An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products

Applied Mathematical Modelling xxx (2014) xxx–xxx

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An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products G. Partha Sarathi, S.P. Sarmah ⇑, M. Jenamani Department of Industrial Engineering and Management, Indian Institute of Technology Kharagpur, India

a r t i c l e

i n f o

Article history: Received 1 July 2011 Received in revised form 26 November 2013 Accepted 3 February 2014 Available online xxxx Keywords: Supply chain coordination Contracts Revenue sharing Quantity discounts Short life-cycle product

a b s t r a c t This paper develops a combined contract model for coordinating a two stage supply chain where the demand at the retailer’s end is price sensitive and stock dependent. It has been shown that proposed coordination mechanism achieves perfect coordination and win–win situation for both the members of the supply chain. Further, an extensive sensitivity analysis is performed to examine the impact of various parameters on supply chain performance. It has been found that stock dependency factor has positive impact on order quantity and subsequently on supply chain performance. The paper has also made a comparative statics analysis to see the impact of certain parameters on the pricing and replenishment policies of the retailer. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Short life cycle products are characterized by a short selling season after which their value reduce drastically. Fashion apparel, electronic goods, personal computers, toys etc. are some of the examples of short life cycle product. The short selling season with uncertain nature of demand poses many challenges to the members of supply chain (SC) dealing with such type of products. For example, the Korean manufacturers of cellular phones introduce more than 50 new models each year, and the average life-cycle of cellular phones in Korea is less than 10 months [1]. The short life-cycle products on one hand, invite the risks of overstocking and understocking while on the other hand, it provides greater opportunities for higher margins [2]. These attractive characteristics of this category of products have prompted the researchers to work in the area of managing supply chains of short life-cycle products. In addition to it, since the selling season of such product is small, it is necessary to ensure the availability of stock in the shelf of the retailer and display of it plays an important role in stimulating the demand. Stock dependent demand has been studied in the inventory literatures for quite some times now (Urban, [3]) but only few studies have explored the effect of stock dependency on the performance of the SC and it has motivated us to take up the study on price sensitive stock dependent demand of short life cycle product in the context of SC. The uncertain nature of demand, faulty planning and poor purchasing practices are some of the reasons that increase the risk of under stoking and over stoking of short life cycle products among the members of the SC. To minimize these risks, various contracts such as buy back (BB), quantity discount (QD), revenue sharing (RS) and quantity flexibility (QF) contract etc. have been cited as coordination mechanism in the SC literature. For example, Agrawal and Seshadri [4] developed risk ⇑ Corresponding author. Tel.: +91 3222 283735. E-mail addresses: [email protected] (G. Partha Sarathi), [email protected] (S.P. Sarmah). http://dx.doi.org/10.1016/j.apm.2014.02.003 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

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free contract for a risk averse retailer under news vendor framework to coordinate the SC. In SC contract literature, BB contracts have been widely studied, but in reality, BB contract has certain limitations. First, the physical handling of returns may be impractical in certain situations [5]. Second, the retailer may not have enough cash to go for one time procurement of the optimum stock [6]. Under such situations it is necessary to explore other contract mechanisms to coordinate the SC. RS is one such mechanism that has gained a lot of attention from both academia and industry [7,1]. Under RS contract, the retailer pays the supplier a wholesale price for each unit purchased and a certain percentage of revenue the retailer generates [8]. This type of contract is attractive to the retailer and can replace BB contract particularly when the items are expensive [6]. There are certain advantages with RS contract over BB contract. First, the physical handling of returns is not required. Second, the retailer obtains goods at a lesser wholesale price and need not have to invest huge capital in procuring the items. It is fond that RS contract is widely employed in video rental industries and e-commerce businesses. In this paper, using newsvendor framework, a combined RS and QD contract model is developed for single manufacturer and single retailer SC dealing with a short life-cycle product considering demand to be price-sensitive and stock dependent. This paper extends the work of Yao et al. [9]. They have studied RS contract using price-setting newsvendor framework. Our work differs from Yao et al. [9] in two respects. First, it captures stock dependency of demand in addition to price-sensitivity. Second, it integrates RS and QD contract. In Section 2, we briefly review the literature relevant to our work. In Section 3, we present the mathematical model developed for coordinating the SC. In Section 4, we illustrate the model through a numerical example. We also provide the results of sensitivity analysis on this section. Section 5 concludes the paper with a brief summary and directions for further research work.

2. Literature review In this section, we have provided a brief review of literature related to our work. Excellent review on SC contract models have been provided by Tsay et al. [10] and Cachon [11] and readers can read those articles to get idea about different types of coordination models. Earlier, many authors have studied SC coordination considering demand as deterministic or constant. But in real practice, it is not true and demand is found to be stochastic in nature. Stochastic model under news vendor framework can be classified as fixed price and price setting. In fixed price case, the market price is considered to be fixed and determines only the optimal order quantity (see [12,13]); whereas in case of price setting, optimal price and order quantity are determined simultaneously (see [9,14–17]). Wei and Choi [18] under mean variance framework, SC coordination is explored through wholesale price and profit sharing scheme. Chiu et al. [19] have studied target sales rebate as a coordination .mechanism for a manufacturer and a risk averse retailer under mean variance framework. Tsay [20] analyzed how sensitivity to risk affects the behaviors and outcomes on both sides of a manufacturer-retailer supply relationship, and how these dynamics are altered by a manufacturer return policy. Modeling of demand is an important component to correctly depict the reality. It is well known that price influences demand and vice versa. To capture this reality, some researchers have introduced a new concept from the principles of thermodynamics to model demand and price relationship. When demand is price sensitive, they consider price to be analogous to temperature. In such models, demand is considered as the potential or the driving force that creates the difference in monetary value or price and for details of such model, one can refer Jaber et al. [21,22]. In recent times, some authors have developed models considering sales effort of the retailer, lead time and stock level at the retailer’s end as a means for enhancing the demand [23–25]. Further, it has been well recognized in the literature that demand of many items at the retailer level is proportional to the amount of inventory displayed [26]. The real-life examples include short life-cycle products such as, sugar, spices, clothes, gift cards etc. [27]. Specifically, the supermarkets, where the products are displayed on the aisles, require inventory control models that take stock level into consideration. To model this situation, mainly two approaches have been adopted in the literature. Demand is expressed as a function of initial stock level or is considered as time dependent stock level [3]. Though in the inventory literature, the importance of stock level has been recognized for quiet a long time, yet it has not received adequate attention from SC management researchers. A coordination model considering a two stage SC with an initial stock level dependent demand was developed by Wang and Gerchak [28]. A coordination model considering stock dependent demand is also developed recently by Zhou et al. [29]. To coordinate a SC, in recent times, combined contract models, i.e., contracts with two or more coordination mechanisms have been proposed in the literature [30,31]. Under fixed price newsvendor framework, Shi and Su [13] developed a combined BB and QD contract model and have shown that contract is self-enforcing. Burnetas et al. [30] have pointed out that QD contract combined with other mechanisms such as BB offers tremendous scope for future research. Recently, Güler and Bilgiç [32] have used a mixed BB and RS contract to coordinate an assembly system. The manufacturer shares its sales revenue with the suppliers. Aydin and Porteous [33] considered two types of rebate for coordinating a two stage SC, one directly to the consumer and another to the retailer and have shown that no members of the SC always favor only one kind of rebate. Demirag et al. [34] also studied two different types of promotion, retailer incentive scheme and customer rebate policy in a two stage SC setting. They have shown that under market uncertainty condition, combined policy performs better and improves the sales and increase the profit of the manufacturer. Considering demand as a function of sales effort of the retailer, Taylor [24] has shown that SC coordination can be achieved through a properly designed target rebate and return contract. Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

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Recently, Chiu et al. [19] have considered additive and multiplicative price dependent demand and shown that combination of wholesale price, channel rebate, and returns together can coordinate a SC. In channel rebate, manufacturer makes payment to the retailer on the basis of his sales to the end customer. The rebate contract has the effect of motivating retailer to lower prices for increasing sales. Here the retailer has to invest a large amount to procure the items from the supplier. The retailer is entitled to get the sales rebate, once he crosses a threshold limit of sales. The inherent deficiency of return policy cannot be overcome by combining with the rebate policy. Moreover, if the item is of short cycle and the value is high, the cash starved retailer may be unwilling to go for optimum stock. For such kind of product, RS contract has the biggest advantage of making higher availability of product in the market and in combination with QD; it can improve the performance of such SC. Also, some authors have used two part tariff contract for coordinating a SC. In two part tariff mechanism, there are two components. One fixed component and another per unit (usage) price. The retailer has to pay the fixed amount first and then to pay the additional charges for the purchase of each unit. Retailer not having enough cash will face difficulty for single period procurement of short life cycle product. Therefore implementation of two part tariff contract under such situation may not be a feasible proposition. 3. Development of the mathematical model We have considered a two stage SC consists of single manufacturer and single retailer dealing with short life-cycle products. The retailer faces a stock dependent and price-sensitive demand which is uncertain in nature. In this model, we develop a coordinating mechanism combining RS and QD contracts. Though, both BB and RS contracts are designed for managing SC risks, there are some inherent differences between them. In case of BB contract, if there is overstocking, the manufacturer bears the risk and buys back the leftover items. Under RS contract, the manufacturer takes the risk of supplying the goods at a lower wholesale price and gets a fraction of the sales revenue. Owing to low wholesale price, the retailer buys more and the risk of understocking is reduced. However, the retailer bears both overage cost, i.e., the cost incurred by the retailer when order quantity is more than the realized demand; and underage cost, i.e., the cost incurred by the retailer when the order quantity is less than the realized demand. Here, we have considered newsvendor model for coordination through contracts. Considering demand D and order quantity Q , the expected number of sales, the expected number of leftover items and the expected shortages at the end of the selling season are given by E½minðQ ; DÞ, E½ðQ  DÞþ  and E½ðD  Q Þþ , respectively. Following notation are used in the development of the model: Decision variables p w Q Other notation D w0 h s m

e l r f ðÞ FðÞ

Pxy

Unit sale price at the retailer end Unit wholesale price Order quantity of the retailer Demand rate of the item Unit wholesale price under PO contract Unit cost of overage Unit cost of underage Unit cost of manufacturing Random component of demand Mean demand Fraction of revenue kept by the retailer Probability density function (pdf) of the random demand Cumulative distribution function (cdf) of the random demand Expected profit of member x under situation y

(Here x can be M; R or T where M stands for manufacturer, R for retailer and T for the total system; and y can be c or dc where c stands for centralized system, and dc for decentralized system.) For the development of the model, following assumptions are made:     

Both the members of the SC are risk neutral. The information of market is a common knowledge. The manufacturer has unlimited capacity and offers product to the retailer before the start of the selling season. The market demand of end item is uncertain, price sensitive and dependent on initial stock. The cost parameters follow some straightforward assumptions to ensure internal consistency: (i) p > w0 > m > 0. (ii) 0 6 r 6 1. (iii) h P 0; s P 0.

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3.1. Demand model The price-sensitivity, stock dependency and uncertainty of demand are captured in the presented demand model. We model demand as a function of initial stock, i.e., the order quantity and demand is considered as a linear additive function as follows:

D ¼ a  bp þ cQ þ e:

3.2. Flow of payments in the supply chain In order to define the profit functions of the individual member and that of the entire SC, it is important to understand the flow of payments in the supply chain. As shown in Fig. 1, under PO contract, the manufacturer produces the product spending a manufacturing cost m per unit and sells the product to the retailer at a wholesale price w per unit. In response to the offer made by the manufacturer, the retailer determines order quantity Q . The retailer sells the products to the end customer at a price p per unit. No returns are allowed in case of leftover items, and no sharing of revenue is permitted when operating under PO contract. The manufacturer spends an amount mQ for producing the products and receives an amount w0 Q from the retailer. Since D is the demand, the expected number of sales at the retailer end can be derived by E½minðQ ; DÞ. It means that the retailer receives an amount pE½minðQ ; DÞ from the customers due to the proceeds of the sale. The expected number of leftover items and the expected number of shortages are given by E½ðQ  DÞþ  and E½ðD  Q Þþ , respectively. The retailer pays a penalty of hE½ðQ  DÞþ  and sE½ðD  Q Þþ  in case of overage and shortage, respectively. The flow of payments with PO contract is shown in Fig. 1. The retailer receives an amount pE½minðQ ; DÞ from the proceeds of the sale. When RS contract is in place, the retailer retains rpE½minðQ ; DÞ and remits the balance ð1  rÞpE½minðQ ; DÞ to the manufacturer as per the contract. The retailer pays a penalty of hE½ðQ  DÞþ  and sE½ðD  Q Þþ  in case of overage and shortage, respectively. The flow of payments under RS contract is shown in Fig. 2. In this study, the combination of QD and RS contracts are used. The combination of QD with RS can be represented in the same way as in Fig. 2 where the ðw; Q Þ pair will be determined as per the model outcome. 3.3. Methodology for solution To analyze and design the proposed combined contract, the following steps are followed: Step 1: Analysis of decentralized non-coordinated SC operating under PO contract In the first step, the optimal pricing and order strategies of the retailer is determined and based on these optimum values, the profits of SC members are computed. Step 2: Design of an equivalent RS contract In this step, manufacturer offers RS contract in place of PO contract and accordingly, the manufacturer reduces the wholesale price to encourage the retailer to opt for RS contract. Step 3: Incorporation of QD contract into the existing RS contract In the existing RS contract, the wholesale price is discounted to entice the retailer to choose for higher order quantity i.e., equal to centralized order quantity. Step 4: Testing the effectiveness of the combined contract model

Fig. 1. Flow of payments in a supply chain under PO contract.

Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

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Fig. 2. Flow of payments in a supply chain under RS contract.

In this step, to measure the effectiveness of the proposed combined contract model, numerical example is carried out and profit of the SC under combined contract and those under PO only contract are compared. The impact of various parameters on SC coordination is investigated by performing extensive sensitivity analysis. 3.4. Non-coordinated decentralized setting Here the problem is analyzed as a Stackelberg game where the manufacturer acts as the leader and the retailer acts as the follower. As a first mover, manufacturer announces his pricing policy and based on it retailer decides his ordering and pricing decisions simultaneously. Manufacturer optimizes his pricing policy considering the rational behavior of the retailer and maximizes his profit while the objective of the retailer is to maximize his own profit corresponding to the pricing policy of the manufacturer. The equilibrium point that specifies the pricing and ordering policies of the retailer is found out by the solution of a two-stage non-cooperative game. Retailer’s profit can be expressed as

PR ¼ Expected sales revenue  Expected purchase cost  Expected overage cost  Expected underage cost: The mathematical expression is as follows

PR ¼ pE½minðQ ; DÞ  w0 Q  hE½ðQ  DÞþ   sE½ðD  Q Þþ :

ð1Þ

Similarly, profit of the manufacturer is given by

PM ¼ ðw0  mÞQ:

ð2Þ

The profit function of the SC is given by

PT ¼ pE½minðQ ; DÞ  mQ  hE½ðQ  DÞþ   sE½ðD  QÞþ :

ð3Þ

Introducing a transformational variable z ¼ Q  ða  bp þ cQ Þ, the order quantity expression becomes

Q¼

a  bp þ z : 1c

ð4Þ

Substituting D ¼ a  bp þ cQ þ e and Q ¼ ða  bp þ cQ Þ þ z, we can have the following identities:

E½ðQ  DÞþ  ¼ z  l þ HðzÞ; E½ðD  Q Þþ  ¼ HðzÞ and subsequently,

E½minðQ ; DÞ ¼ a  bp þ c

  a  bp þ z þ l  HðzÞ: 1c

Since for two whole numbers x and y, the minimum of x and y is given by minðx; yÞ ¼ x  ðx  yÞþ where xþ denotes the maximum of 0 and x. RB Here HðzÞ denotes the loss function and is given by HðzÞ ¼ z ðx  zÞf ðxÞdx. Therefore, the retailer’s profit function given in Eq. (1) can be written as follows.

PR ¼ p½a  bp þ cQ þ ðl  HðzÞÞ  w0 Q  h½z  l þ HðzÞ  s½HðzÞ ¼ p½a  bp þ ðl  HðzÞÞ þ Q ðpc  w0 Þ  h½z  l þ HðzÞ  s½HðzÞ   a  bp þ z ðpc  w0 Þ  h½z  l þ HðzÞ  s½HðzÞ: ¼ p½a  bp þ ðl  HðzÞÞ þ 1c

ð5Þ

Also, the profit function of the retailer given in Eq. (5) can be rearranged as follows

PR ¼ p½a  bp þ ðl  HðzÞÞ þ

  a  bp þ z ðpc  w0 Þ  h½z  ðl  HðzÞÞ  s½l  ðl  HðzÞÞ: 1c

Since E½minðz; eÞ ¼ l  HðzÞ (see Appendix A), the above equation can be written as

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PR ¼ pða  bp þ E½minðe; zÞÞ þ

  a  bp þ z ðpc  w0 Þ  hðz  E½minðe; zÞÞ  sðl  E½minðe; zÞÞ: 1c

After rearranging the terms, the profit function of the retailer can be expressed as

PR ¼ pða  bpÞ þ

  a  bp þ z ðpc  w0 Þ  Hz  sl þ ðp þ h þ sÞE½minðe; zÞ: 1c

ð6Þ

Now, the objective function of the retailer is

Maximise PR :

ð7Þ

Taking the first and second order derivatives of PR with respect to z and p, one gets

@ PR pc  w0  ¼  h þ ðp þ h þ sÞ½1  FðzÞ; @z 1c

ð8Þ

@ 2 PR ¼ ðp þ h þ sÞf ðzÞ; @z2

ð9Þ

@ PR a  2bp þ bw0 þ zc þ ð1  cÞðl  HðzÞÞ ¼ ; 1c @p

ð10Þ

@ 2 PR 2b ¼ ; 1c @p2

ð11Þ

The profit equation of retailer, PR is concave in nature with respect to p for a specific value of z. Under such situation, retailer’s objective function given in 7 can be converted to an optimization problem over a single variable z by first solving for an optimal value of p as a function of z and after that, one can substitute the results back into PR . Exploring the resulting optimal trajectory to maximize PR ðp ; z; eÞ, one can obtain z . From Eq. (10), one gets

p ¼ pðzÞ ¼

1 ½a þ bw0 þ zc þ ð1  cÞðl  HðzÞÞ: 2b

Substituting p ¼ pðzÞ in Eq. (7), the optimization problem becomes a maximization problem over a single variable z. Alternately, one can find the optimal solution of the retailer’s problem by determining p and z by solving the following two response functions iteratively, which are derived from the Eqs. (8) and (10), respectively.

z ¼ F 1

p¼



 p þ sð1  cÞ  w0 ; ð1  cÞðp þ s þ hÞ

1 ½a þ bw0 þ zc þ ð1  cÞðl  HðzÞÞ: 2b

ð12Þ

ð13Þ

The optimum order quantity of the retailer under decentralized scenario can be determined by substituting the values of p and z in Eq. (4). The optimal order quantity of the retailer in a non-coordinated scenario is given as 

Q ¼

a  bp þ z : 1c

ð14Þ

3.5. Decentralized setting with revenue sharing contract In this section, we have considered a decentralized supply chain where the manufacturer intends to reduce the wholesale price to entice the retailer to order more and in turn shares the sales revenue generated by the retailer to just compensate the reduction in wholesale price of the manufacturer so that the profits of the SC members under this contract are same as that under PO contract. It is assumed that the manufacturer reduces the wholesale price from w0 to wrs to entice the retailer to share his revenue. Under decentralized setting, the expected profit of the retailer under RS contract is given by

PR ¼ PR ðQ ¼ Q ; r; w ¼ wrs Þ ¼ rpE½minðQ ; DÞ  wrs Q  hE½ðQ  DÞþ   sE½ðD  Q Þþ :

ð15Þ

Correspondingly, the profit of the manufacturer under RS contract in decentralized setting is given by

PM ¼ PM ðQ ¼ Q; r; w ¼ wrs Þ ¼ ð1  rÞpE½minðQ ; DÞ  wrs Q  mQ :

ð16Þ

The total system profit under RS contract in decentralized setting is the sum of the profit of the retailer and the profit of the manufacturer. Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

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Proposition 1. To facilitate revenue sharing under decentralized system, the manufacturer reduces the wholesale price from w0 to wrs , where, wrs is given by

wrs ¼ w0 

ð1  rÞpE½minðQ dc ; DÞ : Q dc

ð17Þ

Proof. Under decentralized system, the manufacturer will reduce the wholesale price and share revenue generated by the retailer only when his profit is not suffered due to RS contract. Mathematically, this constraint is expressed as

PM ðQ ¼ Q dc ; r; w ¼ wrs Þ  PM ðQ ¼ Q dc ; r ¼ 1; w ¼ w0 Þ P 0; ð1  rÞpdc E½minðQ dc ; DÞ þ wrs Q dc  mQ dc ¼ w0 Q dc  mQ dc P 0; ð1  rÞpdc E½minðQ dc ; DÞ P ðw0  wrs ÞQ dc ; wrs P w0 

ð1  rÞpdc E½minðQ dc ; DÞ : Q dc

In order to incorporate revenue sharing in the decentralized setting, the wholesale price should be decreased from w0 to dc ;DÞ wrs . It is apparent from the above inequality that wrsmin ¼ w0  ð1rÞpdc E½minðQ is the minimum wholesale price that the manQ dc

ufacturer can offer to the retailer to accommodate RS contract. This RS contract with w ¼ wrsmin and revenue sharing fraction r is equivalent to PO contract as it provides the same profits as that of PO contract. In the next section, centralized setting is considered. h 3.6. Centralized setting Centralized setting is equivalent to a situation where the supplier and retailers are considered to be parts of a vertically integrated firm with a single controlling body taking the decision to maximize the overall SC profit. Therefore, the total supply chain profit is the sum of the profits of both the members as shown below

Max PT ¼ PR þ PM ;

ð18Þ

PT ¼ pE½minðQ ; DÞ  mQ  hE½ðQ  DÞþ   sE½ðD  QÞþ  ¼ pða  bp þ cQ þ E½minðz; eÞÞ  mQ  hE½ðz  eÞþ   sE½ðe  zÞþ  ¼ pða  bp þ E½minðz; eÞÞ  ðpc  mÞQ  hE½ðz  eÞþ   sE½ðe  zÞþ : Finally, after simplification, the total profit becomes



PT ¼ pða  bp þ l  HðzÞÞ  ðpc  mÞ

 a  bp þ z  hðz  l þ HðzÞÞ  sðHðzÞÞ: 1c

ð19Þ

Again, the centralized system profit is given by



 a  bp þ z  hðz  l þ HðzÞÞ  sðHðzÞÞ 1c   a  bp þ z  hðz  E½minðz; eÞÞ  sðl  E½minðz; eÞÞ; ¼ pða  bp þ E½minðz; eÞÞ  ðpc  mÞ 1c

PT ¼ pða  bp þ l  HðzÞÞ  ðpc  mÞ

PT ¼ pða  bpÞ  ðpc  mÞ

  a  bp þ z  hz  sl þ ðp þ h þ sÞE½minðz; eÞ: 1c

ð20Þ

Taking the first and second order derivatives of PT with respect to z and p, one gets

@ PT pc  m  h þ ðp þ h þ sÞ½1  FðzÞ; ¼ 1c @z

ð21Þ

@ 2 PT ¼ ðp þ h þ sÞf ðzÞ; @z2

ð22Þ

@ PT a  2bp þ bm þ zc þ ð1  cÞðl  HðzÞÞ ¼ ; 1c @p

ð23Þ

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@ 2 PT 2b ¼ ; 1c @p2

ð24Þ

Again, here PR is concave in p for a specific value of z. Therefore, objective function presented in 18 can be changed to an optimization problem over a single variable z by initially solving it for an optimal value of p as a function of z and then one can substitute the results back into PR . Finding the resulting optimal trajectory to maximize PR ðp ; z; eÞ, one can get z . From the Eq. (17), one gets

p ¼ pðzÞ ¼

1 ½a þ bw0 þ zc þ ð1  cÞðl  HðzÞÞ: 2b

Substituting p ¼ pðzÞ into 18, the optimization problem becomes a maximization problem over a single variable z. Alternately, as discussed earlier, the optimum solution under centralized can be obtained by solving the following two response functions iteratively, which are derived from Eqs. (21) and (23), respectively.

z ¼ F 1

p¼



 p þ sð1  cÞ  m ; ð1  cÞðp þ s þ hÞ

ð25Þ

  1 a þ bm þ zc þ ð1  cÞðl  HðzÞÞ : 2b

ð26Þ

Substituting these optimum values denoted by pc and zc in the Eq. (4), one can obtain the optimum centralized order quantity. Therefore, the optimum order quantity under centralized setting is given by 

Q c ¼

a  bpc þ zc : 1c

ð27Þ

Comparing order quantities under centralized and decentralized settings using the Eqs. (14) and (27), one can easily establish that when Q c > Q dc as w0 > m. Therefore, the manufacturer can use quantity discount mechanism to entice the retailer to opt for higher order quantity, i.e., Q c . Finally, the optimum retail price and order quantity for centralized SC operating under price-sensitive and stock dependent demand environment are given by the following equations (see Appendix C).

pc ¼

1  a þ bm þ zc  ð1  cÞE½ðzc  eÞþ  ; 2b

ð28Þ

Qc ¼

a  bm þ zc 1  þ E ðzc  eÞþ ; 2 2ð1  cÞ

ð29Þ

h

i

pþsð1cÞm . where zc ¼ F 1 ð1cÞðpþhþsÞ In the next section we perform comparative statics analysis to investigate the impact of various factors on the pricing and replenishment decisions under centralized setting using the Eqs. (28) and (29) and derive the following propositions.

3.6.1. Comparative statics analysis Comparative statics is the determination of the changes in the endogenous variables of a model that will result from a change in the exogenous variables or parameters of that model. It is a powerful tool for establishing theoretical deductions. These deductions can be determined by simply differentiating the first order conditions with respect to parameters. Proposition 2. Retail price increases with stock dependency factor under centralized setting.

Proof. It follows from the Eq. (28) that

    @pc @ 1 1 @zc @ ¼ ða þ bm þ zc  ð1  cÞE½ðzc  eÞþ Þ ¼  ðð1  cÞE½ðzc  eÞþ Þ @c 2b 2b @c @c @c       þ 1 @zc @E½ðzc  eÞ  1 @zc @E½ðzc  eÞþ  1  ð1  cÞ þ E½ðzc  eÞþ   ð1  cÞ þ E½ðzc  eÞþ  ¼ ¼ 2b @c @c 2b @zc @c    1 @zc ð1  ð1  cÞFðzc ÞÞ þ E½ðzc  eÞþ  : ¼ 2b @c Since

@zc @c

> 0, Fðzc Þ < 1 and E½ðzc  eÞþ  > 0 (see Appendixes D and E), it follows that

@pc @c

> 0. h

Proposition 3. Order quantity increases with stock dependency factor under centralized setting. Proof. It follows from the Eq. (29) that Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

9

"  # @Q c a  bm 1 @zc 1 @E½ðzc  eÞþ  þ ¼ þ ð1  cÞ  z ð1Þ c 2 2 2 @c @c @c 2ð1  cÞ 2ð1  cÞ "  # a  bm 1 @zc 1 @E½ðzc  eÞþ  @zc ¼ þ þ ð1  cÞ þ z : c 2 @zc @c @c 2ð1  cÞ2 2ð1  cÞ2 Hence,

@Q c @c

> 0, since

@zc @c

> 0 (see Appendix D). h

Proposition 4. Retail price decreases with price-sensitivity factor under centralized setting. Proof. It follows from the Eq. (28) that

    @pc @ 1 @ a þ zc m ð1  cÞ E½ðzc  eÞþ  ¼ ða þ bm þ zc  ð1  cÞE½ðzc  eÞþ Þ ¼ þ  @b 2b @b 2 2b @b 2b ¼

ða þ zc Þ 2

2b

þ

ð1  cÞE½ðzc  eÞþ  2b

2

Since zc > E½ðzc  eÞþ , it follows that

¼

@pc @b

a  zc þ ð1  cÞE½ðzc  eÞþ  2b

2

:

< 0 (see Appendixes D and E).

h

Proposition 5. Order quantity decreases with price-sensitivity factor under centralized setting. Proof. It follows from the Eq. (29) that

@Q c @b

m ¼ 2ð1cÞ < 0.

h

Proposition 6. Retail price increases with primary demand (market base) under centralized setting. Proof. It follows from the Eq. (28) that

@pc @a

1 ¼ 2b > 0.

h

Proposition 7. Order quantity increases with primary demand (market base) under centralized setting. Proof. It follows from the Eq. (29) that

@Q c @a

1 ¼ 2ð1cÞ > 0.

h

Proposition 8. Order quantity increases with retail price under centralized setting. Proof. It follows from the Eqs. (28) and (29) that

@Q c 1 @zc 1 @ ¼ þ ðE½ðzc  eÞþ Þ; @pc 2ð1  cÞ @pc 2 @pc @Q c 1 @zc 1 @ðE½ðzc  eÞþ Þ @zc ¼ þ ; @zc @pc 2ð1  cÞ @pc 2 @pc    @Q c @zc 1 1 @ðE½ðzc  eÞþ Þ : þ ¼ 2ð1  cÞ 2 @zc @pc @pc Since

@zc @pc

> 0 and

@ðE½ðzc eÞþ Þ @zc

> 0, it follows that

@Q c @pc

> 0. h

Proposition 9. Order quantity decreases with manufacturing cost under centralized setting. Proof. It follows from the Eq. (29) that

    @Q c 1 @zc 1 @E½ðzc  eÞþ  1 @zc 1 @E½ðzc  eÞþ  @zc þ þ ¼ ¼ : b þ b þ 2 @m 2ð1  cÞ 2 @zc @m 2ð1  cÞ @m @m @m Since

@zc @m

< 0, it follows that

@Q c @m

< 0 (see Appendix D).

h

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3.7. Coordination under decentralized setting – incorporation of quantity discounts into revenue sharing contract It is worth mentioning here that the retailer will show his interest in raising his order quantity provided he receives discount for ordering larger quantity. In other words, the retailer will opt for centralized order quantity provided his profit is more than or equal to that under decentralized order quantity. In the same line, the manufacturer will be interested for centralized order quantity provided his profit is higher than or equal to that under decentralized-no revenue sharing scenario. Looking at the problem from the perspective of both the members, one can determine the upper and lower bound of wholesale price for coordination (for complete derivation see Appendix B).

w0 6

    1 ðr ½pc E½minðQ c ;DÞ  pdc E½minðQ dc ;DÞ þ wrs Q dc  h E½ðQ c  DÞþ   E½ðQ dc  DÞþ   s E ðD  Q c Þþ  E ðD  Q dc Þþ Þ Qc

and

W0 P

1 ½mðQ c  Q dc Þ þ wrs Q dc þ ð1  rÞ½pdc E½minðQ dc ; DÞ  pc E½minðQ c ; DÞ: Qc

One can determine the wholesale price range ðwmin ; wmax Þ where,

wmin ¼

1 ðmðQ c  Q dc Þ þ wrs Q dc þ ð1  rÞ½pdc E½minðQ dc ; DÞ  pc E½minðQ c ; DÞÞ Qc

ð30Þ

and

wmax ¼

 1

r½pc E½minðQ c ;DÞ  pdc E½minðQ dc ;DÞ þ wrs Q dc  h½E½ðQ c  DÞþ   E½ðQ dc  DÞþ   s½E½ðD  Q c Þþ   E½ðD  Q dc Þþ  : Qc ð31Þ

Any wholesale price chosen from the feasible range will not only coordinate the SC but also allocate the profits arbitrarily to both the members of the SC. The retailer will be interested to choose the centralized order quantity provided the retailer gets some form of incentives from the manufacturer. Therefore, the manufacturer offers incentive in the form of quantity discounts to encourage the retailer to opt for the centralized order quantity. Let w00 be the new reduced wholesale price offered by the manufacturer to the retailer so that the retailer is interested for the centralized order quantity. The expected profit of the retailer under centralized order quantity can be written as

PRc ¼ PR ðQ ¼ Q c ; r; w ¼ w00 Þ ¼ rpc E½minðQ c ; DÞ  w00 Q c  hE½ðQ c  DÞþ   sE½ðD  Q c Þþ :

ð32Þ

The expected profit of the manufacturer can be written as

PMc ¼ PM ðQ ¼ Q c ; r; w ¼ w00 Þ ¼ ð1  rÞpc E½minðQ c ; DÞ þ w00 Q c  mQ c

ð33Þ

The total expected profit under centralized order quantity is obtained by the sum of the profit of the retailer and the manufacturer. The coordination benefit can be computed by assuming that the manufacturer charges the maximum wholesale price under coordinated setting. In that case, w ¼ wmax , and the manufacturer will receive all the benefits leaving the profit of the retailer unaffected from the initial situation. The increase in total profit due to coordination is equal to the increase in the profit of the manufacturer only. Under this situation, the expected profit of the retailer is given as

PRc ¼ PR ðQ ¼ Q c ; r; w ¼ wmax Þ ¼ rpc E½minðQ c ; DÞ  wmax Q c  hE½ðQ c  DÞþ   sE½ðD  Q c Þþ :

ð34Þ

And, the expected profit of the manufacturer is given by

PMc ¼ PM ðQ ¼ Q c ; r; w ¼ wmax Þ ¼ ð1  rÞpc E½minðQ c ; DÞ þ wmax Q c  mQ c :

ð35Þ

The total system expected profit is the sum of the profit of the retailer and the manufacturer and the increase in profit of the manufacturer is equal to the increase in total system profit due to coordination. This extra profit can be shared through certain profit sharing mechanism which is agreeable to both the parties. The model under consideration achieves perfect coordination and leads to a win–win situation for both the entities of the SC. 4. Numerical example Here the coordination model discussed above is illustrated through a numerical example. The SC comprises of one manufacturer and one retailer and the demand at the retailer’s end is assumed to be stock dependent and it is modeled as: D ¼ 200  25p þ 0:1Q þ e where, e is uniformly distributed in the range ½0; 10. The cost parameters are given below:

w0 ¼ 3:25;

m ¼ 1;

s ¼ 0:25;

h ¼ 0:25;

r ¼ 0:65:

To determine the solution for the joint pricing problem and perform sensitivity analysis, all the equations are coded in MATLAB 7. Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

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G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 1 Optimum values. Optimum values

Decentralized setting

Centralized setting

p z Q

5.70 4.79 69.21

4.60 8.34 103.59

Table 2 Profits under different contract model settings. Contract model

Price-only contract Benchmark RS contract Combined RS and QD contract* Combined RS and QD contract**

Profit in rupees

PR

PM

PT

162.40 162.40 162.40 181.57

155.72 155.72 194.06 174.89

318.12 318.12 356.46 356.46

*

In this case, all the coordination benefit goes to the manufacturer. In this case, both retailer and manufacturer are equally powerful and the coordination benefit is shared equally.

**

4.1. Comparison of contracts under different settings Here four settings of the SC are considered and they are: decentralized SC with PO contract, decentralized SC with RS contract that yields the same profits as that of PO contract, decentralized SC with combined RS and QD contract and finally, decentralized coordinated SC with combined RS and QD contract. The optimum values of retail price and order quantity under both decentralized and centralized setting are determined and results are shown in Table 1, and the corresponding profits are given in Table 2. Using Eqs. (12)–(14), the optimal retail price and order quantity under PO contract under decentralized non-coordinated setting are determined and are found to be Rs. 5.7 and 69.21 units, respectively (see Table 1) and the profits of the retailer, the manufacturer and the total SC are Rs. 162.40, 155.72 and 318.12, respectively (see Table 2). The manufacturer wishes to offer RS contract with a reduced wholesale price to encourage the retailer and in turn shares a certain fraction of revenue generated by the retailer. If r ¼ 0:65 is the agreed fraction of revenue retained by the retailer then the rest 0.35 fraction of revenue is given to the manufacturer. With this RS agreement, using the Eq. (17), the manufacturer decreases the wholesale price from Rs. 3.25 to Rs. 1.288 and in turn shares the revenue with r ¼ 0:65. The profits of SC members under this scheme are same as that of PO contract. Similarly, the optimum retail price and order quantity under centralized setting are determined using Eqs. (25)–(27). Table 1 presents the optimum values under decentralized and centralized settings and the total profits under both the situations are found to be Rs. 318.12 and Rs. 356.46, respectively. To improve the SC profits, the manufacturer wishes to adopt QD contract by which the centralized order quantity can be offered at a discounted wholesale price. Using the Eqs. (30) and (31), the wholesale price range 0.9458–1.3159 is determined. When the wholesale price is reduced to 0.9458 then all the coordination benefit goes to the retailer and when it is discounted to 1.3159 entire benefit of coordination goes to the manufacturer. To quantify the benefit of coordination, we assume that the manufacturer discounts to Rs. 1.3159 taking all the benefit of coordination and leaving the retailer’s profit unaffected. The coordination benefit can be shared equally by the manufacturer and the retailer. The profits of both the retailer and the manufacturer after equal division of the coordination benefit are shown in the last row of the Table 2. The results shown in Table 2 reveal that after coordination, the SC profit is improved by 12%. When equal bargaining power is assumed, the retailer’s profit is improved by 11.8% and the manufacturer profit is improved by 12.31% by the combined RS and QD contract. It is observed that both the members of SC are better off by following combined contract. 4.2. Sensitivity analysis Sensitivity analysis is carried out to study the impact of stock dependency, price-sensitivity and demand uncertainty on SC performance. While coordination benefit is defined as the improvement in SC profit, SC performance is defined as the percentage improvement in SC profit due to coordination. We have varied one parameter while keeping all other parameters constant to perform sensitivity analysis.

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G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 3 Impact of stock dependency factor. c

wrs

wmin

wmax

pdc

Q dc

pc

Q c

Coordination benefit

Supply chain performance in percentage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.2878 1.2881 1.2884 1.2888 1.2893 1.2901 1.2911 1.2927 1.2953 1.2975

0.9469 0.9458 0.9445 0.9428 0.9406 0.9377 0.9335 0.9270 0.9159 0.8932

1.3162 1.3159 1.3155 1.3151 1.3145 1.3138 1.3127 1.3110 1.3078 1.2989

5.69 5.70 5.71 5.72 5.74 5.76 5.79 5.85 5.96 6.33

62.0 69.2 78.3 90.1 106.1 128.9 164.3 226.2 361.7 872.6

4.59 4.60 4.62 4.63 4.66 4.69 4.73 4.81 4.96 5.44

92.7 103.6 117.4 135.4 159.9 195.1 250.0 347.2 563.1 1392.0

34.23 38.33 43.55 50.40 59.78 73.38 94.81 133.29 220.65 564.77

12.01 12.05 12.10 12.17 12.26 12.39 12.58 12.86 13.36 14.28

Fig. 3. Impact of stock dependency factor on coordination benefit.

Fig. 4. Impact of stock dependency factor on supply chain performance. Table 4 Impact of price-sensitivity factor. b

wrs

wmin

wmax

pdc

Q dc

pc

Q c

Coordination benefit

Supply chain performance in percentage

15 16 17 18 19 20 21 22 23 24 25

0.3686 0.5119 0.6385 0.7510 0.8518 0.9426 1.0247 1.0995 1.1678 1.2304 1.2881

0.3463 0.4586 0.5540 0.6352 0.7044 0.7633 0.8135 0.8559 0.8916 0.9213 0.9458

0.5613 0.6886 0.7990 0.8954 0.9798 1.0542 1.1198 1.1779 1.2295 1.2752 1.3159

8.45 8.02 7.64 7.30 7.00 6.73 6.49 6.26 6.06 5.87 5.70

88.9 86.9 84.9 82.9 80.9 79.0 77.0 75.1 73.1 71.2 69.2

7.35 6.92 6.54 6.20 5.90 5.63 5.39 5.17 4.96 4.78 4.60

110.1 109.5 108.8 108.1 107.5 106.8 106.2 105.5 104.9 104.2 103.6

23.68 25.17 26.65 28.13 29.60 31.07 32.53 33.99 35.44 36.89 38.33

3.69 4.25 4.86 5.53 6.26 7.04 7.90 8.82 9.81 10.89 12.05

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Coordination Benefit in Rupees

G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

13

40 a=200;c=0.1;[A,B]=[0,10];r=0.65 35

30

25

20 15

20 Price-sensitivity factor, b

25

Fig. 5. Impact of price-sensitivity on coordination benefit.

0.14 Supply chain performance

a=200;c=0.1;[A,B]=[0,10];r=0.65 0.12 0.1 0.08 0.06 0.04 0.02 15

20 Price-sensitivity factor, b

25

Fig. 6. Impact of price-sensitivity on supply chain performance.

Table 5 Impact of demand uncertainty. B

wrs

wmin

wmax

pdc

Q dc

pc

Q c

Coordination benefit

Supply chain performance in percentage

10 20 30 40 50 60 70 80 90 100

1.2881 1.2941 1.2995 1.3041 1.3080 1.3113 1.3140 1.3160 1.3174 1.3182

0.9458 0.9408 0.9366 0.9331 0.9302 0.9278 0.9258 0.9242 0.9228 0.9217

1.3159 1.3181 1.3199 1.3215 1.3227 1.3237 1.3244 1.3247 1.3248 1.3247

5.70 5.78 5.86 5.94 6.02 6.10 6.19 6.27 6.36 6.45

69.2 72.6 76.1 79.7 83.4 87.3 91.2 95.3 99.6 103.9

4.60 4.71 4.81 4.92 5.03 5.13 5.24 5.35 5.45 5.56

103.6 110.1 116.6 123.3 130.1 137.0 143.9 150.9 158.0 165.2

38.33 41.52 44.71 47.89 51.06 54.22 57.35 60.46 63.53 66.57

12.05 12.52 12.92 13.28 13.57 13.82 14.02 14.18 14.29 14.37

(i) Effect of variation of stock dependency factor Considering the demand function D ¼ 200  bp þ cQ þ e where e is uniformly distributed in the range ½0; 10, we vary stock dependency factor c while keeping all other parameters constant. The results are shown in Table 3, Figs. 3 and 4. It can be seen that as the stock dependency factor increased, retail price, order quantity, coordination benefit and SC performance are increased. Examining the Eqs. (14) and (27) for order quantity in both decentralized and centralized setting, we observe that the term ð1  cÞ appears in the denominator, therefore, as c approaches 1, there is obvious increase in coordination benefit and SC performance, but, at c ¼ 1, the solution becomes infeasible. Sudden rise in the slope of both the curves are the evidences of this phenomenon. This phenomenon is quite practical as there can be no situation that guarantees that as soon as the product is displayed it will be sold.

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G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

Coordination Benefit in Rupees

70 a=200;b=25;c=0.1;A=0;r=0.65

65 60 55 50 45 40 35

0

20

40 60 80 Demand Uncertainty, B

100

Fig. 7. Impact of demand uncertainty on coordination benefit.

Supply chain performance

0.145

a=200;b=25;c=0.1;A=0;r=0.65

0.14 0.135 0.13 0.125 0.12 0

20

40 60 80 Demand Uncertainty, B

100

Fig. 8. Impact of demand uncertainty on supply chain performance.

(ii) Effect of variation of price-sensitivity factor We now consider the demand function D ¼ 200  bp þ 0:1Q þ e where e is uniformly distributed in the range ½0; 10 and vary the price-sensitivity factor b, keeping all other parameters constant and the results are shown in Table 4, Figs. 5 and 6. It is clear from the Table 4 that as the price-sensitivity is increased, the retail price and order quantity are decreased as one can see that in the Eqs. (13) and (26), b appears in the denominator. But it may be noted that the coordination benefit and SC performance are improved. It can be interpreted as follows. When the product is more price-sensitive, it negatively affects the demand (please see the above demand function). Since the order quantity is low, the corresponding profits of the channel members will also decrease. Therefore, for such a product, coordination through the proposed contract can help improving their profits. (iii) Effect of variation of demand uncertainty Finally, the impact of the demand uncertainty is investigated considering the demand function D ¼ 200  25p þ 0:1Q þ e where e is uniformly distributed in the range ½0; B. Here, we vary B while keeping all other parameters constant and the results are shown in Table 5, Figs. 7 and 8. A closer look at the Table 5 reveals that as the demand uncertainty increased the retail price, order quantity, coordination benefit and SC performance are increased when the SC operates under the combined RS and QD contract. To interpret this result, one may observe that with discounting, the wholesale price decreases enticing the retailer to order more. As a result, the service level of the retailer goes up and the retailer is better prepared to handle the demand uncertainty, consequently SC performance also improves. Therefore, coordination through contract can be beneficial in an uncertain demand environment.

5. Conclusions In this paper, we have proposed a combined RS and QD contract model to coordinate a decentralized SC operating under price-sensitive and stock dependent demand. Such models are preferred over BB contract where the items are expensive and physical return of unsold items is not practical. We demonstrate that the combined contract model improves the SC perforPlease cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

15

mance and ensures win–win situation to both manufacturer and the retailer. The following conclusions are drawn from the sensitivity analysis. First, as the stock dependency coefficient increases, both the retail price and the order quantity tend to increase. Second, as the price-sensitivity increases, both the retail price and order quantity decrease. Third, as the demand uncertainty increases, both price and order quantity increase. Finally, with respect to stock dependency, price-sensitivity and demand uncertainty, it is found that coordination benefit and SC performance under RS and QD contract increases with the increase of the values. Further, the above observations emphasize the potential of the proposed contract for most SCs dealing with short lifecycle products such as agro products. For example, one can consider the case of a factory supplying flour to a retail chain. The flour like other perishable agro-products has a definite expiry date. If the flour packets are displayed in large quantities in the aisles, it creates an impression of freshly arrived stock, draws attention of the customers, and hence likely to be sold faster. So it is wiser for the factory to provide discount on the existing wholesale price and entice the retailer to buy more. As a result, the sales is likely to go up, increasing retailer’s profit which later on can be shared with the factory. While the service level of the retailer increases, the retailer has to share the risk of salvaging the item and the loss due to expiration if any. On the contrary, when there is no contract, retailer’s service level will be low and factory bears the risk of product expiration. Finally, this work can be extended in many ways. First, the risk neutrality assumption can be relaxed considering other alternative objectives such as expected utility maximization, satisficing or aspiration level objectives, or mean–variance objectives. Second, it is assumed that the demand is influenced by the initial stock level. But, in most instances the demand depends on the current inventory levels and not only on the initial inventory levels. Third, it is assumed that the demand and cost structures are known to all members of the SC, which seldom is true in reality. The current situation can be modeled as a game with incomplete information, considering private information of each player. Fourth, the limitation of two echelon SCs can be extended to include multi level case. Fifth, the single-period setting can be relaxed to multi period situation to represent more than one opportunity for the buyer to procure the items. Lastly, other demand enhancing factors such as retailer’s reputation, location, advertisement, sales promotion, service and effort can be included in the future work. Appendix A Since D ¼ a  bp þ cQ þ e and Q ¼ ða  bp þ cQ Þ þ z, we can have the following identities:

E½minðQ ; DÞ ¼ E½a  bp þ cQ þ minðz; eÞ ¼ a  bp þ cQ þ E½minðz; eÞ ¼ a  bp þ c

  a  bp þ z þ E½minðz; eÞ; 1c

E½ðQ  DÞþ  ¼ E½ðz  eÞþ  ¼ E½z  minðz; eÞ ¼ E½z  E½minðz; eÞ ¼ z  E½minðz; eÞ; E½ðD  Q Þþ  ¼ E½ðe  zÞþ  ¼ E½e  minðe; zÞ ¼ E½e  E½minðe; zÞ ¼ l  E½minðe; zÞ: Rz RB RB RB But, E½minðz; eÞ ¼ A xf ðxÞdx þ z zf ðxÞdx ¼ A xf ðxÞdx  z ðx  zÞf ðxÞdx. RB Therefore, the term E½minðz; eÞ can be written as l  HðzÞ where HðzÞ is defined as HðzÞ ¼ z ðx  zÞf ðxÞdx. Therefore,

E½ðQ  DÞþ  ¼ E½ðz  eÞþ  ¼ z  E½minðz; eÞ ¼ z  ðl  HðzÞÞ ¼ z  l þ HðzÞ; E½ðD  Q Þþ  ¼ E½ðe  zÞþ  ¼ l  E½minðe; zÞ ¼ l  ðl  HðzÞÞ ¼ HðzÞ; E½minðQ ; DÞ ¼ E½a  bp þ cQ þ minðz; eÞ ¼ a  bp þ cQ þ E½minðz; eÞ ¼ a  bp þ c   a  bp þ z þ l  HðzÞ: ¼ a  bp þ c 1c

  a  bp þ z þ E½minðz; eÞ 1c

Appendix B From the perspectives of the manufacturer and the retailer, we have the following constraints, known as participation constraints.

PM ðQ ¼ Q c ; r; w ¼ w0 Þ  PM ðQ ¼ Q dc ; r ¼ 1; w ¼ w0 Þ P 0; PR ðQ ¼ Q c ; r; w ¼ w0 Þ  PR ðQ ¼ Q dc ; r ¼ 1; w ¼ w0 Þ P 0; The above participating constraints can also be written as follows.

PM ðQ ¼ Q c ; r; w ¼ w0 Þ  PM ðQ ¼ Q r ; r; w ¼ wrs Þ P 0 PR ðQ ¼ Q c ; r; w ¼ w0 Þ  PR ðQ ¼ Q r ; r; w ¼ wrs Þ P 0:

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G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

From the manufacturer’s constraint, we get

ð1  rÞpc E½minðQ c ; DÞ þ w0 Q c  mQ c  ½ð1  rÞpdc E½minðQ dc ; DÞ þ wrs Q dc  mQ dc  P 0; W 0 Q c P mðQ c  Q dc Þ þ wrs Q dc þ ð1  rÞ½pdc E½minðQ dc ; DÞ  pc E½minðQ c ; DÞ; 1 ½mðQ c  Q dc Þ þ wrs Q dc þ ð1  rÞ½pdc E½minðQ dc ; DÞ  pc E½minðQ c ; DÞ: Qc

w0 P

Similarly, from the retailer’s constraint, we obtain



rpc E½minðQ c ; DÞ  w0 Q c  hE½ðQ c  DÞþ   sE½ðD  Q c Þþ   rpdc E½minðQ dc ; DÞ  wrs Q dc  hE½ðQ dc  DÞþ   sE½ðD  Q dc Þþ  P 0; w0 Q c 6 r½pc E½minðQ c ; DÞ  pdc E½minðQ dc ; DÞ þ wrs Q dc  h½E½ðQ c  DÞþ   E½ðQ dc  DÞþ   s½E½ðD  Q c Þþ   E½ðD  Q dc Þþ ; w0 6

 1

r½pc E½minðQ c ;DÞ  pdc E½minðQ dc ;DÞ þ wrs Q dc  h½E½ðQ c  DÞþ   E½ðQ dc  DÞþ   s½E½ðD  Q c Þþ   E½ðD  Q dc Þþ  : Qc

From the above two participating constraints, one can find the wholesale price range wmin  wmax which can coordinate the SC and can also facilitate the win–win situation, where,

wmin ¼

1 ½mðQ c  Q dc Þ þ wrs Q dc þ ð1  rÞ½pdc E½minðQ dc ; DÞ  pc E½minðQ c ; DÞ Qc

wmax ¼

 1

/½pc E½minðQ c ;DÞ  pdc E½minðQ dc ;DÞ þ wrs Q dc  h½E½ðQ c  DÞþ   E½ðQ dc  DÞþ   s½E½ðD  Q c Þþ   E½ðD  Q dc Þþ : Qc

and

Appendix C Retail price under centralized setting is given by

1 1 ½a þ bm þ l  Hðzc Þ þ cðzc  l þ Hðzc ÞÞ ¼ ½a þ bm þ czc þ ð1  cÞðl  Hðzc ÞÞ 2b 2b 1 1 ½a þ bm þ czc þ ð1  cÞE½minðzc ; eÞ ¼ ½a þ bm þ czc þ ð1  cÞE½ðzc  ðzc  eÞþ Þ ¼ 2b 2b 1 ½a þ bm þ zc  ð1  cÞE½ðzc  eÞþ : ¼ 2b

pc ¼

Substituting

pc ¼

1 ½a þ bm þ zc  ð1  cÞE½ðzc  eÞþ  2b

c þzc in Q c ¼ abp , we get 1c

Qc ¼

a  bm þ zc 1 þ E½ðzc  eÞþ : 2 2ð1  cÞ

Appendix D pc þsð1cÞm We know that Fðzc Þ ¼ ð1cÞðp . þhþsÞ c

Let

pc þsð1cÞm ð1cÞðpc þhþsÞ

¼ q so that zc ¼ F 1 ðqÞ

!   @zc @zc @ q @F 1 ðqÞ @ q 1 @q 1 @ pc þ sð1  cÞ  m 1 h þ m þ sc > 0; ¼ ¼ ¼ ¼ ¼ @ q @p f ðF 1 ðqÞÞ @p f ðF 1 ðqÞÞ @p ð1  cÞðpc þ h þ sÞ @p @ q @p f ðF 1 ðqÞÞ ð1  cÞðpc þ h þ sÞ2     @zc @zc @ q @F 1 ðqÞ @ q 1 @q 1 @ pc þ sð1  cÞ  m 1 1 ¼ < 0; ¼ ¼ ¼ ¼ @ q @m f ðF 1 ðqÞÞ @m f ðF 1 ðqÞÞ @m ð1  cÞðpc þ h þ sÞ @m @ q @m f ðF 1 ðqÞÞ ð1  cÞðpc þ h þ sÞ   @zc @zc @ q @F 1 ðqÞ @ q 1 @q 1 @ pc þ sð1  cÞ  m 1 ¼ ¼ ¼ ¼ ¼ ð0Þ ¼ 0; 1 1 @q @a @ q @a @a f ðF ðqÞÞ @a f ðF ðqÞÞ @a ð1  cÞðpc þ h þ sÞ f ðF 1 ðqÞÞ

Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003

17

G. Partha Sarathi et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

  @zc @zc @ q @F 1 ðqÞ @ q 1 @q 1 @ pc þ sð1  cÞ  m 1 ¼ ¼ ¼ ¼ ¼ ð0Þ ¼ 0; @q @b @ q @b @b f ðF 1 ðqÞÞ @b f ðF 1 ðqÞÞ @b ð1  cÞðpc þ h þ sÞ f ðF 1 ðqÞÞ !   @zc @zc @ q @F 1 ðqÞ @ q 1 @q 1 @ pc þ sð1  cÞ  m 1 pc  m > 0: ¼ ¼ ¼ ¼ ¼ @ q @c f ðF 1 ðqÞÞ @c f ðF 1 ðqÞÞ @c ð1  cÞðpc þ h þ sÞ @c @ q @c f ðF 1 ðqÞÞ ð1  cÞ2 ðpc þ h þ sÞ Appendix E

E½ðz  eÞþ  ¼

Z A

z

 z Z z Z Z z Z z ðz  xÞf ðxÞdx ¼ ðz  xÞFðxÞ þ FðxÞdx ¼ ½ðz  xÞFðxÞzA þ FðxÞdx ¼ FðxÞdx: FðxÞdx ¼ 0  0 þ

For uniform distribution, E½ðz  eÞþ  ¼ Therefore,

"

R z xA A

BA

dx ¼

h

A

iz

ðxAÞ2 2ðBAÞ

A

A

A

A

2

ðzAÞ ¼ 2ðBAÞ .

#

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Please cite this article in press as: G. Partha Sarathi et al., An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle products, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.003