(119887 minus 119896

2) 120579 (119911)

2119887 minus 119896

2

(39)

Since 2119887 gt 119896

2 if 119887 ge 119896

2 120597119876lowast120597120573 ge 0 if 11989622 lt 119887 lt

119896

2 120597119876lowast120597120573 lt 0 and the order quantity is not related toparameter 120572

(ii) If 2119887 = 1198962 by (16) and (29) we have

119886 minus 2119887119901 + 119896119860 + 120583 + 119887119888 minus (1 + 120573) 120579 (119911) = 0

119860

lowast= 119896 (119901 minus 119888)

(40)

Then the optimal price can be arbitrary one in reasonablescale Furthermore the optimal advertising effort level can bedetermined that is 119860lowast = 119896(119901

lowastminus 119888) In this situation the

loss aversions cannot affect the optimal policy of price andadvertising effort level

(iii) If 2119887 lt 1198962 by (16) and (29) the optimal solutions of119860 and 119901 can be determined respectively that is

119860

lowast=

119896

2119887 minus 119896

2[119886 + 120583 minus 119887119888 minus (1 + 120573) 120579 (119911)]

119901

lowast=

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573)

2119887 minus 119896

2120579 (119911)

(41)

In (41) we have 120573 gt max[(119886 + 120583 minus 119887119888)120579(119911) minus 1 0] because of119860

lowastge 0 119901lowast gt 119888 and 120573 ge 0Therefore when 2119887 lt 1198962 if 120573 satisfies 120573 gt max[(119886 + 120583 minus

119887119888)120579(119911) minus 1 0] then the optimal solution of 119860 and 119901 existsas shown in (41) if not there is no feasible solution In thissituation the advertising effort level 119860lowast and the endogenousprice 119901lowast are related to the stockout loss aversion but not tothe surplus loss aversion

(3) If the advertising effort level 119860 is fixed then theoptimization problem max

119860119911119901119864[119880(119860 119911 119901)] is converted

into max119911119901119864[119880(119911 119901)] and then the optimal solution of the

order quantity factor 119911 and price 119901 can be determined by thefollowing discussion

According to Lemmas 2 and 7 we have119865(119911lowast) = (1+120573)(119901minus119888 + 119904)((1 + 120572)(119888 minus V) + (1 + 120573)(119901 minus 119888 + 119904)) and 119901lowast = 119901(119911) =(12119887)[119896119860 + 119886 + 120583 + 119887119888 minus (1 + 120573)120579(119911)] By substituting 119901lowast =119901(119911) into 119865(119911lowast) optimization problemmax

119911119901119864[119880(119911 119901)] can

be converted into the optimization problem with a singlevariable 119911 that is max

119911119864119880[119860(119901(119911)) 119911 119901(119911)] Furthermore

we haveTheorem 14

Theorem 14 If 119865(120576) satisfies the condition 2119883(119911)

2+

119889119883(119911)119889119911 gt 0 then 119911lowast is the largest 119911 in the region [119872119873]that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0 If 119865(120576) satisfies thecondition 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 and 119886 + 120583 minus 119887119888 + 119896119860 +2119887119904 minus (1 + 120573)(120583 minus 119872) gt 0 then 119911lowast is the unique 119911 in theregion [119872119873] that satisfies 120597119864[119880(119911 119901(119911))]120597119911 = 0

Proof See Appendix

Therefore we know that if the advertising effort level 119860is fixed the optimal policy is to order 119876lowast (119876lowast = 119910(119901

lowast) +

119896119860 + 119911

lowast) units to sell at price 119901lowast where 119911lowast is determinedaccording to Theorem 14 and 119901lowast is specified by Lemma 7 Inthis situation the order quantity 119876lowast and price 119901lowast are relatedto the stockout loss aversion and surplus loss aversion In thefollowing we provide the sensitivity analysis of the effect ofthe loss aversions on the order quantity and the price whenthe unique solution exists

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

Proposition 15 Given 119911lowast the price 119901lowast decreases with param-eter 120573 but it is not related to parameter 120572

Proof Since the order quantity factor 119911lowast is fixed accordingto Lemma 7 we have that the price 119901lowast is not related to theparameter 120572 and have

120597119901

lowast

120597120573

=

minus120579 (119911)

2119887

(42)

Apparently 120597119901lowast120597120573 le 0

Proposition 16 Given 119901

lowast the order quantity factor 119911lowastincreases with parameter 120573 but decreases with parameter 120572

Proof Since the price 119901lowast is fixed according to Lemma 2 wehave that

120597119911

lowast

120597120573

=

(1 + 120572) (119888 minus V) (119901 minus 119888 + 119904)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

120597119911

lowast

120597120572

= minus

(1 + 120573) (119901 minus 119888 + 119904) (119888 minus V)

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]2 119891 (119911)

(43)

Apparently 120597119911lowast120597120573 ge 0 and 120597119911lowast120597120572 le 0

6 Managerial Insights

According to the above analysis we can give the managerialinsights that may be helpful to the decision-makers inpractical joint inventory pricing and advertising decisions Ifthe newsvendor exhibits the loss aversions that is the surplusand stockout loss aversions then hisher decisions on theprice the order quantity and the advertising effort level aredifferent from the ones of the traditional newsvendor withoutconsidering the loss aversions

In the case of the exogenous price (ie the price ofthe product depends on the market) the order quantity ofthe loss-averse newsvendor deviates from the one of thetraditional newsvendor Several interesting insights can beobtained as follows

(a) If the newsvendor is concerned more about the sur-plus loss aversion then hisher order quantity shouldbe less than the one of the traditional newsvendor

(b) If the newsvendor is concernedmore about the stock-out loss aversion then hisher order quantity shouldbe more than the one of the traditional newsvendor

(c) Particularly if the newsvendorrsquos perceptions to thesurplus loss aversion and the stockout loss aversionare the same then hisher order quantity should bethe same as the one of the traditional newsvendor

In the case of the endogenous price (ie the price ofthe product depends on the newsvendor) the loss-aversenewsvendorrsquos decisions on the price order quantity and

advertising effort level will be affected by the price elastic-ity and the advertising sensitive degree Several interestinginsights can be obtained below

(a) If the relation between the price elasticity and theadvertising sensitive degree meets the certain con-dition (see Theorem 9) then the price determinedby the loss-averse newsvendor should be the boundprice

(b) If the demand is considered to be deterministic (iethe demand factor 120576 takes a deterministic value) thenthe loss-averse newsvendorrsquos decisions on the priceand advertising effort level will depend on the priceelasticity and advertising sensitive degree

(c) If the newsvendor determines in advance hisheradvertising effort level then hisher decisions on theprice and order quantity factor will depend on thesurplus and stockout loss aversion degrees

7 Conclusions

In this paper we extend the classical newsvendor andpricing model to integrate advertising decisions and to takethe stockout loss aversion and surplus loss aversion intoaccount We apply a linear utility function to depict thestockout loss aversion and surplus loss aversion and constructtotal utility function of the newsvendor by integrating theloss aversion utility function and the profit function Thenewsvendorrsquos expected utility is maximized by optimiz-ing the order quantity price and advertising effort levelpolicies Then we solve the model under the exogenousprice case and the endogenous price case and obtain theoptimal policy of the order quantity and the advertisingeffort level for exogenous price case and the optimal policyof the order quantity price and advertising effort level insome situations for endogenous price case Furthermore weprovide the sensitivity analysis regarding the loss aversionparameters

We find that the loss-averse solutions are different fromthe loss-neutral solutions since the loss aversion behaviors ofa newsvendor affect the order quantity pricing and adver-tising decisions specifically the order quantity increaseswith the stockout aversion parameter and decreases withthe surplus aversion parameter For the exogenous price theadvertising effort level is not affected by the loss aversions ofthe newsvendor When the degree of stockout loss aversionis equal to the degree of surplus loss aversion the optimalorder quantity is also not affected by the loss aversions Forthe endogenous price under the mild conditions the policyof the order quantity price and the advertising effort level isrelated to the two loss aversions and furthermore the policy isanalyzed and given when the advertising effort level or orderquantity factor is fixed We also find that compared with thenonclassified loss aversion the classified loss aversions thatis surplus loss aversion and stockout loss aversion show theclearer and more specific effects on the order quantity priceand the advertising effort level

Compared with the existing research on loss-aversenewsvendor problem our work classifies the loss aversion

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

into the surplus loss aversion and the stockout loss aversionand emphasizes the analysis of the impacts of two lossaversions on the optimal policy respectively In additionthe advertising effect is taken into account Compared withthe research on the advertising effect our work focuses onthe newsvendor problem and takes the loss aversions intoaccount Our study compensates them by clearly describingand modeling the surplus and stockout loss aversion effectsor by considering the advertising effect

For the further research we will explore and analyzethe behavioral factors which can affect the newsvendorrsquosdecisions and study how to determine the values of thebehavioral parameters It would also be interesting to conductexperimental studies to investigate the parameter scales ofloss aversions

Appendix

Proof of Theorem 8

Proof of (i) According to (22) and (31) we have

119901 =

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2

(A1)

Then according to (15) and (A1) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot [1 minus 119865 (119911)]

(A2)

Let 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 then

119889119903 (119911)

119889119911

=

(1 + 120573)

2

2119887 minus 119896

2[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

sdot 119891 (119911)

(A3)

Furthermore 1198892119903(119911)1198891199112 can be obtained as follows

119889

2119903 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 minus 119896

2119888

2119887 minus 119896

2minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887 minus 119896

2]

119889119891 (119911)

119889119911

=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887 minus 119896

2

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A4)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A4) can be converted into

119889

2119903 (119911)

119889119911

2=

119889119903 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A5)

Then we have

119889

2119903 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119903(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

(2119887 minus 119896

2)119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A6)

According to (A6) if (a) 2119887minus1198962 gt 0 and 2119883(119911)2+119889119883(119911)119889119911 gt0 or (b) 2119887 minus 1198962 lt 0 and 2119883(119911)2 + 119889119883(119911)119889119911 lt 0 then119889

2119903(119911)119889119911

2le 0 and it implies that 119903(119911) has at most two

roots Since 119903(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119903(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119903(119911) has only one root itindicates that 119903(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119903(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119903(119873) = minus(1+120572)(119888minusV) lt 0 and119864119880[119911 119901(119911)]is unimodal if 119889119891(119911)119889119911 ge 119883(119911) if 119903(119872) gt 0 holds that is119886+120583minus119887119888+ (2119887minus 119896

2)119904 minus (1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has

only one root That is if 119886 + 120583 minus 119887119888 + (2119887 minus 1198962)119904 minus (1 + 120573)(120583 minus119872) gt 0 then there exists the uniquely optimal solution Thiscompletes the proof of (ii)

Proof ofTheorem 9 Since 2119887 = 1198962 according to (22) and (29)we have

120597119864 [119880 (119901)]

120597119901

= 119886 + (119896

2minus 2119887) 119901 + 120583 + 119887119888 minus 119896

2119888

minus (1 + 120573) 120579 (119911)

= 119886 + 120583 + 119887119888 minus 119896

2119888 minus (1 + 120573) 120579 (119911)

(A7)

Then according to the (A7) the second-order partialderivative of 119864[119880] with respect to the price 119901 is obtained asfollows

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

120597119864

2[119880 (119901)]

120597119901

2

=

(1 + 120573)

2

(1 + 120572)

2(119888 minus V)2

[(1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904)]3 119891 119865minus1 [(1 + 120573) (119901 minus 119888 + 119904) ((1 + 120572) (119888 minus V) + (1 + 120573) (119901 minus 119888 + 119904))]gt 0

(A8)

Since 1205971198642[119880(119901)]1205971199012 gt 0 the expected utility function119864[119880(119901)] is convex so the maximum of the expected utilitycan be obtained at bound prices

Proof of Theorem 14

Proof of (i) According to (15) and (31) we have

119889119864 [119880 (119911 119901 (119911))]

119889119911

= minus (1 + 120572) (119888 minus V) 119865 (119911) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot [1 minus 119865 (119911)]

(A9)

Let 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 then we have the second-order derivative of 119877(119911) that is

119889119877 (119911)

119889119911

=

(1 + 120573)

2

2119887

[1 minus 119865 (119911)]

2minus (1 + 120572) (119888 minus V)

+ (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

sdot 119891 (119911)

(A10)

Furthermore the second-order derivative of 119877(119911) can beobtained that is

119889

2119877 (119911)

119889119911

2= minus

3 (1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

minus (1 + 120572)

sdot (119888 minus V) + (1 + 120573)

sdot [

119886 + 120583 + 119887119888 + 119896119860

2119887

minus 119888 + 119904 minus

(1 + 120573) 120579 (119911)

2119887

]

119889119891 (119911)

119889119911

=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887

[1 minus 119865 (119911)] [119889119891 (119911) 119889119911]

119891 (119911)

2

+ 3

(A11)

Since 119889119891(119911)119889119911 = [119889119883(119911)119889119911minus119883(119911)2][1minus119865(119911)] where119883(sdot) =119891(sdot)[1 minus 119865(sdot)] then (A11) can be converted into

119889

2119877 (119911)

119889119911

2=

119889119877 (119911) 119889119911

119891 (119911)

119889119891 (119911)

119889119911

minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2+

119889119883 (119911)

119889119911

]

(A12)

Then we have

119889

2119877 (119911)

119889119911

2

100381610038161003816100381610038161003816100381610038161003816119889119877(119911)119889119911=0

= minus

(1 + 120573)

2

[1 minus 119865 (119911)] 119891 (119911)

2119887119883 (119911)

2[2119883 (119911)

2

+

119889119883 (119911)

119889119911

]

(A13)

According to (A13) if 2119883(119911)2 + 119889119883(119911)119889119911 gt 0 then119889

2119877(119911)119889119911

2le 0 and it implies that 119877(119911) has at most two

roots Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 if 119877(119911) hastwo roots the smaller root corresponds to a local minimumof 119864119880[119911 119901(119911)] and the larger one corresponds to a localmaximum of 119864119880[119911 119901(119911)] if 119877(119911) has only one root itindicates that 119877(119911) is from positive to negative and the rootcorresponds to a local maximum of 119864119880[119911 119901(119911)] hence119864119880[119911 119901(119911)] has only one local maximum For two rootssituation the optimal value of 119911 is the larger one of two valuesof 119911 that satisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 For only oneroot situation the optimal value of 119911 is the unique value thatsatisfies 119877(119911) = 119889119864119880[119911 119901(119911)]119889119911 = 0 This completes theproof of (i)

Proof of (ii) Since 119877(119873) = minus(1 + 120572)(119888 minus V) lt 0 and119864119880[119911 119901(119911)] is unimodal if 119889119891(119911)119889119911 ge 119884(119911) on the basisof this if 119877(119872) gt 0 holds that is 119886 + 120583 minus 119887119888 + 119896119860 + 2119887119904 minus(1+120573)(120583minus119872) gt 0 119864119880[119911 119901(119911)] has only one rootThat is if119886+120583minus119887119888+119896119860+2119887119904minus(1+120573)(120583minus119872) gt 0 then there exists theunique optimal solutionThis completes the proof of (ii)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 13

Acknowledgments

The research was partly supported by the National Nat-ural Science Foundation of China (Project nos 7127104971271051 and 71571039) and the Fundamental ResearchFunds for the Central Universities NEU China (Project noN140607001)

References

[1] Y F Chen X M Su and X B Zhao ldquoModeling bounded ratio-nality in capacity allocation games with the quantal responseequilibriumrdquoManagement Science vol 58 no 10 pp 1952ndash19622012

[2] Y F Chen and X B Zhao ldquoDecision bias in capacity allocationgames with uncertain demandrdquo Production and OperationsManagement vol 24 no 4 pp 634ndash646 2015

[3] X Wu and J A Niederhoff ldquoFairness in selling to the newsven-dorrdquo Production and OperationsManagement vol 23 no 11 pp2002ndash2022 2014

[4] F Gino and G Pisano ldquoToward a theory of behavioral opera-tionsrdquoManufacturing and Service Operations Management vol10 no 4 pp 676ndash691 2008

[5] C-Y Lee X Li and M Yu ldquoThe loss-averse newsvendor prob-lem with supply optionsrdquo Naval Research Logistics vol 62 no1 pp 46ndash59 2015

[6] X M Su ldquoBounded rationality in newsvendor modelsrdquoManu-facturing amp Service Operations Management vol 10 no 4 pp566ndash589 2008

[7] M Becker-Peth E Katok and U W Thonemann ldquoDesigningbuyback contracts for irrational but predictable newsvendorsrdquoManagement Science vol 59 no 8 pp 1800ndash1816 2013

[8] P K Kopalle P K Kannan L B Boldt and N Arora ldquoTheimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Retailingvol 88 no 1 pp 102ndash114 2012

[9] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[10] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquoManagement Science vol 53 no 8 pp 1303ndash13142007

[11] Y Liu C Ding C Fan and X Chen ldquoPricing decision underdual-channel structure considering fairness and free-ridingbehaviorrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 536576 10 pages 2014

[12] Y F Ren and R Croson ldquoOverconfidence in newsvendororders an experimental studyrdquoManagement Science vol 59 no11 pp 2502ndash2517 2013

[13] L Chen A G Kok and J D Tong ldquoThe effect of paymentschemes on inventory decisions the role of mental accountingrdquoManagement Science vol 59 no 2 pp 436ndash451 2013

[14] F Herweg ldquoThe expectation-based loss-averse newsvendorrdquoEconomics Letters vol 120 no 3 pp 429ndash432 2013

[15] J Sun and X Xu ldquoCoping with loss aversion in the newsvendormodelrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 851586 11 pages 2015

[16] U Schmidt and H Zank ldquoWhat is loss aversionrdquo The Journalof Risk and Uncertainty vol 30 no 2 pp 157ndash167 2005

[17] C X Wang and S Webster ldquoThe loss-averse newsvendor pro-blemrdquo Omega vol 37 no 1 pp 93ndash105 2009

[18] C X Wang ldquoThe loss-averse newsvendor gamerdquo InternationalJournal of Production Economics vol 124 no 2 pp 448ndash4522010

[19] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 1 pp 1ndash27 1985

[20] G Loomes and R Sugden ldquoDisappointment and dynamic con-sistency in choice under uncertaintyrdquo The Review of EconomicStudies vol 53 no 2 pp 271ndash282 1986

[21] B Koszegi and M Rabin ldquoReference-dependent risk attitudesrdquoAmerican Economic Review vol 97 no 4 pp 1047ndash1073 2007

[22] D YWu and K-Y Chen ldquoSupply chain contract design impactof bounded rationality and individual heterogeneityrdquo Produc-tion and Operations Management vol 23 no 2 pp 253ndash2682014

[23] M Nagarajan and S Shechter ldquoProspect theory and the news-vendor problemrdquoManagement Science vol 60 no 4 pp 1057ndash1062 2014

[24] W Liu S Song B Li and C Wu ldquoA periodic review inventorymodel with loss-averse retailer random supply capacity anddemandrdquo International Journal of Production Research vol 53no 12 pp 3623ndash3634 2015

[25] L J Ma Y X Zhao W L Xue T C E Cheng and H MYan ldquoLoss-averse newsvendor model with two ordering oppor-tunities and market information updatingrdquo International Jour-nal of Production Economics vol 140 no 2 pp 912ndash921 2012

[26] X Xu Z Meng R Shen M Jiang and P Ji ldquoOptimaldecisions for the loss-averse newsvendor problemunderCVaRrdquoInternational Journal of Production Economics vol 164 pp 146ndash159 2015

[27] S Du T Nie C Chu and Y Yu ldquoNewsvendor model for adyadic supply chain with nash bargaining fairness concernsrdquoInternational Journal of Production Research vol 52 no 17 pp5070ndash5085 2014

[28] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[29] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[30] M J Khouja ldquoOptimal ordering discounting and pricing inthe single-period problemrdquo International Journal of ProductionEconomics vol 65 no 2 pp 201ndash216 2000

[31] S A Raza and M Turiac ldquoJoint optimal determination of pro-cess mean production quantity pricing and market segmen-tation with demand leakagerdquo European Journal of OperationalResearch vol 249 no 1 pp 312ndash326 2016

[32] A A Taleizadeh and M Noori-Daryan ldquoPricing manufac-turing and inventory policies for raw material in a three-levelsupply chainrdquo International Journal of Systems Science vol 47no 4 pp 919ndash931 2016

[33] Y Qin R Wang A J Vakharia Y Chen and M M SerefldquoThe newsvendor problem review and directions for futureresearchrdquoEuropean Journal of Operational Research vol 213 no2 pp 361ndash374 2011

[34] A N Sadigh S K Chaharsooghi and M SheikhmohammadyldquoA game theoretic approach to coordination of pricing adver-tising and inventory decisions in a competitive supply chainrdquoJournal of Industrial and Management Optimization vol 12 no1 pp 337ndash355 2016

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Discrete Dynamics in Nature and Society

[35] F El Ouardighi G Feichtinger D Grass R Hartl and P MKort ldquoAutonomous and advertising-dependent lsquoword ofmouthrsquounder costly dynamic pricingrdquo European Journal of OperationalResearch vol 251 no 3 pp 860ndash872 2016

[36] PD Berger andTMagliozzi ldquoOptimal co-operative advertisingdecisions in direct-mail operationsrdquo Journal of the OperationalResearch Society vol 43 no 11 pp 1079ndash1086 1992

[37] S Karray and G Zaccour ldquoEffectiveness of coop advertisingprograms in competitive distribution channelsrdquo InternationalGameTheory Review vol 9 no 2 pp 151ndash167 2007

[38] Z Wu W Zhu and P Crama ldquoThe newsvendor problem withadvertising revenuerdquo Manufacturing and Service OperationsManagement vol 13 no 3 pp 281ndash296 2011

[39] S Karray and S H Amin ldquoCooperative advertising in asupply chain with retail competitionrdquo International Journal ofProduction Research vol 53 no 1 pp 88ndash105 2015

[40] G Aust and U Buscher ldquoCooperative advertising modelsin supply chain management a reviewrdquo European Journal ofOperational Research vol 234 no 1 pp 1ndash14 2014

[41] J Zhang Q Gou L Liang and Z Huang ldquoSupply chain coor-dination through cooperative advertising with reference priceeffectrdquo Omega vol 41 no 2 pp 345ndash353 2013

[42] J Yang J Xie X Deng and H Xiong ldquoCooperative advertisingin a distribution channel with fairness concernsrdquo EuropeanJournal ofOperational Research vol 227 no 2 pp 401ndash407 2013

[43] P Zipkin Foundations of Inventory Management McGraw-HillHigher Education New York NY USA 2000

[44] W Chung S Talluri and R Narasimhan ldquoPrice markdownscheme in amulti-echelon supply chain in a high-tech industryrdquoEuropean Journal of Operational Research vol 215 no 3 pp581ndash589 2011

[45] F Bernstein F Chen and A Federgruen ldquoCoordinating sup-ply chains with simple pricing schemes the role of vendor-managed inventoriesrdquo Management Science vol 52 no 10 pp1483ndash1492 2006

[46] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[47] Y Yu G Q Huang and L Liang ldquoStackelberg game-theoreticmodel for optimizing advertising pricing and inventory poli-cies in vendor managed inventory (VMI) production supplychainsrdquo Computers amp Industrial Engineering vol 57 no 1 pp368ndash382 2009

[48] M Khouja and S S Robbins ldquoLinking advertising and quantitydecisions in the single-period inventory modelrdquo InternationalJournal of Production Economics vol 86 no 2 pp 93ndash105 2003

[49] B Liu X Ma and R Zhang ldquoJoint decision on pricing andadvertising for competing retailers under emergency purchas-ingrdquo Economic Modelling vol 39 no 1 pp 257ndash264 2014

[50] W Chu and P S Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 14 no 4 pp 343ndash359 1995

[51] P S Desai ldquoAdvertising fee in business-format franchisingrdquoManagement Science vol 43 no 10 pp 1401ndash1419 1997

[52] S P Sigue and P Chintagunta ldquoAdvertising strategies in afranchise systemrdquo European Journal of Operational Researchvol 198 no 2 pp 655ndash665 2009

[53] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[54] G Loomes and R Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[55] R Engelbrecht-Wiggans and E Katok ldquoRegret in auctionstheory and evidencerdquo EconomicTheory vol 33 no 1 pp 81ndash1012007

[56] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[57] Z P Fan X Zhang F D Chen and Y Liu ldquoMultiple attributedecision making considering aspiration-levels a method basedon prospect theoryrdquo Computers amp Industrial Engineering vol65 no 2 pp 341ndash350 2013

[58] M Braun and A Muermann ldquoThe impact of regret on thedemand for insurancerdquo Journal of Risk and Insurance vol 71no 4 pp 737ndash767 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of