Impact of flexible contracts on the performance of both retailer and supplier

Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Impact of flexible contracts on the performance of both retailer and supplier Wenbo Cai n, Layek Abdel-Malek 1, Babak Hoseini, Sharareh Rajaei Dehkordi Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology, University Heights, Newark, NJ 07102-1982, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 14 April 2014 Accepted 29 June 2015

In today’s global environment supply chains should be agile and responsive. Ideally, suppliers would like to know the order quantities with enough lead-time to set up the production process and schedule for delivery. However, retailers cannot predict demand accurately and thus both suppliers and retailers may incur significant losses due to under-stocking and/or over-stocking. To attenuate this effect, one of the viable practices is that suppliers impose minimum Order Quantity Commitments (OQC) for regular orders and penalties for reduced orders, while providing retailers with expedited-delivery options at premium costs if needs arise. In this paper, we introduce a methodology to assist in developing inventory management policies to optimize the expected profit. Our model considers a two-period setting where retailers place orders according to pre-sale forecasts but then have the opportunity to update their orders just prior to the season according to the most recent forecasts and the current inventory level. Utilizing the newsvendor model and dynamic programming methods, we obtain the optimal ordering and replenishing policies that maximize the expected profits of retailers. We then analyze the impacts of these flexible terms on the performance of both the retailer and the supplier. & 2015 Elsevier B.V. All rights reserved.

Keywords: Supply chain contract Order Quantity Commitment (OQC) Supply chain coordination Newsvendor Dynamic programming

1. Introduction Today’s global environment necessitates the responsiveness and agility of any efficient supply chain. Matching supply with demand is particularly challenging due to the continual evolution of consumer preferences and market trends, which lead to high inventory costs, markdowns, and lost sales (Johnson, 2001). To elucidate and as an example, in apparel industry, while “overstocking leads to deep markdowns that erode profits, running out of a hot-selling item is a missed opportunity” (Holmes, 2010). Many retailers thus attempt to “chase” the ever-changing consumer demand by pushing their suppliers for faster turnaround on smaller orders, especially ahead of holiday seasons. This allows the retailers to test styles in stores and re-order only if there is an additional demand. On the other hand, suppliers who ideally wish to know the exact order quantities months in advance to set up the production process and schedule for delivery, end up facing even higher risk from scaled back orders or cancellations. With the rising labor and raw material costs, some suppliers would rather turn down large orders from retailers who do not order in n

Corresponding author. Tel.: þ 1 973 596 3338. E-mail addresses: [email protected] (W. Cai), [email protected] (L. Abdel-Malek), [email protected] (B. Hoseini), [email protected] (S. Rajaei Dehkordi). 1 Tel.: þ1 973 596 3648.

advance. As a result, both suppliers and retailers incur significant losses. One of the practices attempted in order to attenuate this effect is to have both parties enter a contract in which the supplier imposes a minimum Order Quantity Commitment (OQC). An example of this practice is Apple. The company “sets a quota for how many iPhones each of the European carriers needs to sell over three years. If quotas are not met, the carrier is obligated to pay Apple for unsold devices” (Chen et al., 2013). Other electronics companies, such as IBM printer division (Bassok et al., 1997) and Hewlett Packard (Tsay and Lovejoy, 1999), use similar contract terms as well. To add flexibility to the supply chains, suppliers also provide the expedited-delivery option at a premium to accommodate sudden surge in consumers’ demand. In the oil and natural gas markets in the US, buyers get a lower rate if they place an order at the beginning of the winter. However, they do have the option of buying extra at a higher cost if needed in case of a colder season than expected (Lian and Deshmukh, 2009). As can be seen from the aforementioned examples, many industries in retail, manufacturing, and other services can experience volatile demand. A testimonial to this also, as well be seen in the literature section, is numerous articles that appeared introducing models and techniques to evaluate the various forms of contracts in their potential of achieving supply chain coordination and consequently enhancing the performance of inventory management policies, as well as to capture the value of information

http://dx.doi.org/10.1016/j.ijpe.2015.06.030 0925-5273/& 2015 Elsevier B.V. All rights reserved.

Please cite this article as: Cai, W., et al., Impact of flexible contracts on the performance of both retailer and supplier. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.06.030i

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sharing and its reduction in bullwhip effect. Motivated by the interest of the community in systems with volatile demand, this work contributes to the literature by incorporating both the minimum OQC requirement and a limited expedited-delivery option in a retailer's dynamic inventory management policies, which to the best of our knowledge and despite its importance has not been sufficiently addressed. Our aim is to fill the gap of understanding whether the combined flexible contract terms can achieve supply chain coordination and improve the performance of both the supplier and the retailer. The introduced approach in this paper adds to the existing body of knowledge in that it investigates the intrinsic relationships between the optimal inventory replenishing levels with these key contract terms in both analytical models and numerical results and shows that the performance of both the supplier and retailer are sensitive to contract terms. Among other distinctions to the published work is the inclusion of forecast signals in the decision-making vis-à-vis the order quantities as well as the study of how the supplier would strategically set the contract parameters to optimize his own performance. More clearly, we address the common situation where a retailer who can place orders in both pre-sale season and (regular) sales season. While the retailer is obliged to order no less than the minimum OQC or pay a penalty for reduced orders in both seasons, in the sales season he also has the option of ordering additional units that can be expedited at a premium cost should such need arises. During the negotiation stage, both parties agree on the unit purchase price, the minimum OQC, and a reducedorder penalty for ordering quantity lower than the minimum OQC. This penalty compensates the cost undertaken by the supplier for reserving necessary factory capacity, labor, and raw materials. The retailer also accepts the cost of expedited-delivery as well as the maximum quantities within which the supplier can provide on this basis. In the scenario where the retailer has high initial inventory in the sales season due to a slow pre-sale season, the retailer may be better off paying for the reduced-order penalty instead. Consequently, three replenishing possibilities emerge in both seasons: 1) order the minimum OQC, 2) order-up-to the optimal inventory level and incur a reduced-order penalty if the ordering quantity is less than the minimum OQC, and 3) place no orders and pay the full penalty. Our proposed methodology contributes to the existing literature by comparing the optimality of these three possibilities and also evaluates the impact of the minimum OQC and the expedited-delivery option on the retailer’s inventory management policies, his two-period expected profit, as well as the supplier’s expected profit. It is noted that in developing these ordering and replenishing policies, we utilize the newsvendor model combined with dynamic programming methods. Our taxonomy in this paper is as follows: Section 2 provides related literature. Section 3 introduces the combined newsvendor model and dynamic programming methods to derive the expected profits for both retailer and supplier and present analytical solutions. Section 4 presents numerical examples as well as sensitivity analyses. We provide managerial insights in Section 5 and conclude in Section 6.

2. Literature review As mentioned in the introduction there is a plethora of articles in the vein of inventory control in the arena of supply chain management. In this section, we present a review of the pertinent areas, which include demand forecast updating, the effect of information sharing, supply chain coordination, and flexible contracts. Also, we reference works in the newsvendor problem since

the developed model utilizes its framework. It is to be noted that in our attempt to categorize the literature overlap may exist. 2.1. Demand forecast updating and the effect of information sharing In the supply chain management arena, volatility of demand creates the bullwhip effect and hence leads to inefficiency of the supply chain. One approach to reduce such effect is demand forecast updating. A number of papers have studied the benefits of incorporating early sales information on retailer’s ordering strategy (Chen et al., 2006; Huang et al., 2005; So and Zheng, 2003; Song et al., 2014; Weng and Parlar, 1999). Moreover, some studies report that a quick response to the updating can benefit the retailer by better matching supply and demand and improve the speed of delivery and hence customer satisfaction (Feng et al., 2005; Fisher and Raman, 1996; Iyer and Bergen, 1997). See Choi and Sethi (2010) for a detailed review. Various papers have demonstrated that information sharing has the potential to improve forecast accuracy and consequently reduce the impact of the bullwhip effect in supply chain performance (Ali et al., 2012; Babai et al., 2013; Barlas and Gunduz, 2011; Dominguez et al., 2014; Hosoda et al., 2008; Trapero et al., 2012; Zhao and Zhao, 2015). Voigt and Inderfurth (2012), however, point out that information sharing may lead to a serious deterioration in the supply chain performance if the supplier's perception of the retailer's signaling strategy is significantly different from the retailer's actual signaling strategy. Our works assumes a simple information structure. That is, at the end of the pre-sale season, the retailer is able to get an either good or bad signal, which can be used to update the demand distribution of the sales season. Although we do not explicitly assume any information sharing between the retailer and the supplier, the latter knows the likelihood each signal will be realized and thus can infer the signal from the retailer's ordering quantities. 2.2. Supply chain coordination and flexible contracts This paper also closely relates to the vast literature of supply chain coordination where the effectiveness of contracts in aligning the supplier and retailer’s incentives are investigated. These contracts include buy-back contracts (Emmons and Gilbert, 1998; Pasternack, 2008; Xiong et al., 2011), sales-rebate contracts (Taylor, 2002; Wong et al., 2009), quantity-discount contracts (Kolay et al., 2004; Krishnan et al., 2004), backup agreements (Eppen and Iyer, 1997), revenue-sharing contracts (Cachon and Lariviere, 2005; Giannoccaro and Pontrandolfo, 2004; Li et al., 2009; Pasternack, 2005), and quantity flexible contracts (Lian and Deshmukh, 2009; Tsay, 1999; Tsay and Lovejoy, 1999; Wang and Tsao, 2006). See Cachon (2003) for more detailed reviews. Two of the most studied flexible contracts in the literature are minimum OQC and expedited delivery. Motivated by a common practice of the electronics industry, Bassok and Anupindi (1997), Chen and Krass (2001), and Bassok and Anupindi (2008) find that the optimal ordering policy for a given a total minimum OQC over a horizon is characterized by the order-up-to policy. The joint impact of minimum order quantity and the reduced-order penalty on the procurement cost and supply chain’s flexibility is subsequently investigated by Das and Abdel-Malek (2003) and Kesen et al. (2010). A number of studies have shown that providing the expedited-delivery option at a premium increases the coordination efficiency of a supply chain and that upward inventory adjustment policies can lead to a reduction of demand variance (Donohue, 2000; Ernst and Cohen, 1992; Moinzadeh and Nahmias, 2000). While Durango-Cohen and Yano (2006) and Hsu and Chen (2011) demonstrate that properly designed contracts can alleviate suppliers' tendency to underproduce and tame retailers from overforecasting, DeYong and

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Cattani (2012) argue that the joint ability of unlimited expediteddelivery and cancellation increases retailer’s expected profits significantly. Our work differentiates itself from the aforementioned papers in that we consider both forms of flexible contracts as well as a limited expedited-delivery option, both of which are more practically realistic assumptions. 2.3. Newsvendor framework in supply chain management The newsvendor problem was first introduced by Arrow et al. (1951), where it is used to study optimal inventory policies for stochastic demands of finished goods. It has been since extended in various directions, including multi-products (Moon and Silver, 2000; Vairaktarakis, 2000), multi-units (Grubbström, 2010), multiperiod (Abdel–Malek and Zanoni, 2011; Linh and Hong, 2009; Pan et al., 2009), capacity and budget constraints (Erlebacher, 2000; Zhang et al., 2009), general or unknown distributions (Godfrey and Powell, 2001; Moon and Choi, 1997; Mostard et al., 2005; Petrović et al., 1996). A common characteristic among all these papers is that the demand is uncertain. Our paper utilizes the newsvendor framework in deriving the retailers' optimal ordering and replenishing strategy. Our work contributes to the existing literature in amalgamating the minimum OQC, the reduced-order penalty, a limited expedited-delivery option, and an observed signal forecast in enhancing the supply chain’s performance, in particular, the retailer’s dynamic inventory management policies. The contribution of this paper also includes showing that the retailer’s ordering and replenishing levels are sensitive to the contract terms, and may lead to an undesirable outcome for both the retailer and the supplier.

3. The model We consider a retailer who procures goods from a supplier and sells them in both a pre-sale season and a sales season. Due to the uncertain demand in both seasons, the retailer incurs a lost sales cost from unfulfilled demand as well as a holding cost for any inventory carried over from the pre-sale season to the sales season. Unsold items at the end of the sales season can be salvaged at a much lower value. At the negotiation stage, the retailer enters a contract by committing to order at least the minimum OQC at the beginning of each season; otherwise a reduced-order penalty would be imposed to reduce the risk taken by the supplier for reserving production capacity, labor and raw materials. In exchange, the supplier offers an expedited-delivery option at a premium in the sales season to help the retailer increase his customer service level. Upon observing the realized demand during the sales season, the retailer may make additional purchase and the products would arrive in time to prevent lost sales. We utilize an extension of a two-period newsvendor model, in which the first period corresponds to the pre-sale season and the second represents the sales season, to examine the retailer’s inventory management policy. Fig. 1 describes the timeline of the interactions between the supplier and the retailer. At the beginning of the pre-sale season (i.e. the first period), the retailer has the following information regarding the product. The unit retail price for the product is $p, the cost of purchasing each item from the supplier is $c. Moreover, the per unit holding cost for products not sold in the pre-sale season and carried over to the sales season (i.e. the second period) is $h, and the per unit cost of lost sales is $l. At the end of the sales season, unsold items can be salvaged at $s per item. We assume that the salvage value is lower than the purchase price (s o c);

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otherwise the retailer can purchase a very large number of products and sell them after the second period at a profit. In order to capture the uncertainty in demand, D1 , in the presale season, we assume it is a random variable and its distribution is unknown. The retailer learns the customers’ demand during the pre-sale season and updates the sales forecast at its end by observing a signal (j) at the end of period 1. If the signal turns out to be favorable (or good), demands in both periods follow probability density function ðf g Þ and cumulative density functions ðF g Þ. Otherwise, demands in both periods are unfavorable (or bad) and they follow probability density function ðf b Þ and cumulative density functions ðF b Þ. Let qg and qb denote the probability that a good and a bad signal is observed, respectively, and qg þqb ¼ 1: The supplier imposes a minimum OQC for each period. Let Q denote the minimum OQC and R1 the retailer’s order quantity for the first period. If no orders is placed, the retailer is charged with a full penalty of $K. If the order quantity is lower than the
minimum  OQC, the retailer is charged with a partial penalty of 1  R1 =Q K. We assume that the penalty cost is lower than the cost of ordering minimum OQC (i.e. K o cQ ); otherwise the retailer would always order at least the minimum OQC in order to avoid the penalty cost. At the end of the first period, the retailer updates the demand forecast and signal j is observed. The retailer then decides the base stock level ðR2j Þ based on the signal. If the order quantity is lower than Q, a (partial) penalty is applied to the supplier. If the actual demand turns out to be more than the stocked inventory in the second period, the retailer may choose to exercise the expediteddelivery option. Each item is charged $e extra, and the maximum order quantity is A. We further assume that c þ e o l to ensure that the lost sale outweighs the procurement price when using the expedited-delivery option. Table 1 provides a summary of the notation we use in our model. In the following sections, we derive the retailer’s optimal inventory management policies for both periods using dynamic programming and backward induction. We start by considering the retailer’s optimal replenishing policy in the second period by comparing the different strategies the retailer has. Then, we move on to the first period and obtain the optimal ordering quantity that maximizes the retailer’s expected two-period profit. Afterwards, we compute the expected two-period profit for both the retailer and the supplier. Moreover, we characterize the supplier’s profitability under a coordinated supply chain. 3.1. Retailer’s optimal order-up-to inventory level in the second period We first compute the order-up-to inventory level that maximizes the retailer’s expected profit while considering the expediteddelivery option. Here, we ignore the minimum OQC restriction and the penalty associated. Let y denote the initial inventory level, and π 2j ðyÞ denote the retailer’s expected profit for the order-up-to R2j policy when signal j is observed. Here, the retailer orders R2j  y at the beginning of the second period, and may choose to expedite upto A units if the realized demand is higher. The retailer’s objective function can be written as follows: Z ∞
 π 2j ðyÞ ¼ p D2j ∧ ðR2j þ AÞdF j D2j 0

Z

þs

R2j 0

−ðe þ cÞ




Z

 

R2j −D2j dF j D2j −c R2j −y ∞

R2j



 A ∧ D2j −R2j dF j D2j −l

Z

∞ R2j þ A





 D2j −R2j −A dF j D2j ;

ð1Þ where D2j ∧ðR2j þ AÞ computes the minimum of D2j and ðR2j þAÞ. The first term in (1) computes the retailer’s expected sales

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Fig. 1. Timeline of interactions.

Table 1 Summary of notation. Parameters p c h l s j qj e A Q K wr we wc Random Variables D1

Retail selling price [$/unit] Cost of purchasing [$/unit] Holding cost [$/unit] Cost of lost sales [$/unit] Salvage value at the end of selling horizon [$/unit]   Signal observed at the end of period 1, j A g; b , g for good and b for bad. Probability of signal j is observed Cost of the expedited-delivery option [$/unit] Expedited-delivery capacity [unit] Minimum OQC [unit] Reduced-order penalty [$] Production cost for regular orders [$/unit] Production cost for expedited-delivery orders [$/unit] Extra cost of reserving capacity for expedited-delivery option [$/unit] Period 1 demand [unit], follows pdf f j and cdf F j with probability qj

D2j

Period 2 demand [unit] when type j signal is observed, follows pdf f j and cdf F j

Decision Variables R1 R2j

The order quantity in the first period [unit] The order-up-to quantity in the second period if type j signal is observed [unit]

revenue, and the second term his expected revenue from salvaging unsold inventory. The third, fourth and fifth terms correspond to his costs of purchase, expected cost of expedited-delivery, and expected cost of lost sales, respectively. Let Rn2j denote the optimal order-up-to inventory level. We present the optimal solution for the retailer’s profit in the following proposition. Proposition 1. The optimal order-up-to inventory level (Rn2j ) in the second period upon signal j is observed solves the following equation:     ðc þ e  sÞ U F j Rn2j þ ðp þ l  c  eÞ U F j Rn2j þ A ¼ p þ l  c: ð2Þ Moreover, Rn2j decreases in A. Proof. Differentiating π 2j ðyÞ with respect to R2j and setting the derivative to zero results in Eq. (2). Because c 4 s and l c  e 4 0 hold by assumption, Rn2j decreases in A. Proposition 1 suggests that when the expedited-delivery option is unavailable (A ¼0), the optimal order-up-to inventory level is Rn2j ¼ F j  1 ððp þ l  cÞ=ðp þ l  sÞÞ  RU2j , which is an upper bound for Rn2j . Conversely, when the option is unlimited (A¼ 1), Rn2j ¼ F j  1 ððeÞ=ðc þ e sÞÞ  RL2j , and this is a lower bound for Rn2j . As long as the expedited-delivery cost is small enough

(e op þ l  c), the optimal order-up-to inventory level decreases in the expedited-delivery capacity (A), and it is bounded above by RU2j and below by RL2j . To shed lights on the relationships between the replenishing strategies with the other governing parameters, let us consider the case where demands from both periods follow uniform distributions. Because the uniform distributions result in the most conservative estimate of uncertainty, the analytical results derived are indicative of worse case scenarios. In the first period, the demand   distribution is unknown. With probability qj , j A g; b , the dis tribution follows U μ  ðδj =2Þ; μ þ ðδj =2Þ . At the end of the first period, a signal j (either good or bad) is observed, and the distribution of the second period demand becomes known. We further assume that and δb 4 δg to reflect that the variance of the demand is smaller if a good signal is observed. It is also assumed that μ  ðδb =2Þ 4 0 such that the demand is non-negative even if an undesirable signal is observed. Further, we assume that the expedited-delivery capacity is small enough ðA o ðc−sÞδj = ðc þ e−sÞÞ to ensure that the sum of the base stock and expedited-delivery capacity does not exceed the upper bound of the demands (i.e., Rn2j þ A o μ þ ðδj =2Þ).

Please cite this article as: Cai, W., et al., Impact of flexible contracts on the performance of both retailer and supplier. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.06.030i

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In this case, the optimal order-up-to inventory level of the retailer is:
 δj ðp þ l cÞ U δj A þ eA : ð3Þ Rn2j ¼ μ  þ 2 pþls Note that the optimal order-up-to inventory level (Rn2j ) decreases in the purchase price (c), but increases in the salvage value (s), the selling price (p) as well as the cost of the lost sales (l). Even though a higher purchase price deters the retailer from placing a large order, the retailer must increase the base stock at the beginning of the second period when the selling price, the cost of lost sales, or the salvage value increases. This is because the expedite-delivery capacity may be insufficient to fulfill excess demand. It should be mentioned that, the optimal order-up-to inventory level increases in expedited-delivery cost (e). That is, the more costly the expedited-delivery option is, the higher level the retailer needs to stock up at the beginning of the second period in order to minimize the use of the option. Further, Rn2g 4 Rn2b if e o c  s while Rn2g rRn2b if eZ c  s. Thus, if a bad signal is observed, the retailer is dealing with a higher variance of the demand when compared to the case of a good signal. The retailer must stock higher at the beginning of the second period because the expedited-delivery cost (e) is higher than the loss from an unsold item (c  s). However, when the expedited-delivery cost is cheap, the retailer can lower the base stock at the beginning of the second period and fulfills the excess demand using the expedite-delivery option. 3.2. Retailer’s optimal replenishing policy in the second period Having established the optimal order-up-to levels in the second period, we now examine the expected profit of three respective strategies, where the retailer selects one of the following three strategies: 1) order the minimum OQC, 2) order-up-to the optimal inventory level, and 3) place no orders. Strategy 1: Order the minimum OQC. Let π ð2j1Þ ðyÞ denote the retailer’s expected profit when he has an initial inventory of y and orders the minimum OQC upon the observation of signal j. It can be computed as follows: π ð2j1Þ ðyÞ ¼ p

Z

∞

0

Z

þs


 D2j ∧ðy þ Q þ A ÞdF j D2j yþQ




0


 yþ Q −D2j dF j D2j Z

−cQ −ðe þ cÞ 

Z −l

∞ yþQ þA

∞

yþQ


 A∧ðD2j −y−Q ÞdF j D2j



 D2j −y−Q −A dF j D2j :

ð4Þ

The terms in Eq. (4) correspond to the retailer’s expected sales revenue, the expected revenue from salvaging unsold items, the cost of purchase, the expected costs of expedited-delivery and lost sales, respectively. Strategy 2: Order-up-to Rn2j . Let π ð2j2Þ ðyÞ denote the retailer’s expected profit under the order-up-to Rn2j policy when he has y units at the beginning of the second period and when signal j is observed. We can compute the retailer’s expected profit as

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follows: Z Rn2j  

 D2j ∧ðRn2j þ AÞdF j D2j þ s Rn2j −D2j dF j D2j 0 0 Z ∞  
 n A∧ðD2j −Rn2j ÞdF j D2j −c R2j −y −ðe þcÞ  Z

π ð2j2Þ ðyÞ ¼ p

∞

Rn2j

!  
 Q ∧ðRn2j −yÞ D2j −Rn2j −A dF j D2j − 1− −l K: Q Rn2j þ A Z

∞

ð5Þ

The first five terms in Eq. (5) are the same as those in Eq. (1). The last term is the reduced-order cost if the ordering quantity, Rn2j  y, is lower than Q . Because π ð2j2Þ ðyÞ is the retailer’s expected profit for the order-up-to Rn2j policy, y must be between 0 and Rn2j . Strategy 3: Place no orders. Let π ð2j3Þ ðyÞ represent the retailer’s second-period expected profit when he places no orders but pay for the full penalty and when signal j is observed: Z ∞ Z ∞

 π ð2j3Þ ðyÞ ¼ p D2j ∧ðy þ AÞdF j D2j −ðeþ cÞ A∧ðD2j −yÞdF j D2j 0

Z þs

y 0





 y−D2j dF j D2j −l

y

Z

∞ yþA





 D2j −y−A dF j D2j −K:

ð6Þ

The first four terms in Eq. (6) correspond to the retailer’s expected sales revenue, expected cost of expedited-delivery, expected revenue from salvaging unsold items, and expected cost of lost sales, respectively. The last term is the full penalty cost. Note that π ð2j2Þ ðyÞ ¼ π ð2j3Þ ðyÞ when the initial inventory is higher than Rn2j . To compare and rank the three profit functions, let us first consider a special case where there is no penalty costs (K ¼ 0). Because Rn2j is the optimal order-up-to inventory level, ordering any other quantities is suboptimal, and thus π ð2j2Þ ðyÞ Z π ð1Þ 2j ðyÞ. Moreover, when the initial inventory level is higher than Rn2j , placing no order is the optimal. Therefore, the optimal replenishing policy when there is no penalty cost is to order-up-to Rn2j when the initial inventory level is lower than Rn2j , and to place no orders otherwise. We therefore focus our discussion on the scenario where the penalty cost is positive (K 4 0). As shown in Appendix A, π ð2j1Þ ðyÞ is ð2Þ concave in y while π 2j ðyÞ is piece-wise linear. Let αj and β j denote the intersections of these two functions, we show that αj ¼ Rn2j  Q and βj solves Eq. (B.1) in Appendix B. Moreover, if Rn2b 4 Rn2g , then αb 4 αg . Consequently, the order-up-to Rn2j policy is better when yr αj and y 4 β j while the order minimum OQC policy is better when αj o yr β j . Also, because δb 4 δg , βb 4 βg . Next, let γ j denote the intersection of the order-up-to Rn2j policy and the order nothing policy. Thus, γ j ¼ Rn2j . Last, let θj denote the intersection of the order minimum OQC policy and the order nothing policy. Because the relationship between γ j and θj depends on the penalty cost, let K tj denote the cutoff valuation at which all three policies gives the retailer the same expected profit in the second period, i.e. βj ¼ γ j ¼ θj ¼ Rn2j . We show that K tj solves Eq. (C.3) in Appendix C. Table 5 provides numerical values of these cutoffs.  In the case of uniform demand distributions U μ  ðδj =2Þ; μ þ ðδj =2Þ, the cutoff valuations for the second period’s optimal replenishing strategy are: αj ¼ Rn2j −Q , βj ¼ Rn2j −Q þ ð2Kδj Þ= %ðQ ðp þ l−sÞÞ, %γ j ¼ Rn2j ¼ μ−ðδj =2Þ þ ððp þ l−cÞ  ðδj −AÞ þ eAÞ=ðp þ l−sÞ, and θj ¼ Rn2 þ ðKδj Þ=ðQ ðp þ l−sÞÞ−ðQ =2Þ. Moreover, if e oc  s, then both α2g 4 α2b and γ 2g 4 γ 2b . If, on the other hand, e Zc  s, then

Table 2 The optimal policy when the penalty cost is small. Initial inventory level

Low ðyr αj Þ

Medium-low ðαj o y r βj Þ

Medium-high ðβj o y r γ j Þ

High ðy 4γ j Þ

Optimal replenishing policy

Order-up-to Rn2j

Order Q

Order-up-to Rn2j

No orders

Pay penalty?

No

No

Partial

Full

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both α2g r α2b and γ 2g r γ 2b . Moreover, β2g r β2b and θ2g r θ2b as long as p þl þ s Z2c  ð2K=Q Þ. Additionally, the cutoff penalty values are K tj ¼ Q 2 ðp þ l−sÞ=ð2δj Þ and K tb oK tg . We now analyze the retailer's replenishing policy based on the following three cases: 1) when the penalty cost is small and 2) when it is large, and 3) when it is medium. Case 1. Small penalty (K oK tb ). Table 2 summarizes the optimal replenishing policy for both observed signals, j ¼ g and b. In this case, αj ¼ Rn2j  Q o β j o θj o γ j ¼ Rn2j , for both j ¼ g and b. When the initial inventory level is low, the ordering quantity, Rn2j  y, exceeds Q . Therefore, the order-up-to Rn2j policy is optimal and no penalty occurs. When the initial inventory level is medium-low, the order minimum OQC policy yields a higher profit because the cost of paying reduced-order penalty outweighs the cost of stocking more than optimal. When the initial inventory is medium-high, however, the order-up-to Rn2j policy becomes superior as ordering the minimum OQC would result in excessive inventory. Therefore, the retailer pays a partial reduced-order penalty instead. Last, placing no orders is optimal when the initial inventory is already higher than sRn2j . Note that the strategy summarized in Table 2 is optimal if Rn2j 4 Q . In the case of Rn2j r Q , αj is negative and thus the low initial inventory region disappears. If Q gets even larger, β j becomes negative and the medium-low region disappears as well. In the small penalty case, the retailer is more likely to adhere to the optimal base stock policy. Case 2. Large penalty (K tg r K). Table 3 summarizes the optimal replenishing policy for both observed signals, j ¼ g and b. In this case, Rn2j  Q ¼ αj o γ j ¼ Rn2j o θj for both j ¼ g and b. Using the same reasoning presented in the small penalty case, the order-up-to the Rn2j policy is optimal when the initial inventory level is low. When the initial inventory is medium (between αj and θj ), however, the order the minimum OQC policy becomes optimal, as the penalty outweighs the risk of stocking more than optimal. When the initial inventory level is too high (greater θj ), the chance of having excess inventory is too high. Thus, the retailer is better off not placing any orders but simply pay for the penalty. Compare to the small penalty case, the retailer is less likely to adhere to the optimal base stock policy. His focus is to avoid as much as possible of paying the penalty. Case 3. Medium penalty (K tb o K o K tg ). Tables 4a and 4b summarize the optimal replenishing policy when the good and bad signal is observed, respectively. In this case, the retailer adopts a different strategy based on the observed signal. If it is a desirable signal, the retailer follows the same replenishing strategy of the small penalty case and is more likely to follow optimal base stock policy. When an undesirable signal is observed, however, the retailer undertakes the optimal replenishing strategy of the large penalty and is more likely to deviate from the optimal base stock policy. Table 3 The optimal policies when the penalty cost is large. Initial inventory level

Low ðyr αj Þ

Medium ðαj o y r θj Þ

High ðy 4 θj Þ

Optimal replenishing policy

Order-up-to Rn2j

Order Q

No orders

Pay penalty?

No

No

Full

3.3. Retailer’s optimal ordering policy in the first period Let π 1j ðR1 Þ denote the retailer’s expected profit for the first period when R1 is ordered and signal j is observed at the end of the first period. We assume that the initial inventory at the beginning of the first period is zero, therefore the retailer’s profit is Z 1 Z 1 π 1j ðR1 Þ ¼ p ðD1 4 R1 ÞdF j ðD1 Þ  cR1  h ðR1  D1 Þ þ dF j ðD1 Þ 0 0

 Z 1 Q 4 R1 K; ð7Þ l ðD1  R1 Þ þ dF j ðD1 Þ  1  Q 0 where ðR1  D1 Þ þ is the non-negative value of R1  D1 . The terms in Eq. (7) represent the retailer’s expected revenue, cost of purchase, holding cost, lost sales, and (partial) penalty if order is less than the minimum OQC,  respectively. S
Let ϕ π 1j ; π 2j denote the retailer’s expected two-period profit when the penalty is small and signal j is observed. Z 1

 ϕS π 1j ; π 2j ¼ π 1j ðR1 Þ þ π ð2j2Þ ðR1 D1 Þ þ dF j ðD1 Þ Z þ Z þ Z þ

R1  α j R1  β j R1  β j R1  γ j R1  γ j 0

R1  α j

π ð2j1Þ ðR1  D1 ÞdF j ðD1 Þ π ð2j2Þ ðR1  D1 ÞdF j ðD1 Þ π ð2j3Þ ðR1  D1 ÞdF j ðD1 Þ:

ð8Þ

The first term in Eq. (8) is the retailer’s expected profit in the first period where signal j is observed. The next four terms of the equation correspond to the retailer’s expected profit of the second period when the initial inventory level ðR1  D1 Þ þ is low, mediumlow, medium-high, and high, respectively. L
Similarly, let ϕ π 1j ; π 2j represent the retailer’s expected profit when the penalty is large and signal j is observed. Z R1

 ϕL π 1j ; π 2j ¼ π 1j ðR1 Þ þ π ð2j2Þ ðR1  D1 Þ þ dF j ðD1 Þ R1  α j

Z þ Z þ

R1  α j R1  θ j R1  θ j 0

π ð2j1Þ ðR1  DÞdF j ðD1 Þ π ð2j3Þ ðR1  DÞdF j ðD1 Þ:

ð9Þ

The first term in Eq. (9) is the retailer’s expected profit in the first period. The next five terms correspond to the retailer’s second-period expected profit when the initial inventory level ðR1  D1 Þ þ is low, medium, and high, respectively. 3.4. Retailer’s expected two-period profit Having characterized the retailer’s optimal ordering policy in the first period, we can now compute his expected two-period profit, which also needs to be done separately for each penalty range: small, medium, and large. Case 1. Small penalty (K o K tb ). Let Π ðR1 Þ denote the retailer’s two-period expected profit function when ordering R1 in the first period and the supplier charges a small penalty. It can be written as follows:
 ΠS ðR1 Þ ¼ qg U ϕS π 1g ; π 2g þ qb U ϕS ðπ 1b ; π 2b Þ: ð10Þ S

In this case, the retailer uses the strategy summarized in Table 2 regardless the signal observed. His expected profit is simply the sum of the weighted two-period expected profits. Let RS1 be the optimal ordering quantity in the first period, it solves Eq. (D.1) in Appendix D and must satisfy Eq. (D.2).

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Table 4a The optimal policy when a good signal is observed. Initial inventory level

Low (y r αg )

Medium-low (αg o y r βg )

Medium-high (βg o y r γ g )

High (y 4γ g )

Optimal replenishing policy

Order-up-to Rn2g

Order Q

Order-up-to Rn2g

No orders

3.5. Supplier’s expected two-period profit under a non-coordinated supply chain

Table 4b The optimal policy when a bad signal is observed. Initial inventory level

Low (y r αb )

Medium (αb o yr θb )

High (y 4θb )

Optimal replenishing policy

Order-up-to Rn2b

Order Q

No orders

The analytical solution for the uniform demand distributions is provided in Eq. (D.3). Notice that as the minimum OQC (Q ) increases, the difference between αj and βj decreases. This means that the possibility of the retailer deviates from the order-up-to Rn2j policy in the second period increases. The retailer thus places a larger order in the first period so that the initial inventory level is less likely to fall into the medium-low region. Conversely, if either the expedited-delivery capacity (A) increases or the associated cost (e) decreases, both αj and βj decrease at the same rate and thus the possibility of deviating from the order-up-to Rn2j policy stays the same. The retailer then places a smaller order in the first period to achieve a low or medium-low initial inventory, which helps him avoid the need of paying any penalty. Case 2. Large penalty (K tg r K). Let Π ðR1 Þ denote the retailer’s two-period expected profit function when ordering R1 in the first period and the supplier charges a large penalty. It can be computed as follows:
 ΠL ðR1 Þ ¼ qg U ϕL π 1g ; π 2g þqb U ϕL ðπ 1b ; π 2b Þ: ð11Þ L

Here, the retailer uses the strategy summarized in Table 3. The first period’s optimal ordering quantity, RL1 , solves Eq. (E.1) and must satisfy Eq. (E.2) in Appendix E. The analytical solution for the uniform demand distributions is provided in Eq. (E.3). We observe that a larger minimum OQC (Q ) increases the difference between αj and θj . As a result the likelihood of deviating from the order-up-to optimal level policy increases. Though the order Q policy may lead to excessive inventory, it does help the retailer to avoid paying the penalty. Thus, the retailer places a smaller order in the first period. As the expedited-delivery capacity increases (A) or the associated cost (e) decreases, the chance of not meeting demand decreases and this also helps the retailer to order less in the first period. Case 3. Medium penalty Let Π ðR1 Þ denote the retailer’s two-period expected profit function when ordering R1 in the first period and the supplier charges a medium penalty. It can be computed as follows:
 ΠM ðR1 Þ ¼ qg U ϕS π 1g ; π 2g þ qb U ϕL ðπ 1b ; π 2b Þ: ð12Þ (K tb o K o K tg ).

M

In order to explore the impact of the contract parameters (Q , K, and A) as well as the retailer’s inventory management policy on the supplier’s profitability in this section, we derive the supplier’s expected two-period profit. Let ψ ð1Þ 2j ðyÞ denote the supplier’s expected profit in the second period when the retailer has an initial inventory of y and orders the minimum OQC. Then, ð1Þ ψ 2j ðyÞ ¼ ðc  wr Þ UQ  wc A

Z

þ ðeþ c  we Þ U

1 yþQ

 :

ð13Þ

The first term in Eq. (13) computes the supplier’s profit for delivering the minimum OQC, and the second term is the cost of reserving the expedited-delivery capacity (A). The third term is the supplier’s expected profit from the expedited-delivery option. The supplier’s expected profit in the second period when the retailer has an initial inventory of y and orders-up-to Rn2j , denoted by ψ ð2Þ 2j ðyÞ, can be written as follows:   n ψ ð2Þ 2j ðyÞ ¼ ðc−wr Þ  R2j −y −wc A þ ðe þ c−we Þ ! ! Z ∞  
 Q ∧ðRn2j −yÞ n þ 1− A∧ D2j −R2j dF j D2j K: ð14Þ  Q Rn2j The first three terms in Eq. (14) are the same as those in Eq. (13). The last term is the supplier’s partial penalty payment if the order is less than Q . Similarly, let ψ ð3Þ 2j ðyÞ denote the supplier’s expected profit in the second period when the retailer places no orders. Then,

Z 1 

 ð3Þ ψ 2j ðyÞ ¼ K þ ðe þ c  we Þ U A4 D2j  y Q dF j D2j  wc A: y

ð15Þ Because no order is placed at the beginning of the second period, the supplier only receives the penalty payment. He has to pay for the cost of reserving the expedited-delivery capacity regardless of whether the expedited-delivery option is used by the retailer. The supplier’s profit for the first period when R1 is ordered, denoted by ψ 1 ðR1 Þ , is simply

ψ 1 ðR1 Þ ¼ ðc  wr ÞR1 :

ð16Þ     and Ψ RL1 denote the supplier’s Finally, let Ψ RS1 , Ψ RM 1 expected two-period profit in the small, medium and large penalty cases, respectively. These profit functions can be computed as follows:     ð17Þ ΨðRS1 Þ ¼ qg U ϕS ψ 1 ðRS1 Þ; ψ 2g þ qb U ϕS ψ 1 ðRS1 Þ; ψ 2b : 





In this case, the retailer uses the strategy summarized in Table 4a (resp. 4b) if a good (resp. bad) signal is observed in the second period. The optimal ordering quantity in the first period, RM 1 , solves Eq. (F.1) and must satisfy Eq. (F.2) in Appendix F. The analytical solution for the uniform demand distribution is provided in Eq. (F.3).



 A4 D2j  y Q dF j D2j







S L M M ΨðRM 1 Þ ¼ qg U ϕ ψ 1 ðR1 Þ; ψ 2g þ qb U ϕ ψ 1 ðR1 Þ; ψ 2b :









ΨðRL1 Þ ¼ qg U ϕL ψ 1 ðRL1 Þ; ψ 2g þ qb U ϕL ψ 1 ðRL1 Þ; ψ 2b : 

ð18Þ ð19Þ



The function ϕ ψ 1 ; ψ 2j follows Eq. (8)   by replacing π 1j with ψ 1 and π 2j with ψ 2j . Similarly, ϕL ψ 1 ; ψ 2j follows Eq. (9). S

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In the case of uniform demand distributions, the optimal produce-up-to level for the second period is
 δj ðp þ l  wr Þ U δj A þ ðwe  wr Þ U A RC2j ¼ μ  þ : ð22Þ pþls 2

3.6. Supplier’s expected two-period profit under a coordinated supply chain In this section, we consider a coordinated supply chain where the supplier sells the product to the consumers directly. In this case, there is no minimum order quantity or penalty imposed, but the supplier does need to decide the size of the expedited-delivery capacity. Let us start by considering the optimal replenishing policy in the second period. Let π C2j denote the second period’s expected profit of the supply chain when signal j is observed. Recall that R2j represent the order-up-to inventory. For a given initial inventory (yo R2j ), we can compute the expected profit as follows: Z ψ C2j ðy; R2j Þ ¼ p

∞

0

Z −l

Z D2j ∧ðR2j þ AÞdF j ðD2j Þ þ s

∞ R2j þ A

Z −we

∞ R2j

Note that the difference between the produce-up-to level in the coordinated supply chain and the order-up-to level in the noncoordinated supply chain is RC2j  Rn2j ¼

0



 D2j −R2j −A dF j ðD2j Þ−wr R2j −y

 A∧ D2j −R2j dF j D2j −wc A:

ð23Þ

Thus, if the expedited-delivery capacity (A) is sufficiently small ðA o Atj ¼ ðc−wr Þδj =ðc þ e−we ÞÞ, the optimal produce-up-to level in the second period is higher; otherwise, the optimal order-up-to level in the second period is higher. C Let ψ C1j ðR1 Þ and Ψ ðR1 Þ denote the supplier’s expected profit for the first period and his two-period expected profit when R1 is produced, respectively. They can be written as the following: Z 1 Z 1 ψ C1j ðR1 Þ ¼ p ðD1 4 R1 ÞdF j ðD1 Þ  wr R1 h ðR1 D1 Þ þ dF j ðD1 Þ 0 0 Z 1 ðD1  R1 Þ þ dF j ðD1 Þ: ð24Þ l


 R2j −D2j dF j ðD2j Þ

R2j

ðc  wr Þ U δj  ðc þ e  we Þ UA : pþls

ð20Þ

0

The terms in Eq. (20) are the supply chain’s expected sales revenue, the expected salvage value from any unsold items, the expected lost sales, the manufacturing cost for regular orders, the cost of providing the expedited-delivery option, and the cost of reserving the expedited-delivery capacity, respectively. The optimal produce-up-to inventory level upon signal j is observed, denoted by RC2j , solves the following equation:     ðwe  sÞ UF j RC2j þ ðp þ l  we Þ U F j RC2j þ A ¼ p þ l  wr :

Ψ ðR1 Þ ¼ C

X j

þ

" qj U ψ C1j ðR1 Þ þ

Z

1 R1  γ j

ψ

C 2j




Z



1 R1  γ j



ψ C2j ðR1  D1 Þ þ ; RC2j dF j ðD1 Þ

#  ðR1  D1 Þ ; ðR1  D1 Þ dF j ðD1 Þ : þ

þ

ð25Þ

The optimal ordering quantity in the first period, RC1 , solves Eq. (G.1) and must satisfy Eq. (G.2) in Appendix G. Moreover, the optimal production quantity in the first period for the uniform distributions is provided in Eq. (G.3). As stated prior, the uniform distributions result in the most conservative estimate of uncertainty, the analytical results derived are indicative of worse case scenarios. Nevertheless, the optimal order quantities in the first period, albeit in closed form, are very difficult to be used to compute the retailer’s and the supplier’s profit or to perform any sensitivity analysis. We thus provide numerical analysis on normal distributions in the next session.

ð21Þ

Comparing the solution of Eq. (21) and that of Eq. (2), we have the following proposition. Proposition 2. There exists a cutoff value for the expediteddelivery capacity (Atj ) such that the optimal produce-up-to inventory levels in the second period of the coordinated supply chain are higher (resp. lower) than the optimal order-up-to inventory levels of the non-coordinated supply chain when the expediteddelivery capacity is lower (resp. higher) than the cutoff.

4. Numerical analysis To generalize our findings, we present numerical solutions of normal demand distributions and conduct sensitivity analysis on governing parameters in this section. The following parameters are used: f g  Nð250; 90Þ and f b  Nð250; 140Þ, p ¼ $20, l ¼ $15, c ¼ $5, h ¼ $3, e ¼ $3, s ¼ $0, qg ¼ qb ¼ 0:5, wr ¼ $3, we ¼ $5, and wc ¼ $1. Table 5 provides the retailer’s replenishing strategy in the second period for Q ¼ 170, and the following scenarios: 1) K ¼ 400, a small penalty; 2) K ¼ 500, a medium penalty; and 3) K ¼ 800, a large penalty. Irrespective of the penalty, Rn2g ¼ 243 and Rn2b ¼ 283. We observe that the higher the penalty is, the more

Proof. The result from Eq. (21) suggests that when the expediteddelivery option is unavailable (A ¼ 0), the optimal produce-up-to inventory level is RC2j ¼ F j  1 ððp þ l  wr Þ=ðp þ l  sÞÞ, which is the upper bound. However, it is greater the upper bound of the optimal order-up-to inventory (RU2j Þ in the non-coordinated supply chain. Conversely, when the option is unlimited (A ¼ 1), Rn2j ¼ F j  1 ððwe  wr Þ=ðwe  sÞÞ and is a lower bound for Rn2j . Moreover, it is less than the lower bound of the optimal order-up-to inventory level (RL2j ). Because both RC2j and Rn2j decrease in the expedited-delivery capacity, there exists a unique cutoff value (Atj ) such that RC2j ¼ Rn2j . Table 5 Retailer’s replenishing strategy in the second period. Penalty 400 500 800

Good signal is observed ½0; 73 Order-up-to 243 ½0; 73 Order-up-to 243 ½0; 73 Order-up-to 243

ð73; 185 Order 170 ð73; 232 Order 170 ð73; 349 Order 170

Bad signal is observed ð185; 243 Order-up-to 243 ð232; 243 Order-up-to 243

þ

243 No order 243 þ No order 349 þ No order

½0; 113 Order-up-to 283 ½0; 113 Order-up-to 283 ½0; 113 Order-up-to 283

ð113; 251 Order 170 ð113; 293 Order 170 ð113; 293 Order 170

ð251; 283 Order-up-to 283

283 þ No order 293 þ No order 438 þ No order

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Fig. 2. Effect of reduced-order penalty on the supply chain’s performance.

Fig. 3. Effect of reduced-order penalty on the performance of the retailer and the supplier.

likely the retailer will deviate from the optimal base stock (orderup-to Rn2j ) policy. 4.1. Effect of the reduced-order penalty Figs. 2 and 3 show the optimal order quantity in the first period and the expected profit of the non-coordinated supply chain, supplier and retailer with respect to the reduced-order penalty (K). As the penalty increases, the supplier’s order quantity in the first period decreases. This is mainly driven by the fact that a higher initial inventory in the first period may lead to the retailer to pay penalty in the second period. Thus, the retailer decreases his order in the first period to avoid being in that situation as much as he can. This may exacerbate his chance of lost sales in the first period, as the expedited-delivery option is only available in the second period. Thus, the performance of both the retailer and the supply chain as a whole decrease. Note that in the small penalty region, the supplier’s profit first increases and then decreases. In both the medium and large penalty regions, the supplier’s profit decreases monotonically. This suggests that for a given minimum OQC and expedited delivery capacity, the optimal penalty is in the small region. 4.2. Effect of the minimum OQC Figs. 4 and 5 provide the expected profits of the supplier, retailer and the non-coordinated supply chain, as well as the first period’s optimal order quantity with respect to increasing minimum OQC. Because K ¼ $400 and c ¼ $5 in this numerical analysis, we start with Q ¼ 90 in order to satisfy the K o cQ condition.

When Q is between 90 and 140, a penalty of $400 is in the large range; when Q is between 140 and 160, the penalty is in the medium range; and when Q is higher than 160, the penalty is in the small range. In the large penalty region, the retailer’s order quantity in the first period decreases as Q increases in order to avoid a high initial inventory in the second period. However, this creates lost sales in the first period and as a result the retailer’s expected profit decreases. The increase in the supplier’s profit in the large penalty region is primarily because of the reduced-order penalty paid by the retailer in the second period. Recall that the retailer is more likely to deviate from the optimal based stock policy in the second period when the penalty is large. This leads to the total supply chain’s expected profit to decrease in the large penalty region. In the small penalty region, the retailer’s order quantity in the first period continues to decrease in Q to create a low initial inventory in the second period. But this strategy leads to lost sales in the first period and consequently makes the retailer’s profit continues to decrease. Because the retailer’s replenishing strategy in the second period is less likely to deviate from the optimal base stock policy, his profit does not decrease as fast as that in the large penalty region. The supplier enjoys the benefit from a larger order size in the second period and thus his profit increases at a higher rate in the small penalty region. The total supply chain’s expected profit then increases slightly in this region. 4.3. Effect of the expedited-delivery capacity Figs. 6 and 7 show the expected profits of the supplier, retailer and the non-coordinated supply chain, as well as the first period’s

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Fig. 4. Effect of minimum OQC on the performance of the retailer and the supplier.

Fig. 5. Effect of minimum OQC on the supply chain’s performance.

Fig. 6. Effect of the expedited-delivery capacity on the supplier's performance and the supply chain's performance.

optimal ordering quantity with respect to increasing expediteddelivery capacity. For any given minimum OQC and reduced-order penalty, the retailer’s expected profit increases in the expediteddelivery capacity while the supplier’s expected profit decreases. This is primarily driven by the fact that the better the expediteddelivery option is, the lower the retailer would order in both periods. The retailer in essence passes the demand risk to the supplier. When compared against the supplier’s expected profit without any flexible terms (Q ¼ 0, K ¼ 0, A ¼ 0), there only exists a small range of the expedited-delivery capacity (A r 20) during which the

supplier’s expected profit is higher. The capacity that provides the highest expected profit for the non-coordinated supply chain; however, is far higher (A ¼ 180). Thus, the expedited-delivery option does not help achieve channel coordination. 4.4. Comparison of the non-coordinated supply and the coordinated supply chain Table 6 compares the optimal order quantity in the first period (RC1 ), the optimal order-up-to levels (RC2g and RC2b ), and the expected two-period profit for the coordinated supply chain under two

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Fig. 7. Effect of expedited-delivery capacity on the retailer's perofrmance.

Table 6 Performance of the coordinated supply chain. Optimal ordering and replenishing quantities RC1 No expedited-delivery option (A ¼ 0) 408 With expedited-delivery option (A ¼ 170) 405

RC2g

RC2b

373 255

441 309

SC profit

$7209 $7345

scenarios: 1) when the expedited-delivery option is unavailable; and 2) when the optimal delivery capacity is available (A ¼ 170). The most noticeable difference between the two cases is that while the order-up-to levels in the second period are much smaller when the expedited-delivery option is available, the channel’s expected profit increases nearly 2%. This verifies that the expedited-delivery option helps the supply chain better meet consumers’ demand in the second period, and as a result increases the performance of the coordinated supply chain. Table 7 provides the optimal order quantity in the first period (Rn1 ), the optimal order-up-to levels (Rn2g and Rn2b ), and the expected two-period profits for the retailer, supplier and the supply chain under different combinations of the contract terms. Using the results of the no flexible terms as a benchmark, we can see that imposing the minimum OQC and reduced-order penalty without providing the expedited-delivery option only benefits the supplier. Recall that when the initial inventory level in the second period is high, the retailer is more likely to deviate from the optimal base stock policy regardless the signal observed. Thus, placing a smaller order in the first period helps the retailer reduce the initial inventory in the second period. However, because the expedited-delivery option is only available in the second period, the retailer incurs lost sales. As a result, his expected profit is lower than the no flexible contract case. In the case of expedited-delivery option is offered, both the order quantity in the first period (Rn1 ) and the optimal order-up-to levels in the second period (Rn2g and Rn2b ) decrease, and the channel profit increases. The results are similar to those of the coordinated supply chain. Upon close examination, we notice that the higher the expedited-delivery capacity, the higher the retailer’s expected profit and the lower the supplier’s expected profit. This is because the retailer is able to pass much of demand risk upstream to the supplier. Moreover, when the expedited-delivery capacity is very small, the retailer’s expected profit is lower compare to the no flexible terms contract but the supplier’s profit is higher. On the other hand, when expedited-delivery capacity is large, the supplier’s expected profit is lower compare to the no flexible terms contract but the retailer receives a higher profit.

When the expedited delivery capacity is in the medium range, both the retailer and the supplier receive a higher expected profit than they would in the no flexible terms contract. This creates a win-win situation for both parties. The channel’s expected profit has a 0:2% increase, much less than the coordinated supply chain. As the expedited-delivery capacity continues to increase, the supplier’s expected profit decreases significantly even though the retailer and the channel have a higher profit. The optimal expedited-delivery capacity that achieves the highest total supply chain profit for the non-coordinated supply chain is A ¼ 180, higher than that in the coordinated supply chain. Moreover, the supplier’s performance suffers nearly 15% loss when compared to the no flexible terms contract. The retailer’s profit increases by 5% while the total supply chain’s gains is close to 2%. This suggests that without some further compensation scheme, such as revenue sharing or cost sharing, the supplier will offer an expedited-delivery capacity that is much lower than the one that achieve the highest profit for the supply chain. To clarify the difference between coordinated and non-coordinated supply chain, we perform sensitivity analysis on the probability of observing a good signal at the end of the first period. Table 8 provides the results of the optimal expedited-delivery capacity ðAC Þ, the optimal order quantity ðRC1 Þ, the optimal order-up-to levels (RC2g and RC2b ), and the expected profit of the coordinated supply chain with varying probability of observing a good signal. Table 9 summarizes similar results of the non-coordinated supply chain. We observe that while the optimal expedited-delivery capacity for both coordinated ðAC Þ and non-coordinated ðAn Þ supply chain decreases in the probability of observing a good signal, An r AC only when the chance of observing a bad signal is very high. Further, the ratio of the channel profit between the non-coordinated and the coordinated supply chain decreases slightly as the probability of observing a good signal increases. The results suggest that the more likely a bad signal occurs, the better the non-coordinated supply chain performs. This mainly because the retailer cannot pass as much demand risk to the supplier as he would when the chance of having a favorable demand is high.

5. Managerial insights The developed model in this work considers the impact of the flexible terms in a contract, including the minimum OQC with a reduced-order penalty as well as an expedited-delivery option, on the retailer’s inventory management policies as well as on the performance of the retailer and the supplier. We summarize the implications of our results in this section. First, we find that given a contract with a minimum OQC and a reduced-order penalty, a retailer would strategically respond to the contract terms by adopting different replenishing strategy in

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Table 7 Performance of the non-coordinated supply chain.

No flexible terms (Q ¼ 0, K ¼ 0, A ¼ 0) Minimum OQC only (Q ¼ 170, K ¼ 400, A ¼ 0) Minimum OQC with the expedited-delivery option Minimum OQC with the expedited-delivery option Minimum OQC with the expedited-delivery option Minimum OQC with the expedited-delivery option

(Q (Q (Q (Q

¼ 170, ¼ 170, ¼ 170, ¼ 170,

K ¼ 400, K ¼ 400, K ¼ 400, K ¼ 400,

A ¼ 10) A ¼ 15) A ¼ 20) A ¼ 180)

Table 8 Effect of the signal on coordinated supply chain. qg AC

0 205

0.2 190

0.4 175

0.6 165

0.8 150

1 140

RC1

438

426

413

399

385

372

RC2g

–

251

257

264

269

274

RC2b SC profit

294

303

312

321

328

–

$7133

$7211

$7298

$7396

$7506

$7626

Table 9 Effect of the signal on non-coordinated supply chain. qg An Rn1 Rn2g

0 190 409 –

0.2 180 397 243

0.4 180 384 243

0.6 175 372 245

0.8 160 361 250

1 150 352 255

Rn2b Retailer’s profit Supplier’s profit SC profit

278 $6004 $1117 $7121

283 $6095 $1095 $7190

283 $6202 $1068 $7270

285 $6315 $1046 $7361

293 $6430 $1035 $7465

– $6561 $1023 $7584

the second period based on his demand forecast update as well as the reduced-order penalty. The higher the penalty is, the more likely the retailer would deviate from the optimal base stock policy in the second period. Thus, a supplier who imposes a large penalty for a given minimum OQC will lead to an undesirable outcome. That is, both the retailer and the supplier suffer in their expected profits when compared to no flexible terms are included in the contract. On the other hand, for a given penalty, increasing the minimum OQC would increase the supplier’s expected profit while decreasing the retailer’s expected profit. Therefore, managers from the supply side have a strong incentive to impose a large minimum OQC while setting a small reduced-order penalty. Second, our analysis demonstrates that the optimal order-up-to level in the second period decreases as the expedited-delivery capacity increases. This is because the retailer not only uses the option to better meet customers’ demand but also passes much of the demand risk to the supplier. Moreover, the optimal order sizes must increase if either the selling price or the cost of lost sales increases as the retailer cannot fully rely on the expedited-delivery option and thus must order more at the beginning of the second period to prevent lost sales. The developed analytical approach reveals that the expedited-delivery option is most valuable when one or more of the following conditions holds: 1) the purchase price is high, 2) the cost of the expedited-delivery is high, or 3) the salvage value is low. It is thus important for retail managers to look for these characters when choosing a supplier. Third, our model shows that the optimal expedited-delivery capacity that achieves the highest supply chain performance exceeds the capacity the supplier is willing to offer, as his profitability would suffer significantly otherwise. Although this suggests that the joint use of the flexible contract terms cannot achieve full

Optimal ordering and replenishing quantities

Expected profit

Rn1

Rn2g

Rn2b

Retailer

Supplier

SC

404 391 390 389 389 378

346 346 338 335 331 243

399 399 392 388 384 283

$5955 $5917 $5948 $5963 $5977 $6259

$1237 $1271 $1255 $1246 $1239 $1055

$7192 $7188 $7203 $7209 $7216 $7314

supply chain coordination, it is important to note that there does exists combinations of the minimum OQC and the expediteddelivery capacity such that both parties are better off when compared against the contract with no flexible terms. To summarize these insights, our analysis shows that the benefit a retailer accrues from the expedited-delivery option often outweighs the loss incurred by the order restriction imposed by the supplier. Thus, it is important for the retail managers to jointly evaluate all three of the contract terms (the minimum OQC, the penalty, and the expedited-delivery capacity) when selecting a supplier. In particular, industries such as the apparel and electronics where upstream competition is high, retail managers may find that a contract with a higher expedited-delivery capacity is superior even if it includes a large minimum OQC, as long as the penalty is relatively small.

6. Conclusion In this paper, we develop a model that amalgamates the flexible contract terms to assess their joint impact on the inventory management policies of a retailer who sells products in both the pre-sale and the sales seasons. The retailer faces not only demand uncertainty but also a minimum OQC requirement imposed by the supplier. More specifically, if the retailer orders less than the stipulated minimum OQC, he would pay a (partial) penalty. To compensate for this restriction, the retailer is offered with an expedited-delivery option in the sales season at a premium should the need arise. The Newsvendor model and dynamic programming approaches are extended to render the optimum ordering and replenishing strategies in both periods based on the governing parameters and the contract terms. One important finding of our analysis exhibits that there exists a range of the expedited-delivery capacity that the supplier provides in which it is beneficial to both parties of the chain. This range depends on the expedited-delivery cost as well as the other flexible terms, such as the minimum OQC and the penalty for any reduced order as shown in the section of numerical analysis. Another major finding that pertains to the minimum OQC restriction is that the penalty specified for orders less than the minimum OQC needs to be significantly smaller than the cost of purchasing the foresaid amount. Otherwise, the profits of both the supplier and the retailer are lower when compared to their profits under the contract with no flexible terms. Our future research includes a couple of directions. First, we plan to extend this model to investigate whether a revenue sharing or a cost-sharing scheme on the expedited-delivery option can lead to better performance for both the supplier and the retailer. Second, we intend to examine how would competition among suppliers and retailers affect the decisions of the involved parties. We believe that answering these questions would facilitate retail managers as well as suppliers in negotiating contracts that would lead to win-win outcomes.

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Acknowledgment

no orders policy as long as the order quantity is less than the minimum OQC (y o Rn2j  Q ). Since γ j ¼ Rn2j is the cut-off between

The authors are indebted to the anonymous reviewers for their insightful comments and suggestions that helped sharpening this work.

π ð2j2Þ and π ð2j3Þ and π ð2j2Þ 4 π ð2j3Þ when y oRn2j , therefore βj r γ j . Last, we show the relationship between the cutoff valuations

 generated by π ð2j1Þ and π ð2j3Þ . Let θj solves π ð2j1Þ θj  π ð2j3Þ θj ¼ 0, i.e.

Appendix A. functions

Structural properties of second period’s profit

We show that the profit functions for ordering minimum OQC ðπ ð2j1Þ ðyÞÞ and ordering nothing ðπ ð2j3Þ ðyÞÞ are concave and the profit function for ordering-up-to Rn2j ðπ ð2j2Þ ðyÞÞ is piece-wise linear. π ð2j1Þ ðyÞ is increasingly concave. dπ

ð1Þ 2j ðyÞ

dy


 ¼ ðe þ cÞ U F j ðy þ Q þ AÞ  F j ðy þ Q Þ þ s U F j ðy þ Q Þ
 þ ðp þ lÞ U 1 F j ðyþ Q þ AÞ Z 0:

d π ð2j1Þ ðyÞ

dπ

θj

¼  ðp þ l  e  cÞ U f j ðy þQ þ AÞ  ðe þ c  sÞ U f j ðy þQ Þ o 0:

θj þ Q


 F j D2j dD2j  ðp þ lÞ


 F j D2j dD2j 

Z θj þ A þ Q θj þ A

Z θj þ A θj


 F j D2j dD2j


 F j D2j dD2j ¼ 0:

!

ðB:3Þ

 In the case of uniform distribution U μ  ðδj =2Þ; μ þ ðδj =2Þ , Q 2

ðA:1Þ

π

þs

Z θj þ Q

θj ¼ Rn2j  þ

2

dy

Z θj þ Q þ A

ðp þ l  cÞQ þK þ ðe þ cÞ

K δj : Q ðp þ l  sÞ

ðB:4Þ

Because θj can be either greater than or less than γ j depending on the penalty (K), we divide our analysis in two cases, namely small penalty and large penalty, and provide the computation of the cutoff valuation Appendix C.

ð2Þ 2j ðyÞ

ð2Þ 2j ðyÞ

dy

is piece-wise linear. 8
Appendix C. ðA:2Þ

π ð2j1Þ ðyÞ is increasingly concave. d π ð2j1Þ ðyÞ dy



 ¼ ðe þ cÞ U F j ðy þ AÞ  F j ðyÞ þ s U F j ðyÞ þ ðp þ lÞ U 1  F j ðy þAÞ Z 0:

d π ð2j1Þ ðyÞ

Calculation of cutoff penalty values (K tj )

The difference between π ð2j1Þ and π ð2j3Þ is Z yþAþQ
 π ð2j1Þ ðyÞ  π ð2j3Þ ðyÞ ¼ ðp þ l  cÞ U Q  ðp þ lÞ F j D2j dD2j þ ðeþ cÞ U

2

dy

¼  ðp þ l  e  cÞ U f j ðy þAÞ  ðeþ c  sÞ U f j ðyÞ o 0:

ðA:3Þ

Z 

yþA

þs

Relationships among cutoff valuations

First, we show the relationship between the cutoff valuations generated by π ð2j1Þ and π ð2j2Þ . Let αj ¼ Rn2j  Q . When the initial inventory in the second period is less than αj , the optimal replenishing strategy is ordering-up-to Rn2j because the ordering

 quantity is greater than Q. Because π ð2j1Þ αj  π ð2j2Þ αj ¼ 0, α is an ð1Þ ð2Þ ð1Þ intersection between π 2j and π 2j . Since π 2j is a concave function and π ð2j2Þ is linear for y 4 αj , we name the second intersection βj , and it solves the following equation: ðeþ cÞ

Z βj þ Q þ A βj þ Q

 ðp þ l Þ þs

Z βj þ Q þ A Rn2j þ A

Z βj þ Q n

R2j


 F j D2j dD2j 

Z

Rn2j þ A n


 F j D2j dD2j

!

2K δj : Q ðp þ l  sÞ

ðB:2Þ

Next, we show the relationship between the cutoff valuations   generated by π ð2j2Þ and π ð2j3Þ . Let γ j ¼ Rn2j . Because π ð2j2Þ γ j    π ð2j3Þ γ j ¼ 0, γ j is an intersection between the two profit functions. Because the order-up-to Rn2j policy meets the optimal inventory level while only incurring a partial penalty, it dominates the place


 F j D2j dD2j þ K:

ðC:1Þ

π ð2j1Þ ðyÞ and π ð2j3Þ ðyÞ. We define that at the cutoff reduced-order penalty, K tj , is the valuation that satisfy βj ¼ θj ¼ Rn2j . Thus,

Z

ðB1Þ

!

Because π ð2j1Þ ð0Þ 4 π ð2j3Þ ð0Þ, there is only one intersection between

 ðe þ c Þ

In the case of uniform distribution U μ  ðδj =2Þ; μ þ ðδj =2Þ ;

βj ¼ Rn2j  Q þ

y

Z

R2j




 F j D2j dD2j


 d  ð1Þ π ðyÞ  π ð2j3Þ ðyÞ ¼  ðp þ l  e cÞ U F j ðy þQ þ AÞ  F j ðy þ AÞ dy 2j
  ðe þ c  sÞ F j ðy þQ Þ  F j ðyÞ o 0: ðC:2Þ


 F j D2j dD2j

! Rn2j  βj þ 1 K ¼ 0: Q

yþQ


 F j D2j dD2j

yþQ

K tj ¼ ðp þ l  cÞQ þ ðp þ lÞ

 
 F j D2j dD2j þ ðp þ l  cÞ U βj þ Q  Rn2j

yþAþQ

y

Z Appendix B.

yþA

Z

Rn2j þ A þ Q

Rn2j þ Q

Rn2j þ A þ Q

Rn2j þ A


 F j D2j dD2j  s


 F j D2j dD2j 

Z

Rn2j þ A Rn2j

Z

Rn2j þ Q

Rn2j


 F j D2j dD2j

!
 F j D2j dD2j :

ðC:3Þ

    At y ¼ Rn2j , π ð2j1Þ Rn2j  π ð2j3Þ Rn2j ¼ K  K tj . When K 4 K tj ,     π ð2j1Þ Rn2j 4 π ð2j3Þ Rn2j . This indicates that the cut-off value θj is greater than γ j ¼ Rn2j .

Appendix D.

Calculation of RS1 in the small penalty case

By differentiating Π ðR1 Þ with respect to R1 and setting it to zero, we get the following equation: h 

X  qj U p þ l  c þ π ð2j2Þ U F j R1  αj S

j

     F j ðR1 Þ þ F j R1  γ j  F j R1  β j :

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      K Q U F j R1  γ j  Q  γ j þ β j UF j R1  βj Q

Z R1  γ j F j ðR1  D1 þAÞ dF j ðD1 Þ þ p þ l  h U F j ðR1 Þ  ðp þ l  e cÞ U 

0

Z þ Z

R1  α j R1  βj R1  γ j

0

!

F j ðR1  D1 ÞdF j ðD1 Þ þ

R1  α j R1  β j

F j ðR1 D1 ÞdF j ðD1 Þ

¼ 0: ðD:1Þ

that maximizes Π ðR1 Þ solves The optimal order quantity Eq. (D.1). We further verify that the second order sufficient condition is satisfied. S

∂2 Π

S

∂R21

¼

 
X h  qj p þ l  c þ π ð2j2Þ U f j R1  αj  f j ðR1 Þ j

    þf j R1  γ j  f j R1  β j :       K Q Uf j R1  γ j  Q  γ j þ β j Uf j R1  βj  ðp þ l  e cÞ U  Q     

 F j γ j þ A U f j R1  γ j þ F j αj þ A þQ U f j R1  αj  Z R1  γ j   
 F j βj þ A þ Q Uf j R1  βj þ f j ðR1  D1 þAÞ dF j D2j 0

Z þ

R1  α j R1  βj

R1  βj




 þF j αj þQ Uf j R1  αj     F j βj þ Q U f j R1  βj  hf j ðR1 Þ o0:

ðD:2Þ



For uniform distribution U μ  ðδj =2Þ; μ þ ðδj =2Þ , RS1 ¼

X j

Z

R1  α j R1  θ j

R1  θ j 0

F j ðR1  D1 ÞdF j ðD1 Þ !

F j ðR1 D1 þ Q ÞdF j ðD1 Þ

 : ¼ 0:

ðE:1Þ

We further verify that the second order sufficient condition is satisfied. "   
 ∂ 2 ΠL ð2Þ  f j ðR1 Þ−f j R1 −αj ¼ ∑ q j −hf j ðR1 Þ þ c−p−l−π 2j 2 ∂R1 j

Z R1 −θj f j ðR1 −D1 þ AÞdF j ðD1 Þ −ðp þ l−e−cÞ  0

 þ F j θj þ A  f j R1 −θj  



þ F j αj þ A þ Q f j R1 −αj −F j θj þ A þ Q f j R1 −θj ! Z R1 −αj f j ðR1 −D þ A þ Q ÞdF j ðD1 Þ −ðe þ c−sÞ þ R1 −θj

Z



 f j ðR1 −D1 Þ  dF j ðD1 Þ þ F j θj f j R1 −θj

 þ F j αj þ Q f j R1 −αj !# Z R1 −αj

 f j ðR1 −D1 þ Q ÞdF j ðD1 Þ o 0: −F j θj þ Q f j R1 −θj þ R1 −θj

0

R1 −θ j

     þ ðs  e  cÞ U F j γ j U f j R1  γ j Z R1  α j Z R1  γ j f j ðR1  D1 ÞdF j ðD1 Þ þ f j ðR1  D1 þQ ÞdF j ðD1 Þ þ


F j ðR1  D1 þ A þ Q ÞdF j ðD1 Þ

 ðeþ c  sÞ U

!

f j ðR1  D1 þ A þ Q ÞdF j ðD1 Þ

0

!

R1  αj

R1  θ j

þ

!#

(RS1 )

þ

Z

F j ðR1  D1 þ A þQ Þ dF j ðD1 Þ  ðe þ c  sÞ U Z

Z

qffiffiffiffiffi 1 △Sj  hδj  ðp þ l  e  cÞA @ A; where qj 2μ  δ j þ pþls 0

 2 δj þhδj △Sj ¼ ðp þ l  e  cÞA  2ðp þ l  sÞ μ  2 

δj hδj þ 2ðp þ l  sÞ U μ  2 


 2K δj 2  Rn2j U ðp þ l  cÞδj  ðp þl e  cÞA þ ðp þ lÞδj þ Q ðp þ l  s Þ



 4K δj K δj 2 Rn2j þ þ 2ðp þ l  sÞ U Rn2j2  Q ðp þ l  s Þ Q ðp þ l  sÞ



 δj 2K δj Rn2j : ðD:3Þ þ μ 2 Q ðp þ l  sÞ



For uniform distribution U μ  ðδj =2Þ; μ þ ðδj =2Þ , qffiffiffiffiffi 0 1 △Lj  hδj  ðp þl  e cÞA X L A; where qj @2μ  δj þ R1 ¼ pþls j

ðE:2Þ



2 δj þ hδj þ 2ðp þ l  sÞ △Lj ¼ ðp þl e  cÞA  2ðp þ l sÞ U μ  2 

 
δj 2 hδj þ ðp þ lÞδj þ Q  Rn2j U ðp þ l  cÞδj  μ 2  ðp þl  e cÞAÞ

  

  δj 2 n2 ðE:3Þ  Q 2  Rn2j  Q U μ  þ 2ðp þ l  sÞ U R2j 2



Appendix E.

Calculation of RL1 in the large penalty case

The optimal order quantity (RL1 ) that maximizes Π ðR1 Þ solves the following:  

X h  qj p þl hF j ðR1 Þ þ p þ l  c þ π ð2j2Þ F j R1  αj

Appendix F.

The optimal order quantity (RM 1 ) that maximizes Π ðR1 Þ solves the following: 
    h   2Þ F g R1  αg þ F g R1  γ g  F g R1  βg  F g ðR1 Þ qg U p þ l  c þ π ð2g M

      K Q U F g R1  γ g  Q  γ g þ β g U F g R1  βg Q

Z R1  γ g F g ðR1  D1 þ AÞ dF g ðD1 Þ þ p þ l  hF g ðR1 Þ  ðp þ l  e  cÞ 0 ! Z



R1  αg

þ

R1  β g

Z

R1  γ g

L

j

  F j ðR1 Þ  ðp þ l  e cÞ: Z R1  θ j F j ðR1  D1 þ AÞdF j ðD1 Þ  0

Calculation of RM 1 in the medium penalty case

0

F g ðR1  D1 þ A þ Q ÞdF g ðD1 Þ  ðeþ c  sÞ U Z

F g ðR1  D1 ÞdF g ðD1 Þ þ

R1  α g R1  β g

!# F g ðR1  D1 þ Q ÞdF g ðD1 Þ

 2Þ p þl c þ π ð2b ðF b ðR1  αb Þ  F b ðR1 ÞÞ þ p þ l  hF b ðR1 Þ Z R1  θb  ðp þ l  e cÞ U F b ðR1  D1 þAÞdF b ðD1 Þ

þ qb U

Z þ

h

0

R1  αb

R1  θ b



F b ðR1  D1 þ A þ Q ÞdF b ðD1 Þ

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Z  ðe þc  sÞ U Z þ

R1  αh

R1  θ h

0

R1  θ h

þ

 F h ðR1  D1 þ Q ÞdF b ðD1 Þ : ¼ 0:

∂2 Π

∂R21

¼ qg U

h

ðF:1Þ

2Þ c  p l  π ð2g




 f g ðR1 Þ f g R1  αg

     f g R1  γ g þ f g R1  β g :       K Q Uf g R1  γ g  Q  γ g þ βg Uf g R1  β g  Q

Z R1  γ   g
 f g ðR1 D1 þ AÞdF g D2g þ F g γ g þ A U  ðp þl  e cÞ 0

 

 f l R1  γ g þF g αg þ A þ Q U f g R1  αg      F g β g þ A þ Q U f g R1  βg ! Z R1  α l þ f g ðR1  D1 þ A þ Q ÞdF g ðD1 Þ  hf g ðR1 Þ  ðeþ c  sÞ U R1  βl

Z

R1  γ l

qb U

R1  α g

R1  βg

h




 f b ðR1 Þ f b ðR1  αb Þ

 hf b ðR1 Þ  ðp þ l  e  cÞ Z R1  θb

 f b ðR1  D1 þ AÞdF b ðD1 Þ þ F b θh þ A U f b R1  θh  0



 þ F b ðαb þ A þ Q Þ Uf b ðR1  αb Þ  F b θb þA þ Q Uf b R1  θb  Z R1  α b f b ðR1  D1 þ A þ Q ÞdF b ðD1 Þ þ R θ

1 b


  ðe þ c sÞ F b θb Uf b R1  θb Z Z R1  θ b f b ðR1  D1 ÞdF b ðD1 Þ þ þ

0




R1  α b R1  θ b

f b ðR1 D1 þ Q ÞdF b ðD1 Þ


 o 0: þ F b ðαb þ Q Þ Uf b ðR1  αb Þ F b θb þ Q U f b R1  θb ðF:2Þ For uniform distribution U μ  ðδj =2Þ; μ þ ðδj =2Þ , pffiffiffiffiffiffiffiffi

 ΔM h ql δl þ qh δh  ðp þ l e  cÞA M ; where R1 ¼ 2μ  ql δl þ qs δs þ pþls    ΔM ¼ h qg δg þqb δb þ Aðp þl e  cÞ  2ðp þ l  sÞ   112 0 qg δg þqb δb @μ  AA 2 h   2 2 þ 2ðp þ l  sÞ U ðp þ lÞ qg δg þqb δb



 δg δ þ qb δ b h μ  b þ qg δ g h μ  2 2


 2K δg n U ðp þ l  cÞδg þ ðe þc  p  lÞA  qg R2g  Q ðp þ l  sÞ

 2  qb Rn2b  Q U ðp þ l  cÞδg δb þ ðe þ c  p  lÞA þ 2ðp þ l  sÞ U





 4K δg K δg n2 Rn2g þ  ql R2g Q ðp þl sÞ Q ðp þ l  sÞ 

Calculation of RC1 for the coordinated supply C

j

Z

R1  RC2j



 ðp þ lÞ U 1  F j ðR1  D1 þ AÞ þ sF j ðR1  D1 Þ
  þ we F j ðR1 D1 þ AÞ F j ðR1  D1 Þ  wr  h dF j ðD1 Þ ¼ 0: þ

0

ðG:1Þ

We further verify that the second order sufficient condition is satisfied. ∂2 Ψ

C

¼

  X h qj ðp þ l  wr Þf j R1  RC2j  ðp þ l þ h  wr Þf j ðR1 Þ j

Z

R1  RC2j



ðwe  p  lÞ U f j ðR1  D1 þ AÞ þ ðs we Þ  f j ðR1  D1 Þ dF j ðD1 Þ     þ ðwe  p  lÞ U F j RC2j þ A U f j R1  RC2j    i þ ðs  we Þ U F j RC2j Uf j R1 RC2j o 0: 

ðF:3Þ

The optimal order quantity ðRC1 Þ that maximizes Ψ ðR1 Þ solves the following:    X h
 qj ðp þ l  wr Þ U 1  F j ðR1 Þ  h F j ðR1 Þ  F j R1  RC2j

þ

f g ðR1  D þ Q ÞdF g ðD1 Þ  þ

2Þ c  p  l  π ð2b

Appendix G. chain

∂R21


 f g ðR1  DÞdF g ðD1 Þ þ f g R1  αg 0  
 F g αg þ Q þ f g R1  γ g       F g γ g  f g R1  β g UF g β g þQ ! Z þ

 

δ 2K δl Rn2g μ g : Q ðp þ l  sÞ 2



 
 δ þ qh Rn2b2  Q 2  Rn2b  Q μ  b : 2

F b ðR1  D1 ÞdF b ðD1 Þ

We further verify that the second order sufficient condition is satisfied. M

15

0

ðG:2Þ



For uniform distribution U μ  ðδj =2Þ; μ þ ðδj =2Þ , qffiffiffiffiffiffi 0 1 △Cj  ðh  we Þδj  ðp þ l  wr ÞA X C A; where q j @2 μ  δ j þ R1 ¼ pþls j





 δj δj △Lj ¼ ðp þ l  we ÞA μ  þ RC2j ðl  2we ÞA þ 2s μ  : ðG:3Þ 2 2

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Please cite this article as: Cai, W., et al., Impact of flexible contracts on the performance of both retailer and supplier. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.06.030i