Research ArticleContract Coordination in Dual Sourcing Supply Chain underSupply Disruption Risk

Tong Shu1 Fang Yang1 Shou Chen1 Shouyang Wang12 Kin Keung Lai34 and Lu Gan5

1Business School Hunan University Changsha Hunan 410082 China2Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing 100080 China3International Business School Shaanxi Normal University Xirsquoan 710062 China4Department of Management Sciences City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong5Office of Humanities and Social Sciences Hunan University Changsha 410082 China

Correspondence should be addressed to Tong Shu shutonghnueducn

Received 26 June 2015 Revised 11 August 2015 Accepted 12 August 2015

Academic Editor Young Hae Lee

Copyright copy 2015 Tong Shu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper explores a coordination model for a three-echelon supply chain including two different manufacturers one distributerand one retailer via the combined option and back contracts And one manufacturer provides the high wholesale price with lowsupply disruption risk and the other is completely the opposite This differs from the previous supply chain coordination modelFirstly supply disruption is added to the three-echelon supply chain Secondly considering the coordination of the supply chainwe deploy the combined option and back contracts which are seldom used in the previous study Furthermore it is interesting thatsupply disruption risk and buyback factor do not affect the distributorrsquos order quantity from themanufacturer who has low productprice and unreliable operating ability while the order quantity increases with the rise of option premium and option strike priceThe distributorrsquos order quantity from the manufacturer which has high product price and reliable operating ability increases withthe rise of supply disruption risk but decreases when the buyback factor option premium and option strike price decrease

1 Introduction

With the growing popularity of the online shopping logis-tics industry in China has shown significant developmentrecently Nevertheless many unexpected changesmay hinderthe normal operation of the supply chain for instance theinsufficient supply of spare parts in Toyota in 1997 theshortage of chips in Apple in 1999 the fire at a supplierof Ericsson in 2000 and the sea earthquake in Miyagi inJapan resulting in disruption of supply of car spare parts on11 March 2011 All of the accidents discussed above haveled to a sense of danger among the people or a great lossto both the local economy and peoplersquos lives Supply chaincan be disrupted by many events such as natural disastersbankruptcy strikes by workers terrorist attacks and policyfailures Supply chain enterprises have to face diverse externalrisks and the internal risks of supply chains are ubiquitousSupply chain enterprises are independent economic entitiesin market pursuing the maximum individual profits and

potential conflicts of interest exist as stance rationalityknowledge background andmindsets of enterprises vary andengender variations in understanding All the players of asupply chain are greatly affected by such demand disruptionswhich can also affect the performance of a supply chainsignificantly and cause irreversible losses to the supply chainThis poses a challenge to managers regarding what can bedone to maintain coordination of the supply chain andreduce the damage The issue of how to tackle the uncertaindisruptions efficiently and effectively has become increasinglysignificant to the managers nowadays To the best of ourknowledge little attention has been paid to such problems incurrent research and how to coordinate the supply chain withinterruption risk is another problem we intend to tackle

With regard to the problem of coordinating the supplychain option contract has been widely used And the optioncontract means that the player orders some product beforethe selling season with certain wholesale price per unit andpurchases the product with another strike price per unit Its

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 473212 10 pageshttpdxdoiorg1011552015473212

2 Mathematical Problems in Engineering

advantage is that the option contract is beneficial to solve thesupply chain in an unstable environment which had beeninvestigated in the previous study Ritchken and Tapiero [1]argued that option contracts could hedge the risk caused byproduct prices and quantity fluctuations but Barnes-Schusteret al [2] proposed a two-stage model in an option contractwhere the use of options could help sellers cope with marketchanges and improve flexibility Burnetas and Ritchken [3]noted that the introduction of options could lead to a risein wholesale prices and retail prices tended to be stableat the same time and therefore manufacturers had to takethe relevant conditions into consideration when they selectoptions Wang and Liu [4] applied Stackelbergrsquos game toanalyzing and determining the conditions to achieve supplychain coordination where retailers with option contractsplay a major role Ning and Dai [5] developed a one-to-many model deploying options to coordinate supply chainenterprises to improve the capacity to cope with marketchanges Shang et al [6] discussed the three-stage processingand ordering strategies in option contracts and showed thatat the moment manufacturers did not have the motivationto speculate and the overall efficiency of the supply chainsystem and its members had increased but the extent ofincrease depended on the negotiating capacity of supplychain enterprises in relation to option purchasing pricesLin and Fan [7] coordinated two-echelon supply chain withuncertainty via option contract Du et al [8] integrated thecontract of wholesale price discounts with option contractsand identified the contract parameters which increase oper-ational efficiency of supply chains and help supply chainmembers share profit increases equally The literature hasthus far shown that the option contract is based on thefixed ordering contract by individual suppliers in the two-stage model Tian et al [9] developed emergency suppliespurchasing model based on capacity option contract withdual purchasing sources Y Luo and Y J Luo [10] studied theagricultural produce supply chains and obtained the optimalorder and supply strategy with circulation loss and optioncontract Liu et al [11] applied option contract to tacklingoverloading problems in the delivery service supply chainLuo et al [12] studied the coordination of a supply chain withdual procurement sources via real-option contract Ma andZeng [13] explored order strategy of retailers with stochasticdemand based on payment in advance and option contract

In addition to the option contract growing importancehas also attached to the buyback contract because of itsadvantage For example it is easy to calculate the parameterof buyback or it is equal to the revenue sharing contract[14] which has received more attention Pasternack [15]proposed that buyback contracts can be used to achievesupply chain coordination Cachon [14] argued that recyclingand compensation could help coordinate supply chain tosome degree Yu et al [16]maintained that supply chains werevery robust under buyback contracts and buyback contractscould be reset to achieve supply chain coordination after anydisruptions or accidents Jia et al [17] testified that in thetwo-echelon supply chain including suppliers and retailersif the inventory cost of retailers is nonlinear the type ofbuyback contract in supply chains depended on retail prices

of suppliers Hu and Wang [18] discussed the coordinatingfunction of buyback contracts in three-echelon supply chainsin the event of an accident on the basis of random demandSome studies integrate buyback contracts with other typesof contracts For instance Hou and Qiu [19] incorporatedbuyback contracts into revenue-sharing contracts while afew studies have combined buyback contracts with optioncontracts Xu et al [20] noted that in restricted buybackcontracts if the quantity of products eligible for buybackwas limited retailers had to choose the optimal retail pricesand ordering quantities and suppliersrsquo profits increasedas the buyback prices rise which was vice versa for theretailers Chen [21] analyzed the impact of sales return onordering quantity and wholesale prices in buyback contractsby applying the Stackelberg game and developed amotivationmechanism for sharing information among supply chainmembers Guang et al [22] investigated a buyback contractin a two-echelon supply chain consisting of a risk-neutralsupplier and a risk-averse retailer and if retailers are risk-averse supply chain can also achieve coordination Thereis an enormous amount of research on buyback contractswhich extends into the three-echelon supply chains whereasthere are only a few models of network supply chains

Most of the research discussed above is concerned withone single option contract or buyback contract based ontwo-echelon supply chains whose external environment isrelatively stable It is seldom to see the use of combinedcontracts to tackle the supply chain with disruption riskunder unstable environment which is not inevitable inpractice The buyback contract has been extensively studiedin the previous research which has received considerableattention in practice Certainly the option contract has itsown advantage and thus we choose the combined contractTherefore this paper attempts to address these problemsFirstly integrating the advantages of the option contract andthe buyback contract this paper applies the two contractstogether to coordinating the supply chain Secondly to becloser to the real-life environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the recent situation wecannot neglect the disruption risk factors in the supply chainConsequently supply disruption risk factors are consideredin this paper And the buyback contract is deployed tostimulate and lead the retailers to increase ordering quantitythe distributor shares the partial risk engendered by demanduncertainty a balance can be struck between marginalrevenue and marginal cost of distributors and retailers Thedisruption risk can be hedged by distributors who selectoptions

2 Model Description

21 Notations and Assumptions Here the single three-echelon supply chain includes two manufacturers one dis-tributor and one retailer among which Manufacturer 1 canreduce supplies of low-price products and supply disruptionis likely to occur Manufacturer 2rsquos products have a relativelyhigher price but they are stable and reliable The upstream

Mathematical Problems in Engineering 3

Manufacturer 1

Manufacturer 2

The distributor The retailerBuyback contract

Option contract

Figure 1 Research network chart

enterprises provide single products for downstream enter-prises that are mutually independent without cross-echelonrelations Before sales the manufacturer and the distribu-tor offer contracts to the downstream enterprises retailersdetermine the ordering quantity in terms of the marketdemand and the contract provided by the distributor Atthe same time the distributor determines their orderingquantity according to the ordering quantity of retailers andthe contract of the manufacturers (see Figure 1) Manufac-turer 1 provides products for the distributor according tothe wholesale prices 119908

119898 as the supply of Manufacturer 1

is likely to be disrupted the distributor will determine theordering quantity based on the option contract offered byManufacturer 2 in order to ensure more stable sourcing oftheir products Before selling seasons the distributors reserve1199021units of products from Manufacturer 1 at the wholesale

price they reserve units of option purchasing quantity fromManufacturer 2 At the initial stage of the selling seasonthe distributor buys products within the option purchasingquantity 119902

2at a certain price fromManufacturer 2 on the basis

of the disrupted information obtained from Manufacturer 1in order to stimulate the product ordering from the retailerthe distributor is able to provide a buyback contract (120573 119908

119889)

for the retailer The distributor sells products at price 119908119889per

unit and after the selling season unsold products are boughtback at 120573-fold of the distributorrsquos price

At the same time the symbols used in the modes areshown in Notations The superscript lowast denotes the optimumordering value of retailers in allied contracts

The following are the hypotheses used for building andtesting the models

Assumption 1 Participants in supply chains are completelyrational and they are risk-neutral

Assumption 2 All distribution functions are two-echelon anddifferential and there are strict single inverse functions

Assumption 3 With 119908119889gt 119908119898gt 119888 gt V the limited profits

of Manufacturer 1 are ensured and profits of distributors arealso guaranteed

Assumption 4 With 119908119889gt 119890 + ℎ gt 119908

119898 119890 + V lt 119908

119898 validity of

the option contract is guaranteed

Assumption 5 With 119903 gt 119908119889 120573119908119889gt V the retailerrsquos profits are

ensured and validity of buyback contract is guaranteed

Assumption 6 Manufacturers play leading roles and thedistributer acts as the follower

22 OptimumOrdering Strategies of Centralized Supply ChainIn centralized supply chains the manufacturers the distrib-utor and the retailer are considered as a whole and theobjective is tomaximize the overall profits of the supply chainregardless of the internal transference of payments betweenmember enterprises There is no ldquodual-marginalized effectrdquoin supply chains and it is a typical newsvendor model

Below is the overall profit of supply chains when disrup-tions occur to Manufacturer 1

Π = 119903min (1199022 119909) + Vmax (119902

2minus 119909 0) minus 119888119902

2

minus 119892max (119909 minus 1199022 0)

(1)

Below is the overall profit of supply chain when disrup-tions do not happen to Manufacturer 1

Π = 119903min (1199021+ 1199022 119909) + Vmax (119902

1+ 1199022minus 119909 0)

minus 119888 (1199021+ 1199022) minus 119892max (119909 minus 119902

1minus 1199022 0)

(2)

Now below is the overall profit expected of supply chain

119864 (Π) = 119901 [119903min (1199022 119909) + Vmax (119902

2minus 119909 0) minus 119888119902

2

minus 119892max (119909 minus 1199022 0)] + (1 minus 119901) [119903119898119894119899 (119902

1+ 1199022 119909)

+ Vmax (1199021+ 1199022minus 119909 0) minus 119888 (119902

1+ 1199022)

minus 119892max (119909 minus 1199021minus 1199022 0)]

(3)

WithMax(119864(Π)) and 1199021 1199022ge 0 the expected profit is the

concave function of 1199021and 119902

2 making 120597119864(Π)120597119902

1= 0 and

120597119864(Π)1205971199022= 0

The optimum ordering quantity can be derived based onthe above equations

Theorem 1 The optimum ordering quantity is

119876lowast

119888= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) (4)

Proof Equation (3) can be written as

4 Mathematical Problems in Engineering

119864 (Π) = 119901 [119903 (120583 + int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) + V(119902

2minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) minus 119888119902

2

minus 119892(1199022minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909)] + (1 minus 119901) [119903(120583 + int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)

+ V(1199021+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909) minus 119888 (119902

1+ 1199022) minus 119892(119902

1+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)]

120597119864 (Π)

1205971199021

= 0

120597119864 (Π)

1205971199022

= 0

997904rArr

(1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888) = 0

119901 ((119903 minus 119908 + 119892)int

+infin

1199022

119891 (119909) 119889119909 + (119908 minus 119888) + (1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888)) = 0

997904rArr

(119903 minus 119908 + 119892) 119865 (1199022) + (119908 minus 119888) = 0

(119903 minus 119908 + 119892) 119865 (1199021+ 1199022) + (119908 minus 119888) = 0

997904rArr 119865 (1199022) = 119865 (119902

1+ 1199022) =

119903 + 119892 minus 119888

119903 + 119892 minus V

1199022= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) 1199021= 0

(5)

Then we can get Theorem 1 This is the end of the proof

3 Coordination of the Decentralized SupplyChain via Allied Contracts

Manufacturers play a leading role in supply chains The opti-mal wholesale price and option contracts can be determinedby the possible responses of the distributors and distributorsfollow manufacturers The distributorrsquos optimal orderingquantity is determined by the manufacturerrsquos informationThen the distributor plays a leading role and the Stackelberggame comes into play between the distributor and the retailerThe decision is made through Stackelberg reverse inductionThe sequence of steps in the option order is as follows

(a) At the beginning the distributor reserves the future1199022fromManufacturer 2

(b) The distributor is informed about the disruption fromManufacturer 1

(c) The distributor obtains the ordering informationfrom the retailer

(d) The distributor invokes some option contracts fromManufacturer 2

(e) Manufacturer 2 satisfies the distributorrsquos demands

31 Distributorsrsquo Decision-Making Process Calculating thefirst-order value of the distributorrsquos profit we can get the

optimal order quantity from the manufacturer Below isthe distributorsrsquo profit when disruptions occur in whichthe notations 119901 119890 and119891 denote the probability of supplydisruptions the order price per unit and the strike price perunit in the option contract

Π119889= (119908119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2

(6)

Below is the distributorrsquos profit without disruptions

Π119889= 119908119889min (119902

1 119909)

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022

(7)

Now below is the expected profit of the distributor

119864 (Π119889) = 119901 [(119908

119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2] + (1 minus 119901)

sdot [119908119889min (119902

1 119909)

Mathematical Problems in Engineering 5

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022]

(8)

With 119901 lt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) the optimum ordering quantity

of the supply chain system is

119876lowast

119889= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V + 120573119908

119889

) (9)

The proof of 119876lowast119889 119902lowast1 and 119902

lowast

2is similar to that of (4) to

which detailer process has been added For brevity it is notnecessary to recalculate it

With 119901 gt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+119892minusℎminusV+120573119908

119889) the optimumordering quantity of the

supply chain system is to satisfy the solution of 120597[prod119889]1205971199021=

0 120597[prod119889]1205971199022= 0119876lowast

119889gt 119865minus1

((119908119889minus119908119898+ 119892 minus ℎ)(119908

119889minus V + 119892 minus

ℎ + 120573119908119889)) and119876lowast

119889gt 119865minus1

((119908119889minus 119890 + 119892 minus ℎ minus 119901(119908

119889minus V + 119892 minus ℎ +

120573119908119889))(119908119889minus V + 119892 minus ℎ + 120573119908

119889)(1 minus 119901))

The optimal conditions of KKT (the Karush-Kuhn-Tucker) are based on the ideas proposed by Karush [23] andKuhn and Tucker [24] which indicates that a linear program-ming problem is able to have the necessary and sufficientconditions for the best solution equivalent to a Lagrangemultiplication in a broad sense One of the conditions ofKKT (the Karush-Kuhn-Tucker) is that the optimum mustbe a possible solution and satisfy the restrictive conditions ofinequality and equation

Testifying The target function is the strict concave functionand its restriction is linearity and there is only one optimalsolution which is achieved through the condition of KKT

120597Π

1205971199021

+ 1205821= (1 minus 119901) [119908

119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022)] + 120582

1= 0

(10)

120597Π

1205971199022

+ 1205822= 119908119889+ 119892 minus 119890 minus ℎ minus 119901 (119908

119889minus V + 120573119908

119889minus ℎ

+ 119892) 119865 (1199022) minus (1 minus 119901) (119908

119889minus V + 120573119908

119889minus ℎ + 119892) 119865 (119902

1

+ 1199022) + 1205822= 0

(11)

12058211199021= 0 (12)

12058221199022= 0 (13)

1205821 1205822 1199021 1199022ge 0 (14)

The four possibilities to be analyzed are (a) 1205821 1205822gt 0 (b)

1205821gt 0 120582

2= 0 (c) 120582

1= 0 120582

2gt 0 and (d) 120582

1= 0 120582

2= 0

Case (a) With 1205821 1205822gt 0 from (11) and (12) 119902

1 1199022= 0 can

be derived now from (10) 1205821= minus(1 minus 119901)(119908

119889+ 119892 minus 119908

119898) lt 0

is known which is in conflict with the hypothesis and thus itis not the optimum

Case (b)With 1205821gt 0 and 120582

2= 0 119902

1= 0 is derived from (11)

and now 119865(1199022) = (119903+119892minus119890minusℎ)(119903+119892minusℎ) is derived from (11)

When it is incorporated into (10)1205821= minus(1minus119901)[(119890+ℎ)minus119908

119898] lt

0 is derived which is inconsistent with the hypothesis Thusthe solution is not the optimum

Case (c)With 1205821= 0 and 120582

2gt 0 119902

2= 0 is derived from (12)

and now 119865(1199021) = (119908

119889+119892minus119908

119898)(119908119889+119892minus V+120573119908

119889) is derived

from (10) It is incorporated into (11) and 1205822= (1minus119901)(119908

119889+119892minus

ℎminusV+120573119908119889)((119908119889+119892minus119908

119898)(119908119889+119892minusV+120573119908

119889))minus(119908

119889+119892minusℎminus119890)

When 119901 lt 1 minus (119908119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V + 120573119908

119889)(119908119889+ 119892 minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is satisfied 120582

2gt 0 is satisfied

and now 119902lowast

1= 119865minus1

((119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus V + 120573119908

119889)) and

119902lowast

2= 0 is the optimal solution and all the conditions of KKT

are satisfied

Case (d) As the only optimal solution is obtained thecorollary is that 119901 gt 1 minus (119908

119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V +

120573119908119889)(119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is the optimal

solution with 1205821= 0 and 120582

2= 0 and the conditions of KKT

are satisfiedTheoptimal solution has satisfied 120597[prod119889]1205971199021= 0

and 120597[prod119889]1205971199022= 0 with 119902lowast

1gt 0 and 119902lowast

2gt 0

32 The Retailerrsquos Decision-Making Process Calculating thefirst derivation value of the retailerrsquos profit we can get theoptimal order quantity from the distributor and the optimalwholesale price given by the distributor to the retailer Belowis the retailerrsquos profit with disruptions

Π119903= 119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0)

minus 119892max (119909 minus 1199022 0) minus 119908

1198891199022

(15)

Below is the retailerrsquos profit without disruptions

Π119903= 119903min (119902

1+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)

(16)

Now below is the expected retailerrsquos profit

119864 (Π119903) = 119901 [119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0) minus 119892

sdotmax (119909 minus 1199022 0) minus 119908

1198891199022] + (1 minus 119901) [119903

sdotmin (1199021+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)]

(17)

When the above equation is calculated for 1199021and 119902

2

119865(119902lowast

2) = (119903 + 119892 minus 119908

119889)(119903 + 119892 minus 120573119908

119889) 119902lowast1= 0 are derived

and below is the optimum ordering quantity of the retailer

119876lowast

119903= 119865minus1

(

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

) (18)

33 Coordination of the Decentralized Supply Chain Herethe option contract is combined with buyback contract and

6 Mathematical Problems in Engineering

the three-echelon supply chain is coordinated and optimizedthrough designing the corresponding parameters In theallied contracts the following conditions must be satisfied toachieve complete coordination of the supply chain

Proposition 2 In the allied contract consisting of the optioncontract and the buyback contract if complete coordination isto be achieved in the supply chain then the contract parametersmust satisfy the following cases

Case 1 Consider the following

119901 lt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(19)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (20)

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V minus 120573119908

119889

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(21)

Case 2 Consider the following

119901 gt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(22)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (23)

max(119908119889minus 119908119898+ 119892 minus ℎ

119908119889minus V + 119892 minus ℎ + 120573119908

119889

119908119889minus 119890 + 119892 minus ℎ minus 119901 (119908

119889minus V + 119892 minus ℎ + 120573119908

119889)

(119908119889minus V + 119892 minus ℎ + 120573119908

119889) (1 minus 119901)

)

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(24)

Equations (19) and (22) are two different sets of circum-stances where disruptions are likely to occur in (20) and (23)the validity of contract coordination under different risks isensured in (21) and (24) the effective circulation of supplychain products under different risks is ensured In order toguarantee the validity and the continuity of supply chainoperations the ordering quantity purchased by distributorsfrom manufacturers has to be more than or equal to thatpurchased by the retailer from the distributor Only if theordering quantity of the retailer is consistent with that of thecentralized decision can coordination be achieved

Proposition 3 When the three-stage supply chain is coordi-nated the buyback factor 120573must satisfy the following equation120573 = ((119903 + 119892)(119908

119889+ V minus 119888) minus V119908

119889)119908119889(119903 + 119892 minus 119888)

Corollary 4 When (119908119889+119892+120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt ℎ(119908

119889+

119892 minus 119908119898) the reliable Manufacturer 2 is always deployed

Corollary 5 The increase of the ordering quantity fromManufacturer 1 does not follow the rise of the disruption risksand the ordering quantity fromManufacture 2 does not declinewith the rise of the disruption risks

Corollary 6 The larger the buyback factor 120573 is the larger theordering quantity from retailer is The ordering quantity bydistributors fromManufacturer 1 does not increase with the riseof 120573 and the ordering quantity from Manufacturer 2 does notdecline with the rise of 120573

Corollary 7 The ordering quantity by distributors fromMan-ufacturer 1 does not decline with the rise of option purchasingprices and option strike prices and the ordering quantity by thedistributor does not increase with the rise of option premiumand the option strike prices

Testifying It can be deduced from (20) and (23)

Proof of Corollary 4 When (119908119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt

ℎ(119908119889+119892minus119908

119898) then 1minus(119908

119889+119892minus119890minusℎ)(119908

119889+119892minusV+120573119908

119889)(119908119889+

119892 minus119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) = ((119908

119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus

119908119898)minusℎ(119908

119889+119892minus119908

119898))(119908119889+119892minus119908

119898)(119908119889+119892minusℎminusV+120573119908

119889) lt 0

from 119901 ge 0 119901 gt 1 minus (119908119889+119892minus 119890 minus ℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+

119892 minus 119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is derived from Case (d) in

Section 31 119902lowast2gt 0 is derived and the test is completed

Proof of Corollary 5 With 1199021015840

1= 119889119902

lowast

1(119901)120597119901 and 119902

1015840

2=

119889119902lowast

2(119901)120597119901 from 120597prod

1198891205971199021= 0 the following can be derived

119908119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022) = 0

(25)

As the first derivation of 119901 is carried out with 120597prod1198891205971199021=

0 ℎ11990210158401119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(1199021015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved Thus the plus and minus signs of 11990210158401and 1199021015840

2are

the opposite As the first derivation of 119901 is conducted bysubstituting (25) into 120597prod

1198891205971199022= 0 (119908

119889minus ℎ + 120573119908

119889minus V +

119892)[minus119865(1199022) + 119901119902

1015840

2119891(1199022) + 119865(119902

1+ 1199022)] + (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is

achieved To satisfy the equation with 11990210158401le 0 1199021015840

2ge 0 and the

test is completed

Proof of Corollary 6 With 119876lowast

119903= 119865minus1

((119903 + 119892 minus 119908119889)(119903 +

119892 minus 120573119908119889)) the more the buyback factor 120573 is the greater

(119903 + 119892 minus 119908119889)(119903 + 119892 minus 120573119908

119889) is Also since 119865(119909) is continuous

and differentiable within intervals and it is strictly increasingthe ordering quantity of the retailer 119876lowast

119903increases with the

rise of the buyback factor 120573 With 1199021015840

1= 119889119902

lowast

1(119901)120597120573 and

1199021015840

2= 119889119902lowast

2(119901)120597120573 the first derivation of 120573 is carried out for

120597prod1198891205971199021= 0 and ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+

1199021015840

2)119891(1199021+ 1199022) + 119908119889119865(1199021+ 1199022) = 0 is achieved As such the

sign of 11990210158401is not positive Substituting (25) into 120597prod

1198891205971199022= 0

for the first derivation of 120573 119901119908119889119865(1199022) + 119901(119908

119889minusℎ+120573119908

119889minus V+

119892)1199021015840

2119891(1199022) minus (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is achieved To satisfy the

equation the sign of 11990210158401is not positive and 1199021015840

2lt 0 is known

and hence 11990210158401le 0 1199021015840

2lt 0 and the test is completed

Proof of Corollary 7 With 11990210158401= 119889119902lowast

1(119901)120597119890 and 1199021015840

2= 119889119902lowast

2(119901)

120597119890 (25) is obtained from 120597prod1198891205971199021= 0

The first derivation 119890 is carried out for 120597prod1198891205971199021= 0 and

ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved As such the plus and minus signs of 11990210158401 11990210158402are

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

advantage is that the option contract is beneficial to solve thesupply chain in an unstable environment which had beeninvestigated in the previous study Ritchken and Tapiero [1]argued that option contracts could hedge the risk caused byproduct prices and quantity fluctuations but Barnes-Schusteret al [2] proposed a two-stage model in an option contractwhere the use of options could help sellers cope with marketchanges and improve flexibility Burnetas and Ritchken [3]noted that the introduction of options could lead to a risein wholesale prices and retail prices tended to be stableat the same time and therefore manufacturers had to takethe relevant conditions into consideration when they selectoptions Wang and Liu [4] applied Stackelbergrsquos game toanalyzing and determining the conditions to achieve supplychain coordination where retailers with option contractsplay a major role Ning and Dai [5] developed a one-to-many model deploying options to coordinate supply chainenterprises to improve the capacity to cope with marketchanges Shang et al [6] discussed the three-stage processingand ordering strategies in option contracts and showed thatat the moment manufacturers did not have the motivationto speculate and the overall efficiency of the supply chainsystem and its members had increased but the extent ofincrease depended on the negotiating capacity of supplychain enterprises in relation to option purchasing pricesLin and Fan [7] coordinated two-echelon supply chain withuncertainty via option contract Du et al [8] integrated thecontract of wholesale price discounts with option contractsand identified the contract parameters which increase oper-ational efficiency of supply chains and help supply chainmembers share profit increases equally The literature hasthus far shown that the option contract is based on thefixed ordering contract by individual suppliers in the two-stage model Tian et al [9] developed emergency suppliespurchasing model based on capacity option contract withdual purchasing sources Y Luo and Y J Luo [10] studied theagricultural produce supply chains and obtained the optimalorder and supply strategy with circulation loss and optioncontract Liu et al [11] applied option contract to tacklingoverloading problems in the delivery service supply chainLuo et al [12] studied the coordination of a supply chain withdual procurement sources via real-option contract Ma andZeng [13] explored order strategy of retailers with stochasticdemand based on payment in advance and option contract

In addition to the option contract growing importancehas also attached to the buyback contract because of itsadvantage For example it is easy to calculate the parameterof buyback or it is equal to the revenue sharing contract[14] which has received more attention Pasternack [15]proposed that buyback contracts can be used to achievesupply chain coordination Cachon [14] argued that recyclingand compensation could help coordinate supply chain tosome degree Yu et al [16]maintained that supply chains werevery robust under buyback contracts and buyback contractscould be reset to achieve supply chain coordination after anydisruptions or accidents Jia et al [17] testified that in thetwo-echelon supply chain including suppliers and retailersif the inventory cost of retailers is nonlinear the type ofbuyback contract in supply chains depended on retail prices

of suppliers Hu and Wang [18] discussed the coordinatingfunction of buyback contracts in three-echelon supply chainsin the event of an accident on the basis of random demandSome studies integrate buyback contracts with other typesof contracts For instance Hou and Qiu [19] incorporatedbuyback contracts into revenue-sharing contracts while afew studies have combined buyback contracts with optioncontracts Xu et al [20] noted that in restricted buybackcontracts if the quantity of products eligible for buybackwas limited retailers had to choose the optimal retail pricesand ordering quantities and suppliersrsquo profits increasedas the buyback prices rise which was vice versa for theretailers Chen [21] analyzed the impact of sales return onordering quantity and wholesale prices in buyback contractsby applying the Stackelberg game and developed amotivationmechanism for sharing information among supply chainmembers Guang et al [22] investigated a buyback contractin a two-echelon supply chain consisting of a risk-neutralsupplier and a risk-averse retailer and if retailers are risk-averse supply chain can also achieve coordination Thereis an enormous amount of research on buyback contractswhich extends into the three-echelon supply chains whereasthere are only a few models of network supply chains

Most of the research discussed above is concerned withone single option contract or buyback contract based ontwo-echelon supply chains whose external environment isrelatively stable It is seldom to see the use of combinedcontracts to tackle the supply chain with disruption riskunder unstable environment which is not inevitable inpractice The buyback contract has been extensively studiedin the previous research which has received considerableattention in practice Certainly the option contract has itsown advantage and thus we choose the combined contractTherefore this paper attempts to address these problemsFirstly integrating the advantages of the option contract andthe buyback contract this paper applies the two contractstogether to coordinating the supply chain Secondly to becloser to the real-life environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the recent situation wecannot neglect the disruption risk factors in the supply chainConsequently supply disruption risk factors are consideredin this paper And the buyback contract is deployed tostimulate and lead the retailers to increase ordering quantitythe distributor shares the partial risk engendered by demanduncertainty a balance can be struck between marginalrevenue and marginal cost of distributors and retailers Thedisruption risk can be hedged by distributors who selectoptions

2 Model Description

21 Notations and Assumptions Here the single three-echelon supply chain includes two manufacturers one dis-tributor and one retailer among which Manufacturer 1 canreduce supplies of low-price products and supply disruptionis likely to occur Manufacturer 2rsquos products have a relativelyhigher price but they are stable and reliable The upstream

Mathematical Problems in Engineering 3

Manufacturer 1

Manufacturer 2

The distributor The retailerBuyback contract

Option contract

Figure 1 Research network chart

enterprises provide single products for downstream enter-prises that are mutually independent without cross-echelonrelations Before sales the manufacturer and the distribu-tor offer contracts to the downstream enterprises retailersdetermine the ordering quantity in terms of the marketdemand and the contract provided by the distributor Atthe same time the distributor determines their orderingquantity according to the ordering quantity of retailers andthe contract of the manufacturers (see Figure 1) Manufac-turer 1 provides products for the distributor according tothe wholesale prices 119908

119898 as the supply of Manufacturer 1

is likely to be disrupted the distributor will determine theordering quantity based on the option contract offered byManufacturer 2 in order to ensure more stable sourcing oftheir products Before selling seasons the distributors reserve1199021units of products from Manufacturer 1 at the wholesale

price they reserve units of option purchasing quantity fromManufacturer 2 At the initial stage of the selling seasonthe distributor buys products within the option purchasingquantity 119902

2at a certain price fromManufacturer 2 on the basis

of the disrupted information obtained from Manufacturer 1in order to stimulate the product ordering from the retailerthe distributor is able to provide a buyback contract (120573 119908

119889)

for the retailer The distributor sells products at price 119908119889per

unit and after the selling season unsold products are boughtback at 120573-fold of the distributorrsquos price

At the same time the symbols used in the modes areshown in Notations The superscript lowast denotes the optimumordering value of retailers in allied contracts

The following are the hypotheses used for building andtesting the models

Assumption 1 Participants in supply chains are completelyrational and they are risk-neutral

Assumption 2 All distribution functions are two-echelon anddifferential and there are strict single inverse functions

Assumption 3 With 119908119889gt 119908119898gt 119888 gt V the limited profits

of Manufacturer 1 are ensured and profits of distributors arealso guaranteed

Assumption 4 With 119908119889gt 119890 + ℎ gt 119908

119898 119890 + V lt 119908

119898 validity of

the option contract is guaranteed

Assumption 5 With 119903 gt 119908119889 120573119908119889gt V the retailerrsquos profits are

ensured and validity of buyback contract is guaranteed

Assumption 6 Manufacturers play leading roles and thedistributer acts as the follower

22 OptimumOrdering Strategies of Centralized Supply ChainIn centralized supply chains the manufacturers the distrib-utor and the retailer are considered as a whole and theobjective is tomaximize the overall profits of the supply chainregardless of the internal transference of payments betweenmember enterprises There is no ldquodual-marginalized effectrdquoin supply chains and it is a typical newsvendor model

Below is the overall profit of supply chains when disrup-tions occur to Manufacturer 1

Π = 119903min (1199022 119909) + Vmax (119902

2minus 119909 0) minus 119888119902

2

minus 119892max (119909 minus 1199022 0)

(1)

Below is the overall profit of supply chain when disrup-tions do not happen to Manufacturer 1

Π = 119903min (1199021+ 1199022 119909) + Vmax (119902

1+ 1199022minus 119909 0)

minus 119888 (1199021+ 1199022) minus 119892max (119909 minus 119902

1minus 1199022 0)

(2)

Now below is the overall profit expected of supply chain

119864 (Π) = 119901 [119903min (1199022 119909) + Vmax (119902

2minus 119909 0) minus 119888119902

2

minus 119892max (119909 minus 1199022 0)] + (1 minus 119901) [119903119898119894119899 (119902

1+ 1199022 119909)

+ Vmax (1199021+ 1199022minus 119909 0) minus 119888 (119902

1+ 1199022)

minus 119892max (119909 minus 1199021minus 1199022 0)]

(3)

WithMax(119864(Π)) and 1199021 1199022ge 0 the expected profit is the

concave function of 1199021and 119902

2 making 120597119864(Π)120597119902

1= 0 and

120597119864(Π)1205971199022= 0

The optimum ordering quantity can be derived based onthe above equations

Theorem 1 The optimum ordering quantity is

119876lowast

119888= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) (4)

Proof Equation (3) can be written as

4 Mathematical Problems in Engineering

119864 (Π) = 119901 [119903 (120583 + int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) + V(119902

2minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) minus 119888119902

2

minus 119892(1199022minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909)] + (1 minus 119901) [119903(120583 + int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)

+ V(1199021+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909) minus 119888 (119902

1+ 1199022) minus 119892(119902

1+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)]

120597119864 (Π)

1205971199021

= 0

120597119864 (Π)

1205971199022

= 0

997904rArr

(1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888) = 0

119901 ((119903 minus 119908 + 119892)int

+infin

1199022

119891 (119909) 119889119909 + (119908 minus 119888) + (1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888)) = 0

997904rArr

(119903 minus 119908 + 119892) 119865 (1199022) + (119908 minus 119888) = 0

(119903 minus 119908 + 119892) 119865 (1199021+ 1199022) + (119908 minus 119888) = 0

997904rArr 119865 (1199022) = 119865 (119902

1+ 1199022) =

119903 + 119892 minus 119888

119903 + 119892 minus V

1199022= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) 1199021= 0

(5)

Then we can get Theorem 1 This is the end of the proof

3 Coordination of the Decentralized SupplyChain via Allied Contracts

Manufacturers play a leading role in supply chains The opti-mal wholesale price and option contracts can be determinedby the possible responses of the distributors and distributorsfollow manufacturers The distributorrsquos optimal orderingquantity is determined by the manufacturerrsquos informationThen the distributor plays a leading role and the Stackelberggame comes into play between the distributor and the retailerThe decision is made through Stackelberg reverse inductionThe sequence of steps in the option order is as follows

(a) At the beginning the distributor reserves the future1199022fromManufacturer 2

(b) The distributor is informed about the disruption fromManufacturer 1

(c) The distributor obtains the ordering informationfrom the retailer

(d) The distributor invokes some option contracts fromManufacturer 2

(e) Manufacturer 2 satisfies the distributorrsquos demands

31 Distributorsrsquo Decision-Making Process Calculating thefirst-order value of the distributorrsquos profit we can get the

optimal order quantity from the manufacturer Below isthe distributorsrsquo profit when disruptions occur in whichthe notations 119901 119890 and119891 denote the probability of supplydisruptions the order price per unit and the strike price perunit in the option contract

Π119889= (119908119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2

(6)

Below is the distributorrsquos profit without disruptions

Π119889= 119908119889min (119902

1 119909)

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022

(7)

Now below is the expected profit of the distributor

119864 (Π119889) = 119901 [(119908

119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2] + (1 minus 119901)

sdot [119908119889min (119902

1 119909)

Mathematical Problems in Engineering 5

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022]

(8)

With 119901 lt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) the optimum ordering quantity

of the supply chain system is

119876lowast

119889= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V + 120573119908

119889

) (9)

The proof of 119876lowast119889 119902lowast1 and 119902

lowast

2is similar to that of (4) to

which detailer process has been added For brevity it is notnecessary to recalculate it

With 119901 gt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+119892minusℎminusV+120573119908

119889) the optimumordering quantity of the

supply chain system is to satisfy the solution of 120597[prod119889]1205971199021=

0 120597[prod119889]1205971199022= 0119876lowast

119889gt 119865minus1

((119908119889minus119908119898+ 119892 minus ℎ)(119908

119889minus V + 119892 minus

ℎ + 120573119908119889)) and119876lowast

119889gt 119865minus1

((119908119889minus 119890 + 119892 minus ℎ minus 119901(119908

119889minus V + 119892 minus ℎ +

120573119908119889))(119908119889minus V + 119892 minus ℎ + 120573119908

119889)(1 minus 119901))

The optimal conditions of KKT (the Karush-Kuhn-Tucker) are based on the ideas proposed by Karush [23] andKuhn and Tucker [24] which indicates that a linear program-ming problem is able to have the necessary and sufficientconditions for the best solution equivalent to a Lagrangemultiplication in a broad sense One of the conditions ofKKT (the Karush-Kuhn-Tucker) is that the optimum mustbe a possible solution and satisfy the restrictive conditions ofinequality and equation

Testifying The target function is the strict concave functionand its restriction is linearity and there is only one optimalsolution which is achieved through the condition of KKT

120597Π

1205971199021

+ 1205821= (1 minus 119901) [119908

119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022)] + 120582

1= 0

(10)

120597Π

1205971199022

+ 1205822= 119908119889+ 119892 minus 119890 minus ℎ minus 119901 (119908

119889minus V + 120573119908

119889minus ℎ

+ 119892) 119865 (1199022) minus (1 minus 119901) (119908

119889minus V + 120573119908

119889minus ℎ + 119892) 119865 (119902

1

+ 1199022) + 1205822= 0

(11)

12058211199021= 0 (12)

12058221199022= 0 (13)

1205821 1205822 1199021 1199022ge 0 (14)

The four possibilities to be analyzed are (a) 1205821 1205822gt 0 (b)

1205821gt 0 120582

2= 0 (c) 120582

1= 0 120582

2gt 0 and (d) 120582

1= 0 120582

2= 0

Case (a) With 1205821 1205822gt 0 from (11) and (12) 119902

1 1199022= 0 can

be derived now from (10) 1205821= minus(1 minus 119901)(119908

119889+ 119892 minus 119908

119898) lt 0

is known which is in conflict with the hypothesis and thus itis not the optimum

Case (b)With 1205821gt 0 and 120582

2= 0 119902

1= 0 is derived from (11)

and now 119865(1199022) = (119903+119892minus119890minusℎ)(119903+119892minusℎ) is derived from (11)

When it is incorporated into (10)1205821= minus(1minus119901)[(119890+ℎ)minus119908

119898] lt

0 is derived which is inconsistent with the hypothesis Thusthe solution is not the optimum

Case (c)With 1205821= 0 and 120582

2gt 0 119902

2= 0 is derived from (12)

and now 119865(1199021) = (119908

119889+119892minus119908

119898)(119908119889+119892minus V+120573119908

119889) is derived

from (10) It is incorporated into (11) and 1205822= (1minus119901)(119908

119889+119892minus

ℎminusV+120573119908119889)((119908119889+119892minus119908

119898)(119908119889+119892minusV+120573119908

119889))minus(119908

119889+119892minusℎminus119890)

When 119901 lt 1 minus (119908119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V + 120573119908

119889)(119908119889+ 119892 minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is satisfied 120582

2gt 0 is satisfied

and now 119902lowast

1= 119865minus1

((119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus V + 120573119908

119889)) and

119902lowast

2= 0 is the optimal solution and all the conditions of KKT

are satisfied

Case (d) As the only optimal solution is obtained thecorollary is that 119901 gt 1 minus (119908

119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V +

120573119908119889)(119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is the optimal

solution with 1205821= 0 and 120582

2= 0 and the conditions of KKT

are satisfiedTheoptimal solution has satisfied 120597[prod119889]1205971199021= 0

and 120597[prod119889]1205971199022= 0 with 119902lowast

1gt 0 and 119902lowast

2gt 0

32 The Retailerrsquos Decision-Making Process Calculating thefirst derivation value of the retailerrsquos profit we can get theoptimal order quantity from the distributor and the optimalwholesale price given by the distributor to the retailer Belowis the retailerrsquos profit with disruptions

Π119903= 119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0)

minus 119892max (119909 minus 1199022 0) minus 119908

1198891199022

(15)

Below is the retailerrsquos profit without disruptions

Π119903= 119903min (119902

1+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)

(16)

Now below is the expected retailerrsquos profit

119864 (Π119903) = 119901 [119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0) minus 119892

sdotmax (119909 minus 1199022 0) minus 119908

1198891199022] + (1 minus 119901) [119903

sdotmin (1199021+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)]

(17)

When the above equation is calculated for 1199021and 119902

2

119865(119902lowast

2) = (119903 + 119892 minus 119908

119889)(119903 + 119892 minus 120573119908

119889) 119902lowast1= 0 are derived

and below is the optimum ordering quantity of the retailer

119876lowast

119903= 119865minus1

(

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

) (18)

33 Coordination of the Decentralized Supply Chain Herethe option contract is combined with buyback contract and

6 Mathematical Problems in Engineering

the three-echelon supply chain is coordinated and optimizedthrough designing the corresponding parameters In theallied contracts the following conditions must be satisfied toachieve complete coordination of the supply chain

Proposition 2 In the allied contract consisting of the optioncontract and the buyback contract if complete coordination isto be achieved in the supply chain then the contract parametersmust satisfy the following cases

Case 1 Consider the following

119901 lt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(19)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (20)

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V minus 120573119908

119889

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(21)

Case 2 Consider the following

119901 gt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(22)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (23)

max(119908119889minus 119908119898+ 119892 minus ℎ

119908119889minus V + 119892 minus ℎ + 120573119908

119889

119908119889minus 119890 + 119892 minus ℎ minus 119901 (119908

119889minus V + 119892 minus ℎ + 120573119908

119889)

(119908119889minus V + 119892 minus ℎ + 120573119908

119889) (1 minus 119901)

)

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(24)

Equations (19) and (22) are two different sets of circum-stances where disruptions are likely to occur in (20) and (23)the validity of contract coordination under different risks isensured in (21) and (24) the effective circulation of supplychain products under different risks is ensured In order toguarantee the validity and the continuity of supply chainoperations the ordering quantity purchased by distributorsfrom manufacturers has to be more than or equal to thatpurchased by the retailer from the distributor Only if theordering quantity of the retailer is consistent with that of thecentralized decision can coordination be achieved

Proposition 3 When the three-stage supply chain is coordi-nated the buyback factor 120573must satisfy the following equation120573 = ((119903 + 119892)(119908

119889+ V minus 119888) minus V119908

119889)119908119889(119903 + 119892 minus 119888)

Corollary 4 When (119908119889+119892+120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt ℎ(119908

119889+

119892 minus 119908119898) the reliable Manufacturer 2 is always deployed

Corollary 5 The increase of the ordering quantity fromManufacturer 1 does not follow the rise of the disruption risksand the ordering quantity fromManufacture 2 does not declinewith the rise of the disruption risks

Corollary 6 The larger the buyback factor 120573 is the larger theordering quantity from retailer is The ordering quantity bydistributors fromManufacturer 1 does not increase with the riseof 120573 and the ordering quantity from Manufacturer 2 does notdecline with the rise of 120573

Corollary 7 The ordering quantity by distributors fromMan-ufacturer 1 does not decline with the rise of option purchasingprices and option strike prices and the ordering quantity by thedistributor does not increase with the rise of option premiumand the option strike prices

Testifying It can be deduced from (20) and (23)

Proof of Corollary 4 When (119908119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt

ℎ(119908119889+119892minus119908

119898) then 1minus(119908

119889+119892minus119890minusℎ)(119908

119889+119892minusV+120573119908

119889)(119908119889+

119892 minus119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) = ((119908

119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus

119908119898)minusℎ(119908

119889+119892minus119908

119898))(119908119889+119892minus119908

119898)(119908119889+119892minusℎminusV+120573119908

119889) lt 0

from 119901 ge 0 119901 gt 1 minus (119908119889+119892minus 119890 minus ℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+

119892 minus 119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is derived from Case (d) in

Section 31 119902lowast2gt 0 is derived and the test is completed

Proof of Corollary 5 With 1199021015840

1= 119889119902

lowast

1(119901)120597119901 and 119902

1015840

2=

119889119902lowast

2(119901)120597119901 from 120597prod

1198891205971199021= 0 the following can be derived

119908119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022) = 0

(25)

As the first derivation of 119901 is carried out with 120597prod1198891205971199021=

0 ℎ11990210158401119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(1199021015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved Thus the plus and minus signs of 11990210158401and 1199021015840

2are

the opposite As the first derivation of 119901 is conducted bysubstituting (25) into 120597prod

1198891205971199022= 0 (119908

119889minus ℎ + 120573119908

119889minus V +

119892)[minus119865(1199022) + 119901119902

1015840

2119891(1199022) + 119865(119902

1+ 1199022)] + (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is

achieved To satisfy the equation with 11990210158401le 0 1199021015840

2ge 0 and the

test is completed

Proof of Corollary 6 With 119876lowast

119903= 119865minus1

((119903 + 119892 minus 119908119889)(119903 +

119892 minus 120573119908119889)) the more the buyback factor 120573 is the greater

(119903 + 119892 minus 119908119889)(119903 + 119892 minus 120573119908

119889) is Also since 119865(119909) is continuous

and differentiable within intervals and it is strictly increasingthe ordering quantity of the retailer 119876lowast

119903increases with the

rise of the buyback factor 120573 With 1199021015840

1= 119889119902

lowast

1(119901)120597120573 and

1199021015840

2= 119889119902lowast

2(119901)120597120573 the first derivation of 120573 is carried out for

120597prod1198891205971199021= 0 and ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+

1199021015840

2)119891(1199021+ 1199022) + 119908119889119865(1199021+ 1199022) = 0 is achieved As such the

sign of 11990210158401is not positive Substituting (25) into 120597prod

1198891205971199022= 0

for the first derivation of 120573 119901119908119889119865(1199022) + 119901(119908

119889minusℎ+120573119908

119889minus V+

119892)1199021015840

2119891(1199022) minus (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is achieved To satisfy the

equation the sign of 11990210158401is not positive and 1199021015840

2lt 0 is known

and hence 11990210158401le 0 1199021015840

2lt 0 and the test is completed

Proof of Corollary 7 With 11990210158401= 119889119902lowast

1(119901)120597119890 and 1199021015840

2= 119889119902lowast

2(119901)

120597119890 (25) is obtained from 120597prod1198891205971199021= 0

The first derivation 119890 is carried out for 120597prod1198891205971199021= 0 and

ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved As such the plus and minus signs of 11990210158401 11990210158402are

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Manufacturer 1

Manufacturer 2

The distributor The retailerBuyback contract

Option contract

Figure 1 Research network chart

enterprises provide single products for downstream enter-prises that are mutually independent without cross-echelonrelations Before sales the manufacturer and the distribu-tor offer contracts to the downstream enterprises retailersdetermine the ordering quantity in terms of the marketdemand and the contract provided by the distributor Atthe same time the distributor determines their orderingquantity according to the ordering quantity of retailers andthe contract of the manufacturers (see Figure 1) Manufac-turer 1 provides products for the distributor according tothe wholesale prices 119908

119898 as the supply of Manufacturer 1

is likely to be disrupted the distributor will determine theordering quantity based on the option contract offered byManufacturer 2 in order to ensure more stable sourcing oftheir products Before selling seasons the distributors reserve1199021units of products from Manufacturer 1 at the wholesale

price they reserve units of option purchasing quantity fromManufacturer 2 At the initial stage of the selling seasonthe distributor buys products within the option purchasingquantity 119902

2at a certain price fromManufacturer 2 on the basis

of the disrupted information obtained from Manufacturer 1in order to stimulate the product ordering from the retailerthe distributor is able to provide a buyback contract (120573 119908

119889)

for the retailer The distributor sells products at price 119908119889per

unit and after the selling season unsold products are boughtback at 120573-fold of the distributorrsquos price

At the same time the symbols used in the modes areshown in Notations The superscript lowast denotes the optimumordering value of retailers in allied contracts

The following are the hypotheses used for building andtesting the models

Assumption 1 Participants in supply chains are completelyrational and they are risk-neutral

Assumption 2 All distribution functions are two-echelon anddifferential and there are strict single inverse functions

Assumption 3 With 119908119889gt 119908119898gt 119888 gt V the limited profits

of Manufacturer 1 are ensured and profits of distributors arealso guaranteed

Assumption 4 With 119908119889gt 119890 + ℎ gt 119908

119898 119890 + V lt 119908

119898 validity of

the option contract is guaranteed

Assumption 5 With 119903 gt 119908119889 120573119908119889gt V the retailerrsquos profits are

ensured and validity of buyback contract is guaranteed

Assumption 6 Manufacturers play leading roles and thedistributer acts as the follower

22 OptimumOrdering Strategies of Centralized Supply ChainIn centralized supply chains the manufacturers the distrib-utor and the retailer are considered as a whole and theobjective is tomaximize the overall profits of the supply chainregardless of the internal transference of payments betweenmember enterprises There is no ldquodual-marginalized effectrdquoin supply chains and it is a typical newsvendor model

Below is the overall profit of supply chains when disrup-tions occur to Manufacturer 1

Π = 119903min (1199022 119909) + Vmax (119902

2minus 119909 0) minus 119888119902

2

minus 119892max (119909 minus 1199022 0)

(1)

Below is the overall profit of supply chain when disrup-tions do not happen to Manufacturer 1

Π = 119903min (1199021+ 1199022 119909) + Vmax (119902

1+ 1199022minus 119909 0)

minus 119888 (1199021+ 1199022) minus 119892max (119909 minus 119902

1minus 1199022 0)

(2)

Now below is the overall profit expected of supply chain

119864 (Π) = 119901 [119903min (1199022 119909) + Vmax (119902

2minus 119909 0) minus 119888119902

2

minus 119892max (119909 minus 1199022 0)] + (1 minus 119901) [119903119898119894119899 (119902

1+ 1199022 119909)

+ Vmax (1199021+ 1199022minus 119909 0) minus 119888 (119902

1+ 1199022)

minus 119892max (119909 minus 1199021minus 1199022 0)]

(3)

WithMax(119864(Π)) and 1199021 1199022ge 0 the expected profit is the

concave function of 1199021and 119902

2 making 120597119864(Π)120597119902

1= 0 and

120597119864(Π)1205971199022= 0

The optimum ordering quantity can be derived based onthe above equations

Theorem 1 The optimum ordering quantity is

119876lowast

119888= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) (4)

Proof Equation (3) can be written as

4 Mathematical Problems in Engineering

119864 (Π) = 119901 [119903 (120583 + int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) + V(119902

2minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) minus 119888119902

2

minus 119892(1199022minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909)] + (1 minus 119901) [119903(120583 + int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)

+ V(1199021+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909) minus 119888 (119902

1+ 1199022) minus 119892(119902

1+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)]

120597119864 (Π)

1205971199021

= 0

120597119864 (Π)

1205971199022

= 0

997904rArr

(1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888) = 0

119901 ((119903 minus 119908 + 119892)int

+infin

1199022

119891 (119909) 119889119909 + (119908 minus 119888) + (1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888)) = 0

997904rArr

(119903 minus 119908 + 119892) 119865 (1199022) + (119908 minus 119888) = 0

(119903 minus 119908 + 119892) 119865 (1199021+ 1199022) + (119908 minus 119888) = 0

997904rArr 119865 (1199022) = 119865 (119902

1+ 1199022) =

119903 + 119892 minus 119888

119903 + 119892 minus V

1199022= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) 1199021= 0

(5)

Then we can get Theorem 1 This is the end of the proof

3 Coordination of the Decentralized SupplyChain via Allied Contracts

Manufacturers play a leading role in supply chains The opti-mal wholesale price and option contracts can be determinedby the possible responses of the distributors and distributorsfollow manufacturers The distributorrsquos optimal orderingquantity is determined by the manufacturerrsquos informationThen the distributor plays a leading role and the Stackelberggame comes into play between the distributor and the retailerThe decision is made through Stackelberg reverse inductionThe sequence of steps in the option order is as follows

(a) At the beginning the distributor reserves the future1199022fromManufacturer 2

(b) The distributor is informed about the disruption fromManufacturer 1

(c) The distributor obtains the ordering informationfrom the retailer

(d) The distributor invokes some option contracts fromManufacturer 2

(e) Manufacturer 2 satisfies the distributorrsquos demands

31 Distributorsrsquo Decision-Making Process Calculating thefirst-order value of the distributorrsquos profit we can get the

optimal order quantity from the manufacturer Below isthe distributorsrsquo profit when disruptions occur in whichthe notations 119901 119890 and119891 denote the probability of supplydisruptions the order price per unit and the strike price perunit in the option contract

Π119889= (119908119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2

(6)

Below is the distributorrsquos profit without disruptions

Π119889= 119908119889min (119902

1 119909)

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022

(7)

Now below is the expected profit of the distributor

119864 (Π119889) = 119901 [(119908

119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2] + (1 minus 119901)

sdot [119908119889min (119902

1 119909)

Mathematical Problems in Engineering 5

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022]

(8)

With 119901 lt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) the optimum ordering quantity

of the supply chain system is

119876lowast

119889= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V + 120573119908

119889

) (9)

The proof of 119876lowast119889 119902lowast1 and 119902

lowast

2is similar to that of (4) to

which detailer process has been added For brevity it is notnecessary to recalculate it

With 119901 gt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+119892minusℎminusV+120573119908

119889) the optimumordering quantity of the

supply chain system is to satisfy the solution of 120597[prod119889]1205971199021=

0 120597[prod119889]1205971199022= 0119876lowast

119889gt 119865minus1

((119908119889minus119908119898+ 119892 minus ℎ)(119908

119889minus V + 119892 minus

ℎ + 120573119908119889)) and119876lowast

119889gt 119865minus1

((119908119889minus 119890 + 119892 minus ℎ minus 119901(119908

119889minus V + 119892 minus ℎ +

120573119908119889))(119908119889minus V + 119892 minus ℎ + 120573119908

119889)(1 minus 119901))

The optimal conditions of KKT (the Karush-Kuhn-Tucker) are based on the ideas proposed by Karush [23] andKuhn and Tucker [24] which indicates that a linear program-ming problem is able to have the necessary and sufficientconditions for the best solution equivalent to a Lagrangemultiplication in a broad sense One of the conditions ofKKT (the Karush-Kuhn-Tucker) is that the optimum mustbe a possible solution and satisfy the restrictive conditions ofinequality and equation

Testifying The target function is the strict concave functionand its restriction is linearity and there is only one optimalsolution which is achieved through the condition of KKT

120597Π

1205971199021

+ 1205821= (1 minus 119901) [119908

119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022)] + 120582

1= 0

(10)

120597Π

1205971199022

+ 1205822= 119908119889+ 119892 minus 119890 minus ℎ minus 119901 (119908

119889minus V + 120573119908

119889minus ℎ

+ 119892) 119865 (1199022) minus (1 minus 119901) (119908

119889minus V + 120573119908

119889minus ℎ + 119892) 119865 (119902

1

+ 1199022) + 1205822= 0

(11)

12058211199021= 0 (12)

12058221199022= 0 (13)

1205821 1205822 1199021 1199022ge 0 (14)

The four possibilities to be analyzed are (a) 1205821 1205822gt 0 (b)

1205821gt 0 120582

2= 0 (c) 120582

1= 0 120582

2gt 0 and (d) 120582

1= 0 120582

2= 0

Case (a) With 1205821 1205822gt 0 from (11) and (12) 119902

1 1199022= 0 can

be derived now from (10) 1205821= minus(1 minus 119901)(119908

119889+ 119892 minus 119908

119898) lt 0

is known which is in conflict with the hypothesis and thus itis not the optimum

Case (b)With 1205821gt 0 and 120582

2= 0 119902

1= 0 is derived from (11)

and now 119865(1199022) = (119903+119892minus119890minusℎ)(119903+119892minusℎ) is derived from (11)

When it is incorporated into (10)1205821= minus(1minus119901)[(119890+ℎ)minus119908

119898] lt

0 is derived which is inconsistent with the hypothesis Thusthe solution is not the optimum

Case (c)With 1205821= 0 and 120582

2gt 0 119902

2= 0 is derived from (12)

and now 119865(1199021) = (119908

119889+119892minus119908

119898)(119908119889+119892minus V+120573119908

119889) is derived

from (10) It is incorporated into (11) and 1205822= (1minus119901)(119908

119889+119892minus

ℎminusV+120573119908119889)((119908119889+119892minus119908

119898)(119908119889+119892minusV+120573119908

119889))minus(119908

119889+119892minusℎminus119890)

When 119901 lt 1 minus (119908119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V + 120573119908

119889)(119908119889+ 119892 minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is satisfied 120582

2gt 0 is satisfied

and now 119902lowast

1= 119865minus1

((119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus V + 120573119908

119889)) and

119902lowast

2= 0 is the optimal solution and all the conditions of KKT

are satisfied

Case (d) As the only optimal solution is obtained thecorollary is that 119901 gt 1 minus (119908

119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V +

120573119908119889)(119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is the optimal

solution with 1205821= 0 and 120582

2= 0 and the conditions of KKT

are satisfiedTheoptimal solution has satisfied 120597[prod119889]1205971199021= 0

and 120597[prod119889]1205971199022= 0 with 119902lowast

1gt 0 and 119902lowast

2gt 0

32 The Retailerrsquos Decision-Making Process Calculating thefirst derivation value of the retailerrsquos profit we can get theoptimal order quantity from the distributor and the optimalwholesale price given by the distributor to the retailer Belowis the retailerrsquos profit with disruptions

Π119903= 119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0)

minus 119892max (119909 minus 1199022 0) minus 119908

1198891199022

(15)

Below is the retailerrsquos profit without disruptions

Π119903= 119903min (119902

1+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)

(16)

Now below is the expected retailerrsquos profit

119864 (Π119903) = 119901 [119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0) minus 119892

sdotmax (119909 minus 1199022 0) minus 119908

1198891199022] + (1 minus 119901) [119903

sdotmin (1199021+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)]

(17)

When the above equation is calculated for 1199021and 119902

2

119865(119902lowast

2) = (119903 + 119892 minus 119908

119889)(119903 + 119892 minus 120573119908

119889) 119902lowast1= 0 are derived

and below is the optimum ordering quantity of the retailer

119876lowast

119903= 119865minus1

(

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

) (18)

33 Coordination of the Decentralized Supply Chain Herethe option contract is combined with buyback contract and

6 Mathematical Problems in Engineering

the three-echelon supply chain is coordinated and optimizedthrough designing the corresponding parameters In theallied contracts the following conditions must be satisfied toachieve complete coordination of the supply chain

Proposition 2 In the allied contract consisting of the optioncontract and the buyback contract if complete coordination isto be achieved in the supply chain then the contract parametersmust satisfy the following cases

Case 1 Consider the following

119901 lt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(19)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (20)

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V minus 120573119908

119889

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(21)

Case 2 Consider the following

119901 gt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(22)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (23)

max(119908119889minus 119908119898+ 119892 minus ℎ

119908119889minus V + 119892 minus ℎ + 120573119908

119889

119908119889minus 119890 + 119892 minus ℎ minus 119901 (119908

119889minus V + 119892 minus ℎ + 120573119908

119889)

(119908119889minus V + 119892 minus ℎ + 120573119908

119889) (1 minus 119901)

)

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(24)

Equations (19) and (22) are two different sets of circum-stances where disruptions are likely to occur in (20) and (23)the validity of contract coordination under different risks isensured in (21) and (24) the effective circulation of supplychain products under different risks is ensured In order toguarantee the validity and the continuity of supply chainoperations the ordering quantity purchased by distributorsfrom manufacturers has to be more than or equal to thatpurchased by the retailer from the distributor Only if theordering quantity of the retailer is consistent with that of thecentralized decision can coordination be achieved

Proposition 3 When the three-stage supply chain is coordi-nated the buyback factor 120573must satisfy the following equation120573 = ((119903 + 119892)(119908

119889+ V minus 119888) minus V119908

119889)119908119889(119903 + 119892 minus 119888)

Corollary 4 When (119908119889+119892+120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt ℎ(119908

119889+

119892 minus 119908119898) the reliable Manufacturer 2 is always deployed

Corollary 5 The increase of the ordering quantity fromManufacturer 1 does not follow the rise of the disruption risksand the ordering quantity fromManufacture 2 does not declinewith the rise of the disruption risks

Corollary 6 The larger the buyback factor 120573 is the larger theordering quantity from retailer is The ordering quantity bydistributors fromManufacturer 1 does not increase with the riseof 120573 and the ordering quantity from Manufacturer 2 does notdecline with the rise of 120573

Corollary 7 The ordering quantity by distributors fromMan-ufacturer 1 does not decline with the rise of option purchasingprices and option strike prices and the ordering quantity by thedistributor does not increase with the rise of option premiumand the option strike prices

Testifying It can be deduced from (20) and (23)

Proof of Corollary 4 When (119908119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt

ℎ(119908119889+119892minus119908

119898) then 1minus(119908

119889+119892minus119890minusℎ)(119908

119889+119892minusV+120573119908

119889)(119908119889+

119892 minus119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) = ((119908

119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus

119908119898)minusℎ(119908

119889+119892minus119908

119898))(119908119889+119892minus119908

119898)(119908119889+119892minusℎminusV+120573119908

119889) lt 0

from 119901 ge 0 119901 gt 1 minus (119908119889+119892minus 119890 minus ℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+

119892 minus 119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is derived from Case (d) in

Section 31 119902lowast2gt 0 is derived and the test is completed

Proof of Corollary 5 With 1199021015840

1= 119889119902

lowast

1(119901)120597119901 and 119902

1015840

2=

119889119902lowast

2(119901)120597119901 from 120597prod

1198891205971199021= 0 the following can be derived

119908119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022) = 0

(25)

As the first derivation of 119901 is carried out with 120597prod1198891205971199021=

0 ℎ11990210158401119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(1199021015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved Thus the plus and minus signs of 11990210158401and 1199021015840

2are

the opposite As the first derivation of 119901 is conducted bysubstituting (25) into 120597prod

1198891205971199022= 0 (119908

119889minus ℎ + 120573119908

119889minus V +

119892)[minus119865(1199022) + 119901119902

1015840

2119891(1199022) + 119865(119902

1+ 1199022)] + (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is

achieved To satisfy the equation with 11990210158401le 0 1199021015840

2ge 0 and the

test is completed

Proof of Corollary 6 With 119876lowast

119903= 119865minus1

((119903 + 119892 minus 119908119889)(119903 +

119892 minus 120573119908119889)) the more the buyback factor 120573 is the greater

(119903 + 119892 minus 119908119889)(119903 + 119892 minus 120573119908

119889) is Also since 119865(119909) is continuous

and differentiable within intervals and it is strictly increasingthe ordering quantity of the retailer 119876lowast

119903increases with the

rise of the buyback factor 120573 With 1199021015840

1= 119889119902

lowast

1(119901)120597120573 and

1199021015840

2= 119889119902lowast

2(119901)120597120573 the first derivation of 120573 is carried out for

120597prod1198891205971199021= 0 and ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+

1199021015840

2)119891(1199021+ 1199022) + 119908119889119865(1199021+ 1199022) = 0 is achieved As such the

sign of 11990210158401is not positive Substituting (25) into 120597prod

1198891205971199022= 0

for the first derivation of 120573 119901119908119889119865(1199022) + 119901(119908

119889minusℎ+120573119908

119889minus V+

119892)1199021015840

2119891(1199022) minus (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is achieved To satisfy the

equation the sign of 11990210158401is not positive and 1199021015840

2lt 0 is known

and hence 11990210158401le 0 1199021015840

2lt 0 and the test is completed

Proof of Corollary 7 With 11990210158401= 119889119902lowast

1(119901)120597119890 and 1199021015840

2= 119889119902lowast

2(119901)

120597119890 (25) is obtained from 120597prod1198891205971199021= 0

The first derivation 119890 is carried out for 120597prod1198891205971199021= 0 and

ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved As such the plus and minus signs of 11990210158401 11990210158402are

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

119864 (Π) = 119901 [119903 (120583 + int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) + V(119902

2minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909) minus 119888119902

2

minus 119892(1199022minus 120583 minus int

+infin

1199022

(1199022minus 119909)119891 (119909) 119889119909)] + (1 minus 119901) [119903(120583 + int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)

+ V(1199021+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909) minus 119888 (119902

1+ 1199022) minus 119892(119902

1+ 1199022minus 120583 minus int

+infin

1199021+1199022

(1199021+ 1199022minus 119909)119891 (119909) 119889119909)]

120597119864 (Π)

1205971199021

= 0

120597119864 (Π)

1205971199022

= 0

997904rArr

(1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888) = 0

119901 ((119903 minus 119908 + 119892)int

+infin

1199022

119891 (119909) 119889119909 + (119908 minus 119888) + (1 minus 119901) ((119903 minus 119908 + 119892)int

+infin

1199021+1199022

119891 (119909) 119889119909) + (119908 minus 119888)) = 0

997904rArr

(119903 minus 119908 + 119892) 119865 (1199022) + (119908 minus 119888) = 0

(119903 minus 119908 + 119892) 119865 (1199021+ 1199022) + (119908 minus 119888) = 0

997904rArr 119865 (1199022) = 119865 (119902

1+ 1199022) =

119903 + 119892 minus 119888

119903 + 119892 minus V

1199022= 119865minus1

(

119903 + 119892 minus 119888

119903 + 119892 minus V) 1199021= 0

(5)

Then we can get Theorem 1 This is the end of the proof

3 Coordination of the Decentralized SupplyChain via Allied Contracts

Manufacturers play a leading role in supply chains The opti-mal wholesale price and option contracts can be determinedby the possible responses of the distributors and distributorsfollow manufacturers The distributorrsquos optimal orderingquantity is determined by the manufacturerrsquos informationThen the distributor plays a leading role and the Stackelberggame comes into play between the distributor and the retailerThe decision is made through Stackelberg reverse inductionThe sequence of steps in the option order is as follows

(a) At the beginning the distributor reserves the future1199022fromManufacturer 2

(b) The distributor is informed about the disruption fromManufacturer 1

(c) The distributor obtains the ordering informationfrom the retailer

(d) The distributor invokes some option contracts fromManufacturer 2

(e) Manufacturer 2 satisfies the distributorrsquos demands

31 Distributorsrsquo Decision-Making Process Calculating thefirst-order value of the distributorrsquos profit we can get the

optimal order quantity from the manufacturer Below isthe distributorsrsquo profit when disruptions occur in whichthe notations 119901 119890 and119891 denote the probability of supplydisruptions the order price per unit and the strike price perunit in the option contract

Π119889= (119908119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2

(6)

Below is the distributorrsquos profit without disruptions

Π119889= 119908119889min (119902

1 119909)

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022

(7)

Now below is the expected profit of the distributor

119864 (Π119889) = 119901 [(119908

119889minus ℎ)min (119902

2 119909) minus 119892max (119909 minus 119902

2 0)

minus (120573119908119889minus V)max (119902

2minus 119909 0) minus 119890119902

2] + (1 minus 119901)

sdot [119908119889min (119902

1 119909)

Mathematical Problems in Engineering 5

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022]

(8)

With 119901 lt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) the optimum ordering quantity

of the supply chain system is

119876lowast

119889= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V + 120573119908

119889

) (9)

The proof of 119876lowast119889 119902lowast1 and 119902

lowast

2is similar to that of (4) to

which detailer process has been added For brevity it is notnecessary to recalculate it

With 119901 gt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+119892minusℎminusV+120573119908

119889) the optimumordering quantity of the

supply chain system is to satisfy the solution of 120597[prod119889]1205971199021=

0 120597[prod119889]1205971199022= 0119876lowast

119889gt 119865minus1

((119908119889minus119908119898+ 119892 minus ℎ)(119908

119889minus V + 119892 minus

ℎ + 120573119908119889)) and119876lowast

119889gt 119865minus1

((119908119889minus 119890 + 119892 minus ℎ minus 119901(119908

119889minus V + 119892 minus ℎ +

120573119908119889))(119908119889minus V + 119892 minus ℎ + 120573119908

119889)(1 minus 119901))

The optimal conditions of KKT (the Karush-Kuhn-Tucker) are based on the ideas proposed by Karush [23] andKuhn and Tucker [24] which indicates that a linear program-ming problem is able to have the necessary and sufficientconditions for the best solution equivalent to a Lagrangemultiplication in a broad sense One of the conditions ofKKT (the Karush-Kuhn-Tucker) is that the optimum mustbe a possible solution and satisfy the restrictive conditions ofinequality and equation

Testifying The target function is the strict concave functionand its restriction is linearity and there is only one optimalsolution which is achieved through the condition of KKT

120597Π

1205971199021

+ 1205821= (1 minus 119901) [119908

119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022)] + 120582

1= 0

(10)

120597Π

1205971199022

+ 1205822= 119908119889+ 119892 minus 119890 minus ℎ minus 119901 (119908

119889minus V + 120573119908

119889minus ℎ

+ 119892) 119865 (1199022) minus (1 minus 119901) (119908

119889minus V + 120573119908

119889minus ℎ + 119892) 119865 (119902

1

+ 1199022) + 1205822= 0

(11)

12058211199021= 0 (12)

12058221199022= 0 (13)

1205821 1205822 1199021 1199022ge 0 (14)

The four possibilities to be analyzed are (a) 1205821 1205822gt 0 (b)

1205821gt 0 120582

2= 0 (c) 120582

1= 0 120582

2gt 0 and (d) 120582

1= 0 120582

2= 0

Case (a) With 1205821 1205822gt 0 from (11) and (12) 119902

1 1199022= 0 can

be derived now from (10) 1205821= minus(1 minus 119901)(119908

119889+ 119892 minus 119908

119898) lt 0

is known which is in conflict with the hypothesis and thus itis not the optimum

Case (b)With 1205821gt 0 and 120582

2= 0 119902

1= 0 is derived from (11)

and now 119865(1199022) = (119903+119892minus119890minusℎ)(119903+119892minusℎ) is derived from (11)

When it is incorporated into (10)1205821= minus(1minus119901)[(119890+ℎ)minus119908

119898] lt

0 is derived which is inconsistent with the hypothesis Thusthe solution is not the optimum

Case (c)With 1205821= 0 and 120582

2gt 0 119902

2= 0 is derived from (12)

and now 119865(1199021) = (119908

119889+119892minus119908

119898)(119908119889+119892minus V+120573119908

119889) is derived

from (10) It is incorporated into (11) and 1205822= (1minus119901)(119908

119889+119892minus

ℎminusV+120573119908119889)((119908119889+119892minus119908

119898)(119908119889+119892minusV+120573119908

119889))minus(119908

119889+119892minusℎminus119890)

When 119901 lt 1 minus (119908119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V + 120573119908

119889)(119908119889+ 119892 minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is satisfied 120582

2gt 0 is satisfied

and now 119902lowast

1= 119865minus1

((119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus V + 120573119908

119889)) and

119902lowast

2= 0 is the optimal solution and all the conditions of KKT

are satisfied

Case (d) As the only optimal solution is obtained thecorollary is that 119901 gt 1 minus (119908

119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V +

120573119908119889)(119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is the optimal

solution with 1205821= 0 and 120582

2= 0 and the conditions of KKT

are satisfiedTheoptimal solution has satisfied 120597[prod119889]1205971199021= 0

and 120597[prod119889]1205971199022= 0 with 119902lowast

1gt 0 and 119902lowast

2gt 0

32 The Retailerrsquos Decision-Making Process Calculating thefirst derivation value of the retailerrsquos profit we can get theoptimal order quantity from the distributor and the optimalwholesale price given by the distributor to the retailer Belowis the retailerrsquos profit with disruptions

Π119903= 119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0)

minus 119892max (119909 minus 1199022 0) minus 119908

1198891199022

(15)

Below is the retailerrsquos profit without disruptions

Π119903= 119903min (119902

1+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)

(16)

Now below is the expected retailerrsquos profit

119864 (Π119903) = 119901 [119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0) minus 119892

sdotmax (119909 minus 1199022 0) minus 119908

1198891199022] + (1 minus 119901) [119903

sdotmin (1199021+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)]

(17)

When the above equation is calculated for 1199021and 119902

2

119865(119902lowast

2) = (119903 + 119892 minus 119908

119889)(119903 + 119892 minus 120573119908

119889) 119902lowast1= 0 are derived

and below is the optimum ordering quantity of the retailer

119876lowast

119903= 119865minus1

(

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

) (18)

33 Coordination of the Decentralized Supply Chain Herethe option contract is combined with buyback contract and

6 Mathematical Problems in Engineering

the three-echelon supply chain is coordinated and optimizedthrough designing the corresponding parameters In theallied contracts the following conditions must be satisfied toachieve complete coordination of the supply chain

Proposition 2 In the allied contract consisting of the optioncontract and the buyback contract if complete coordination isto be achieved in the supply chain then the contract parametersmust satisfy the following cases

Case 1 Consider the following

119901 lt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(19)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (20)

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V minus 120573119908

119889

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(21)

Case 2 Consider the following

119901 gt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(22)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (23)

max(119908119889minus 119908119898+ 119892 minus ℎ

119908119889minus V + 119892 minus ℎ + 120573119908

119889

119908119889minus 119890 + 119892 minus ℎ minus 119901 (119908

119889minus V + 119892 minus ℎ + 120573119908

119889)

(119908119889minus V + 119892 minus ℎ + 120573119908

119889) (1 minus 119901)

)

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(24)

Equations (19) and (22) are two different sets of circum-stances where disruptions are likely to occur in (20) and (23)the validity of contract coordination under different risks isensured in (21) and (24) the effective circulation of supplychain products under different risks is ensured In order toguarantee the validity and the continuity of supply chainoperations the ordering quantity purchased by distributorsfrom manufacturers has to be more than or equal to thatpurchased by the retailer from the distributor Only if theordering quantity of the retailer is consistent with that of thecentralized decision can coordination be achieved

Proposition 3 When the three-stage supply chain is coordi-nated the buyback factor 120573must satisfy the following equation120573 = ((119903 + 119892)(119908

119889+ V minus 119888) minus V119908

119889)119908119889(119903 + 119892 minus 119888)

Corollary 4 When (119908119889+119892+120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt ℎ(119908

119889+

119892 minus 119908119898) the reliable Manufacturer 2 is always deployed

Corollary 5 The increase of the ordering quantity fromManufacturer 1 does not follow the rise of the disruption risksand the ordering quantity fromManufacture 2 does not declinewith the rise of the disruption risks

Corollary 6 The larger the buyback factor 120573 is the larger theordering quantity from retailer is The ordering quantity bydistributors fromManufacturer 1 does not increase with the riseof 120573 and the ordering quantity from Manufacturer 2 does notdecline with the rise of 120573

Corollary 7 The ordering quantity by distributors fromMan-ufacturer 1 does not decline with the rise of option purchasingprices and option strike prices and the ordering quantity by thedistributor does not increase with the rise of option premiumand the option strike prices

Testifying It can be deduced from (20) and (23)

Proof of Corollary 4 When (119908119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt

ℎ(119908119889+119892minus119908

119898) then 1minus(119908

119889+119892minus119890minusℎ)(119908

119889+119892minusV+120573119908

119889)(119908119889+

119892 minus119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) = ((119908

119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus

119908119898)minusℎ(119908

119889+119892minus119908

119898))(119908119889+119892minus119908

119898)(119908119889+119892minusℎminusV+120573119908

119889) lt 0

from 119901 ge 0 119901 gt 1 minus (119908119889+119892minus 119890 minus ℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+

119892 minus 119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is derived from Case (d) in

Section 31 119902lowast2gt 0 is derived and the test is completed

Proof of Corollary 5 With 1199021015840

1= 119889119902

lowast

1(119901)120597119901 and 119902

1015840

2=

119889119902lowast

2(119901)120597119901 from 120597prod

1198891205971199021= 0 the following can be derived

119908119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022) = 0

(25)

As the first derivation of 119901 is carried out with 120597prod1198891205971199021=

0 ℎ11990210158401119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(1199021015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved Thus the plus and minus signs of 11990210158401and 1199021015840

2are

the opposite As the first derivation of 119901 is conducted bysubstituting (25) into 120597prod

1198891205971199022= 0 (119908

119889minus ℎ + 120573119908

119889minus V +

119892)[minus119865(1199022) + 119901119902

1015840

2119891(1199022) + 119865(119902

1+ 1199022)] + (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is

achieved To satisfy the equation with 11990210158401le 0 1199021015840

2ge 0 and the

test is completed

Proof of Corollary 6 With 119876lowast

119903= 119865minus1

((119903 + 119892 minus 119908119889)(119903 +

119892 minus 120573119908119889)) the more the buyback factor 120573 is the greater

(119903 + 119892 minus 119908119889)(119903 + 119892 minus 120573119908

119889) is Also since 119865(119909) is continuous

and differentiable within intervals and it is strictly increasingthe ordering quantity of the retailer 119876lowast

119903increases with the

rise of the buyback factor 120573 With 1199021015840

1= 119889119902

lowast

1(119901)120597120573 and

1199021015840

2= 119889119902lowast

2(119901)120597120573 the first derivation of 120573 is carried out for

120597prod1198891205971199021= 0 and ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+

1199021015840

2)119891(1199021+ 1199022) + 119908119889119865(1199021+ 1199022) = 0 is achieved As such the

sign of 11990210158401is not positive Substituting (25) into 120597prod

1198891205971199022= 0

for the first derivation of 120573 119901119908119889119865(1199022) + 119901(119908

119889minusℎ+120573119908

119889minus V+

119892)1199021015840

2119891(1199022) minus (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is achieved To satisfy the

equation the sign of 11990210158401is not positive and 1199021015840

2lt 0 is known

and hence 11990210158401le 0 1199021015840

2lt 0 and the test is completed

Proof of Corollary 7 With 11990210158401= 119889119902lowast

1(119901)120597119890 and 1199021015840

2= 119889119902lowast

2(119901)

120597119890 (25) is obtained from 120597prod1198891205971199021= 0

The first derivation 119890 is carried out for 120597prod1198891205971199021= 0 and

ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved As such the plus and minus signs of 11990210158401 11990210158402are

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

+ (119908119889minus ℎ)max min (119909 minus 119902

1 1199022) 0

+ (V minus 120573119908119889)max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

1198981199021minus 1198901199022]

(8)

With 119901 lt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) the optimum ordering quantity

of the supply chain system is

119876lowast

119889= 119902lowast

1+ 119902lowast

2= 119865minus1

(

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V + 120573119908

119889

) (9)

The proof of 119876lowast119889 119902lowast1 and 119902

lowast

2is similar to that of (4) to

which detailer process has been added For brevity it is notnecessary to recalculate it

With 119901 gt 1minus (119908119889+119892minus119890minusℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+119892minus

119908119898)(119908119889+119892minusℎminusV+120573119908

119889) the optimumordering quantity of the

supply chain system is to satisfy the solution of 120597[prod119889]1205971199021=

0 120597[prod119889]1205971199022= 0119876lowast

119889gt 119865minus1

((119908119889minus119908119898+ 119892 minus ℎ)(119908

119889minus V + 119892 minus

ℎ + 120573119908119889)) and119876lowast

119889gt 119865minus1

((119908119889minus 119890 + 119892 minus ℎ minus 119901(119908

119889minus V + 119892 minus ℎ +

120573119908119889))(119908119889minus V + 119892 minus ℎ + 120573119908

119889)(1 minus 119901))

The optimal conditions of KKT (the Karush-Kuhn-Tucker) are based on the ideas proposed by Karush [23] andKuhn and Tucker [24] which indicates that a linear program-ming problem is able to have the necessary and sufficientconditions for the best solution equivalent to a Lagrangemultiplication in a broad sense One of the conditions ofKKT (the Karush-Kuhn-Tucker) is that the optimum mustbe a possible solution and satisfy the restrictive conditions ofinequality and equation

Testifying The target function is the strict concave functionand its restriction is linearity and there is only one optimalsolution which is achieved through the condition of KKT

120597Π

1205971199021

+ 1205821= (1 minus 119901) [119908

119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022)] + 120582

1= 0

(10)

120597Π

1205971199022

+ 1205822= 119908119889+ 119892 minus 119890 minus ℎ minus 119901 (119908

119889minus V + 120573119908

119889minus ℎ

+ 119892) 119865 (1199022) minus (1 minus 119901) (119908

119889minus V + 120573119908

119889minus ℎ + 119892) 119865 (119902

1

+ 1199022) + 1205822= 0

(11)

12058211199021= 0 (12)

12058221199022= 0 (13)

1205821 1205822 1199021 1199022ge 0 (14)

The four possibilities to be analyzed are (a) 1205821 1205822gt 0 (b)

1205821gt 0 120582

2= 0 (c) 120582

1= 0 120582

2gt 0 and (d) 120582

1= 0 120582

2= 0

Case (a) With 1205821 1205822gt 0 from (11) and (12) 119902

1 1199022= 0 can

be derived now from (10) 1205821= minus(1 minus 119901)(119908

119889+ 119892 minus 119908

119898) lt 0

is known which is in conflict with the hypothesis and thus itis not the optimum

Case (b)With 1205821gt 0 and 120582

2= 0 119902

1= 0 is derived from (11)

and now 119865(1199022) = (119903+119892minus119890minusℎ)(119903+119892minusℎ) is derived from (11)

When it is incorporated into (10)1205821= minus(1minus119901)[(119890+ℎ)minus119908

119898] lt

0 is derived which is inconsistent with the hypothesis Thusthe solution is not the optimum

Case (c)With 1205821= 0 and 120582

2gt 0 119902

2= 0 is derived from (12)

and now 119865(1199021) = (119908

119889+119892minus119908

119898)(119908119889+119892minus V+120573119908

119889) is derived

from (10) It is incorporated into (11) and 1205822= (1minus119901)(119908

119889+119892minus

ℎminusV+120573119908119889)((119908119889+119892minus119908

119898)(119908119889+119892minusV+120573119908

119889))minus(119908

119889+119892minusℎminus119890)

When 119901 lt 1 minus (119908119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V + 120573119908

119889)(119908119889+ 119892 minus

119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is satisfied 120582

2gt 0 is satisfied

and now 119902lowast

1= 119865minus1

((119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus V + 120573119908

119889)) and

119902lowast

2= 0 is the optimal solution and all the conditions of KKT

are satisfied

Case (d) As the only optimal solution is obtained thecorollary is that 119901 gt 1 minus (119908

119889+ 119892 minus 119890 minus ℎ)(119908

119889+ 119892 minus V +

120573119908119889)(119908119889+ 119892 minus 119908

119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is the optimal

solution with 1205821= 0 and 120582

2= 0 and the conditions of KKT

are satisfiedTheoptimal solution has satisfied 120597[prod119889]1205971199021= 0

and 120597[prod119889]1205971199022= 0 with 119902lowast

1gt 0 and 119902lowast

2gt 0

32 The Retailerrsquos Decision-Making Process Calculating thefirst derivation value of the retailerrsquos profit we can get theoptimal order quantity from the distributor and the optimalwholesale price given by the distributor to the retailer Belowis the retailerrsquos profit with disruptions

Π119903= 119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0)

minus 119892max (119909 minus 1199022 0) minus 119908

1198891199022

(15)

Below is the retailerrsquos profit without disruptions

Π119903= 119903min (119902

1+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)

(16)

Now below is the expected retailerrsquos profit

119864 (Π119903) = 119901 [119903min (119902

2 119909) + 120573119908

119889max (119902

2minus 119909 0) minus 119892

sdotmax (119909 minus 1199022 0) minus 119908

1198891199022] + (1 minus 119901) [119903

sdotmin (1199021+ 1199022 119909) + 120573119908

119889max (119902

1+ 1199022minus 119909 0)

minus 119892max (119909 minus 1199021minus 1199022 0) minus 119908

119889(1199021+ 1199022)]

(17)

When the above equation is calculated for 1199021and 119902

2

119865(119902lowast

2) = (119903 + 119892 minus 119908

119889)(119903 + 119892 minus 120573119908

119889) 119902lowast1= 0 are derived

and below is the optimum ordering quantity of the retailer

119876lowast

119903= 119865minus1

(

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

) (18)

33 Coordination of the Decentralized Supply Chain Herethe option contract is combined with buyback contract and

6 Mathematical Problems in Engineering

the three-echelon supply chain is coordinated and optimizedthrough designing the corresponding parameters In theallied contracts the following conditions must be satisfied toachieve complete coordination of the supply chain

Proposition 2 In the allied contract consisting of the optioncontract and the buyback contract if complete coordination isto be achieved in the supply chain then the contract parametersmust satisfy the following cases

Case 1 Consider the following

119901 lt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(19)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (20)

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V minus 120573119908

119889

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(21)

Case 2 Consider the following

119901 gt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(22)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (23)

max(119908119889minus 119908119898+ 119892 minus ℎ

119908119889minus V + 119892 minus ℎ + 120573119908

119889

119908119889minus 119890 + 119892 minus ℎ minus 119901 (119908

119889minus V + 119892 minus ℎ + 120573119908

119889)

(119908119889minus V + 119892 minus ℎ + 120573119908

119889) (1 minus 119901)

)

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(24)

Equations (19) and (22) are two different sets of circum-stances where disruptions are likely to occur in (20) and (23)the validity of contract coordination under different risks isensured in (21) and (24) the effective circulation of supplychain products under different risks is ensured In order toguarantee the validity and the continuity of supply chainoperations the ordering quantity purchased by distributorsfrom manufacturers has to be more than or equal to thatpurchased by the retailer from the distributor Only if theordering quantity of the retailer is consistent with that of thecentralized decision can coordination be achieved

Proposition 3 When the three-stage supply chain is coordi-nated the buyback factor 120573must satisfy the following equation120573 = ((119903 + 119892)(119908

119889+ V minus 119888) minus V119908

119889)119908119889(119903 + 119892 minus 119888)

Corollary 4 When (119908119889+119892+120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt ℎ(119908

119889+

119892 minus 119908119898) the reliable Manufacturer 2 is always deployed

Corollary 5 The increase of the ordering quantity fromManufacturer 1 does not follow the rise of the disruption risksand the ordering quantity fromManufacture 2 does not declinewith the rise of the disruption risks

Corollary 6 The larger the buyback factor 120573 is the larger theordering quantity from retailer is The ordering quantity bydistributors fromManufacturer 1 does not increase with the riseof 120573 and the ordering quantity from Manufacturer 2 does notdecline with the rise of 120573

Corollary 7 The ordering quantity by distributors fromMan-ufacturer 1 does not decline with the rise of option purchasingprices and option strike prices and the ordering quantity by thedistributor does not increase with the rise of option premiumand the option strike prices

Testifying It can be deduced from (20) and (23)

Proof of Corollary 4 When (119908119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt

ℎ(119908119889+119892minus119908

119898) then 1minus(119908

119889+119892minus119890minusℎ)(119908

119889+119892minusV+120573119908

119889)(119908119889+

119892 minus119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) = ((119908

119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus

119908119898)minusℎ(119908

119889+119892minus119908

119898))(119908119889+119892minus119908

119898)(119908119889+119892minusℎminusV+120573119908

119889) lt 0

from 119901 ge 0 119901 gt 1 minus (119908119889+119892minus 119890 minus ℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+

119892 minus 119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is derived from Case (d) in

Section 31 119902lowast2gt 0 is derived and the test is completed

Proof of Corollary 5 With 1199021015840

1= 119889119902

lowast

1(119901)120597119901 and 119902

1015840

2=

119889119902lowast

2(119901)120597119901 from 120597prod

1198891205971199021= 0 the following can be derived

119908119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022) = 0

(25)

As the first derivation of 119901 is carried out with 120597prod1198891205971199021=

0 ℎ11990210158401119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(1199021015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved Thus the plus and minus signs of 11990210158401and 1199021015840

2are

the opposite As the first derivation of 119901 is conducted bysubstituting (25) into 120597prod

1198891205971199022= 0 (119908

119889minus ℎ + 120573119908

119889minus V +

119892)[minus119865(1199022) + 119901119902

1015840

2119891(1199022) + 119865(119902

1+ 1199022)] + (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is

achieved To satisfy the equation with 11990210158401le 0 1199021015840

2ge 0 and the

test is completed

Proof of Corollary 6 With 119876lowast

119903= 119865minus1

((119903 + 119892 minus 119908119889)(119903 +

119892 minus 120573119908119889)) the more the buyback factor 120573 is the greater

(119903 + 119892 minus 119908119889)(119903 + 119892 minus 120573119908

119889) is Also since 119865(119909) is continuous

and differentiable within intervals and it is strictly increasingthe ordering quantity of the retailer 119876lowast

119903increases with the

rise of the buyback factor 120573 With 1199021015840

1= 119889119902

lowast

1(119901)120597120573 and

1199021015840

2= 119889119902lowast

2(119901)120597120573 the first derivation of 120573 is carried out for

120597prod1198891205971199021= 0 and ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+

1199021015840

2)119891(1199021+ 1199022) + 119908119889119865(1199021+ 1199022) = 0 is achieved As such the

sign of 11990210158401is not positive Substituting (25) into 120597prod

1198891205971199022= 0

for the first derivation of 120573 119901119908119889119865(1199022) + 119901(119908

119889minusℎ+120573119908

119889minus V+

119892)1199021015840

2119891(1199022) minus (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is achieved To satisfy the

equation the sign of 11990210158401is not positive and 1199021015840

2lt 0 is known

and hence 11990210158401le 0 1199021015840

2lt 0 and the test is completed

Proof of Corollary 7 With 11990210158401= 119889119902lowast

1(119901)120597119890 and 1199021015840

2= 119889119902lowast

2(119901)

120597119890 (25) is obtained from 120597prod1198891205971199021= 0

The first derivation 119890 is carried out for 120597prod1198891205971199021= 0 and

ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved As such the plus and minus signs of 11990210158401 11990210158402are

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

the three-echelon supply chain is coordinated and optimizedthrough designing the corresponding parameters In theallied contracts the following conditions must be satisfied toachieve complete coordination of the supply chain

Proposition 2 In the allied contract consisting of the optioncontract and the buyback contract if complete coordination isto be achieved in the supply chain then the contract parametersmust satisfy the following cases

Case 1 Consider the following

119901 lt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(19)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (20)

119908119889+ 119892 minus 119908

119898

119908119889+ 119892 minus V minus 120573119908

119889

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(21)

Case 2 Consider the following

119901 gt 1 minus

(119908119889+ 119892 minus 119890 minus ℎ) (119908

119889+ 119892 minus V + 120573119908

119889)

(119908119889+ 119892 minus 119908

119898) (119908119889+ 119892 minus ℎ minus V + 120573119908

119889)

(22)

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

=

119903 + 119892 minus 119888

119903 + 119892 minus V (23)

max(119908119889minus 119908119898+ 119892 minus ℎ

119908119889minus V + 119892 minus ℎ + 120573119908

119889

119908119889minus 119890 + 119892 minus ℎ minus 119901 (119908

119889minus V + 119892 minus ℎ + 120573119908

119889)

(119908119889minus V + 119892 minus ℎ + 120573119908

119889) (1 minus 119901)

)

ge

119903 + 119892 minus 119908119889

119903 + 119892 minus 120573119908119889

(24)

Equations (19) and (22) are two different sets of circum-stances where disruptions are likely to occur in (20) and (23)the validity of contract coordination under different risks isensured in (21) and (24) the effective circulation of supplychain products under different risks is ensured In order toguarantee the validity and the continuity of supply chainoperations the ordering quantity purchased by distributorsfrom manufacturers has to be more than or equal to thatpurchased by the retailer from the distributor Only if theordering quantity of the retailer is consistent with that of thecentralized decision can coordination be achieved

Proposition 3 When the three-stage supply chain is coordi-nated the buyback factor 120573must satisfy the following equation120573 = ((119903 + 119892)(119908

119889+ V minus 119888) minus V119908

119889)119908119889(119903 + 119892 minus 119888)

Corollary 4 When (119908119889+119892+120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt ℎ(119908

119889+

119892 minus 119908119898) the reliable Manufacturer 2 is always deployed

Corollary 5 The increase of the ordering quantity fromManufacturer 1 does not follow the rise of the disruption risksand the ordering quantity fromManufacture 2 does not declinewith the rise of the disruption risks

Corollary 6 The larger the buyback factor 120573 is the larger theordering quantity from retailer is The ordering quantity bydistributors fromManufacturer 1 does not increase with the riseof 120573 and the ordering quantity from Manufacturer 2 does notdecline with the rise of 120573

Corollary 7 The ordering quantity by distributors fromMan-ufacturer 1 does not decline with the rise of option purchasingprices and option strike prices and the ordering quantity by thedistributor does not increase with the rise of option premiumand the option strike prices

Testifying It can be deduced from (20) and (23)

Proof of Corollary 4 When (119908119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus119908

119898) lt

ℎ(119908119889+119892minus119908

119898) then 1minus(119908

119889+119892minus119890minusℎ)(119908

119889+119892minusV+120573119908

119889)(119908119889+

119892 minus119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) = ((119908

119889+ 119892 + 120573119908

119889minus V)(119890 + ℎ minus

119908119898)minusℎ(119908

119889+119892minus119908

119898))(119908119889+119892minus119908

119898)(119908119889+119892minusℎminusV+120573119908

119889) lt 0

from 119901 ge 0 119901 gt 1 minus (119908119889+119892minus 119890 minus ℎ)(119908

119889+119892minus V+120573119908

119889)(119908119889+

119892 minus 119908119898)(119908119889+ 119892 minus ℎ minus V + 120573119908

119889) is derived from Case (d) in

Section 31 119902lowast2gt 0 is derived and the test is completed

Proof of Corollary 5 With 1199021015840

1= 119889119902

lowast

1(119901)120597119901 and 119902

1015840

2=

119889119902lowast

2(119901)120597119901 from 120597prod

1198891205971199021= 0 the following can be derived

119908119889+ 119892 minus 119908

119898minus ℎ119865 (119902

1)

minus (119908119889minus ℎ minus V + 120573119908

119889+ 119892) 119865 (119902

1+ 1199022) = 0

(25)

As the first derivation of 119901 is carried out with 120597prod1198891205971199021=

0 ℎ11990210158401119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(1199021015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved Thus the plus and minus signs of 11990210158401and 1199021015840

2are

the opposite As the first derivation of 119901 is conducted bysubstituting (25) into 120597prod

1198891205971199022= 0 (119908

119889minus ℎ + 120573119908

119889minus V +

119892)[minus119865(1199022) + 119901119902

1015840

2119891(1199022) + 119865(119902

1+ 1199022)] + (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is

achieved To satisfy the equation with 11990210158401le 0 1199021015840

2ge 0 and the

test is completed

Proof of Corollary 6 With 119876lowast

119903= 119865minus1

((119903 + 119892 minus 119908119889)(119903 +

119892 minus 120573119908119889)) the more the buyback factor 120573 is the greater

(119903 + 119892 minus 119908119889)(119903 + 119892 minus 120573119908

119889) is Also since 119865(119909) is continuous

and differentiable within intervals and it is strictly increasingthe ordering quantity of the retailer 119876lowast

119903increases with the

rise of the buyback factor 120573 With 1199021015840

1= 119889119902

lowast

1(119901)120597120573 and

1199021015840

2= 119889119902lowast

2(119901)120597120573 the first derivation of 120573 is carried out for

120597prod1198891205971199021= 0 and ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+

1199021015840

2)119891(1199021+ 1199022) + 119908119889119865(1199021+ 1199022) = 0 is achieved As such the

sign of 11990210158401is not positive Substituting (25) into 120597prod

1198891205971199022= 0

for the first derivation of 120573 119901119908119889119865(1199022) + 119901(119908

119889minusℎ+120573119908

119889minus V+

119892)1199021015840

2119891(1199022) minus (1 minus 119901)ℎ119902

1015840

1119891(1199021) = 0 is achieved To satisfy the

equation the sign of 11990210158401is not positive and 1199021015840

2lt 0 is known

and hence 11990210158401le 0 1199021015840

2lt 0 and the test is completed

Proof of Corollary 7 With 11990210158401= 119889119902lowast

1(119901)120597119890 and 1199021015840

2= 119889119902lowast

2(119901)

120597119890 (25) is obtained from 120597prod1198891205971199021= 0

The first derivation 119890 is carried out for 120597prod1198891205971199021= 0 and

ℎ1199021015840

1119891(1199021) + (119908

119889minus ℎ + 120573119908

119889minus V + 119892)(119902

1015840

1+ 1199021015840

2)119891(1199021+ 1199022) = 0

is achieved As such the plus and minus signs of 11990210158401 11990210158402are

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1

119903 119888 119890 ℎ 119908119898

119908119889

V 119892 119901 120573 120583 120590

150 25 5 45 40 80 5 10 01 01 1350 380

Table 2 The optimal ordering quantity under decentralization andeach contract

Decentralized Buyback contract Option contractEconomicorderingquantity

1365367393 1375084489 1849519365

the opposite Substituting (25) into 120597prod1198891205971199022= 0 for the first

derivation of 119890 minus11990111990210158402119891(1199022)(119908119889minus ℎ + 120573119908

119889minus V + 119892) + (1 minus

119901)ℎ1199021015840

1119891(1199021) = 1 is achieved To satisfy the equation 1199021015840

1ge 0

1199021015840

2le 0 is known and the test is completedLikewise with 1199021015840

1= 119889119902lowast

1(119901)120597ℎ and 1199021015840

2= 119889119902lowast

2(119901)120597ℎ from

120597prod1198891205971199021= 0 (25) is derived

The first derivation of ℎ is carried out for 120597prod1198891205971199021= 0

and119865(1199021)+ℎ1199021015840

1119891(1199021)+(119908119889minusℎ+120573119908

119889minusV+119892)(1199021015840

1+1199021015840

2)119891(1199021+1199022) =

0 is achieved As such the plus and minus signs of 11990210158401 11990210158402are

the opposite Substituting (25) into 120597Π1198891205971199022= 0 for the first

derivation of ℎminus11990111990210158402119891(1199022)(119908119889minusℎ+120573119908

119889minusV+119892)+119901119865(119902

2)+(1minus

119901)[119865(1199021) + ℎ119902

1015840

1119891(1199021)] = 1 is achieved To satisfy the equation

1199021015840

1ge 0 1199021015840

2le 0 is known and the test is completed

4 Numerical Analysis

We deploy MATLAB to do the simulation Similar to Li et al[25] andTian et al [9] it is suitable to suppose that themarketdemand is subject to the normal distribution119873(1350 380

2

)and the relevant parameters are as in Table 1

The optimal ordering quantity under decentralizationand each contract is shown in Table 2

The optimal profit under decentralization and contractcoordination is shown in Table 3

Tables 2 and 3 show that the ordering quantity and profitsin the centralized mode are smaller than those of buybackcontracts and option contracts among which the orderingquantity of distributors in the option contracts is more thanthose of distributors in the buyback contracts The reasonmight be that the option purchases from the manufacturercannot be exercised according to the real-life circumstancesand the coordinated contracts play a role in optimization

If other parameters are definite the possibility of occur-rence of different disruption risks buyback factors optionstrike prices and changes of option premium will have animpact on ordering quantity (Figures 2 to 5)

Figure 2 shows that the ordering quantity of distributorsfrom Manufacturer 1 does not decline with the rise ofdisruption risks because the cost of products offered byManufacturer 1 is always lower than that of Manufacturer 2When the disruptions do not occur in practice distributorsearn relatively higher profits from the low-cost products byManufacturer 1 which can mitigate the possible loss causedby the disruption of Manufacturer 1

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

q1q2

Disruption risks p of Manufacturer 1

Figure 2 The impact of different disruption risks on orderingquantity

0

500

1000

1500

2000

01 02 03 04 05 06 07 08 09

Ord

erin

g qu

antit

y

Buyback factor 120573

Qr

q1q2

q1 + q2

Figure 3 The impact of different buyback factors 120573 on orderingquantity

It can be seen from Figure 3 that the ordering quantityof the retailerrsquos increases with increase of the buyback factorDistributors share the partial risk of surplus inventory withthe retailer which stimulates the ordering from the retailerto some extent The ordering quantity of distributors fromManufacturer 1 does not changewith the variation of buybackfactor and the ordering quantity of Manufacturer 2 doesnot decline with the rise of the buyback factor The largerthe buyback factor is the higher the buyback cost of thedistributor is The profit margin tends to be smaller andthe order of products with higher prices will decline Whenthe buyback price is beyond a certain degree stockout ispreferred

Figures 4 and 5 show that the ordering quantity ofthe distributor from Manufacturer 1 increases when optionstrike prices and purchasing prices rise whereas the orderingquantity by the distributor from Manufacturer 2 decreaseswith the rise of option strike prices and option premiumWith the rise of the option premium distributors tend toreduce the ordering quantity from Manufacturer 2 and theordering fromManufacturer 1 increases to obtain the revenuewhen there is no risk in ordering fromManufacturer 1

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 3 The optimal profits with decentralized model and contract coordination

Supply chain type Retailerrsquos profits Distributorrsquos profits Overall profits of manufacturers Total profits of supply chainDecentralized model 777458466 409610218 3413418483 1528410533Contract coordination 782599824 452689764 3713792283 1606668817

0

500

1000

1500

2000

35 375 40 425 45 475 50 525 55 575

Ord

erin

g qu

antit

y

Option strike prices hq1q2

Figure 4 The impact of different option strike prices 119890 on orderingquantity

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Ord

erin

g qu

antit

y

Option premium e

q1q2

Figure 5 The impact of different option premium 119890 on orderingquantity

5 Conclusions

The main findings of this paper are as follows Firstlyintegrating the advantages of the option contract and thebuyback contract this paper applies these two contractstogether to coordinating the supply chain Secondly to becloser to the practical environment we consider the three-echelon supply chain model with two different suppliers(called dual sourcing purchase) one distributor and oneretailer In addition considering the practical situation inrecent years we cannot ignore the disruption risk factors inthe supply chain Consequently supply disruption risk factors

are considered in this paper And the buyback contract isused to stimulate and lead the retailer to increase orderingquantity the distributor shares the partial risk engenderedby demand uncertainty a balance can be struck betweenmarginal revenue andmarginal cost of the distributor and theretailer The disruption risk can be hedged by the distributorwho selects options

This study has investigated the three-echelon supplychain mode with random demand where distributors areable to choose one of twomanufacturers one has lower-priceproducts but disruptions are more likely to occur and theother has stable supply but its price of products is relativelyhigher The proposed model incorporates an option contractand a buyback contract whose coordination can help achievethe optimal ordering strategies It is revealed that increasedordering from a stable source can mitigate the disruptionrisks in supply chains suggesting greater adaptability androbustness in optimization of the operation of supply chainsIn dual sourcing purchasing disruption risks do not affect theordering quantity from enterprises with lower-price productsand unstable operations whereas the ordering quantity ofoption contracts from enterprises with high-price productsand stable supply will increase accordingly In specific cir-cumstances it is likely to choose suppliers with relativelystable operations but relatively higher prices The buybackfactor option premium and option strike prices influencethe ordering decisions by the distributor When the buybackfactor option purchasing prices and option strike pricesare greater the distributor reduces the ordering quantity ofproducts which are stable but costly and tends to buy unstableand cheaper products which tend to increase with the riseof option premium and option strike prices but will not beaffected by the buyback efficiency The buyback factor canstimulate the order from the retailerThese conclusionsmightprovide important references for supply chainmemberswhenthey make decisions

Here it is assumed that the risks of supply chain enter-prises are neutral and it is worth investigating the circum-stances with different risk preferences In addition there aremany measures to mitigate the supply chain disruptionsbut we only consider the dual sourcing purchasing strategywhereas other different strategies can also be addressed infuture research to develop the many-to-many models Thedisruption risks and demand uncertainties can be estimatedand further research can consider the impact engendered byestimation errors of measurement or different strategies inmisjudgment on profits of supply chain enterprises

Notations

1199021 The ordering quantity of distributors fromManufacturer 1

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1199022 The option purchasing quantity of

distributors fromManufacturer 2119890 Option premiumℎ Option strike prices119901 The probability of disruptions for

Manufacturer 1 (0 lt 119901 lt 1)

119888 Production cost of the two manufacturers119908119898 Wholesale prices provided by Manufacturer 1

for distributors119908119889 Wholesale prices provided by distributors 1

for the retailer119903 Sale prices of the retailer120573 Buyback price factor of distributors

(0 lt 120573 lt 1)

119892 V Shortage cost of the retailer and commoditysalvage respectively

119909 The random demand of the retailerrsquos marketthe random variable is continuous

119891(119909) The random demand probability densityfunction of the retailerrsquos market randomdemand cumulative distribution function ofthe retailerrsquos market

119865(119909) The random demand probability distributionfunction of the retailerrsquos market 119865(119909) iscontinuous and differentiable withinintervals and it is strictly increasing as119865(0) = 0

Π The overall profit of centralized supply chainwithout contract

119864(Π) The expected overall profit of centralizedsupply chains without contracts

119876 Ordering quantity of the retailer119876lowast

119888 The optimum ordering quantity of the

centralized supply chain system withoutcontract

119876lowast

119863 The optimum ordering quantity of the

decentralized supply chain system withoutcontracts

Π119903 Profits of the retailer

Π119889 Profits of the distributor

119864(Π119903) Expected profits of the retailer

119864(Π119889) Expected profits of the distributor

Conflict of Interests

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

Acknowledgment

This paper is financially supported by the Natural Sci-ence Foundation of China (Grant no 71172194 Grant no71390330 Grant no 71390331 and Grant no 71221001)

References

[1] PH Ritchken andC S Tapiero ldquoContingent claims contractingfor purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingamp Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoManagement Science vol 51 no 4 pp 566ndash580 2005

[4] X L Wang and L W Liu ldquoCoordination in a retailer-ledsupply chain through option contractrdquo International Journal ofProduction Economics vol 110 no 1 pp 115ndash127 2007

[5] Z Ning and J J Dai ldquoThe application of options in supply chainrisk managementrdquo System Engineering Theory amp Practice vol25 no 7 pp 49ndash54 2005

[6] W F Shang M Qi and Z Y Zhang ldquoOption contracts forperishable commodities with forecast updating and shortagedelivery postponedrdquo Chinese Journal of Management vol 9 no6 pp 908ndash912 2012

[7] L I Lin and T J Fan ldquoCoordination by option contract intwo-echelon supply chain with uncertaintyrdquo Journal of SystemsEngineering vol 27 no 6 pp 812ndash822 2012

[8] RDuA Banerjee and S LKim ldquoCoordination of two-echelonsupply chains using wholesale price discount and credit optionrdquoInternational Journal of Production Economics vol 143 no 2 pp327ndash334 2013

[9] J Tian H Q Zhang and Y L Wang ldquoEmergency suppliespurchasing model based on capacity option contract with dualpurchasing sourcesrdquo System EngineeringTheory amp Practice vol33 no 9 pp 2212ndash2219 2013

[10] Y Luo and Y J Luo ldquoStudy on order and supply strategy ofagricultural produce supply chains with circulation loss andoption contract consideredrdquo Logistics Technology vol 33 no 3pp 384ndash388 2014

[11] X Liu Q L Gou L Alwan and L Liang ldquoOption contracts asolution for overloading problems in the delivery service supplychainrdquo Journal of the Operational Research Society 2015

[12] M L Luo G Li C L J Wan and R Qu ldquoSupply chaincoordination with dual procurement sources via real-optioncontractrdquo Computers amp Industrial Engineering vol 80 pp 274ndash283 2015

[13] Z H Ma and J M Zeng ldquoOrder strategy of retailers withstochastic demand based on payment in advance and optioncontractrdquo Journal of Shanghai Maritime University vol 36 no5 pp 25ndash32 2015

[14] G P Cachon ldquoThe allocation of inventory risk in a supplychain push pull and advance-purchase discount contractsrdquoManagement Science vol 50 no 2 pp 222ndash238 2004

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] H Yu J Chen and G Yu ldquoSupply chain coordination underdisruptions with buy back contractrdquo System EngineeringTheoryamp Practice vol 25 no 8 pp 38ndash43 2005

[17] T Jia Y Xu and J L Chen ldquoBuy back policies retailerpromotions with inventories and supply chain coordinationrdquoForecasting vol 21 no 6 pp 591ndash597 2006

[18] J S Hu and H Wang ldquoThe price discount contract analysis ofthree-level supply chain under disruptionrdquo Chinese Journal ofManagement Science vol 15 no 3 pp 103ndash107 2007

[19] L L Hou and W H Qiu ldquoCoordinating the three-level supplychain with combined contracts under demand uncertaintyrdquoJournal of Beijing University of Aeronautics and Astronautics(Social Sciences Edition) vol 21 no 1 pp 1ndash5 2008

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[20] Z Xu D L Zhu and W G Zhu ldquoBuy back contract designin a supply chain under price-dependent demandrdquo Journal ofSystems Engineering vol 24 no 2 pp 173ndash177 2009

[21] J Chen ldquoThe impact of sharing customer returns informationin a supply chain with and without a buyback policyrdquo EuropeanJournal of Operational Research vol 213 no 3 pp 478ndash4882011

[22] X Guang X Deng Y H Qin and Q Wu ldquoBuyback contractcoordinating supply chain incorporated risk aversionrdquoResearchJournal of Applied Sciences Engineering and Technology vol 5no 5 pp 1744ndash1749 2013

[23] W Karush Minima of functions of several variables withinequalities as side conditions [MS dissertation] Department ofMathematics University of Chicago Chicago Ill USA 1939

[24] H W Kuhn and A W Tucker ldquoNonlinear programmingrdquo inProceedings of the 2nd Berkeley Symposium on MathematicalStatistics and Probability pp 481ndash492 University of CaliforniaPress Berkeley Berkeley Calif USA 1951

[25] J C Li Y W Zhou Y G Zhong and J S Guo ldquoOptimalordering strategies for seasonal products based on spectrumrisk measure and option contractrdquo System Engineering Theoryamp Practice vol 33 no 10 pp 2486ndash2496 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of