Research Article Research on Coordination of Fresh Produce

Research ArticleResearch on Coordination of Fresh Produce Supply Chain in BigMarket Sales Environment

Juning Su1 Jiebing Wu2 and Chenguang Liu1

1 School of Economics and Management Xirsquoan University of Technology Xirsquoan 710048 China2 College of Public Administration Zhejiang University Hangzhou 310058 China

Correspondence should be addressed to Jiebing Wu wujiebinggmailcom

Received 23 October 2013 Accepted 20 November 2013 Published 9 February 2014

Academic Editors P A D Castro and X-l Luo

Copyright copy 2014 Juning Su et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we propose two decision models for decentralized and centralized fresh produce supply chains with stochastic supplyand demand and controllable transportation time The optimal order quantity and the optimal transportation time in these twosupply chain systems are derived To improve profits in a decentralized supply chain based on analyzing the risk taken by eachparticipant in the supply chain we design a set of contracts which can coordinate this type of fresh produce supply chain withstochastic supply and stochastic demand and controllable transportation time as well We also obtain a value range of contractparameters that can increase profits of all participants in the decentralized supply chain The expected profits of the decentralizedsetting and the centralized setting are compared with respect to given numerical examples Furthermore the sensitivity analyses ofthe deterioration rate factor and the freshness factor are performedThe results of numerical examples show that the transportationtime is shorter the order quantity is smaller the total profit of whole supply chain is less and the possibility of cooperation betweensupplier and retailer is higher for the fresh produce which is more perishable and its quality decays more quickly

1 Introduction

Fresh produce such as fresh fruits fresh vegetables freshflowers and live seafood characteristically has a randomlifetime in the postharvest period The decaydeteriorationrisk creates huge uncertainties for the effective supply anddemand of fresh produce As a result both suppliers andretailers involved in the supply chain could suffer substantiallosses For this reason coordination of the supply chain playsan important role in the supply chain management of freshproduce [1] This is especially true in the ldquobig market salesrdquoenvironment A market condition in which the supplier andretailer are far apart is referred to as a ldquobig market salesrdquoenvironment in this paper

Trade in fresh produce has been among themost dynamicareas of the international agriculture trade stimulated byrising incomes and growing consumer interest in productvariety freshness convenience and year-round availabilityAdvances in production postharvest handling processing

and logistical technologies along with increased levels ofinternational investment have played a facilitating roleChina is currently the worldrsquos largest producer of fruits andvegetables [2] China exported 4795 million tons of fruit and973 million tons of vegetables to foreign markets in 2011Thebig market sales mode is common in the international freshproduce trade because the supplier and retailer are far fromone another In such an environment quantity deteriorationquality decay and the transportation costs of fresh producein delivering the product from its origin to the target salesmarket become considerations that cannot be ignored in thesupply chain

This paper attempts to analysis coordination of the freshproduce supply chain in a big market sales environmentWhen the supply chain involves long-distance transportationshortening logistics times may decrease the possible productdecaydeterioration but it may require expensive transporta-tion Therefore the supply chain for fresh produce needsto determine the best tradeoff between the associated costs

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 873980 12 pageshttpdxdoiorg1011552014873980

2 The Scientific World Journal

and benefits to optimize the total profit Decaydeteriorationdecreases not only the effective supply but also the freshnesslevel that might impact market demand (ie market demanddepends on the freshness level) The research questionsare threefold how should a decision-maker optimize orderquantity and logistics time How are the optimal orderquantity and logistics time determined Could one developan effective mechanism under which both parties can bebetter off In so doing we first give a quantitative descriptionabout the perishability properties of fresh produce and therelationship between decaydeterioration losses and timeThen we analyze quantitatively transportation costs in abig market sales environment and the relationship betweentransportation costs and time Based on this we investigatedecision-making models under both decentralized and cen-tralized supply chain settings and analyze solutions to themodels Further based on a comparative analysis of the risksand benefits faced by the parties we put forward a designidea for a coordinated scheme Finally a combined incen-tive contract is developed using supply chain coordinationtheory

The remainder of this paper is organized as follows InSection 2 we review the related literature In Section 3 wedescribe the problem and the notation In Section 4 weconstruct decision models of decentralized and centralizedsupply chains for fresh produce and optimal order quantityand logistics time in the two supply chain systems arederived Section 5 is dedicated to the development of acoordination contract in the fresh produce supply chainSection 6 illustrates the sensitivity analysis of parameters inthe models Section 7 concludes and outlines areas for futureresearch

2 Literature Review

This research is most related to the literature in two differentareas models for perishable product ordering and inventorycontrol and coordination models for a perishable productsupply chain

Numerous models for managing the ordering and inven-tory of perishable products have been developed Ghare andSchrader [3] were the first researchers to consider decayinginventory They develop an economic order quantity modelfor products in which the number of usable units is subjectto exponential decay when demand is constant Covert andPhilip [4] extend this model to use a Weibull distributionto describe item deterioration Nahmias [5] computes theoptimal inventory policy for a product with a multiperiodshelf life when ordering holding run-out and out-datingcosts are considered Tadikamalla [6] examines the model foritems with gamma distribution deterioration Elasayed andTeresi [7] developed the optimal order level for deterioratingitems Nahmias [8] gives a thorough review of the earlyliterature on perishable inventory and classifies the perishableproduct into fixed lifetime and random lifetime types Fixedlifetime perishable items are those that can be stored fora specified fixed time and after that time they must bediscarded However random lifetime perishable items are

those that will be discarded after an uncertain expirationtime Subsequently Kalpakam and Arivarignan [9] Rau et al[10] and Ghosh et al [11] consider ordering andor inventorymodels where the items deteriorate continuously at a constantperishing rate Wee [12] studies perishable product orderingstrategy under quantity discounts and buyback with theassumption that the deterioration rate obeys a two-parameterWeibull distribution Halim et al [13] discuss economic orderquantities in cases where the product deterioration rate isfuzzy

In the most early research on perishable inventoryperishability is defined as the number of units of product thatare outdated (perish) thus decay is not in terms of qualityor value However in recent studies some papers proposemodels that address deterioration in terms of reduction ofproduct quantity and degradation of quality and value overtime Weiss [14] examines a situation where the value of anitem decreases nonlinearly the longer it is held in stock Fuji-wara andPerera [15] developEOQmodels for inventoryman-agement under the assumption that product value diminishesover time according to an exponential distribution Morerecently Ferguson et al [16] apply Weissrsquo model to optimalorder quantities for perishable goods in small-to-mediumsize grocery stores with delivery surcharges Blackburn andScudder [17] using the productrsquos marginal value of time(MVT) the rate at which the product loses value over timedevelop a model to minimize lost value In the most recentstudy Sainathan [18] examines both inventory and pricingcontrol of perishable products by taking into account thatthe quality of the ldquonewrdquo product is higher than that ofthe ldquooldrdquo product Using this information we extend theconventional EOQ model to the ordering model for freshproduce in a big market sales environment which is char-acterized by simultaneously considering quantity loss andquality decay including logistics costs and random demandand supply the optimal order quantity and logistics time arederived

To date ordering and inventory control models forperishable products have been extended to the supply chainYang and Wee [19] developed an integrated deterioratinginventory model for both buyers and vendors the integratedapproach results in an impressive costreduction comparedwith independent decisions by the buyer Sarker et al [20]develop supply chain models to determine an optimal order-ing policy for deteriorating items under inflation permissibledelay of payment and allowable shortage Rau et al [10]develop a multiechelon inventory model for a deterioratingitem and to derive an optimal joint total cost from anintegrated supply chain perspective among the supplier theproducer and the buyer Subsequently a coordination modelof the supply chain for perishable product is investigatedPoole et al [21] survey a fruit vendor and retailer in Spainand obtain some important factors that affect fresh producesupply chain cooperation with an empirical analysis Weng[22] developed a framework to address the problem of coor-dinating decisions of the manufacturer and the distributoroperating tomeet price-sensitive randomdemand for a prod-uct with a short product life cycle Ferguson and Ketzenberg[23] examine the value of information sharing considering

The Scientific World Journal 3

Order quantityis decided by

retailer

Retailer choose atransportation

transportation

solution

Supplier delivers product

Long-distance

Retailer sellsthe product

Marketdemand is

satisfied

Produce reaches marketthe quality loss and

quality decay are known

Figure 1 Timeline of decisions in the fresh produce supply chain

a supplier sharing age-dependent information with retailersfor perishable products Ketzenberg and Ferguson [24] stud-ied the value of sharing the retailerrsquos information on agingand demand with the supplier Further Xu [25] investigatesthe optimal ordering and pricing decisions of suppliers anddistributors considering the uncertainty of long-distancetransportation Nahmias [26] recently reviews the literatureon the perishable goods supply chain with models thatconsider different aspects (eg random versus deterministiclifetime stochastic versus deterministic demand)

Althoughmany ordering and inventorymodels of perish-able products have been developed there are few reports thatstudy the supply and demand coordination problem of freshproduce from the perspective of the supply chain Amongmany reports the focus is on perishable products with fixedlifetimes (ie the marketing life or shelf life of the productsalthough it is relatively short but is a certain period) suchas high technology products fashion apparel medicinesand food products but few discuss perishable products withrandom lifetime (ie the duration in which the products arealive or useful is relatively short but is a uncertain period)such as fresh produce However fresh produce for examplefresh fruits and fresh vegetables can be found anywhere aspart of our daily life As a result there is a continued needto understand fresh produce and to investigate the impactof perishability and the random life of products on supplychain decisions Thus this study particularly aims to developordering models for such a system to explore optimal coordi-nation contract in the supply chain considering deteriorationand long-distance transportation in the context of big marketsales Our study differs from existing studies mainly in thefollowing three aspects First we derive random effectivesupply rate expressions for fresh produce based on our insightthat there is a variety of other complex factors affecting thelife-span of fresh produce beyond simply time Second wedevelop a joint decision model that includes ordering anddelivering decision-making Third we design a combinedcontract form that can achieve supply chain coordination forfresh produce To our knowledge this is the first reportedstudy that examines both ordering and delivery control offresh produce taking into account random demand andsupply and variable transportation time

3 Problem Description andModel Assumptions

31 Problem Description This paper studies a two-levelsupply chain system for fresh produce consisting of a supplier(such as a production base or a professional cooperative inthe original) and a retailer (such as a dealer or exporter inthe distant market) Such a supply chain is a specific case ofserial supply chains [27]The supplier cleans sorts packagesprocesses and supplies fresh produce in accordance withan order from the retailer The supplier has a very largesupply capacity and cannot run out of stock The retaileris responsible for shipping the fresh produce to distantmarket to sell The transportation market is well developedfrom the place of origin to the sale market and there is awide variety of logistics solutions to choose from for theretailer

Because fresh produce is a class of highly perishableproducts the longer the transportation time during the long-distance transportation process is the more the deterioratingproduces are This results in a more quantity loss of freshproduce and a smaller effective supply of fresh producereaching the target market Meanwhile the longer the trans-portation time is the more the serious quality decay of freshproduce is and the smaller the freshness of fresh produceis This has a greater impact on market demand Thusthe retailer is motivated to shorten the logistics time Theretailer can control transportation time by selecting differenttransportation solutionsHowever shortening transportationtime tends to increase transportation costs because an extraurgent transportation expense is paid And the more thetransportation time is compressed the faster the urgentdelivery costs rise Thus the decision-maker needs to weighthe two aspects of transportation time and economy Thedecision-making sequence for the supply chain of freshproduce is shown in Figure 1

As a kind of perishable product we assume that thesalvage value of unsold fresh produce is zero and theshortage cost of fresh produce is not taken into accountFurthermore we assume that both parties of supply chainshare information are risk-neutral and pursuing expectedprofit maximization


32 Notation and Assumptions Consider the following

119888119904 unit production cost of the supplier

119908 wholesale price of the supplier119902 order quantity of the retailer a decision variable119905 transportation time from origin to sales point adecision variable for the retailer (retailer controlstransportation time by choosing different transporta-tion methods) We assume that 119905 isin [119905

119897

119905119906

] where 119905119906 isthe normal transportation time and 119905

119897 is theminimumpossible transportation time

119901 the retail price of the retailer120579(119905) the deterioration rate of fresh produce it increases

with transportation time 120579(119905) isin [0 1] 1205791015840(119905) gt 0 label1 minus 120579(119905) as119898(119905) indicates the effective supply factor offresh produce

119872(119905 1205761) effective supply rate when product reaches the targetmarket it is decreased by the transportation time andinfluenced by the random factor 120576

1119872(119905 120576

1) isin [0 1]

120582(119905) freshness level of the fresh produce As transportationtime lengthens freshness declines 0 le 120582(119905) le 11205821015840

(119905) lt 0 The freshness level equals 1 when the freshproduce is in its freshest state We assume that thefreshness of the produces that retailer loads is 1 Fromthe beginning of transportation the freshness level offresh produces decreases gradually

119888 normal transportation fee With given starting andending points the normal transportation cost isrelated to the quantities of fresh produce transporteddecided as a unit transportation fee 119888 multiplied byorder quantity 119902

V(119905) urgent transportation cost this refers to extra costcaused by the retailer choosing urgent transporta-tion so as to shorten the transportation timeto reduce deterioration loss during transportationUrgent transportation costs are related to the degreeof compression of the transportation time

119888(119905) total transportation cost119863(119901 120582(119905) 120576

2) market demand of the product influenced by

product price 119901 freshness level 120582(119905) and randomfactor 120576

2

Π119904 the expected profit of the supplier

Π119903 the expected profit of the retailer

Π total expected profit of the supply chain

In the followingmodels subscript119891 indicates a decentral-ized uncoordinated system subscript 119895 indicates a centralizedsupply chain system subscript 119909 indicates a decentralizedcoordinated system and superscript lowast indicates the optimalvalue

We propose several basic assumptions before modeling

Assumption 1 119872(119905 1205761) = 119898(119905)120576

1 1205761is a random factor of

continuous distribution its mean is 1 and the probabilitydensity function and distribution function are 119892(119909) and119866(119909)

respectively This function indicates the effects of other ran-dom factors apart from transportation time (eg temperatureor humidity and human factors such as handling loadingand unloading theft and loss) on the fresh produce effectivesupply rate

Assumption 2 Themarket demand for fresh produce is eitheraffected by the sales price or the freshness level of produceWith the same price the higher the freshness level of produceis the bigger the market demand is With the same freshnesslevel of produce the lower the price is the bigger the marketdemand is Take a multiplicative form of market demand119863(119901 120582(119905)) = 119886119901

minus119896

120582(119905)1205762 where 120576

2is a random factor of

continuous distribution its mean is 1 and the probabilitydensity function and distribution function are 119891(119909) and119865(119909) respectively The function indicates the effects of otherrandom factors except for price and freshness level onmarketdemand The 119886 measures the size of the market and is aconstant The 119896 is price elasticity 119896 gt 1 so that the lowerthe price the greater the demand

Assumption 3 Consider 119888(119905) = 119888119902 + V(119905) V(119905) = (12)V(119905119906 minus119905)2 where V is time cost coefficient Transportation cost

includes two parts normal transportation costs and urgenttransportation costsThe latter one increases quickly with thedegree of time compression

4 Decisions in a Decentralized andCentralized Supply Chain

41 Model and Solutions for a Decentralized Supply ChainIn a decentralized supply chain the retailer makes decisionsindependently about the order quantity and transportationtime by the way of maximizing its profit The suppliermakes the choice between accepting or rejecting the retailerrsquosdecisions Then the expected profit functions of retailersupplier and supply chain system in a decentralized supplychain are respectively as follows

Π119903119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(1)

Π119904119891

= 119902119908 minus 119902119888119904 (2)

Π119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119888minus 119888 (119905)

(3)

In the case where both effective supply and marketdemand follow a random distribution the optimal decisionfor the retailer should aim to maximize the expected profitstatistically There are two random variables 120576

1and 120576

2 in

the retailerrsquos profit function (1) When the fresh producesreach the distant markets the effect of random variable 120576

1on

effective supply of fresh produces can be observed (Figure 1)Then (1) can be solved in two steps first we fix 120576

1and only

consider the effect of random variable 1205762on the retailerrsquos

profit Then we consider the effect of random variable 1205761

on the retailerrsquos profit Thus in the decentralized setting the


retailerrsquos model is maxΠ119903119891(119902 119905) = 119864

1205761

[Π119903119891(119902 119905 | 120576

1)] when 120576

1

equals 1205851

Π119903119891(119902 119905 | 120585

1)

= 119901119864 min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(4)

According to Petruzzi and Dada [28] define an inventoryfactor 119911 = 119898(119905)120585

1119902[119886119901

minus119896

120582(119905)] substituting it into (4) then(4) can be rewritten as

Π119903119891(119911 | 119902 119905 120585

1)

= (

119911119886120582 (119905)

119898 (119905) 1205851119902

)

1119896

1198641205762

min119898 (119905) 1205851119902

119898 (119905) 1205851119902

119911

1205762

minus 119902119908 minus 119888119902 minus V (119905) (5)

According to Ferguson and Ketzenberg [23] optimalinventory factor 119911

0 is uniquely decided by equation int

119911

0

(119896 minus

1)119909119891(119909)119889119909 = 119911[1 minus 119865(119911)] It can be observed that 1199110has

no relationship with 119902 or 119905 therefore by substituting 1199110and

simplifying (5) we get

Π119903119891(119902 119905 | 120585

1) = (119911

0119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times 1198641205762

min1

1205762

1199110

minus 119902119908 minus 119888119902 minus V (119905)

= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times (1 minus int

1199110

0

(1 minus

119909

1199110

)119891 (119909) 119889119909)


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

119896 (1 minus 119865 (1199110))

119896 minus 1


Based on (6) and using random variable 1205761 we can derive the

expected profit function of the retailer as follows

Π119903119891(119902 119905) = 119864

1205761

[Π119903119891(119902 119905 | 120576

1)]

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(7)

where 1198600= (1198861199110)1119896

[1 minus 119865(1199110)]1198641205761

1205761

(119896minus1)119896

Proposition 4 In decentralized supply chain when the trans-portation time of fresh produce from the origin to the sales pointis given by 119905 the retailerrsquos optimal order quantity is 119902lowast

119903119891(119905) =

120582(119905)119898(119905)119896minus1

(1198600(119908 + 119888))

119896

Proof In (7) we fix 119905 and obtain a first-order differential anda second-order differential with regard to 119902

120597Π119903119891

120597119902

= 119860119900120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus1119896

minus 119908 minus 119888

1205972

Π119903119891

1205971199022

= minus

1

119896

1198600120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus(119896+1)119896

(8)

It can be observed that 1205972Π1199031198911205971199022

lt 0 so Π119903119891

is strictly aconcave function with regard to 119902 For a given 119905 there exists a119902lowast

119903119891(119905) that makesΠ

119903119891maximal at this point Let 120597Π

119903119891120597119902 = 0

then we can obtain

119902lowast

119903119891(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

(119908 + 119888)

)

119896

(9)

Substituting 119902lowast

119903119891(119905) into (7) the retailerrsquos expected profit

functionwith regard to transportation time 119905 can be obtained

Π119903119891(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905) (10)

On the one hand compressing the transportation timecan reduce product deterioration and increase the effectivesupply on reaching themarketmore fresh product and biggermarket demand can increase the retailerrsquos income simulta-neously On the other hand compressing the transportationtime will increase the urgent transportation costsThus theremust exist an optimal transportation time that maximizes theretailerrsquos profit

The first-order differential of Π119903119891(119905) with regard to 119905 is

119889Π119903119891(119905)

119889119905

=

119860119896

0(119896 minus 1) 120582 (119905)119898(119905)

119896minus2

1198981015840

(119905) + 119860119896

01205821015840

(119905)119898(119905)119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V1015840 (119905)

(11)

Because we cannot judge whether 119889Π119903119891(119905)119889119905 is greater than

zero and cannot know the points which make the valueof 119889Π

119903119891(119905)119889119905 equal to zero also the monotonicity and the

stationary points of the function Π119903119891(119905) cannot be known

and so it is impossible to directly get optimal transporttime 119905

lowast

119903119891 However according to different situations we can

finally get 119905lowast119903119891by classification discussion The 119905

lowast

119903119891is given by

Proposition 5 as follows

Proposition 5 In the decentralized supply chain the retailerrsquosoptimal transportation time can be obtained according to thefollowing approach

when 119889Π119903119891(119905)119889119905 lt 0 119905lowast

119903119891= 119905119897

when 119889Π119903119891(119905)119889119905 gt 0 119905lowast

119903119891= 119905119906

otherwise 119905lowast

119903119891= argmax Π

119903119891(119905119897

) Π119903119891(1199051) Π119903119891(1199052)

Π119903119891(119905119899) Π119903119891(119905119906

) in which 1199051 1199052 1199053 119905119899 is the

solution set of the equation 119889Π119903119891(119905)119889119905 = 0



119903119891into (9) we can obtain the retailerrsquos

optimal order quantity

119902lowast

119903119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(

1198600

119908 + 119888

)

119896

(12)



optimal expected profit under a decentralized system

Πlowast

119903119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

(13)


119903119891into (2) we can obtain the supplierrsquos

optimal profit

Πlowast

119904119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904) (14)

The expected profit of the whole supply chain is

Πlowast

119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

+ 120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904)

(15)

42 The Model and Solutions of Centralized Supply Chain Incentralized supply chain the supplier and the retailer are aninterest unit and they cooperate closely sharing informationwith each other and pursuing total profit maximization astheir objective The expected profit of supply chain is

Π119895(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

(16)

Similar to procedure in Section 41 the model of central-ized supply chain can be written as

maxΠ119895(119902 119905) = 119864

1205761

Π119895(119902 119905 | 120576

1)

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896

minus 119902119888119904minus 119888119902 minus V (119905)

(17)

Similar to methods in Section 41 the optimal orderquantity of centralized supply chain can be obtained asProposition 6

Proposition 6 In the centralized supply chain when thetransportation time of fresh produce from the origin to the salespoint is given as 119905 the optimal order quantity of supply chain is

119902lowast

119895(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

119888119904+ 119888

)

119896

(18)


119895(119905) into (17) the expected profit function

of the supply chain with regard to transportation time 119905 canbe obtained

Π119895(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905) (19)

Similarly optimal transportation time 119905lowast119895can be obtained

as Proposition 7

Proposition 7 In the centralized supply chain the optimaltransportation time of supply chain can be obtained accordingto the following approach

(1) if 119889Π119895(119905)119889119905 lt 0 then 119905

lowast

119895= 119905119897

(2) if 119889Π119895(119905)119889119905 gt 0 then 119905

lowast

119895= 119905119906

(3) else one solves the equation 119889Π119895(119905)119889119905 = 0 and then

labels the solution set as 119879119895= 1199051198951 1199051198952 1199051198953 119905119895119899

Then 119905lowast

119895= argmax Π

119895(119905119897

) Π119895(1199051198951) Π119895(1199051198952) Π

119895(119905119895119899)

Π119895(119905119906

)


119895into (18) we can obtain the optimal order

quantity under a centralized supply chain

119902lowast

119895= 120582 (119905

lowast

119895)119898(119905lowast

119895)

119896minus1

[

1198600

119888119904+ 119888

]

119896

(20)


119895into (19) we can obtain the optimal

expected profit under a centralized supply chain

Πlowast

119895=

119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905lowast119895) (21)

Because the models are too complex the explicit formu-lations of the optimal transport time and those of the optimalorder quantity in the decentralized setting and centralizedsetting cannot be obtained But we can know that the optimaldecisions of the retailer in the decentralized setting are differ-ent from those of the whole supply chain in the centralizedsetting by intuitively observing The optimal decisions ofthese two situations will be compared via numerical examplein Section 61

Now we compare the optimal order quantity in thedecentralized setting with that in the centralized settingbased on assuming that the optimal transport times inthese two situations all are 119905

119906 In Proposition 4 119902lowast

119903119891=

120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119908 + 119888))

119896 when 119905 = 119905119906 In Proposition 6

119902lowast

119895= 120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119888119904+119888))119896 when 119905 = 119905

119906 Because119908 gt 119888119904

119902lowast

119903119891lt 119902lowast

119895when 119905 = 119905

119906That is the optimal order quantity of theretailer in decentralized setting is less than that of the wholesupply chain in the centralized setting

We compare the total profit of supply chain in the decen-tralized setting and that in the centralized setting Because the(119902lowast

119895 119905lowast

119895) is the point which maximizes the function Π

119895(119902 119905)

there exists Π119895(119902 119905) le Π

119895(119902lowast

119895 119905lowast

119895) for any point (119902 119905) It also

holds for Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) Comparing (3) with (16)

it is obvious that the total profit function forms of supplychain in the decentralized and centralized setting are exactlyconsistent so it holds thatΠ

119891(119902lowast

119903119891 119905lowast

119903119891) = Π119895(119902lowast

119903119891 119905lowast

119903119891) Because

Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) and Π

119891(119902lowast

119903119891 119905lowast

119903119891) = Π

119895(119902lowast

119903119891 119905lowast

119903119891)

Π119891(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) That is the total profit of supply

chain in decentralized setting is less than that in centralizedsetting


5 Coordination of a DecentralizedSupply Chain

In reality the decentralized supply chain is more commonThus it is necessary to implement a coordinationmechanismfor a decentralized supply chain so as to the decisionsfor order quantity and transportation time made from thepoint of view of the retailer are consistent with the opti-mal decisions for the supply chain to realize supply chainoptimization In designing the coordination mechanism thedecisions of centralized setting are often used as a benchmarkfor the decentralized system to reach coordination

51 Design of the Coordination Contract Supply chain con-tracts are a common supply chain coordination mechanismThe design principles of the supply chain coordinationmech-anism are risk sharing and revenue sharingThen we have toanalyze the risks borne by the supplier and the retailer in adecentralized system It is obvious that in an uncoordinateddecentralized systems the risks of supply andmarket demanduncertainty caused by product deterioration are both passedon to the retailer so it is necessary for the supplier to sharesome of the risks in designing a coordination contract tomotivate the retailer to order more products According tothis thinking this paper proposes the following combinedcontracts

(1) A Wholesale Price Discount Contract Because of thedecaying of fresh produce the effective supply of productdecreasesWe can consider this as an increase in procurementcost for the retailer (or as an increase in wholesale price forthe supplier) in disguise A wholesale price discount contractwould be adopted tomake the supplier share some risk causedby the deterioration of the produce and this would stimulatethe retailer to order more produces

The design idea of the wholesale price discount contractis that the supplier adopts cost-plus pricing method todetermine the list wholesale price which means that the listwholesale price equals 119888

119904to add a 120593 proportion of marginal

profit of per unit product in supply chain When the salesprice of fresh produce is 119901 the supply chain marginal profitobtained from per unit of produce is 119901 minus 119888 minus 119888

119904 Then the list

wholesale price of supplier can be written as 1199080= 119888119904+ 120593(119901 minus

119888 minus 119888119904) When the produces reach the target market and part

of produces decay the wholesale price should be cut downbased on the list wholesale price so as to make the suppliershare some part of losses from produce deterioration If thereal effective supply rate is 119898(119905)120585

1 the deterioration loss of

unit produce is 119901 minus 119898(119905)1205851119901 Given the deterioration loss

share ratio of the supplier is 120593 then the wholesale price ofsupplier will reduce120593(119901minus119898(119905)120585

1119901) based on the list wholesale

price 1199080 So the specific form of the wholesale price function

offered by the supplier is

119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)

Substituting the expression of1199080into (22) we can obtain

the wholesale price discount contract as follows

119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)

The wholesale price discount contract connects the inter-ests of supplier with the interests of retailer by establishingrelationships between wholesale price and retail price So thesupplier shares the risks with the retailer together under awholesale price discount contract(2) Unsaleable Produce Subsidy Contract Uncertain marketdemand brings an unmarketable product risk This risk isborne by the retailer when there is not coordination contractWe design a contract in which the supplier shares some riskof unsaleable produce by providing a certain percentage ofsubsidies for losses due to unsold produce Because freshproduce is perishable we assume that the salvage value forsurplus produce is zero at the end of the sales period Forevery unsold produce the retailer will lose119901 and the supplierrenders 120593119901 Then the amount of subsidy 119904 is

119904 = 120593119901 (24)

(3) Cost-Compensating Contract Urgent transportation canshorten transportation time so it can reduce the deteriora-tion of the fresh produce and can keep the fresh producefresh All these effects benefit product sales but the retailerneeds to pay the extra urgent transportation costs Thereforewe propose a cost-compensating contract that makes thesupplier provide a portion of 120593 towards the retailerrsquos urgenttransportation costs The symbol 119911 indicates the amount ofcompensation given by the supplier The form of the cost-compensation contract is

119911 = 120593V (119905) (25)

52 Analysis of Decision-Making and Coordination underCombined Contracts

Proposition 8 In coordination with combined contracts119908(119905) = 119888

119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)

the retailerrsquos optimal order quantity and optimal transportationtime are consistent with optimal decisions of the centralizedsupply chain

Proof With the combined contracts which consist of thethree contracts above the expected profit function of theretailer can be transformed intoΠ119903119909

(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)

The above function can be expanded as follows

Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761

[119898 (119905) 1205851119901 minus 119908 (119905) minus 119888] 119902 minus (119901 minus 119904) 119864

1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761

(1 minus 120593) [119898 (119905) 1205851119901 minus 119888119904minus 119888] 119902 minus 119901119864

1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)

Obviously under the combined contracts the optimaldecisions of the retailer are suboptimization of the decisionsof the entire supply chain Proposition 8 is proven

It can be demonstrated that the retailerrsquos optimal profit is(1minus120593)Π

lowast

119895 and the supplierrsquos optimal profit is 120593Πlowast

119895This shows

that 120593 not only represents the proportion that the suppliershares of the risks of the supply chain under a combinedcontract but also represents the proportion that the supplierobtaines of the total profits of the entire supply chain Thisillustrates that the combined contracts designed for a freshproduce supply chain embody a profit distribution principlein risk market that is the greater the risk the greater thereturns

Proposition 9 The combined contracts can achieve perfectcoordination of the supply chain for fresh produce when 120593

belongs [120593min 120593max] where

120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

minus (119896 minus 1) (119888119904+ 119888)119896minus1V (119905lowast

119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

minus (119896 minus 1) (119908 + 119888)119896minus1V (119905lowast

119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)

Proof Proposition 8 illustrates that under combined con-tracts decentralized supply chain performance reaches theperformance level of the centralized system To realize Paretoimprovement with the members of the supply chain bothaccepting this contract the following two conditions must besatisfied Π

lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891

Through mathematical derivation the following resultscan be obtained

120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)

Because 120593max minus 120593min = 1 minus Πlowast

119903119891Πlowast

119895minus Πlowast

119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast

119895gt 0 the interval [120593min 120593max] exists and

Proposition 9 is proven

When 120593 = 120593min all increased profits in supply chaincoordination are occupied by the retailer while the profitincrement for the supplier is 0 The retailer is absolutelydominant in the supply chain In contrast when 120593 = 120593maxall increased profits in supply chain coordination flow tothe supplier while the profit increment for the retailer is0 and the supplier is in the dominant position of supplychain Thus when the value of 120593 is given in [120593min 120593max] thepurpose can achieve that the coordination profit of supplychain is discretionarily allotted between the parties of thesupply chainThe practical value of 120593 depends on the relativebargaining power of the parties

6 Numerical Examples

The model expressions in Sections 4 and 5 are complexand we cannot obtain the explicit solutions In order toillustrate the proposed models we give numerical examplesas follows Consider a fresh produce supply chain with thefollowing characteristics 119888

119904= 4 119908 = 6 119901 = 12 The

deterioration characteristics accord with a traditional three-parameter Weibull function 120579(119905) = 120572120573(119905 minus 120574)

120573minus1 where 120572 =

01 120573 = 11 and 120574 = 05 Freshness function is 120582(119905) = 120582119905

0


1205820

= 0999 The parameters of the transportation methodfrom the origin to market are as follows 119888 = 1 119905119906 = 10119905119897

= 5 and V = 500 Other values are as follows 119896 = 2119886 = 500000 120576

1isin 119880[0 2] 120576

2isin 119880[0 2] According to Ferguson

andKetzenberg [23] the optimal inventory factor satisfies thefollowing equation

1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)

61 Solution of theModels Substituting these parameters intothe models and computing with Matlab we can then obtainthe optimal decisions and each partyrsquos profits in decentralizedand centralized systems (Table 1)

It can be concluded fromTable 1 that the order quantity inthe centralized system is larger than that in the decentralizedsystem the transport time in the centralized system isshorter than that in the decentralized system and the overallprofit of the whole supply chain in the centralized systemis higher than that in the decentralized system Howeverthe implementation conditions of the centralized system areharsh it is common to see the decentralized system in realityAfter introducing the combined contracts proposed in thispaper the coordination conditions can be calculated as 120593 isin

[02041 02857] in which contract can be accepted by boththe supplier and the retailer and it can make the overallprofit of the decentralized supply chain reach the level of thatof the centralized supply chain When contract parameter 120593gets value in this range the changes in the profits and itsincrements of the retailer and the supplier after coordinationwith the parameter 120593 are shown in Table 2

It can be concluded from Table 2 that the profits of thesupplier and the retailer after coordination increase thanthose before coordination when 120593 is within the scope ofvalid value of it With an increasing of the value of 120593 theprofit increments of the retailer are declining while the profitincrements of the supplier are increasing and the increasedprofits of supply chain after coordination transfer from theretailer to the supplier gradually These verify that the com-bined contract can coordinate fresh produce supply chaineffectively and the combined contract can flexibly allocate theincreased profits of supply chain after coordination betweenthe supplier and the retailer in an arbitrary ratio when 120593 iswithin the scope of valid value of it

62 Sensitivity Analysis of the Models To further analyze theadaptability of the models and to provide more managementimplications for the fresh produce supply chain in practicein this section we aim to analyze the impact of severalimportant model parameters on decision-making results

621 Influence of Deterioration for Fresh Produce on Decision-Making The perishability nature is one of the most impor-tant characteristics of fresh produce In the context of bigmarket sales quantity loss of fresh produce caused by longdistance transportation occurs due to decay How does thedeterioration characteristic of the fresh produce affect thedecisions in the supply chain In this section we performa sensitivity analysis of parameter 120572 which comes from the

Table 1 Optimal decisions and profits in decentralized and central-ized systems

119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast

Decentralized decision 11472 996 8030 2294 10324Centralized decision 22486 994 mdash mdash 11242Δ119902Δ119905ΔΠ 11014 002 mdash mdash 918

Table 2The profits and their increments of the retailer and supplierafter coordination

120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918

deterioration rate function The larger the value of 120572 is themore perishable the produce is and the more the quantityloss of the fresh produce during transportation is Whenthe value of 120572 changes in [01 02] the optimal decisions ofdecentralized and centralized systems and the value range ofsupply chain coordination parameter 120593 are shown in Table 3Figure 2 shows that the profits of all parties and supply chainchange with 120572 either in decentralized system or in centralizedsystem

We can make the following conclusions by analyzingTable 3 and Figure 2

Observation 1 Whether in a decentralized or centralizedsystem the more perishable the produce is the shorter thetransportation time is and the smaller the order quantityis This observation can be explained because the moreperishable the produce is the greater the potential loss is andthe decision-maker is therefore more cautious

Observation 2 As 120572 becomes larger the profits of every partyand total profit of supply chain tend to decrease in a decen-tralized system also the profit of the entire supply chaindecreases gradually in a centralized systemThis observationillustrates that the more perishable the produce becomes theweaker the profitability of supply chain is

Observation 3 For supply chain coordination as 120572 becomesbigger the lower and upper limits of the contract parameter120593 increase at the same time however the upper limitincreases faster than the lower limit and the value range of120593 becomes bigger and biggerThis observation illustrates thatthe more perishable the produce is the higher the possibilityof cooperation between supplier and retailer is

622 Influence of Quality Decay for Fresh Produce onDecision-Making Apart from its perishable nature freshnessis another important characteristic of fresh produce Sohow does freshness affect decisions in the supply chainPreviously we used a freshness level function to describe


Table 3 Optimal decisions in each supply chain system with deterioration factor 120572

120572

Decentralized system Centralized system Coordination system119902lowast

119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820

Table 4 Optimal decisions in each supply chain system with freshness factor 1205820

1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier

Decentralized systemCentralized system

120572

Figure 2 Profits of supply chain parties with deterioration factor 120572

the freshness of fresh produce The parameter 1205820in the

function indicates fresh-keeping performance The larger 1205820

is the easier the produce is to keep fresh When the value of1205820changes in [0995 0999] we analyze decision results of

different supply chain systems and coordination conditionsin Table 4 Figure 3 shows that the profits of all partiesand supply chain change with freshness factor 120582

0either in

decentralized system or in centralized systemWe can make the following conclusions by analyzing

Table 4 and Figure 3

Observation 4 As freshness factor 1205820becomes bigger the

optimal order quantity and transportation time both becomebigger in both decentralized and centralized supply chainsThis observation illustrates that the easier the produce retainsfresh the more of the produce the retailer tends to order andtherefore the retailer selects a cheaper transportationmethod

Observation 5 As freshness factor 1205820becomes larger the

profits of every party and total profit of supply chain increase

02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820

Figure 3 Profits of supply chain parties with freshness factor 1205820

at the same time in the decentralized system and the profitof the centralized system increases also This observationillustrates that a produce that decays more slowly is beneficialto all parties in the supply chain


lower and upper limits of contracts parameter 120593 decreasesimultaneously however the upper limit decreases fasterso the value range of 120593 becomes smaller This observationillustrates that the faster the quality of produce decays thehigher the possibility of cooperation between supplier andretailer is

7 Conclusions

In the context of the rapid development of modern agricul-ture and logistics the ldquobig market salesrdquo model of fresh pro-duce sales has prevailedThis paper constructs a deteriorationrate function and freshness function for fresh produce that


depend on the transport time in long-distance transporta-tion It is assumed that effective supply is an indeterminatevariable influenced by deterioration rate and random factorsand that market demand is a random variable influencedby price and freshness level random factor as well Basedon these assumptions decision models of decentralized andcentralized supply chains are built and we present a solutionalgorithm for the models By analyzing numerical exampleswe find that order quantity in a centralized system is higherthan that in a decentralized system while transportationtime in a centralized system is shorter than that in adecentralized system and the total profit of the supply chainin a centralized system is higher than that in a decentralizedsystem We design a combined contract to coordinate thedecentralized supply chainwhich consists of awholesale pricediscount contract an unsaleable produce subsidy contractand a cost-compensating contract A mathematical deriva-tion demonstrates that the combined contracts can effectivelycoordinate a two-level supply chain of fresh produce wheresupply and demand both conform to a time-varying randomdistribution and can discretionarily allot the coordinationprofit of the supply chain between the supplier and retailerThen we provide the conditions with which the coordinationof supply chain is achieved Using a sensitivity analysis oftwo important parameters (deterioration rate 120572 and freshnessfactor 120582

0) we draw the conclusion that the more perishable

the produce is the faster the quality of produce decays andthe higher the possibility of cooperation between supplier andretailer is These conclusions provide a better understandingof fresh produce supply chain management practices

Our study makes some contributions to the under-standing of integrated optimization of more than that in adecentralized procurement and logistics in the fresh producesupply chain Another contribution of our work is the designof a combined contract which ensures that both parties arebetter off by coordinating in a situation where both theeffective supply and the market demand of the produceare random freshness deterioration rate and transportationcosts are sensitive to time and market demand is sensitive tofreshness level

Although this study provides several managerial implica-tions for fresh produce supply chains the paper only studiesquantity loss and quality decay of fresh produce caused bylong-distance transportation and assumes that freshness onlyaffects market demand Freshness also affects the marketprice of fresh produce and the value of fresh producewould therefore be reduced over long-distance transporta-tion Additionally the implementation of the combinedcontracts proposed in this paper requires that there is mutualcooperation and information sharing between the supplierand retailer Such conditions are difficult in practiceThus wesuggest further research to study supply chain coordination offresh produce under conditions of asymmetric information

Conflict of Interests

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (71171161 71371153 and 71273227)the Project of Humanities and Social Science Research ofEducation Ministry of China (13YJA630077) the ResearchFund for theDoctoral Program ofHigher Education of China(20126118110018) the Natural Science Foundation of ShaanxiProvince (2011JM9004) and the Fund of the Ministry ofEducation of Shaanxi Province (11JK0168)

References

[1] X Cai J Chen Y Xiao and X Xu ldquoOptimization andcoordination of fresh product supply chains with freshness-keeping effortrdquo Production and OperationsManagement vol 19no 3 pp 261ndash278 2010

[2] M A Aksoy and J C Beghin Global Agricultural Trade andDeveloping Countries World Bank Publications WashingtonDC USA 2004

[3] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventoryrdquo Journal of Industrial Engineering vol 14no 5 pp 238ndash243 1963

[4] R P Covert and G C Philip ldquoAn EOQ model for items withweibull distribution deteriorationrdquoAIIE Transactions vol 5 no4 pp 323ndash326 1973

[5] S Nahmias ldquoOptimal ordering policies for perishableinventory-IIrdquo Operations Research vol 23 no 4 pp 735ndash7491975

[6] P R Tadikamalla ldquoAn EOQ inventory model for items withgamma distribution deteriorationrdquo AIIE Transactions vol 10no 1 pp 100ndash103 1978

[7] E A Elasayed and C Teresi ldquoAnalysis of inventory systemswith deteriorating itemsrdquo International Journal of ProductionResearch vol 21 no 4 pp 449ndash460 1983

[8] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[9] S Kalpakam and G Arivarignan ldquoA continuous review perish-able inventorymodelrdquo Statistics vol 19 no 3 pp 389ndash398 1988

[10] H RauM-YWu andH-MWee ldquoIntegrated inventorymodelfor deteriorating items under a multi-echelon supply chainenvironmentrdquo International Journal of Production Economicsvol 86 no 2 pp 155ndash168 2003

[11] S KGhosh S Khanra andK S Chaudhuri ldquoOptimal price andlot size determination for a perishable product under conditionsof finite production partial backordering and lost salerdquoAppliedMathematics and Computation vol 217 no 13 pp 6047ndash60532011

[12] H-M Wee ldquoDeteriorating inventory model with quantity dis-count pricing and partial backorderingrdquo International Journalof Production Economics vol 59 no 1ndash3 pp 511ndash518 1999

[13] K A Halim B C Giri and K S Chaudhuri ldquoFuzzy economicorder quantity model for perishable items with stochasticdemand partial backlogging and fuzzy deterioration raterdquoInternational Journal of Operational Research vol 3 no 1-2 pp77ndash96 2008

[14] H J Weiss ldquoEconomic order quantity models with nonlinearholding costsrdquo European Journal of Operational Research vol 9no 1 pp 56ndash60 1982

[15] O Fujiwara and U L J S R Perera ldquoEOQ models for con-tinuously deteriorating products using linear and exponential


penalty costsrdquoEuropean Journal ofOperational Research vol 70no 1 pp 104ndash114 1993

[16] M Ferguson V Jayaraman and G C Souza ldquoNote an applica-tion of the EOQmodel with nonlinear holding cost to inventorymanagement of perishablesrdquo European Journal of OperationalResearch vol 180 no 1 pp 485ndash490 2007

[17] J Blackburn and G Scudder ldquoSupply chain strategies forperishable products the case of fresh producerdquo Production andOperations Management vol 18 no 2 pp 129ndash137 2009

[18] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquoProduction and OperationsManagement vol 22 no 5 pp 1157ndash1181 2013

[19] P-C Yang and H-M Wee ldquoEconomic ordering policy of dete-riorated item for vendor and buyer an integrated approachrdquoProduction Planning and Control vol 11 no 5 pp 474ndash4802000

[20] B R Sarker AMM Jamal and SWang ldquoSupply chainmodelsfor perishable products under inflation and permissible delay inpaymentrdquo Computers amp Operations Research vol 27 no 1 pp59ndash75 2000

[21] N D Poole F J del Campo Gomis J F Julia Igual and F VidalGimenez ldquoFormal contracts in fresh produce marketsrdquo FoodPolicy vol 23 no 2 pp 131ndash142 1998

[22] Z K Weng ldquoThe power of coordinated decisions for short-life-cycle products in a manufacturing and distribution supplychainrdquo IIE Transactions vol 31 no 11 pp 1037ndash1049 1999

[23] M Ferguson and M E Ketzenberg ldquoInformation sharing toimprove retail product freshness of perishablesrdquo Production andOperations Management vol 15 no 1 pp 57ndash73 2006

[24] M Ketzenberg and M E Ferguson ldquoManaging slow-movingperishables in the grocery industryrdquo Production and OperationsManagement vol 17 no 5 pp 513ndash521 2008

[25] X Xu Optimal decisions in a time-sensitive supply chain withperishable products [PhD thesis] The Chinese University ofHong Kong Hong Kong 2006

[26] S Nahmias Perishable Inventory Systems Springer New YorkNY USA 2011

[27] Y Yin C Liu and I Kaku ldquoCooperation and leadership policiesin a serial supply chainrdquo Journal of Manufacturing Systems vol30 no 1 pp 1ndash7 2011

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and benefits to optimize the total profit Decaydeteriorationdecreases not only the effective supply but also the freshnesslevel that might impact market demand (ie market demanddepends on the freshness level) The research questionsare threefold how should a decision-maker optimize orderquantity and logistics time How are the optimal orderquantity and logistics time determined Could one developan effective mechanism under which both parties can bebetter off In so doing we first give a quantitative descriptionabout the perishability properties of fresh produce and therelationship between decaydeterioration losses and timeThen we analyze quantitatively transportation costs in abig market sales environment and the relationship betweentransportation costs and time Based on this we investigatedecision-making models under both decentralized and cen-tralized supply chain settings and analyze solutions to themodels Further based on a comparative analysis of the risksand benefits faced by the parties we put forward a designidea for a coordinated scheme Finally a combined incen-tive contract is developed using supply chain coordinationtheory

The remainder of this paper is organized as follows InSection 2 we review the related literature In Section 3 wedescribe the problem and the notation In Section 4 weconstruct decision models of decentralized and centralizedsupply chains for fresh produce and optimal order quantityand logistics time in the two supply chain systems arederived Section 5 is dedicated to the development of acoordination contract in the fresh produce supply chainSection 6 illustrates the sensitivity analysis of parameters inthe models Section 7 concludes and outlines areas for futureresearch

2 Literature Review

This research is most related to the literature in two differentareas models for perishable product ordering and inventorycontrol and coordination models for a perishable productsupply chain

Numerous models for managing the ordering and inven-tory of perishable products have been developed Ghare andSchrader [3] were the first researchers to consider decayinginventory They develop an economic order quantity modelfor products in which the number of usable units is subjectto exponential decay when demand is constant Covert andPhilip [4] extend this model to use a Weibull distributionto describe item deterioration Nahmias [5] computes theoptimal inventory policy for a product with a multiperiodshelf life when ordering holding run-out and out-datingcosts are considered Tadikamalla [6] examines the model foritems with gamma distribution deterioration Elasayed andTeresi [7] developed the optimal order level for deterioratingitems Nahmias [8] gives a thorough review of the earlyliterature on perishable inventory and classifies the perishableproduct into fixed lifetime and random lifetime types Fixedlifetime perishable items are those that can be stored fora specified fixed time and after that time they must bediscarded However random lifetime perishable items are

those that will be discarded after an uncertain expirationtime Subsequently Kalpakam and Arivarignan [9] Rau et al[10] and Ghosh et al [11] consider ordering andor inventorymodels where the items deteriorate continuously at a constantperishing rate Wee [12] studies perishable product orderingstrategy under quantity discounts and buyback with theassumption that the deterioration rate obeys a two-parameterWeibull distribution Halim et al [13] discuss economic orderquantities in cases where the product deterioration rate isfuzzy

In the most early research on perishable inventoryperishability is defined as the number of units of product thatare outdated (perish) thus decay is not in terms of qualityor value However in recent studies some papers proposemodels that address deterioration in terms of reduction ofproduct quantity and degradation of quality and value overtime Weiss [14] examines a situation where the value of anitem decreases nonlinearly the longer it is held in stock Fuji-wara andPerera [15] developEOQmodels for inventoryman-agement under the assumption that product value diminishesover time according to an exponential distribution Morerecently Ferguson et al [16] apply Weissrsquo model to optimalorder quantities for perishable goods in small-to-mediumsize grocery stores with delivery surcharges Blackburn andScudder [17] using the productrsquos marginal value of time(MVT) the rate at which the product loses value over timedevelop a model to minimize lost value In the most recentstudy Sainathan [18] examines both inventory and pricingcontrol of perishable products by taking into account thatthe quality of the ldquonewrdquo product is higher than that ofthe ldquooldrdquo product Using this information we extend theconventional EOQ model to the ordering model for freshproduce in a big market sales environment which is char-acterized by simultaneously considering quantity loss andquality decay including logistics costs and random demandand supply the optimal order quantity and logistics time arederived

To date ordering and inventory control models forperishable products have been extended to the supply chainYang and Wee [19] developed an integrated deterioratinginventory model for both buyers and vendors the integratedapproach results in an impressive costreduction comparedwith independent decisions by the buyer Sarker et al [20]develop supply chain models to determine an optimal order-ing policy for deteriorating items under inflation permissibledelay of payment and allowable shortage Rau et al [10]develop a multiechelon inventory model for a deterioratingitem and to derive an optimal joint total cost from anintegrated supply chain perspective among the supplier theproducer and the buyer Subsequently a coordination modelof the supply chain for perishable product is investigatedPoole et al [21] survey a fruit vendor and retailer in Spainand obtain some important factors that affect fresh producesupply chain cooperation with an empirical analysis Weng[22] developed a framework to address the problem of coor-dinating decisions of the manufacturer and the distributoroperating tomeet price-sensitive randomdemand for a prod-uct with a short product life cycle Ferguson and Ketzenberg[23] examine the value of information sharing considering



retailer


transportation

solution


Long-distance


Marketdemand is

satisfied














119897

119905119906






1119872(119905 120576

1) isin [0 1]








2






Assumption 1 119872(119905 1205761) = 119898(119905)120576





minus119896

120582(119905)1205762 where 120576







Π119903119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(1)

Π119904119891

= 119902119908 minus 119902119888119904 (2)

Π119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119888minus 119888 (119905)

(3)


1and 120576

2 in


1on


1and only






1205761

[Π119903119891(119902 119905 | 120576

1)] when 120576

1

equals 1205851

Π119903119891(119902 119905 | 120585

1)

= 119901119864 min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(4)


1119902[119886119901

minus119896


Π119903119891(119911 | 119902 119905 120585

1)

= (

119911119886120582 (119905)

119898 (119905) 1205851119902

)

1119896

1198641205762

min119898 (119905) 1205851119902

119898 (119905) 1205851119902

119911

1205762




119911

0

(119896 minus




Π119903119891(119902 119905 | 120585

1) = (119911

0119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times 1198641205762

min1

1205762

1199110


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times (1 minus int

1199110

0

(1 minus

119909

1199110

)119891 (119909) 119889119909)


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

119896 (1 minus 119865 (1199110))

119896 minus 1




Π119903119891(119902 119905) = 119864

1205761

[Π119903119891(119902 119905 | 120576

1)]

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(7)

where 1198600= (1198861199110)1119896

[1 minus 119865(1199110)]1198641205761

1205761

(119896minus1)119896


119903119891(119905) =

120582(119905)119898(119905)119896minus1

(1198600(119908 + 119888))

119896


120597Π119903119891

120597119902

= 119860119900120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus1119896


1205972

Π119903119891

1205971199022

= minus

1

119896

1198600120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus(119896+1)119896

(8)


lt 0 so Π119903119891


119903119891(119905) that makesΠ


119903119891120597119902 = 0

then we can obtain

119902lowast

119903119891(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

(119908 + 119888)

)

119896

(9)




Π119903119891(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905) (10)



119889Π119903119891(119905)

119889119905

=

119860119896

0(119896 minus 1) 120582 (119905)119898(119905)

119896minus2

1198981015840

(119905) + 119860119896

01205821015840

(119905)119898(119905)119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V1015840 (119905)

(11)






lowast



lowast

119903119891is given by



when 119889Π119903119891(119905)119889119905 lt 0 119905lowast

119903119891= 119905119897

when 119889Π119903119891(119905)119889119905 gt 0 119905lowast

119903119891= 119905119906


119903119891= argmax Π

119903119891(119905119897

) Π119903119891(1199051) Π119903119891(1199052)

Π119903119891(119905119899) Π119903119891(119905119906

) in which 1199051 1199052 1199053 119905119899 is the






119902lowast

119903119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(

1198600

119908 + 119888

)

119896

(12)




Πlowast

119903119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

(13)



optimal profit

Πlowast

119904119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904) (14)


Πlowast

119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

+ 120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904)

(15)


Π119895(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

(16)


maxΠ119895(119902 119905) = 119864

1205761

Π119895(119902 119905 | 120576

1)

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(17)



119902lowast

119895(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

119888119904+ 119888

)

119896

(18)




Π119895(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905) (19)


as Proposition 7


(1) if 119889Π119895(119905)119889119905 lt 0 then 119905

lowast

119895= 119905119897

(2) if 119889Π119895(119905)119889119905 gt 0 then 119905

lowast

119895= 119905119906



Then 119905lowast

119895= argmax Π

119895(119905119897

) Π119895(1199051198951) Π119895(1199051198952) Π

119895(119905119895119899)

Π119895(119905119906

)




119902lowast

119895= 120582 (119905

lowast

119895)119898(119905lowast

119895)

119896minus1

[

1198600

119888119904+ 119888

]

119896

(20)




Πlowast

119895=

119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905lowast119895) (21)




119903119891=

120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119908 + 119888))


119902lowast

119895= 120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119888119904+119888))119896 when 119905 = 119905

119906 Because119908 gt 119888119904

119902lowast

119903119891lt 119902lowast

119895when 119905 = 119905



119895 119905lowast


119895(119902 119905)


119895(119902lowast

119895 119905lowast



119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast



119891(119902lowast

119903119891 119905lowast

119903119891) = Π119895(119902lowast

119903119891 119905lowast

119903119891) Because

Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) and Π

119891(119902lowast

119903119891 119905lowast

119903119891) = Π

119895(119902lowast

119903119891 119905lowast

119903119891)

Π119891(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast





















119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)



119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)


119904 = 120593119901 (24)


119911 = 120593V (119905) (25)



119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)



(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)


Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











retailer


transportation

solution


Long-distance


Marketdemand is

satisfied














119897

119905119906






1119872(119905 120576

1) isin [0 1]








2






Assumption 1 119872(119905 1205761) = 119898(119905)120576





minus119896

120582(119905)1205762 where 120576







Π119903119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(1)

Π119904119891

= 119902119908 minus 119902119888119904 (2)

Π119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119888minus 119888 (119905)

(3)


1and 120576

2 in


1on


1and only






1205761

[Π119903119891(119902 119905 | 120576

1)] when 120576

1

equals 1205851

Π119903119891(119902 119905 | 120585

1)

= 119901119864 min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(4)


1119902[119886119901

minus119896


Π119903119891(119911 | 119902 119905 120585

1)

= (

119911119886120582 (119905)

119898 (119905) 1205851119902

)

1119896

1198641205762

min119898 (119905) 1205851119902

119898 (119905) 1205851119902

119911

1205762




119911

0

(119896 minus




Π119903119891(119902 119905 | 120585

1) = (119911

0119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times 1198641205762

min1

1205762

1199110


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times (1 minus int

1199110

0

(1 minus

119909

1199110

)119891 (119909) 119889119909)


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

119896 (1 minus 119865 (1199110))

119896 minus 1




Π119903119891(119902 119905) = 119864

1205761

[Π119903119891(119902 119905 | 120576

1)]

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(7)

where 1198600= (1198861199110)1119896

[1 minus 119865(1199110)]1198641205761

1205761

(119896minus1)119896


119903119891(119905) =

120582(119905)119898(119905)119896minus1

(1198600(119908 + 119888))

119896


120597Π119903119891

120597119902

= 119860119900120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus1119896


1205972

Π119903119891

1205971199022

= minus

1

119896

1198600120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus(119896+1)119896

(8)


lt 0 so Π119903119891


119903119891(119905) that makesΠ


119903119891120597119902 = 0

then we can obtain

119902lowast

119903119891(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

(119908 + 119888)

)

119896

(9)




Π119903119891(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905) (10)



119889Π119903119891(119905)

119889119905

=

119860119896

0(119896 minus 1) 120582 (119905)119898(119905)

119896minus2

1198981015840

(119905) + 119860119896

01205821015840

(119905)119898(119905)119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V1015840 (119905)

(11)






lowast



lowast

119903119891is given by



when 119889Π119903119891(119905)119889119905 lt 0 119905lowast

119903119891= 119905119897

when 119889Π119903119891(119905)119889119905 gt 0 119905lowast

119903119891= 119905119906


119903119891= argmax Π

119903119891(119905119897

) Π119903119891(1199051) Π119903119891(1199052)

Π119903119891(119905119899) Π119903119891(119905119906

) in which 1199051 1199052 1199053 119905119899 is the






119902lowast

119903119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(

1198600

119908 + 119888

)

119896

(12)




Πlowast

119903119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

(13)



optimal profit

Πlowast

119904119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904) (14)


Πlowast

119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

+ 120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904)

(15)


Π119895(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

(16)


maxΠ119895(119902 119905) = 119864

1205761

Π119895(119902 119905 | 120576

1)

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(17)



119902lowast

119895(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

119888119904+ 119888

)

119896

(18)




Π119895(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905) (19)


as Proposition 7


(1) if 119889Π119895(119905)119889119905 lt 0 then 119905

lowast

119895= 119905119897

(2) if 119889Π119895(119905)119889119905 gt 0 then 119905

lowast

119895= 119905119906



Then 119905lowast

119895= argmax Π

119895(119905119897

) Π119895(1199051198951) Π119895(1199051198952) Π

119895(119905119895119899)

Π119895(119905119906

)




119902lowast

119895= 120582 (119905

lowast

119895)119898(119905lowast

119895)

119896minus1

[

1198600

119888119904+ 119888

]

119896

(20)




Πlowast

119895=

119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905lowast119895) (21)




119903119891=

120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119908 + 119888))


119902lowast

119895= 120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119888119904+119888))119896 when 119905 = 119905

119906 Because119908 gt 119888119904

119902lowast

119903119891lt 119902lowast

119895when 119905 = 119905



119895 119905lowast


119895(119902 119905)


119895(119902lowast

119895 119905lowast



119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast



119891(119902lowast

119903119891 119905lowast

119903119891) = Π119895(119902lowast

119903119891 119905lowast

119903119891) Because

Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) and Π

119891(119902lowast

119903119891 119905lowast

119903119891) = Π

119895(119902lowast

119903119891 119905lowast

119903119891)

Π119891(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast





















119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)



119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)


119904 = 120593119901 (24)


119911 = 120593V (119905) (25)



119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)



(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)


Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119897

119905119906






1119872(119905 120576

1) isin [0 1]








2






Assumption 1 119872(119905 1205761) = 119898(119905)120576





minus119896

120582(119905)1205762 where 120576







Π119903119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(1)

Π119904119891

= 119902119908 minus 119902119888119904 (2)

Π119891(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119888minus 119888 (119905)

(3)


1and 120576

2 in


1on


1and only






1205761

[Π119903119891(119902 119905 | 120576

1)] when 120576

1

equals 1205851

Π119903119891(119902 119905 | 120585

1)

= 119901119864 min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(4)


1119902[119886119901

minus119896


Π119903119891(119911 | 119902 119905 120585

1)

= (

119911119886120582 (119905)

119898 (119905) 1205851119902

)

1119896

1198641205762

min119898 (119905) 1205851119902

119898 (119905) 1205851119902

119911

1205762




119911

0

(119896 minus




Π119903119891(119902 119905 | 120585

1) = (119911

0119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times 1198641205762

min1

1205762

1199110


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times (1 minus int

1199110

0

(1 minus

119909

1199110

)119891 (119909) 119889119909)


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

119896 (1 minus 119865 (1199110))

119896 minus 1




Π119903119891(119902 119905) = 119864

1205761

[Π119903119891(119902 119905 | 120576

1)]

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(7)

where 1198600= (1198861199110)1119896

[1 minus 119865(1199110)]1198641205761

1205761

(119896minus1)119896


119903119891(119905) =

120582(119905)119898(119905)119896minus1

(1198600(119908 + 119888))

119896


120597Π119903119891

120597119902

= 119860119900120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus1119896


1205972

Π119903119891

1205971199022

= minus

1

119896

1198600120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus(119896+1)119896

(8)


lt 0 so Π119903119891


119903119891(119905) that makesΠ


119903119891120597119902 = 0

then we can obtain

119902lowast

119903119891(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

(119908 + 119888)

)

119896

(9)




Π119903119891(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905) (10)



119889Π119903119891(119905)

119889119905

=

119860119896

0(119896 minus 1) 120582 (119905)119898(119905)

119896minus2

1198981015840

(119905) + 119860119896

01205821015840

(119905)119898(119905)119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V1015840 (119905)

(11)






lowast



lowast

119903119891is given by



when 119889Π119903119891(119905)119889119905 lt 0 119905lowast

119903119891= 119905119897

when 119889Π119903119891(119905)119889119905 gt 0 119905lowast

119903119891= 119905119906


119903119891= argmax Π

119903119891(119905119897

) Π119903119891(1199051) Π119903119891(1199052)

Π119903119891(119905119899) Π119903119891(119905119906

) in which 1199051 1199052 1199053 119905119899 is the






119902lowast

119903119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(

1198600

119908 + 119888

)

119896

(12)




Πlowast

119903119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

(13)



optimal profit

Πlowast

119904119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904) (14)


Πlowast

119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

+ 120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904)

(15)


Π119895(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

(16)


maxΠ119895(119902 119905) = 119864

1205761

Π119895(119902 119905 | 120576

1)

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(17)



119902lowast

119895(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

119888119904+ 119888

)

119896

(18)




Π119895(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905) (19)


as Proposition 7


(1) if 119889Π119895(119905)119889119905 lt 0 then 119905

lowast

119895= 119905119897

(2) if 119889Π119895(119905)119889119905 gt 0 then 119905

lowast

119895= 119905119906



Then 119905lowast

119895= argmax Π

119895(119905119897

) Π119895(1199051198951) Π119895(1199051198952) Π

119895(119905119895119899)

Π119895(119905119906

)




119902lowast

119895= 120582 (119905

lowast

119895)119898(119905lowast

119895)

119896minus1

[

1198600

119888119904+ 119888

]

119896

(20)




Πlowast

119895=

119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905lowast119895) (21)




119903119891=

120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119908 + 119888))


119902lowast

119895= 120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119888119904+119888))119896 when 119905 = 119905

119906 Because119908 gt 119888119904

119902lowast

119903119891lt 119902lowast

119895when 119905 = 119905



119895 119905lowast


119895(119902 119905)


119895(119902lowast

119895 119905lowast



119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast



119891(119902lowast

119903119891 119905lowast

119903119891) = Π119895(119902lowast

119903119891 119905lowast

119903119891) Because

Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) and Π

119891(119902lowast

119903119891 119905lowast

119903119891) = Π

119895(119902lowast

119903119891 119905lowast

119903119891)

Π119891(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast





















119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)



119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)


119904 = 120593119901 (24)


119911 = 120593V (119905) (25)



119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)



(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)


Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205761

[Π119903119891(119902 119905 | 120576

1)] when 120576

1

equals 1205851

Π119903119891(119902 119905 | 120585

1)

= 119901119864 min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119908 minus 119888 (119905)

(4)


1119902[119886119901

minus119896


Π119903119891(119911 | 119902 119905 120585

1)

= (

119911119886120582 (119905)

119898 (119905) 1205851119902

)

1119896

1198641205762

min119898 (119905) 1205851119902

119898 (119905) 1205851119902

119911

1205762




119911

0

(119896 minus




Π119903119891(119902 119905 | 120585

1) = (119911

0119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times 1198641205762

min1

1205762

1199110


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

times (1 minus int

1199110

0

(1 minus

119909

1199110

)119891 (119909) 119889119909)


= (1199110119886120582 (119905))

1119896

[119898 (119905) 1205851119902](119896minus1)119896

119896 (1 minus 119865 (1199110))

119896 minus 1




Π119903119891(119902 119905) = 119864

1205761

[Π119903119891(119902 119905 | 120576

1)]

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(7)

where 1198600= (1198861199110)1119896

[1 minus 119865(1199110)]1198641205761

1205761

(119896minus1)119896


119903119891(119905) =

120582(119905)119898(119905)119896minus1

(1198600(119908 + 119888))

119896


120597Π119903119891

120597119902

= 119860119900120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus1119896


1205972

Π119903119891

1205971199022

= minus

1

119896

1198600120582(119905)1119896

119898(119905)(119896minus1)119896

119902minus(119896+1)119896

(8)


lt 0 so Π119903119891


119903119891(119905) that makesΠ


119903119891120597119902 = 0

then we can obtain

119902lowast

119903119891(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

(119908 + 119888)

)

119896

(9)




Π119903119891(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905) (10)



119889Π119903119891(119905)

119889119905

=

119860119896

0(119896 minus 1) 120582 (119905)119898(119905)

119896minus2

1198981015840

(119905) + 119860119896

01205821015840

(119905)119898(119905)119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V1015840 (119905)

(11)






lowast



lowast

119903119891is given by



when 119889Π119903119891(119905)119889119905 lt 0 119905lowast

119903119891= 119905119897

when 119889Π119903119891(119905)119889119905 gt 0 119905lowast

119903119891= 119905119906


119903119891= argmax Π

119903119891(119905119897

) Π119903119891(1199051) Π119903119891(1199052)

Π119903119891(119905119899) Π119903119891(119905119906

) in which 1199051 1199052 1199053 119905119899 is the






119902lowast

119903119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(

1198600

119908 + 119888

)

119896

(12)




Πlowast

119903119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

(13)



optimal profit

Πlowast

119904119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904) (14)


Πlowast

119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

+ 120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904)

(15)


Π119895(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

(16)


maxΠ119895(119902 119905) = 119864

1205761

Π119895(119902 119905 | 120576

1)

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(17)



119902lowast

119895(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

119888119904+ 119888

)

119896

(18)




Π119895(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905) (19)


as Proposition 7


(1) if 119889Π119895(119905)119889119905 lt 0 then 119905

lowast

119895= 119905119897

(2) if 119889Π119895(119905)119889119905 gt 0 then 119905

lowast

119895= 119905119906



Then 119905lowast

119895= argmax Π

119895(119905119897

) Π119895(1199051198951) Π119895(1199051198952) Π

119895(119905119895119899)

Π119895(119905119906

)




119902lowast

119895= 120582 (119905

lowast

119895)119898(119905lowast

119895)

119896minus1

[

1198600

119888119904+ 119888

]

119896

(20)




Πlowast

119895=

119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905lowast119895) (21)




119903119891=

120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119908 + 119888))


119902lowast

119895= 120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119888119904+119888))119896 when 119905 = 119905

119906 Because119908 gt 119888119904

119902lowast

119903119891lt 119902lowast

119895when 119905 = 119905



119895 119905lowast


119895(119902 119905)


119895(119902lowast

119895 119905lowast



119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast



119891(119902lowast

119903119891 119905lowast

119903119891) = Π119895(119902lowast

119903119891 119905lowast

119903119891) Because

Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) and Π

119891(119902lowast

119903119891 119905lowast

119903119891) = Π

119895(119902lowast

119903119891 119905lowast

119903119891)

Π119891(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast





















119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)



119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)


119904 = 120593119901 (24)


119911 = 120593V (119905) (25)



119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)



(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)


Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119902lowast

119903119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(

1198600

119908 + 119888

)

119896

(12)




Πlowast

119903119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

(13)



optimal profit

Πlowast

119904119891= 120582 (119905

lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904) (14)


Πlowast

119891=

119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

(119896 minus 1) (119908 + 119888)119896minus1

minus V (119905lowast119903119891)

+ 120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1

[

1198600

119908 + 119888

]

119896

(119908 minus 119888119904)

(15)


Π119895(119902 119905) = 119901119864 min 119872 (119905) 119902 119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

(16)


maxΠ119895(119902 119905) = 119864

1205761

Π119895(119902 119905 | 120576

1)

=

119896

119896 minus 1

1198600119902(119896minus1)119896

120582(119905)1119896

119898(119905)(119896minus1)119896


(17)



119902lowast

119895(119905) = 120582 (119905)119898(119905)

119896minus1

(

1198600

119888119904+ 119888

)

119896

(18)




Π119895(119905) =

119860119896

0120582 (119905)119898(119905)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905) (19)


as Proposition 7


(1) if 119889Π119895(119905)119889119905 lt 0 then 119905

lowast

119895= 119905119897

(2) if 119889Π119895(119905)119889119905 gt 0 then 119905

lowast

119895= 119905119906



Then 119905lowast

119895= argmax Π

119895(119905119897

) Π119895(1199051198951) Π119895(1199051198952) Π

119895(119905119895119899)

Π119895(119905119906

)




119902lowast

119895= 120582 (119905

lowast

119895)119898(119905lowast

119895)

119896minus1

[

1198600

119888119904+ 119888

]

119896

(20)




Πlowast

119895=

119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1

(119896 minus 1) (119888119904+ 119888)119896minus1

minus V (119905lowast119895) (21)




119903119891=

120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119908 + 119888))


119902lowast

119895= 120582(119905119906

)119898(119905119906

)119896minus1

(1198600(119888119904+119888))119896 when 119905 = 119905

119906 Because119908 gt 119888119904

119902lowast

119903119891lt 119902lowast

119895when 119905 = 119905



119895 119905lowast


119895(119902 119905)


119895(119902lowast

119895 119905lowast



119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast



119891(119902lowast

119903119891 119905lowast

119903119891) = Π119895(119902lowast

119903119891 119905lowast

119903119891) Because

Π119895(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast

119895) and Π

119891(119902lowast

119903119891 119905lowast

119903119891) = Π

119895(119902lowast

119903119891 119905lowast

119903119891)

Π119891(119902lowast

119903119891 119905lowast

119903119891) le Π

119895(119902lowast

119895 119905lowast





















119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)



119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)


119904 = 120593119901 (24)


119911 = 120593V (119905) (25)



119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)



(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)


Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























119908 (119905) = 1199080minus 120593 [119901 minus 119898 (119905) 120585

1119901] (22)



119908 (119905) = 119888119904+ 120593 [119898 (119905) 120585

1119901 minus 119888 minus 119888

119904] (23)


119904 = 120593119901 (24)


119911 = 120593V (119905) (25)



119904+ 120593[119898(119905)120585

1119901 minus 119888 minus 119888

119904] 119904 = 120593119901 and 119911 = 120593V(119905)



(119902 119905) = 1198641205761

[Π119903119909

(119902 119905 | 1205761)]

= 1198641205761

1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)]

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911

(26)


Π119903119909

(119902 119905)

= 1198641205761

119901 119898 (119905) 1205851119902 minus 1198641205762

119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus 119902119908 (119905) minus 119888 (119905) + 1199041198641205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

+ 119911


= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus V (119905) + 120593V (119905)

= 1198641205761


1205762

times 119898 (119905) 1205851119902 minus 119863 [119901 120582 (119905)]

+

minus (1 minus 120593) V (119905)

= (1 minus 120593) 1198641205761

times 1199011198641205762

min 119898 (119905) 1205851119902119863 [119901 120582 (119905)] minus 119902119888

119904minus 119888 (119905)

= (1 minus 120593) 1198641205761

Π1198952(119902 119905 | 120576

1)

= (1 minus 120593)Π119895(119902 119905)

(27)



lowast


119895This shows




120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus (((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ])

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(28)


lowast

119903119909= (1 minus 120593)Π

lowast

119895ge Πlowast

119903119891 and Π

lowast

119904119909= 120593Πlowast

119895ge Πlowast

119904119891


120593min

= 119860119896

0[120582 (119905lowast

119903119891)119898 (119905

lowast

119903119891) (119888119904+ 119888)]

119896minus1

(119908 minus 119888119904) (119896 minus 1)

times ((119908 + 119888)119896

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

120593max

= 1 minus ((119888119904+ 119888)119896minus1

[119860119896

0120582 (119905lowast

119903119891)119898(119905lowast

119903119891)

119896minus1


119903119891) ]

times ((119908 + 119888)119896minus1

[119860119896

0120582 (119905lowast

119895)119898(119905lowast

119895)

119896minus1


119895) ])

minus1

)

(29)


119903119891Πlowast


119904119891Πlowast

119895= [Π

lowast

119895minus

(Πlowast

119903119891+ Πlowast

119904119891)]Πlowast






119904= 4 119908 = 6 119901 = 12 The


120573minus1 where 120572 =


0


1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205820



1isin 119880[0 2] 120576



1199110=

4

119896 + 1

119865 (1199110) =

2

119896 + 1

(30)








119902lowast

119905lowast

Πlowast

119903Πlowast

119904Πlowast



120593 Πlowast

119903119909ΔΠ119903= Πlowast


119903119891Πlowast

119904119909ΔΠ119904= Πlowast


119904119891

02041 8948 918 2294 002245 8718 688 2524 23002449 8489 459 2753 45902653 8259 229 2983 68902857 8030 0 3212 918









120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

010 11472 9957 22486 9940 02041 02857 00816012 11106 9952 21769 9933 02042 02858 00816014 10740 9947 21051 9926 02042 02859 00817016 10373 9942 20334 9919 02044 02861 00817018 10007 9937 19617 9912 02045 02864 00819020 9641 9932 18899 9905 02048 02868 00820


1205820


119903119891119905lowast

119903119891119902lowast

119895119905lowast

119895120593min 120593max Δ120593

0995 11027 990 21620 985 02041 02858 008170996 11137 991 21832 988 02040 02856 008160997 11247 993 22047 990 02040 02856 008160998 11359 994 22266 992 02039 02854 008150999 11472 996 22486 994 02037 02850 00813

02000400060008000

1000012000

012 014 016 018 0201

Profi

t

RetailerSupplier


120572






0either in







02000400060008000

1000012000

0995 0996 0997 0998 0999

Profi

t

RetailerSupplier


1205820





7 Conclusions










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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