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The impacts of the full returns policy on a supply chain with information asymmetry
European Journal of Operational Research 180 (2007) 630–647 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
The impacts of the full returns policy on a supply chain with information asymmetry Xiaohang Yue a
a,*
, Srinivasan Raghunathan
b
School of Business Administration, The University of Wisconsin at Milwaukee, Milwaukee, WI 53201, United States b School of Management, The University of Texas at Dallas, Richardson, TX 75083, United States Received 19 January 2004; accepted 3 April 2006 Available online 13 July 2006
Abstract In contrast to the existing return policies literature assuming that information is symmetrical between the manufacturer and the retailer, we study the full returns policy’s impact on supply chains with information asymmetry. We first study the case that the base level of the demand follows a discrete distribution with two states. We find that the retailer benefits from the full returns policy in all circumstances, while the manufacturer and the supply chain are better off under some conditions. We then consider the situation in which the base level of the demand is a type of AR(1) process. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Returns policy; Information asymmetry; Information sharing; Supply chains; Decisions under uncertainty
1. Introduction A manufacturer’s returns policy is a common feature in the distribution of many products, such as books, CDs, and fashion apparel. An obvious explanation for a returns policy is risk-sharing; that is, the retailer returns the unsold products to the manufacturer or the manufacturer offers a credit on all unsold products to the retailer. A returns policy is an instrument for channel coordination (i.e., for arriving at the maximum efficiency for the total supply chain). There has been substantial research on why manufacturers might accept returns from the retailers, including Pasternack (1985), Kandel (1996), Padmanabhan and Png (1997), Webster and Weng (2000), Glenn and Puterman (2001), Emmons and Gilbert (1998), and Lee et al. (2000a). Researchers have pointed out that a returns policy, in the form of full returns, partial returns, and consignment, may affect the manufacturer’s profit under some conditions in two ways. The returns policy shifts the burden of demand uncertainty from the retailer to the manufacturer, thus encouraging the retailer to increase stock. Increased stock intensifies retailer distribution, which benefits the manufacturer as the returns policy helps to overcome the double
*
Corresponding author. E-mail addresses: [email protected] (X. Yue), [email protected] (S. Raghunathan).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.04.032
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marginalization problem. However, a returns policy causes the retailer to order stock according to the high demand, which is excessive relative to an integrated channel. At some point, when uncertainty is sufficiently large, the manufacturer is worse off with a returns policy than with a no returns policy. Additionally, a returns policy has an impact on the supply chain players’ price structures, thus influencing their profits. Currently, most of the previous literature assumes that there is no asymmetric information between the manufacturer and the retailer. It means, at all times, the manufacturer and the retailer have equal information about retail demand for the product. Apparently, this assumption is usually not true. In reality, each supply chain player (the manufacturer or the retailer) has a different perspective on the uncertain demand. Typically, they use different data or methodologies to make their own respective forecasts. In this study, we will analyze the impact of a returns policy on both the manufacturer’s and retailer’s performance under the assumption that there exists asymmetric information between the two supply chain players. In this situation, a returns policy enables the manufacturer to learn the previous period’s demand information through the return of unsold products so that the manufacturer could incorporate the updated information into its forecast process for the next period’s demand. Therefore, we can see that while a returns policy passes the burden of demand uncertainty from the retailer to the manufacturer, it enables the manufacturer to obtain useful demand information at the same time. Naturally, questions may arise, such as, what is the overall effect of a returns policy or information sharing on both supply chain players’ profits? Under what conditions is the retailer or the manufacturer better or worse off? In this paper, we will address these questions. The rest of the paper is organized as follows: in Section 2, we review the related literature. In Section 3, we discuss the model framework. In Sections 4 and 5, we analyze the impact of the returns policy on the manufacturer and the retailer when the primary demand follows a discrete distribution with two states as well as follows a continuous distribution. We conclude the paper in Section 6. 2. Literature review There is considerable research on why manufacturers might accept returns from retailers. Almost all of the literature assumes that there is no information asymmetry between supply chain players (i.e., manufacturers share the same demand information with retailers). We review the research chronologically. Pasternack (1985) examines the pricing decision of a manufacturer of a product with a limited shelf life. He presents a model of buy-back as a means to overcome double marginalization, assuming that the supplier and retailer use the same demand distribution function. He proves that channel coordination can be achieved when the manufacturer offers partial credit for all unsold goods. He also shows that limiting returns to a fixed percentage of sales may be an optimal policy in a single-retailer setting but is suboptimal in a multi-retailer setting. Kandel (1996) extends Pasternack (1985) by modeling price sensitivity in end-consumer demand. Specifically, he presents a model of buy-back and consignment contracts and analyzes two extremes: either the supplier or the retailer dictates the wholesale price, the buy-back price and the order quantity. Emmons and Gilbert (1998) build on the work of Pasternack (1985) by analyzing a partial credit/full returns policy with the generalization that the probability distribution of end-customer demand depends on the retail price. Given uniformly distributed demand, they show that a rebate for leftover product increases the expected profit of the manufacturer and the retailer whenever the manufacturer’s price exceeds a threshold value. Padmanabhan and Png (1997) address returns policies from the manufacturer’s perspective. They study the effect of a full returns policy when demand is deterministic and uncertain. They demonstrate that a full returns policy can improve manufacturer profitability by intensifying retail price competition. Webster and Weng (2000) take the viewpoint of a manufacturer selling a short life-cycle product to a single risk-neutral retailer and describe returns policies that, when compared to no returns, satisfy two conditions: (1) the retailer’s expected profit is increased and (2) the manufacturer’s profit is at least as large as when no returns are allowed. They assume that the retailer’s price is exogenous, and derive conditions that a returns policy will result in higher expected manufacturer profit. Glenn and Puterman (2001) present two types of return policies: buy-back and consignment, by which the supplier induces the retailer to accept additional units of a product when the retailer is less optimistic about demand. In the buy-back returns policy, the
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wholesale price is assumed to be the same. They show that the consignment policy is more robust to the supplier’s uncertainty about the retailer’s ordering policy. Sarvary and Padmanabhan (2001) analyze the returns policy’s role in learning demand of a new product. They consider a dual monopoly with a manufacturer and a retailer by assuming that the retail price is fixed and demand is random but identical across periods. A variation of the returns policy is called price protection, which is a commonly used practice between manufacturers and retailers in the personal computer industry, motivated by drastic declines of product values during the product life cycle. It is a form of rebate given by the manufacturer to the retailer for units unsold at the retailer when the price drops during the product life cycle. Lee et al. (2000a) explore the use of price protection with a two-period model. They show that price protection is an instrument for channel coordination. For products with long manufacturing lead times, a properly chosen price protection credit coordinates the channel. For products with shorter manufacturing lead times, price protection alone cannot guarantee channel coordination when wholesale prices are exogenous. However, when the price protection credit is set endogenously together with the wholesale prices, channel coordination is restored. Taylor (2001) extends Lee et al. (2000a) by combing price protection, midlife returns, and end-of-life returns in declining price environments. He shows that if the wholesale prices and the return rebates are set properly, midlife and end-of-life returns achieve channel coordination. However, the manufacturer may be worse off as a result of coordination. If price protection is used in addition to returns and the terms are set properly, then both coordination and a win–win outcome are guaranteed. A related stream of research deals with information sharing within the supply chains. This literature discuss how a manufacturer can elicit information from retailers through inventory, lead time, and shortage allocation policies (Bourland et al., 1996; Gavirneni et al., 1999; Gallego et al., 2000; Cachon and Fisher, 2000). Bourland et al. (1996) show that information sharing offers significant benefits to the manufacturer and retailer when the degree to which their ordering cycles are out of phase is high. Gavirneni et al. (1999) study the holding and penalty cost of a finite capacity supplier facing demands from a single retailer following a (s, S) policy. By considering various types of demand distributions in their numerical experiments, they examine the conditions under which gaining information about the retailer’s inventory level is beneficial. Lee et al. (2000b) study the benefit of demand information sharing when the underlying demand process faced by the retailer is an AR(1) process. They assume that the manufacturer and the retailer incur linear holding and backlogging costs, experience constant lead times, and follow base-stock policies. Chen et al. (2000) quantify the bullwhip effect for a simple two-stage supply chain and demonstrate that centralizing demand information can significantly reduce the increase in variability. Cachon and Fisher (2000) study the value of sharing data in a model with one supplier, N identical retailers, and stationary stochastic consumer demand. They conclude that implementing information technology to accelerate and smooth the physical flow of goods through a supply chain is significantly more valuable than using information technology to expand the flow of information. Gallego et al. (2000) shows that delay base-stock policies can capture a significant portion of the benefits of demand information sharing. They compare the costs under fixed and random delay base-stock policies against the cost under demand information sharing. In this study, we will consider a supply chain consisting of a manufacturer and a retailer, where the demand information is asymmetrical between them. We will analyze the impacts of a returns policy as well as information sharing on the profit of the manufacturer and the retailer, respectively. 3. Model framework Consider a supply chain consisting of a manufacturer and a retailer. The retailer faces a random demand function and has the objective of determining its retail price and order quantity so as to maximize the expected profit. The manufacturer sets up its wholesale price and production quantity so as to maximize its expected profit. The demand function is assumed to be Dt = at bpt, where Dt is the demand at the period t, at is the base level primary demand at the period t, pt is the retailer’s price at period t, and b is the slope. All parameters are positive. To capture uncertainty in market demand resulting from changes in economic and business conditions, we assume that at is a random variable. The setting for this research is retailing of products with
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limited shelf lives, that is, the product is ‘‘perishable’’ (i.e., at the end of each period, all unsold units have zero salvage value). The manufacturer is assumed to move first by choosing wholesale price, wt, based on the available information about the retailer’s demand. The manufacturer behaves like a Stackelberg leader. For simplicity, we restrict the manufacturer to two options: a full returns policy and a no returns policy. Under a full returns policy, the manufacturer will give the retailer a full refund of the wholesale price for any quantity of unsold items. The alternative is a no returns policy, in which case the retailer must bear the risk of potential overstocking or understocking. The retailer then determines its price and order quantity. At the end of period t, the retailer knows at but the manufacturer does not. We assume that (1) all the agents are assumed to be risk neutral, (2) asymmetric information exists between the manufacturer and retailer with respect to the previous period’s primary demand, and (3) any unsold product at the end of the season bears no salvage value or disposal cost. Similarly, in the case of shortages, unsatisfied demand carries no additional penalty except for the products. Zero salvage value or holding cost and zero shortage penalty assumptions are appropriate reflections of reality. Note in the Stackelberg mode of game, one firm will act as the leader in the sense that it will announce its decision first. Then the other firm will follow and announce its own. The sequence of moves is shown in Fig. 1. 4. Discrete model With no returns policy, the retailer and manufacturer each have profit, as shown below: pR t ¼ E½wt qt þ p t minfqt ; Dt g;
ð1Þ
pM t ¼ E½ðwt cÞqt ;
M where pR t and pt are the retailer’s expected profit and the manufacturer’s expected profit, respectively, at period t. c is the manufacturer’s production unit cost, and wt is manufacturer’s wholesale price at period t. We assume that at can only be two values: aH and aL. aH denotes high primary demand, while aL denotes low primary demand. Let k denote the probability of primary demand at the period t is high (low), given that primary demand at the period t 1 is high (low). Specifically, we assume that P(at = aH j at1 = aH) = P(at = aL j at1 = aL) = k, and P(at = aH j at1 = aL) = P(at = aL j at1 = aH) = 1 k. We symbol the relationship in a matrix format shown as below:
at1 n at aH
aH k
aL
1k k
aL 1k
We assume k P 0.5, and all the values of aH, aL, k, b are common knowledge to both the manufacturer and the retailer. The demand uncertainty is reflected in the sense that no one knows at in the beginning of period t. However, both supply chain players know that at can be either aH or aL.
Stage 1
Stage 2
Stage 3
Manufacturer sets its wholesale price wt
Retailer sets its price pt and orders quantity qt
Retailer receives the order from the manufacturer, and the demand is fulfilled Fig. 1. Sequence of events and decisions.
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Thus, Dt ¼ at bpt ; at 2 faH ; aL g;
aH > aL :
Our model is somewhat related to Padmanabhan and Png’s (1997) model. There are three major differences between ours and Padmanabhan and Png’s (1997). First, Padmanabhan and Png (1997) assume that all parties have equal and full information about demand at the time of setting the retail price. We assume that demand information is asymmetric between the supply chain players. Second, Padmanabhan and Png (1997) assume that the state of demand is revealed before the retailer has set the retail price. In our model, we modify this assumption by assuming that the retailer has set the retail price before the demand uncertainty is resolved. This reflects a common situation in practice also. Third, Padmanabhan and Png (1997) only address the full returns policy’s impact on the manufacturer. We analyze the full returns policy’s impact on both the manufacturer and the retailer. 4.1. No returns policy In this scenario, the retailer and the manufacturer maximize pR t ¼ E ½p t minðDt ; qt Þ wt qt pM t
s:t: aL bpt 6 qt 6 aH bpt ;
¼ E½ðwt cÞqt ;
respectively. We show that the following results hold. Remark 1. With a no returns policy, pR t , the retailer’s profit function is convex in qt. Thus, qt should be one of the two values; aL bpt or aH bpt (proof is in Appendix A).
Since it is analytically difficult to obtain the retailer’s optimal order quantity at each period, and since the main focus of this study is to analyze the full returns policy’s informational role in learning the primary demand, we simplify the case and assume that the retailer has two possible heuristic order policies (Policies 1 and 2). Although these heuristic policies are not guaranteed to be optimal all the time, they are at least good and reasonable policies. Policy 1. When at1 = aH, the retailer’s order quantity is qt = aH bpt; when at1 = aL, the retailer’s order quantity is qt = aL bpt. (Note: Policy 1 is reasonable. This is because we assume k P 0.5. Thus, if at1 = aH, there is higher chance that the optimal order quantity is aH bpt instead of aL bpt. By the same reasoning, if at1 = aL, there is higher chance that the optimal order quantity is aL bpt instead of aH bpt.) Policy 2. When at1 = aH, the retailer’s order quantity is qt = aL bpt; when at1 = aL, the retailer’s order quantity is qt = aL bpt. (Note: Policy 2 is a type of conservative order policy, which is to order the minimum amount so that all the products can be sold out.) Evidently, the manufacturer is able to deduce which policy the retailer is using by tracking the retailer’s order history. Observation 1.1. For the no returns policy case under Policy 1, at the beginning of period t, manufacturer can infer at2 from qt1. Proof. If at2 = aH, the retailer would order qt1 = aH bpt1 at the period t 1. Likewise, if at2 = aL, the retailer would order qt1 = aL bpt1 at the period t 1. Knowing qt1 and pt1 at the beginning the period t, the manufacturer can figure out the value of at2. h Observation 1.2. For the no returns policy case under Policy 2, at the beginning of period t, the manufacturer cannot infer at2 from qt1.
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Proof. Whatever the value of at2, the retailer’s order is qt1 = aL bpt1 at the period t 1. Thus, the manufacturer cannot infer at2 from qt1. Also based on the subsequent proof of Result 2, wt has the same value regardless of at2 = aH or at2 = aL. Thus, the manufacturer actually does not need to know at2 when making decisions. h Observation 2. For the full returns policy case under Policy 1, at the beginning of period t, manufacturer can infer at1. Proof. The manufacturer knows Dt1 exactly, as Dt1 = qt1 – returned quantity at the period t. As Dt1 = at1 bPt1, the manufacturer can infer at1. h The above observations show that the demand information is asymmetric between the two supply chain players; that is, in the no returns policy case, at the beginning of period t, while the retailer knows at1, the manufacturer only knows at2. Therefore, because it lacks the most recent demand information, the manufacturer has incentive to obtain this piece of information and incorporate it into its decision making process. A full returns policy would allow the manufacturer to learn about this information. Note that this is a major difference between our study and prior studies. Prior studies usually assume that both players have equal information about the demand. Tables 1–3 summarize the results for all different scenarios. All proofs are in Appendix A. Result 1. Table 1 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with a no returns policy under Policy 1. Result 2. Table 2 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with a no returns policy under Policy 2. 4.2. Full returns policy In a full returns policy case, since the manufacturer accepts returns, the retailer will always order the maximum quantity needed to fulfill demand. Thus, the retailer’s and the manufacturer’s expected profit functions are as follows:
Table 1 Results under Policy 1 (no returns) Variable
Value
wt
kð2kÞaH þð1kÞ2 aL þbc 2b
(if at2 = aH)
k2 aL þð1k2 ÞaH þbc 2b
(if at2 = aL)
pt
qt
aL ð3kÞð1kÞþaH ð4kÞkþbc 4b aL ½3ð2kÞkþaH ð2kÞkþbc 4b aL ½2kð2kÞþaH ½1þð2kÞkþbc 4b aL ð2þk2 ÞþaH ð1k2 Þþbc 4b H ð4kÞkbc aH þ aL ð3kÞð1kÞa 4 H ð2kÞkbc aL þ aL ½3ð2kÞka 4 aL ½2kð2kÞþaH ½3ð2kÞkbc 4 aL ð2k2 ÞaH ð1k2 Þbc 4
(if at2 = aH and at1 = aH) (if at2 = aH and at1 = aL) (if at2 = aL and at1 = aH) (if at2 = aL and at1 = aL) (if at2 = aH and at1 = aH) (if at2 = aH and at1 = aL) (if at2 = aL and at1 = aH) (if at2 = aL and at1 = aL)
pR t
7a2H þ4aH aL þ5a2L bcð6aH 2aL Þþ2b2 c2 þðaH aL Þk½4ð3aH aL þ2bcÞ2kð3aH 5aL þ2bcÞþ2k3 ðaH aL Þ 32b
pM t
2b2 c2 2aL bcþa2L ½12ð1kÞkð2k2 Þþ2aH ½bcð2þkÞaL ð1kÞð14kþ2k3 Þþa2H ½3þ2kð3þkð2þkk2 ÞÞ 16b
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Table 2 Results under Policy 2 (no returns) Variable
Value
wt
aL þbc 2b 3aL þbc 4b aL bc 4b ðaL bcÞ2 16b ðaL bcÞ2 8b
pt qt pR t pM t
pR t ¼ E ½ðp t wt ÞDt ; pM t ¼ E ½wt Dt cðaH bp t Þ: Result 3. Table 3 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer in the full returns policy case. From the above three results, we can see that prices wt and pt are increasing functions of aH and/or aL. This implies that the higher the primary demand, the more optimistic the player so that he would set a higher price. It is also observed in Table 2 that k has nothing to do with the optimal decision variables and expected profits. The reason is intuitively clear: the demand correlation information does not play any role in decision making when Policy 2 is used. 4.3. Comparison between the no returns policy and the full returns policy Now we would like to assess the impact of the full returns policy on each player’s performance. The value of the full returns policy for the manufacturer and the retailer is easily determined from the profit expressions given in Tables 1–3. Before all the results of comparison are presented, it is noted that full returns policy affects the retailer’s profit and the manufacturer’s profit in the following three ways: (1) The returns policy can reduce the chance of stock-out for both the manufacturer and the retailer by encouraging the retailer to order more stock, which is beneficial to both the manufacturer and the retailer. We term this effect as the first effect. (2) The full returns policy shifts the burden of demand uncertainty from the retailer to the manufacturer, which is obviously desirable for the retailer, but is undesirable for the manufacturer. We term this effect as the second effect. Table 3 Results under the full returns policy Variable
Value
wt
kaH þð1kÞaL þbc 2b kaL þð1kÞaH þbc 2b 3ð1kÞaL þ3kaH þbc 4b 3kaL þ3ð1kÞaH þbc 4b aL ½3ð1kÞþaH ð43kÞbc 4 aL ð3kÞþaH ð3kþ1Þbc 4 ½bcaL ð1kÞaH k2 þ½bcaH ð1kÞaL k2 32b a2H þa2L 10aH bcþ6aL bcþ2b2 c2 2ðaH aL Þ2 kþ2ðaH aL Þ2 k2 16b
pt
qt pR t pM t
(if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL)
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(3) The full returns policy enables the retailer to indirectly disclose the previous period’s demand information to the manufacturer, thus creating an opportunity for the manufacturer to use this information to set a high wholesale price, which is desirable for the manufacturer, but undesirable for the retailer. We term this effect as the third effect. Both the retailer and the manufacturer face the tradeoff of these three effects when the full returns policy is adopted. We prove the following two results (proofs are omitted, as they are simply based on the comparisons of the expected profits in Tables 1–3). Result 4. If the full returns policy is adopted to replace Policy 1 (no returns), we can prove: (a) The retailer’s profit is greater with a full returns policy than with a no returns policy. However, the manufacturer’s profit is less with a full returns policy than with a no returns policy. This is also true even when c = 0. (b) The total supply chain profit is greater with a full returns policy than with a no returns policy if and only if aH ð1 kÞð2 k þ k2 k3 Þ þ aL ð2 kð1 þ kð4 2k þ k2 ÞÞÞ 2bcð2 þ 3k k2 Þ P 0: (c) The difference in the retailer’s profit between the full returns policy and no returns policy (Policy 1) is decreasing in k, and the difference in the manufacturer’s profit between the no returns policy (Policy 1) and the full returns policy is decreasing in k. Result 4 shows that the retailer is better off and the manufacturer is worse off when Policy 1 is replaced with the full returns policy. This is to say that, for the retailer, the positive side (the first and second effect) dominates the downside (the third effect) when the full returns policy is adopted. However, an opposite phenomenon occurs for the manufacturer. The downside dominates the positive side. This is even true even when c (production cost) = 0. This result is largely attributed to our assumption that there is no salvage value in overstocking. Thus, the value of the returns policy could be underestimated. Under the conditions in (b), the retailer has the incentive and capability to have the manufacturer adopt the full returns policy and to compensate the manufacturer for any resultant losses. However, when k increases, both the manufacturer’s and the retailer’s profits under the full returns policy are approaching their counterparts under Policy 1 (see Fig. 2). This is because the third effect becomes more valuable for the manufacturer, and the manufacturer therefore strategically makes better use of the information against the retailer. This is to say that the full returns policy becomes increasingly more attractive to the manufacturer and the no returns policy (Policy 1) becomes increasingly less attractive to the retailer when k
The Retailer’s Profit ( aH = 200 , a L=150 , β = 4 , c =1)
The Manufacturer’s Profit ( aH = 200 , aL=150 , β= 4 , c =1 ) 1050
400
Under a no returns policy
1025
Under a full returns policy 1000
300
975 200
950
100
925
Under a no returns policy
Under a full returns policy
900 0 0.5
0.6
0.7
λ
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
λ
Fig. 2. k versus the retailer’s and the manufacturer’s profits under the full returns policy and the no returns policy (Policy 1).
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increases. This result makes sense, as the retailer’s information becomes more valuable with the increase of k, the indicator of demand correlation between the present and previous periods. Result 5. If the full returns policy is adopted to replace Policy 2 (no returns): (a) The retailer’s profit is greater with a full returns policy than with a no returns policy. The manufacturer’s profit can be greater or smaller with a full returns policy than with a no returns policy. However, if c = 0, both the retailer and the manufacturer’s profits are greater with a full returns policy than with a no returns policy. (b) The total supply chain profit will increase if and only if aH ð3 6ð1 kÞkÞ þ aL ð3 þ 6ð1 kÞkÞ 22bc P 0: (c) The differences in both the retailer’s and the manufacturer’s profits between the full returns policy and no returns policy are increasing in k. Similar to Result 4, Result 5 shows that if the manufacturer changes to adopt the full returns policy, the retailer’s profit would increase; however, the manufacturer could be better off or worse off (depending on ð1kÞkþaH ð12ð1kÞkÞ the sign of ðaH aL Þ½aL 10bcþ2aL16b ). This implies that the manufacturer should adopt the full returns policy if the sign is positive. If the condition in (b) is satisfied, the retailer has the incentive and capability to have the manufacturer adopt the full returns policy and compensate the manufacturer for any resultant losses. When k increases, the manufacturer’s profit under the full returns policy increases faster than under Policy 2 (see Fig. 3). This is expected as explained in Result 4 (the information indirectly shared by retailer due to the returns policy becomes more valuable to the manufacturer). However, when k increases, the gap between the retailer’s profit under the full returns policy and under the no returns policy (Policy 2) widens. This result is in contrast with what we have in Result 4. It implies that the full returns policy becomes more attractive than the no returns policy (Policy 2) for both the manufacturer and the retailer when k increases. 4.4. Information sharing under the no returns policy In order to seek out and assess the value of information relayed through the returns policy, we will consider the situation in which the retailer voluntarily shares the information of the previous period (at1) with the manufacturer. In this case, intuitively, the manufacturer would benefit from such information sharing, as the manufacturer could incorporate the information into its forecasting and planning process. Again, this is a Bayesian Stackelberg game. Similar to what we did in the no returns policy and full returns policy analysis, we derive the optimal decisions shown in Tables 4 and 5.
The Retailer’s Profit ( a H = 200 , aL=150 , β = 4 , c =1)
The Manufacturer’s Profit ( a H = 200 , aL=150 , β = 4 , c =1)
500 900
450 850
400
Under a full returns policy
Under a full returns policy 800
350
Under a no returns policy
750
300
Under a no returns policy
250
0.5
0.6
0.7
0.8
λ
0.9
700
1
0.5
0.6
0.7
0.8
0.9
1
λ
Fig. 3. k versus the retailer’s and the manufacturer’s profits under the full returns policy and the no returns policy (Policy 1).
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Table 4 Information sharing under Policy 1 Variable
Value
wt
aH ð2kÞaL ð1kÞþbc 2b aL þbc 2b ð1kÞaL þð2þkÞaH þbc 4b 3aL þbc 4b aL ð1kÞþaH ð2kÞbc 4 aL bc 4 2 2ð6aH þ6aH aL þa2L 2aH bcþb2 c2 Þþ2ðaH aL Þð6aH þaL þbcÞkþðaH aL Þ2 k2 32b ðaL bcÞ2 þ½aL þbcaH ð2kÞaL k2 16b
pt qt pR t pM t
(if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL)
Table 5 Information sharing under Policy 2 (no returns) Variable
Value
wt
aL þbc 2b 3aL þbc 4b aL bc 4 ðaL bcÞ2 16b ðaL bcÞ2 8b
pt qt pR pM
Result 6. Table 4 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with information sharing under Policy 1 (no returns). Result 7. Table 5 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with information sharing under Policy 2 (no returns). Note: The result is the same as the result in Table 2. 4.5. Comparison between no information sharing and information sharing (under the no returns policy) Based on the above two tables, we can assess information sharing’s impact on each supply chain player. Thus, we have the following two results (proofs are omitted). Result 8. For the scenario, the retailer uses the Policy 1. (a) The manufacturer’s profit is greater with information sharing than without information sharing. However, the retailer’s profit can be greater or smaller (depending on the sign of {2bc(1 2k) aH(1 + k)(5 2k2) + aL[3 + k(9 2k(1 + k))]}) with information sharing than without information sharing. (b) The total supply chain profit is greater with information sharing than without information sharing if and only if ð1 þ kÞð2bcð1 þ 2kÞ þ aH ð1 þ kÞð3 þ 2ð2 þ kÞkÞ aL ð1 þ kð3 þ 2ð1 þ kÞkÞÞÞ P 0: (c) The difference in the retailer’s profit between the scenarios of information sharing and no information sharing can be increasing or decreasing in k; however, the difference in the manufacturer’s profit is decreasing in k. The positive value of information sharing to the manufacturer is intuitive because the retailer’s information contributes to increasing the accuracy of the manufacturer’s demand forecast. On the other hand, the retailer
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might suffer from information sharing. This is because the manufacturer could use the information strategically to maximize its own profit at the expense of the retailer. Specifically, the manufacturer may set a higher wholesale price, which, we know, is undesirable to the retailer. Result 8 also implies that the retailer is willing to enter into an ex ante information-sharing contract with the manufacturer if and only if the sign of the expression in (a) is positive. This represents a win–win situation for both the manufacturer and the retailer. With the condition in (b) satisfied, the manufacturer has the incentive and capability to induce the retailer to share his information and compensate the retailer for any resultant losses. We also observe that when k increases, the value of information sharing for the manufacturer decreases. This is due to the effect of diminishing return of margin, so that the value of the retailer’s information decreases. Result 9. In the case when the retailer uses Policy 2, the profit difference between the information sharing and no information sharing is 0. This result is intuitive, as information sharing has no effect on the manufacturer and the retailer under the Policy 2. 5. Continuous model In this section, we assume that at is a type of non-standard auto-correlated AR(1) process. Specifically, we let at = d + qat1 + et, d > 0, 1 < q < 1. et is a random variable and uniformly distributed on the range [B, B]. To be different from the standard AR(1) process where et is normally distributed, we call it uniform AR(1) process. 5.1. No returns policy A convenient expression for profit is obtained by substituting Dt = yt + et, and, consistent with Petruzzi and Dada (1999), by defining zt = qt yt and yt = d + qat1 bPt: ! P t ðy t þ et Þ wt ðy t þ zt Þ if et 6 zt pR ¼ : ð2Þ P t ðy t þ zt Þ wt ðy t þ zt Þ if et > zt This transformation of variables provides an alternative interpretation of the order decision: if the choice of zt is larger than the realized value of et, then leftovers occur; if the choice of zt is smaller than the realized value of et, then shortages occur. The corresponding optimal stocking and pricing policy is to order qt = yt + zt units to sell at the price pt, where zt and pt maximize expected profit. The expected profits are: Z zt Z B pR ðz Þ ¼ p ½y þ e f ðe Þ de þ pt ½y t þ zt f ðet Þ det wt ðy t þ zt Þ; t t t t t t t B
pM t
zt
¼ ðwt cÞqt :
Since the manufacturer is the Stackelberg leader and the retailer is the follower, we derive the optimal solution by deriving optimal pt and qt followed by optimal wt. Therefore, we first consider the retailer’s expected profit, pR t ðzt Þ: opR t ðzt Þ ¼ wt þ pt ½1 F ðzt Þ ¼ 0; ozt opR t ðzt Þ ¼ 2bðP 0 pt Þ hðzt Þ ¼ 0; oP t RB þbwt where, hðzt Þ ¼ zt ðet zt Þf ðet Þ det , P 0 ¼ dþqat1 . 2b
ð3Þ ð4Þ
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Thus, Eqs. (3) and (4) are solved first, we have hðzt Þ ; 2b wt ¼ pt ð1 F ðzt ÞÞ:
pt ¼ P 0
ð5Þ ð6Þ
Thus, qt ¼ zt þ y t ¼ zt þ d þ qat1 bpt : Then, we plug (7) into equation
pM t
ð7Þ
opM ¼ ðwt cÞqt , solving equation t ¼ 0, then we have owt
3
ðB zt ÞP 0
ðB zt Þ ¼ 2Bwt : 8bB
ð8Þ
By solving (5)–(8), we can obtain the optimal zt, wt, pt and qt. 5.2. The full returns policy case In this case, the manufacturer sets a wholesale price wt, and gives the retailer a full refund for unsold stock. Since the manufacturer accepts returns, the retailer always orders the maximum quantity, qH aR bpt + B, where aR = d + qat1. Therefore, pR t ¼ E½ðp t wt ÞDt ; pM t ¼ E½wt Dt cqH : Thus, pR t ¼ ðp t wt Þ pM t
¼
Z
Z
B
ðy t þ et Þf ðet Þ det ¼ ðpt wt ÞðaR bpt Þ;
ð9Þ
B
B
wt ðy t þ et Þf ðet Þ det cðaR bpt þ BÞ ¼ ðwt cÞðaR bpt Þ cB:
ð10Þ
B
opR aR þ bwt t . ¼ 0, thus P t ¼ opt 2b aR þ bwt opM aR þ bc aR bc þ 4B and retailer’s order quantity qH ¼ . into t ¼ 0, we have wt ¼ Putting pt ¼ 2b 4 2b owt Observation 3. For the no returns policy case, at the beginning of period t, the manufacturer can infer at2 from qt1. We first solve
Proof. In Eq. (7): qt1 ¼ zt1 þ d þ qat2 bP t1 , as the manufacturer knows qt1 and P t1 . He can also learn qt1 from Eq. (6). Therefore, the manufacturer can infer at2. h Observation 4. For the full returns policy case, at the beginning of period t, the manufacturer can infer at1. Proof. The manufacturer knows pt1 and Dt1 exactly (because Dt1 = qt1 – returned quantity at the period t). As Dt1 = at1 bpt1, manufacturer can infer at1. h 5.3. A numerical example Now we present a numerical example to illustrate the returns policy’s impact on supply chain performance for the continuous case. Specifically, we will show the magnitude of the profits as a function of the demand parameters (namely, q, B), and c (the production cost). Note that we assume that et is uniformly distributed among the range [B, B].
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For our simulations, we use the following parameter values: d = 50, b = 1, q = {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}, B = {2, 4, 6, 8, 10}, and c = {0, 1, 5, 10, 15, 18}. For every value of the specific parameter analyzed, we average the profits over all other parameters’ various combinations. 5.3.1. Impact of q Figs. 4 and 5 show that, regardless of whether a returns policy is used or not, both the manufacturer’s and the retailer’s profits increase with q. This is because the higher q, the higher correlation between the present and previous demands; thus the manufacturer or the retailer could better forecast the present demand based on the previous demand, which leads to higher profits for both players. For the manufacturer’s side, when q is small, the manufacturer’s profit under a no returns policy is greater than under a full returns policy. However, when q goes over some value, the full returns policy begins to prevail. The reason behind this phenomenon rests in the tradeoff between the manufacturer’s burden of taking back the unsold products and the effect of information sharing. A full returns policy benefits the manufacturer by allowing the manufacturer to obtain the previous demand information at the expense of bearing the burden of demand uncertainty. When q is small, the effect of taking the burden dominates the effect of information sharing. As q increases, the effect of information sharing begins to dominate. On the retailer’s side, the retailer’s profit under a no returns policy dominates the profit under a full returns policy all the time. This is because information sharing allows the manufacturer to strategically exploit the
8000 With no return policy
Manufacturer Profit
7000
With full return policy
6000 5000 4000 3000 2000 1000 0 0
0.2
0.4
ρ
0.6
0.8
Fig. 4. q versus the manufacturer’s profits.
8000 With no return policy
Retailer Profit
7000
With full return policy
6000 5000 4000 3000 2000 1000 0 0
0.2
0.4
ρ
0.6
Fig. 5. q versus the retailer’s profits.
0.8
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643
Manufacturer Profit
300 250 200 150 100
With no return policy
50
With full return policy
0 2
4
6
8
10
B Fig. 6. B versus the manufacturer’s profits.
250
Retailer Profit
200 150 100 With no return policy 50
With full return policy
0 2
4
6
8
10
B Fig. 7. B versus the retailer’s profits.
information to maximize its own profit at the expense of the retailer. This effect dominates the effect of the retailer’s shifting burden of unsold products to the manufacturer for our given set of data. 5.3.2. Impact of B Figs. 6 and 7 show the impact of B on the retailer’s and manufacturer’s profits, respectively. For the manufacturer, with and without the returns policy, profits go down in B. However, when B is small, the manufacturer’s profit under the full returns policy is higher than under the no returns policy. When B increases, the full returns policy begins to prevail. This is because a higher B implies a higher demand uncertainty, which has a negative affect on the manufacturer. The same reasoning applies for the retailer under the no returns policy. For the case of a full returns policy, the manufacturer suffers not only the demand uncertainty’s negative impact on pricing, but also the increasing risk of having higher production cost. Consequently, its profit decreases faster than the profit under a no returns policy. In contrast, the retailer’s profit under the full returns policy remains almost identical (level) all the time when B increases. This is because the retailer has shifted the burden of demand uncertainty to the manufacturer, so B has no impact on its profit. We also notice that the no returns policy’s profit dominates the full returns policy’s profit. This indicates that given the set of data, for the retailer, the effect of information sharing dominates the effect of shifting the burden of demand uncertainty to the manufacturer. 5.3.3. Impact of c Figs. 8 and 9 show the impacts of c. As expected, both the manufacturer’s and the retailer’s profits, regardless of whether a full returns policy is used or not, decrease with c. This is because when c goes up, both the manufacturer and the retailer realize smaller profit margins and smaller demand, thus resulting in a drop in
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Manufacturer Profit
644
450 400 350 300 250 200 150 100 50 0
With no return policy With full return policy 0
5
10
15
c Fig. 8. c versus the manufacturer’s profits.
350
Retailer Profit
300 250 200 150 100
With no return policy
50
With full return policy
0 0
5
10
15
c Fig. 9. c versus the manufacturer’s profits.
profits. The manufacturer’s profit under the full returns policy is greater than that under a no returns policy when c is small. When c increases, the no returns policy begins to dominate the full returns policy. This is expected again as the manufacturer bears the burden of taking unsold products in the full returns policy case, and the effect of the burden becomes larger and dominates the effect of information sharing when c increases. On the other hand, the retailer’s profit under the full returns policy decreases at a slower rate than under the no returns policy. This is because the retailer shifts the burden of demand uncertainty to the manufacturer. Therefore, the retailer experiences a smaller decrease in profit in the case of full returns policy. 6. Conclusion The returns policy literature often assumes (either implicitly or explicitly) that the manufacturer and the retailer have equal demand information. While possibly appropriate in some settings, information symmetry is not very solid assumption. In reality, the manufacture and the retailer usually have different perspectives on demand information because of different data sourcing and methodology used. Therefore, in this study, we analyze the impact of full returns policy as well as information sharing on the manufacturer and the retailer under information assymmetry. We first study the case of the base level of the demand following a discrete distribution with two states. We find that while the retailer always benefits from the full returns policy, the manufacturer would lose when the full returns policy is adopted by the retailer to replace Policy 1 (no returns); and may gain or lose when the full returns policy is adopted to replace Policy 2 (no returns). We also find out the conditions under which the total supply chain profit is higher with a full returns policy than with a no returns policy. We then address the case of the base level of the demand following a continuous distribution (uniform AR(1) process), and provide a numerical example to illustrate the returns policy’s impact on supply chain performance.
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Appendix A Proof for Remark 1. After some algebra, we can prove
o2 pR t > 0. o2 qt
h
Proof for Result 1 pM t ¼ ðwt cÞE½qt : Suppose at2 = aH: Based on results in Appendix B, 2aH aH bwt aL bwt E½qt ¼ k þ ð1 kÞ ; 2 2 2 kð2 kÞaH þ ð1 kÞ aL þ bc by solving the first order condition for the manufactrer’s maxWe can get wt ¼ 2b imization problem. Similarly, the case of at2 = aL can be analyzed. h Proof of Result 2. The proof of procedure is similar to that of result 1. h Proof of Result 3. Full Returns Policy Case 1: Suppose at1 = aH pR t ¼ ðp t wt ÞðaL bpÞð1 kÞ þ ðpt wt ÞðaH bpÞk; aH k þ ð1 kÞaL þ bwt by solving the first order condition for the retailer’s maximization We can obtain pt ¼ 2b problem. pM t ¼ wt ½ðaH bpt Þk þ ð1 kÞðaL bp t Þ cðaH bp t Þ; By taking the FOC and setting it to zero, we can get wt ¼
aL ð1 kÞ þ aH k þ bc : 2b
In the similar way, the case at1 = aL can be analyzed. Then, after some algebra, we can get the results. Proof of Result 6 Case 1: Suppose at1 = aH pR t ¼ kðaH bp t Þp t þ ð1 kÞðaL bp t Þp t ðaH bp t Þwt ; opR t ¼ kðaH 2bpt Þ þ ð1 kÞðaL 2bpt Þ þ bwt ¼ 0: opt Thus, kaH þ ð1 kÞaL þ bwt ; 2b qt ¼ aH bpt ; pt ¼
pM t ¼ E½ðwt cÞqt ; opM t ¼ 0; owt aH ð2 kÞ aL ð1 kÞ þ bc : wt ¼ 2b In the similar way, the case (at1 = aL) can be analyzed.
h
h
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Proof of Result 7. The proof procedure is similar to that of Result 4, and the result is exactly the same as the one in Table 2. h Appendix B Case 1: at1 = aH. Case 1.1: qt = aL bpt (Policy 2) pR t ¼ ðaL bp t Þðpt wt Þ; Taking FOC (first order condition) and setting it to zero, we get aL þ bw ; 2b aL bw : qt ¼ 2 Case 1.2: qt = aH bpt (Policy 1) pt ¼
pR t ¼ kðaH bp t Þp t þ ð1 kÞðaL bp t Þp t ðaH bp t Þwt ; Taking FOC and setting it to zero, we get pt ¼
kaH þ ð1 kÞaL þ bwt : 2b
Let aH ¼ kaH þ ð1 kÞaL : Thus, 2aH aH bwt : 2 Case 2: Suppose at1 = aL. If qt = aL bpt (Policies 1 and 2) qt ¼
ðB:1Þ
aL þ bw ; 2b aL bw : qt ¼ 2 pt ¼
References Bourland, K.E., Powell, S.G., Pyke, D.F., 1996. Exploiting timely demand information to reduce inventories. European Journal of Operational Research 92, 239–253. Cachon, G., Fisher, M., 2000. Supply chain inventory management and the value of shared information. Management Science 46 (8), 1032–1048. Chen, F., Drezner, Z., Ryan, J., Simchi-Levi, D., 2000. Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times and information. Management Science 46 (3), 436–443. Emmons, H., Gilbert, S., 1998. The role of returns policies in pricing and inventory decisions for catalogue goods. Management Science 44 (2), 276–283. Gallego, G., Huang, Y., Leung, Y., 2000. When to share demand information in a simple supply chain. Working paper, Department of Industrial Engineering and Operations Research, Columbia University. Gavirneni, S., Kapuscinski, R., Tayur, S., 1999. Value of information in capacitated supply chains. Management Science 45 (1), 16– 24. Glenn, David, Puterman, Martin, 2001. Returns offers for less optimistic retailers. Working paper, Faculty of Commerce and Business Administration, University of British Columbia. Kandel, Eugene, 1996. The right to return. Journal of Law and Economics XXXIX (April), 329–356.
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Lee, Hau, Padmanabhan, V., Taylor, T.A., Whang, S., 2000a. Price protection in the personal computer industry. Management Science 46 (4), 467–482. Lee, H., So, V.K., Tang, C., 2000b. The value of information sharing in a two-level supply chain. Management Science 46 (5), 626–643. Padmanabhan, V., Png, I.P.L., 1997. Manufacturer’s returns policies and retail competition. Marketing Science 16 (4), 81–94. Pasternack, B., 1985. Optimal pricing and return policies for perishable commodities. Marketing Science 4 (2), 166–176. Petruzzi, N., Dada, M., 1999. Pricing and the newsvendor problem: A review with extensions. Operations Research 47 (2), 183–193. Sarvary, M., Padmanabhan, V., 2001. The information role of manufacturer returns policies: How they can help in learning the demand. Marketing Letters 12 (41), 341–350. Taylor, T., 2001. Channel coordination under price protection, midlife returns, and end-of-life returns in dynamic markets. Management Science 47 (9), 1220–1234. Webster, Scott, Weng, Z. Kevin, 2000. A risk-free perishable item returns policy. Manufacturing & Service Operations Management 2 (1), 100–106.
Production, Manufacturing and Logistics
The impacts of the full returns policy on a supply chain with information asymmetry Xiaohang Yue a
a,*
, Srinivasan Raghunathan
b
School of Business Administration, The University of Wisconsin at Milwaukee, Milwaukee, WI 53201, United States b School of Management, The University of Texas at Dallas, Richardson, TX 75083, United States Received 19 January 2004; accepted 3 April 2006 Available online 13 July 2006
Abstract In contrast to the existing return policies literature assuming that information is symmetrical between the manufacturer and the retailer, we study the full returns policy’s impact on supply chains with information asymmetry. We first study the case that the base level of the demand follows a discrete distribution with two states. We find that the retailer benefits from the full returns policy in all circumstances, while the manufacturer and the supply chain are better off under some conditions. We then consider the situation in which the base level of the demand is a type of AR(1) process. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Returns policy; Information asymmetry; Information sharing; Supply chains; Decisions under uncertainty
1. Introduction A manufacturer’s returns policy is a common feature in the distribution of many products, such as books, CDs, and fashion apparel. An obvious explanation for a returns policy is risk-sharing; that is, the retailer returns the unsold products to the manufacturer or the manufacturer offers a credit on all unsold products to the retailer. A returns policy is an instrument for channel coordination (i.e., for arriving at the maximum efficiency for the total supply chain). There has been substantial research on why manufacturers might accept returns from the retailers, including Pasternack (1985), Kandel (1996), Padmanabhan and Png (1997), Webster and Weng (2000), Glenn and Puterman (2001), Emmons and Gilbert (1998), and Lee et al. (2000a). Researchers have pointed out that a returns policy, in the form of full returns, partial returns, and consignment, may affect the manufacturer’s profit under some conditions in two ways. The returns policy shifts the burden of demand uncertainty from the retailer to the manufacturer, thus encouraging the retailer to increase stock. Increased stock intensifies retailer distribution, which benefits the manufacturer as the returns policy helps to overcome the double
*
Corresponding author. E-mail addresses: [email protected] (X. Yue), [email protected] (S. Raghunathan).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.04.032
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marginalization problem. However, a returns policy causes the retailer to order stock according to the high demand, which is excessive relative to an integrated channel. At some point, when uncertainty is sufficiently large, the manufacturer is worse off with a returns policy than with a no returns policy. Additionally, a returns policy has an impact on the supply chain players’ price structures, thus influencing their profits. Currently, most of the previous literature assumes that there is no asymmetric information between the manufacturer and the retailer. It means, at all times, the manufacturer and the retailer have equal information about retail demand for the product. Apparently, this assumption is usually not true. In reality, each supply chain player (the manufacturer or the retailer) has a different perspective on the uncertain demand. Typically, they use different data or methodologies to make their own respective forecasts. In this study, we will analyze the impact of a returns policy on both the manufacturer’s and retailer’s performance under the assumption that there exists asymmetric information between the two supply chain players. In this situation, a returns policy enables the manufacturer to learn the previous period’s demand information through the return of unsold products so that the manufacturer could incorporate the updated information into its forecast process for the next period’s demand. Therefore, we can see that while a returns policy passes the burden of demand uncertainty from the retailer to the manufacturer, it enables the manufacturer to obtain useful demand information at the same time. Naturally, questions may arise, such as, what is the overall effect of a returns policy or information sharing on both supply chain players’ profits? Under what conditions is the retailer or the manufacturer better or worse off? In this paper, we will address these questions. The rest of the paper is organized as follows: in Section 2, we review the related literature. In Section 3, we discuss the model framework. In Sections 4 and 5, we analyze the impact of the returns policy on the manufacturer and the retailer when the primary demand follows a discrete distribution with two states as well as follows a continuous distribution. We conclude the paper in Section 6. 2. Literature review There is considerable research on why manufacturers might accept returns from retailers. Almost all of the literature assumes that there is no information asymmetry between supply chain players (i.e., manufacturers share the same demand information with retailers). We review the research chronologically. Pasternack (1985) examines the pricing decision of a manufacturer of a product with a limited shelf life. He presents a model of buy-back as a means to overcome double marginalization, assuming that the supplier and retailer use the same demand distribution function. He proves that channel coordination can be achieved when the manufacturer offers partial credit for all unsold goods. He also shows that limiting returns to a fixed percentage of sales may be an optimal policy in a single-retailer setting but is suboptimal in a multi-retailer setting. Kandel (1996) extends Pasternack (1985) by modeling price sensitivity in end-consumer demand. Specifically, he presents a model of buy-back and consignment contracts and analyzes two extremes: either the supplier or the retailer dictates the wholesale price, the buy-back price and the order quantity. Emmons and Gilbert (1998) build on the work of Pasternack (1985) by analyzing a partial credit/full returns policy with the generalization that the probability distribution of end-customer demand depends on the retail price. Given uniformly distributed demand, they show that a rebate for leftover product increases the expected profit of the manufacturer and the retailer whenever the manufacturer’s price exceeds a threshold value. Padmanabhan and Png (1997) address returns policies from the manufacturer’s perspective. They study the effect of a full returns policy when demand is deterministic and uncertain. They demonstrate that a full returns policy can improve manufacturer profitability by intensifying retail price competition. Webster and Weng (2000) take the viewpoint of a manufacturer selling a short life-cycle product to a single risk-neutral retailer and describe returns policies that, when compared to no returns, satisfy two conditions: (1) the retailer’s expected profit is increased and (2) the manufacturer’s profit is at least as large as when no returns are allowed. They assume that the retailer’s price is exogenous, and derive conditions that a returns policy will result in higher expected manufacturer profit. Glenn and Puterman (2001) present two types of return policies: buy-back and consignment, by which the supplier induces the retailer to accept additional units of a product when the retailer is less optimistic about demand. In the buy-back returns policy, the
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wholesale price is assumed to be the same. They show that the consignment policy is more robust to the supplier’s uncertainty about the retailer’s ordering policy. Sarvary and Padmanabhan (2001) analyze the returns policy’s role in learning demand of a new product. They consider a dual monopoly with a manufacturer and a retailer by assuming that the retail price is fixed and demand is random but identical across periods. A variation of the returns policy is called price protection, which is a commonly used practice between manufacturers and retailers in the personal computer industry, motivated by drastic declines of product values during the product life cycle. It is a form of rebate given by the manufacturer to the retailer for units unsold at the retailer when the price drops during the product life cycle. Lee et al. (2000a) explore the use of price protection with a two-period model. They show that price protection is an instrument for channel coordination. For products with long manufacturing lead times, a properly chosen price protection credit coordinates the channel. For products with shorter manufacturing lead times, price protection alone cannot guarantee channel coordination when wholesale prices are exogenous. However, when the price protection credit is set endogenously together with the wholesale prices, channel coordination is restored. Taylor (2001) extends Lee et al. (2000a) by combing price protection, midlife returns, and end-of-life returns in declining price environments. He shows that if the wholesale prices and the return rebates are set properly, midlife and end-of-life returns achieve channel coordination. However, the manufacturer may be worse off as a result of coordination. If price protection is used in addition to returns and the terms are set properly, then both coordination and a win–win outcome are guaranteed. A related stream of research deals with information sharing within the supply chains. This literature discuss how a manufacturer can elicit information from retailers through inventory, lead time, and shortage allocation policies (Bourland et al., 1996; Gavirneni et al., 1999; Gallego et al., 2000; Cachon and Fisher, 2000). Bourland et al. (1996) show that information sharing offers significant benefits to the manufacturer and retailer when the degree to which their ordering cycles are out of phase is high. Gavirneni et al. (1999) study the holding and penalty cost of a finite capacity supplier facing demands from a single retailer following a (s, S) policy. By considering various types of demand distributions in their numerical experiments, they examine the conditions under which gaining information about the retailer’s inventory level is beneficial. Lee et al. (2000b) study the benefit of demand information sharing when the underlying demand process faced by the retailer is an AR(1) process. They assume that the manufacturer and the retailer incur linear holding and backlogging costs, experience constant lead times, and follow base-stock policies. Chen et al. (2000) quantify the bullwhip effect for a simple two-stage supply chain and demonstrate that centralizing demand information can significantly reduce the increase in variability. Cachon and Fisher (2000) study the value of sharing data in a model with one supplier, N identical retailers, and stationary stochastic consumer demand. They conclude that implementing information technology to accelerate and smooth the physical flow of goods through a supply chain is significantly more valuable than using information technology to expand the flow of information. Gallego et al. (2000) shows that delay base-stock policies can capture a significant portion of the benefits of demand information sharing. They compare the costs under fixed and random delay base-stock policies against the cost under demand information sharing. In this study, we will consider a supply chain consisting of a manufacturer and a retailer, where the demand information is asymmetrical between them. We will analyze the impacts of a returns policy as well as information sharing on the profit of the manufacturer and the retailer, respectively. 3. Model framework Consider a supply chain consisting of a manufacturer and a retailer. The retailer faces a random demand function and has the objective of determining its retail price and order quantity so as to maximize the expected profit. The manufacturer sets up its wholesale price and production quantity so as to maximize its expected profit. The demand function is assumed to be Dt = at bpt, where Dt is the demand at the period t, at is the base level primary demand at the period t, pt is the retailer’s price at period t, and b is the slope. All parameters are positive. To capture uncertainty in market demand resulting from changes in economic and business conditions, we assume that at is a random variable. The setting for this research is retailing of products with
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limited shelf lives, that is, the product is ‘‘perishable’’ (i.e., at the end of each period, all unsold units have zero salvage value). The manufacturer is assumed to move first by choosing wholesale price, wt, based on the available information about the retailer’s demand. The manufacturer behaves like a Stackelberg leader. For simplicity, we restrict the manufacturer to two options: a full returns policy and a no returns policy. Under a full returns policy, the manufacturer will give the retailer a full refund of the wholesale price for any quantity of unsold items. The alternative is a no returns policy, in which case the retailer must bear the risk of potential overstocking or understocking. The retailer then determines its price and order quantity. At the end of period t, the retailer knows at but the manufacturer does not. We assume that (1) all the agents are assumed to be risk neutral, (2) asymmetric information exists between the manufacturer and retailer with respect to the previous period’s primary demand, and (3) any unsold product at the end of the season bears no salvage value or disposal cost. Similarly, in the case of shortages, unsatisfied demand carries no additional penalty except for the products. Zero salvage value or holding cost and zero shortage penalty assumptions are appropriate reflections of reality. Note in the Stackelberg mode of game, one firm will act as the leader in the sense that it will announce its decision first. Then the other firm will follow and announce its own. The sequence of moves is shown in Fig. 1. 4. Discrete model With no returns policy, the retailer and manufacturer each have profit, as shown below: pR t ¼ E½wt qt þ p t minfqt ; Dt g;
ð1Þ
pM t ¼ E½ðwt cÞqt ;
M where pR t and pt are the retailer’s expected profit and the manufacturer’s expected profit, respectively, at period t. c is the manufacturer’s production unit cost, and wt is manufacturer’s wholesale price at period t. We assume that at can only be two values: aH and aL. aH denotes high primary demand, while aL denotes low primary demand. Let k denote the probability of primary demand at the period t is high (low), given that primary demand at the period t 1 is high (low). Specifically, we assume that P(at = aH j at1 = aH) = P(at = aL j at1 = aL) = k, and P(at = aH j at1 = aL) = P(at = aL j at1 = aH) = 1 k. We symbol the relationship in a matrix format shown as below:
at1 n at aH
aH k
aL
1k k
aL 1k
We assume k P 0.5, and all the values of aH, aL, k, b are common knowledge to both the manufacturer and the retailer. The demand uncertainty is reflected in the sense that no one knows at in the beginning of period t. However, both supply chain players know that at can be either aH or aL.
Stage 1
Stage 2
Stage 3
Manufacturer sets its wholesale price wt
Retailer sets its price pt and orders quantity qt
Retailer receives the order from the manufacturer, and the demand is fulfilled Fig. 1. Sequence of events and decisions.
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Thus, Dt ¼ at bpt ; at 2 faH ; aL g;
aH > aL :
Our model is somewhat related to Padmanabhan and Png’s (1997) model. There are three major differences between ours and Padmanabhan and Png’s (1997). First, Padmanabhan and Png (1997) assume that all parties have equal and full information about demand at the time of setting the retail price. We assume that demand information is asymmetric between the supply chain players. Second, Padmanabhan and Png (1997) assume that the state of demand is revealed before the retailer has set the retail price. In our model, we modify this assumption by assuming that the retailer has set the retail price before the demand uncertainty is resolved. This reflects a common situation in practice also. Third, Padmanabhan and Png (1997) only address the full returns policy’s impact on the manufacturer. We analyze the full returns policy’s impact on both the manufacturer and the retailer. 4.1. No returns policy In this scenario, the retailer and the manufacturer maximize pR t ¼ E ½p t minðDt ; qt Þ wt qt pM t
s:t: aL bpt 6 qt 6 aH bpt ;
¼ E½ðwt cÞqt ;
respectively. We show that the following results hold. Remark 1. With a no returns policy, pR t , the retailer’s profit function is convex in qt. Thus, qt should be one of the two values; aL bpt or aH bpt (proof is in Appendix A).
Since it is analytically difficult to obtain the retailer’s optimal order quantity at each period, and since the main focus of this study is to analyze the full returns policy’s informational role in learning the primary demand, we simplify the case and assume that the retailer has two possible heuristic order policies (Policies 1 and 2). Although these heuristic policies are not guaranteed to be optimal all the time, they are at least good and reasonable policies. Policy 1. When at1 = aH, the retailer’s order quantity is qt = aH bpt; when at1 = aL, the retailer’s order quantity is qt = aL bpt. (Note: Policy 1 is reasonable. This is because we assume k P 0.5. Thus, if at1 = aH, there is higher chance that the optimal order quantity is aH bpt instead of aL bpt. By the same reasoning, if at1 = aL, there is higher chance that the optimal order quantity is aL bpt instead of aH bpt.) Policy 2. When at1 = aH, the retailer’s order quantity is qt = aL bpt; when at1 = aL, the retailer’s order quantity is qt = aL bpt. (Note: Policy 2 is a type of conservative order policy, which is to order the minimum amount so that all the products can be sold out.) Evidently, the manufacturer is able to deduce which policy the retailer is using by tracking the retailer’s order history. Observation 1.1. For the no returns policy case under Policy 1, at the beginning of period t, manufacturer can infer at2 from qt1. Proof. If at2 = aH, the retailer would order qt1 = aH bpt1 at the period t 1. Likewise, if at2 = aL, the retailer would order qt1 = aL bpt1 at the period t 1. Knowing qt1 and pt1 at the beginning the period t, the manufacturer can figure out the value of at2. h Observation 1.2. For the no returns policy case under Policy 2, at the beginning of period t, the manufacturer cannot infer at2 from qt1.
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Proof. Whatever the value of at2, the retailer’s order is qt1 = aL bpt1 at the period t 1. Thus, the manufacturer cannot infer at2 from qt1. Also based on the subsequent proof of Result 2, wt has the same value regardless of at2 = aH or at2 = aL. Thus, the manufacturer actually does not need to know at2 when making decisions. h Observation 2. For the full returns policy case under Policy 1, at the beginning of period t, manufacturer can infer at1. Proof. The manufacturer knows Dt1 exactly, as Dt1 = qt1 – returned quantity at the period t. As Dt1 = at1 bPt1, the manufacturer can infer at1. h The above observations show that the demand information is asymmetric between the two supply chain players; that is, in the no returns policy case, at the beginning of period t, while the retailer knows at1, the manufacturer only knows at2. Therefore, because it lacks the most recent demand information, the manufacturer has incentive to obtain this piece of information and incorporate it into its decision making process. A full returns policy would allow the manufacturer to learn about this information. Note that this is a major difference between our study and prior studies. Prior studies usually assume that both players have equal information about the demand. Tables 1–3 summarize the results for all different scenarios. All proofs are in Appendix A. Result 1. Table 1 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with a no returns policy under Policy 1. Result 2. Table 2 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with a no returns policy under Policy 2. 4.2. Full returns policy In a full returns policy case, since the manufacturer accepts returns, the retailer will always order the maximum quantity needed to fulfill demand. Thus, the retailer’s and the manufacturer’s expected profit functions are as follows:
Table 1 Results under Policy 1 (no returns) Variable
Value
wt
kð2kÞaH þð1kÞ2 aL þbc 2b
(if at2 = aH)
k2 aL þð1k2 ÞaH þbc 2b
(if at2 = aL)
pt
qt
aL ð3kÞð1kÞþaH ð4kÞkþbc 4b aL ½3ð2kÞkþaH ð2kÞkþbc 4b aL ½2kð2kÞþaH ½1þð2kÞkþbc 4b aL ð2þk2 ÞþaH ð1k2 Þþbc 4b H ð4kÞkbc aH þ aL ð3kÞð1kÞa 4 H ð2kÞkbc aL þ aL ½3ð2kÞka 4 aL ½2kð2kÞþaH ½3ð2kÞkbc 4 aL ð2k2 ÞaH ð1k2 Þbc 4
(if at2 = aH and at1 = aH) (if at2 = aH and at1 = aL) (if at2 = aL and at1 = aH) (if at2 = aL and at1 = aL) (if at2 = aH and at1 = aH) (if at2 = aH and at1 = aL) (if at2 = aL and at1 = aH) (if at2 = aL and at1 = aL)
pR t
7a2H þ4aH aL þ5a2L bcð6aH 2aL Þþ2b2 c2 þðaH aL Þk½4ð3aH aL þ2bcÞ2kð3aH 5aL þ2bcÞþ2k3 ðaH aL Þ 32b
pM t
2b2 c2 2aL bcþa2L ½12ð1kÞkð2k2 Þþ2aH ½bcð2þkÞaL ð1kÞð14kþ2k3 Þþa2H ½3þ2kð3þkð2þkk2 ÞÞ 16b
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Table 2 Results under Policy 2 (no returns) Variable
Value
wt
aL þbc 2b 3aL þbc 4b aL bc 4b ðaL bcÞ2 16b ðaL bcÞ2 8b
pt qt pR t pM t
pR t ¼ E ½ðp t wt ÞDt ; pM t ¼ E ½wt Dt cðaH bp t Þ: Result 3. Table 3 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer in the full returns policy case. From the above three results, we can see that prices wt and pt are increasing functions of aH and/or aL. This implies that the higher the primary demand, the more optimistic the player so that he would set a higher price. It is also observed in Table 2 that k has nothing to do with the optimal decision variables and expected profits. The reason is intuitively clear: the demand correlation information does not play any role in decision making when Policy 2 is used. 4.3. Comparison between the no returns policy and the full returns policy Now we would like to assess the impact of the full returns policy on each player’s performance. The value of the full returns policy for the manufacturer and the retailer is easily determined from the profit expressions given in Tables 1–3. Before all the results of comparison are presented, it is noted that full returns policy affects the retailer’s profit and the manufacturer’s profit in the following three ways: (1) The returns policy can reduce the chance of stock-out for both the manufacturer and the retailer by encouraging the retailer to order more stock, which is beneficial to both the manufacturer and the retailer. We term this effect as the first effect. (2) The full returns policy shifts the burden of demand uncertainty from the retailer to the manufacturer, which is obviously desirable for the retailer, but is undesirable for the manufacturer. We term this effect as the second effect. Table 3 Results under the full returns policy Variable
Value
wt
kaH þð1kÞaL þbc 2b kaL þð1kÞaH þbc 2b 3ð1kÞaL þ3kaH þbc 4b 3kaL þ3ð1kÞaH þbc 4b aL ½3ð1kÞþaH ð43kÞbc 4 aL ð3kÞþaH ð3kþ1Þbc 4 ½bcaL ð1kÞaH k2 þ½bcaH ð1kÞaL k2 32b a2H þa2L 10aH bcþ6aL bcþ2b2 c2 2ðaH aL Þ2 kþ2ðaH aL Þ2 k2 16b
pt
qt pR t pM t
(if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL)
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(3) The full returns policy enables the retailer to indirectly disclose the previous period’s demand information to the manufacturer, thus creating an opportunity for the manufacturer to use this information to set a high wholesale price, which is desirable for the manufacturer, but undesirable for the retailer. We term this effect as the third effect. Both the retailer and the manufacturer face the tradeoff of these three effects when the full returns policy is adopted. We prove the following two results (proofs are omitted, as they are simply based on the comparisons of the expected profits in Tables 1–3). Result 4. If the full returns policy is adopted to replace Policy 1 (no returns), we can prove: (a) The retailer’s profit is greater with a full returns policy than with a no returns policy. However, the manufacturer’s profit is less with a full returns policy than with a no returns policy. This is also true even when c = 0. (b) The total supply chain profit is greater with a full returns policy than with a no returns policy if and only if aH ð1 kÞð2 k þ k2 k3 Þ þ aL ð2 kð1 þ kð4 2k þ k2 ÞÞÞ 2bcð2 þ 3k k2 Þ P 0: (c) The difference in the retailer’s profit between the full returns policy and no returns policy (Policy 1) is decreasing in k, and the difference in the manufacturer’s profit between the no returns policy (Policy 1) and the full returns policy is decreasing in k. Result 4 shows that the retailer is better off and the manufacturer is worse off when Policy 1 is replaced with the full returns policy. This is to say that, for the retailer, the positive side (the first and second effect) dominates the downside (the third effect) when the full returns policy is adopted. However, an opposite phenomenon occurs for the manufacturer. The downside dominates the positive side. This is even true even when c (production cost) = 0. This result is largely attributed to our assumption that there is no salvage value in overstocking. Thus, the value of the returns policy could be underestimated. Under the conditions in (b), the retailer has the incentive and capability to have the manufacturer adopt the full returns policy and to compensate the manufacturer for any resultant losses. However, when k increases, both the manufacturer’s and the retailer’s profits under the full returns policy are approaching their counterparts under Policy 1 (see Fig. 2). This is because the third effect becomes more valuable for the manufacturer, and the manufacturer therefore strategically makes better use of the information against the retailer. This is to say that the full returns policy becomes increasingly more attractive to the manufacturer and the no returns policy (Policy 1) becomes increasingly less attractive to the retailer when k
The Retailer’s Profit ( aH = 200 , a L=150 , β = 4 , c =1)
The Manufacturer’s Profit ( aH = 200 , aL=150 , β= 4 , c =1 ) 1050
400
Under a no returns policy
1025
Under a full returns policy 1000
300
975 200
950
100
925
Under a no returns policy
Under a full returns policy
900 0 0.5
0.6
0.7
λ
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
λ
Fig. 2. k versus the retailer’s and the manufacturer’s profits under the full returns policy and the no returns policy (Policy 1).
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increases. This result makes sense, as the retailer’s information becomes more valuable with the increase of k, the indicator of demand correlation between the present and previous periods. Result 5. If the full returns policy is adopted to replace Policy 2 (no returns): (a) The retailer’s profit is greater with a full returns policy than with a no returns policy. The manufacturer’s profit can be greater or smaller with a full returns policy than with a no returns policy. However, if c = 0, both the retailer and the manufacturer’s profits are greater with a full returns policy than with a no returns policy. (b) The total supply chain profit will increase if and only if aH ð3 6ð1 kÞkÞ þ aL ð3 þ 6ð1 kÞkÞ 22bc P 0: (c) The differences in both the retailer’s and the manufacturer’s profits between the full returns policy and no returns policy are increasing in k. Similar to Result 4, Result 5 shows that if the manufacturer changes to adopt the full returns policy, the retailer’s profit would increase; however, the manufacturer could be better off or worse off (depending on ð1kÞkþaH ð12ð1kÞkÞ the sign of ðaH aL Þ½aL 10bcþ2aL16b ). This implies that the manufacturer should adopt the full returns policy if the sign is positive. If the condition in (b) is satisfied, the retailer has the incentive and capability to have the manufacturer adopt the full returns policy and compensate the manufacturer for any resultant losses. When k increases, the manufacturer’s profit under the full returns policy increases faster than under Policy 2 (see Fig. 3). This is expected as explained in Result 4 (the information indirectly shared by retailer due to the returns policy becomes more valuable to the manufacturer). However, when k increases, the gap between the retailer’s profit under the full returns policy and under the no returns policy (Policy 2) widens. This result is in contrast with what we have in Result 4. It implies that the full returns policy becomes more attractive than the no returns policy (Policy 2) for both the manufacturer and the retailer when k increases. 4.4. Information sharing under the no returns policy In order to seek out and assess the value of information relayed through the returns policy, we will consider the situation in which the retailer voluntarily shares the information of the previous period (at1) with the manufacturer. In this case, intuitively, the manufacturer would benefit from such information sharing, as the manufacturer could incorporate the information into its forecasting and planning process. Again, this is a Bayesian Stackelberg game. Similar to what we did in the no returns policy and full returns policy analysis, we derive the optimal decisions shown in Tables 4 and 5.
The Retailer’s Profit ( a H = 200 , aL=150 , β = 4 , c =1)
The Manufacturer’s Profit ( a H = 200 , aL=150 , β = 4 , c =1)
500 900
450 850
400
Under a full returns policy
Under a full returns policy 800
350
Under a no returns policy
750
300
Under a no returns policy
250
0.5
0.6
0.7
0.8
λ
0.9
700
1
0.5
0.6
0.7
0.8
0.9
1
λ
Fig. 3. k versus the retailer’s and the manufacturer’s profits under the full returns policy and the no returns policy (Policy 1).
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Table 4 Information sharing under Policy 1 Variable
Value
wt
aH ð2kÞaL ð1kÞþbc 2b aL þbc 2b ð1kÞaL þð2þkÞaH þbc 4b 3aL þbc 4b aL ð1kÞþaH ð2kÞbc 4 aL bc 4 2 2ð6aH þ6aH aL þa2L 2aH bcþb2 c2 Þþ2ðaH aL Þð6aH þaL þbcÞkþðaH aL Þ2 k2 32b ðaL bcÞ2 þ½aL þbcaH ð2kÞaL k2 16b
pt qt pR t pM t
(if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL) (if at1 = aH) (if at1 = aL)
Table 5 Information sharing under Policy 2 (no returns) Variable
Value
wt
aL þbc 2b 3aL þbc 4b aL bc 4 ðaL bcÞ2 16b ðaL bcÞ2 8b
pt qt pR pM
Result 6. Table 4 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with information sharing under Policy 1 (no returns). Result 7. Table 5 shows the optimal prices, order quantity, and expected profits for the manufacturer and retailer with information sharing under Policy 2 (no returns). Note: The result is the same as the result in Table 2. 4.5. Comparison between no information sharing and information sharing (under the no returns policy) Based on the above two tables, we can assess information sharing’s impact on each supply chain player. Thus, we have the following two results (proofs are omitted). Result 8. For the scenario, the retailer uses the Policy 1. (a) The manufacturer’s profit is greater with information sharing than without information sharing. However, the retailer’s profit can be greater or smaller (depending on the sign of {2bc(1 2k) aH(1 + k)(5 2k2) + aL[3 + k(9 2k(1 + k))]}) with information sharing than without information sharing. (b) The total supply chain profit is greater with information sharing than without information sharing if and only if ð1 þ kÞð2bcð1 þ 2kÞ þ aH ð1 þ kÞð3 þ 2ð2 þ kÞkÞ aL ð1 þ kð3 þ 2ð1 þ kÞkÞÞÞ P 0: (c) The difference in the retailer’s profit between the scenarios of information sharing and no information sharing can be increasing or decreasing in k; however, the difference in the manufacturer’s profit is decreasing in k. The positive value of information sharing to the manufacturer is intuitive because the retailer’s information contributes to increasing the accuracy of the manufacturer’s demand forecast. On the other hand, the retailer
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might suffer from information sharing. This is because the manufacturer could use the information strategically to maximize its own profit at the expense of the retailer. Specifically, the manufacturer may set a higher wholesale price, which, we know, is undesirable to the retailer. Result 8 also implies that the retailer is willing to enter into an ex ante information-sharing contract with the manufacturer if and only if the sign of the expression in (a) is positive. This represents a win–win situation for both the manufacturer and the retailer. With the condition in (b) satisfied, the manufacturer has the incentive and capability to induce the retailer to share his information and compensate the retailer for any resultant losses. We also observe that when k increases, the value of information sharing for the manufacturer decreases. This is due to the effect of diminishing return of margin, so that the value of the retailer’s information decreases. Result 9. In the case when the retailer uses Policy 2, the profit difference between the information sharing and no information sharing is 0. This result is intuitive, as information sharing has no effect on the manufacturer and the retailer under the Policy 2. 5. Continuous model In this section, we assume that at is a type of non-standard auto-correlated AR(1) process. Specifically, we let at = d + qat1 + et, d > 0, 1 < q < 1. et is a random variable and uniformly distributed on the range [B, B]. To be different from the standard AR(1) process where et is normally distributed, we call it uniform AR(1) process. 5.1. No returns policy A convenient expression for profit is obtained by substituting Dt = yt + et, and, consistent with Petruzzi and Dada (1999), by defining zt = qt yt and yt = d + qat1 bPt: ! P t ðy t þ et Þ wt ðy t þ zt Þ if et 6 zt pR ¼ : ð2Þ P t ðy t þ zt Þ wt ðy t þ zt Þ if et > zt This transformation of variables provides an alternative interpretation of the order decision: if the choice of zt is larger than the realized value of et, then leftovers occur; if the choice of zt is smaller than the realized value of et, then shortages occur. The corresponding optimal stocking and pricing policy is to order qt = yt + zt units to sell at the price pt, where zt and pt maximize expected profit. The expected profits are: Z zt Z B pR ðz Þ ¼ p ½y þ e f ðe Þ de þ pt ½y t þ zt f ðet Þ det wt ðy t þ zt Þ; t t t t t t t B
pM t
zt
¼ ðwt cÞqt :
Since the manufacturer is the Stackelberg leader and the retailer is the follower, we derive the optimal solution by deriving optimal pt and qt followed by optimal wt. Therefore, we first consider the retailer’s expected profit, pR t ðzt Þ: opR t ðzt Þ ¼ wt þ pt ½1 F ðzt Þ ¼ 0; ozt opR t ðzt Þ ¼ 2bðP 0 pt Þ hðzt Þ ¼ 0; oP t RB þbwt where, hðzt Þ ¼ zt ðet zt Þf ðet Þ det , P 0 ¼ dþqat1 . 2b
ð3Þ ð4Þ
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Thus, Eqs. (3) and (4) are solved first, we have hðzt Þ ; 2b wt ¼ pt ð1 F ðzt ÞÞ:
pt ¼ P 0
ð5Þ ð6Þ
Thus, qt ¼ zt þ y t ¼ zt þ d þ qat1 bpt : Then, we plug (7) into equation
pM t
ð7Þ
opM ¼ ðwt cÞqt , solving equation t ¼ 0, then we have owt
3
ðB zt ÞP 0
ðB zt Þ ¼ 2Bwt : 8bB
ð8Þ
By solving (5)–(8), we can obtain the optimal zt, wt, pt and qt. 5.2. The full returns policy case In this case, the manufacturer sets a wholesale price wt, and gives the retailer a full refund for unsold stock. Since the manufacturer accepts returns, the retailer always orders the maximum quantity, qH aR bpt + B, where aR = d + qat1. Therefore, pR t ¼ E½ðp t wt ÞDt ; pM t ¼ E½wt Dt cqH : Thus, pR t ¼ ðp t wt Þ pM t
¼
Z
Z
B
ðy t þ et Þf ðet Þ det ¼ ðpt wt ÞðaR bpt Þ;
ð9Þ
B
B
wt ðy t þ et Þf ðet Þ det cðaR bpt þ BÞ ¼ ðwt cÞðaR bpt Þ cB:
ð10Þ
B
opR aR þ bwt t . ¼ 0, thus P t ¼ opt 2b aR þ bwt opM aR þ bc aR bc þ 4B and retailer’s order quantity qH ¼ . into t ¼ 0, we have wt ¼ Putting pt ¼ 2b 4 2b owt Observation 3. For the no returns policy case, at the beginning of period t, the manufacturer can infer at2 from qt1. We first solve
Proof. In Eq. (7): qt1 ¼ zt1 þ d þ qat2 bP t1 , as the manufacturer knows qt1 and P t1 . He can also learn qt1 from Eq. (6). Therefore, the manufacturer can infer at2. h Observation 4. For the full returns policy case, at the beginning of period t, the manufacturer can infer at1. Proof. The manufacturer knows pt1 and Dt1 exactly (because Dt1 = qt1 – returned quantity at the period t). As Dt1 = at1 bpt1, manufacturer can infer at1. h 5.3. A numerical example Now we present a numerical example to illustrate the returns policy’s impact on supply chain performance for the continuous case. Specifically, we will show the magnitude of the profits as a function of the demand parameters (namely, q, B), and c (the production cost). Note that we assume that et is uniformly distributed among the range [B, B].
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For our simulations, we use the following parameter values: d = 50, b = 1, q = {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}, B = {2, 4, 6, 8, 10}, and c = {0, 1, 5, 10, 15, 18}. For every value of the specific parameter analyzed, we average the profits over all other parameters’ various combinations. 5.3.1. Impact of q Figs. 4 and 5 show that, regardless of whether a returns policy is used or not, both the manufacturer’s and the retailer’s profits increase with q. This is because the higher q, the higher correlation between the present and previous demands; thus the manufacturer or the retailer could better forecast the present demand based on the previous demand, which leads to higher profits for both players. For the manufacturer’s side, when q is small, the manufacturer’s profit under a no returns policy is greater than under a full returns policy. However, when q goes over some value, the full returns policy begins to prevail. The reason behind this phenomenon rests in the tradeoff between the manufacturer’s burden of taking back the unsold products and the effect of information sharing. A full returns policy benefits the manufacturer by allowing the manufacturer to obtain the previous demand information at the expense of bearing the burden of demand uncertainty. When q is small, the effect of taking the burden dominates the effect of information sharing. As q increases, the effect of information sharing begins to dominate. On the retailer’s side, the retailer’s profit under a no returns policy dominates the profit under a full returns policy all the time. This is because information sharing allows the manufacturer to strategically exploit the
8000 With no return policy
Manufacturer Profit
7000
With full return policy
6000 5000 4000 3000 2000 1000 0 0
0.2
0.4
ρ
0.6
0.8
Fig. 4. q versus the manufacturer’s profits.
8000 With no return policy
Retailer Profit
7000
With full return policy
6000 5000 4000 3000 2000 1000 0 0
0.2
0.4
ρ
0.6
Fig. 5. q versus the retailer’s profits.
0.8
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643
Manufacturer Profit
300 250 200 150 100
With no return policy
50
With full return policy
0 2
4
6
8
10
B Fig. 6. B versus the manufacturer’s profits.
250
Retailer Profit
200 150 100 With no return policy 50
With full return policy
0 2
4
6
8
10
B Fig. 7. B versus the retailer’s profits.
information to maximize its own profit at the expense of the retailer. This effect dominates the effect of the retailer’s shifting burden of unsold products to the manufacturer for our given set of data. 5.3.2. Impact of B Figs. 6 and 7 show the impact of B on the retailer’s and manufacturer’s profits, respectively. For the manufacturer, with and without the returns policy, profits go down in B. However, when B is small, the manufacturer’s profit under the full returns policy is higher than under the no returns policy. When B increases, the full returns policy begins to prevail. This is because a higher B implies a higher demand uncertainty, which has a negative affect on the manufacturer. The same reasoning applies for the retailer under the no returns policy. For the case of a full returns policy, the manufacturer suffers not only the demand uncertainty’s negative impact on pricing, but also the increasing risk of having higher production cost. Consequently, its profit decreases faster than the profit under a no returns policy. In contrast, the retailer’s profit under the full returns policy remains almost identical (level) all the time when B increases. This is because the retailer has shifted the burden of demand uncertainty to the manufacturer, so B has no impact on its profit. We also notice that the no returns policy’s profit dominates the full returns policy’s profit. This indicates that given the set of data, for the retailer, the effect of information sharing dominates the effect of shifting the burden of demand uncertainty to the manufacturer. 5.3.3. Impact of c Figs. 8 and 9 show the impacts of c. As expected, both the manufacturer’s and the retailer’s profits, regardless of whether a full returns policy is used or not, decrease with c. This is because when c goes up, both the manufacturer and the retailer realize smaller profit margins and smaller demand, thus resulting in a drop in
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Manufacturer Profit
644
450 400 350 300 250 200 150 100 50 0
With no return policy With full return policy 0
5
10
15
c Fig. 8. c versus the manufacturer’s profits.
350
Retailer Profit
300 250 200 150 100
With no return policy
50
With full return policy
0 0
5
10
15
c Fig. 9. c versus the manufacturer’s profits.
profits. The manufacturer’s profit under the full returns policy is greater than that under a no returns policy when c is small. When c increases, the no returns policy begins to dominate the full returns policy. This is expected again as the manufacturer bears the burden of taking unsold products in the full returns policy case, and the effect of the burden becomes larger and dominates the effect of information sharing when c increases. On the other hand, the retailer’s profit under the full returns policy decreases at a slower rate than under the no returns policy. This is because the retailer shifts the burden of demand uncertainty to the manufacturer. Therefore, the retailer experiences a smaller decrease in profit in the case of full returns policy. 6. Conclusion The returns policy literature often assumes (either implicitly or explicitly) that the manufacturer and the retailer have equal demand information. While possibly appropriate in some settings, information symmetry is not very solid assumption. In reality, the manufacture and the retailer usually have different perspectives on demand information because of different data sourcing and methodology used. Therefore, in this study, we analyze the impact of full returns policy as well as information sharing on the manufacturer and the retailer under information assymmetry. We first study the case of the base level of the demand following a discrete distribution with two states. We find that while the retailer always benefits from the full returns policy, the manufacturer would lose when the full returns policy is adopted by the retailer to replace Policy 1 (no returns); and may gain or lose when the full returns policy is adopted to replace Policy 2 (no returns). We also find out the conditions under which the total supply chain profit is higher with a full returns policy than with a no returns policy. We then address the case of the base level of the demand following a continuous distribution (uniform AR(1) process), and provide a numerical example to illustrate the returns policy’s impact on supply chain performance.
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Appendix A Proof for Remark 1. After some algebra, we can prove
o2 pR t > 0. o2 qt
h
Proof for Result 1 pM t ¼ ðwt cÞE½qt : Suppose at2 = aH: Based on results in Appendix B, 2aH aH bwt aL bwt E½qt ¼ k þ ð1 kÞ ; 2 2 2 kð2 kÞaH þ ð1 kÞ aL þ bc by solving the first order condition for the manufactrer’s maxWe can get wt ¼ 2b imization problem. Similarly, the case of at2 = aL can be analyzed. h Proof of Result 2. The proof of procedure is similar to that of result 1. h Proof of Result 3. Full Returns Policy Case 1: Suppose at1 = aH pR t ¼ ðp t wt ÞðaL bpÞð1 kÞ þ ðpt wt ÞðaH bpÞk; aH k þ ð1 kÞaL þ bwt by solving the first order condition for the retailer’s maximization We can obtain pt ¼ 2b problem. pM t ¼ wt ½ðaH bpt Þk þ ð1 kÞðaL bp t Þ cðaH bp t Þ; By taking the FOC and setting it to zero, we can get wt ¼
aL ð1 kÞ þ aH k þ bc : 2b
In the similar way, the case at1 = aL can be analyzed. Then, after some algebra, we can get the results. Proof of Result 6 Case 1: Suppose at1 = aH pR t ¼ kðaH bp t Þp t þ ð1 kÞðaL bp t Þp t ðaH bp t Þwt ; opR t ¼ kðaH 2bpt Þ þ ð1 kÞðaL 2bpt Þ þ bwt ¼ 0: opt Thus, kaH þ ð1 kÞaL þ bwt ; 2b qt ¼ aH bpt ; pt ¼
pM t ¼ E½ðwt cÞqt ; opM t ¼ 0; owt aH ð2 kÞ aL ð1 kÞ þ bc : wt ¼ 2b In the similar way, the case (at1 = aL) can be analyzed.
h
h
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Proof of Result 7. The proof procedure is similar to that of Result 4, and the result is exactly the same as the one in Table 2. h Appendix B Case 1: at1 = aH. Case 1.1: qt = aL bpt (Policy 2) pR t ¼ ðaL bp t Þðpt wt Þ; Taking FOC (first order condition) and setting it to zero, we get aL þ bw ; 2b aL bw : qt ¼ 2 Case 1.2: qt = aH bpt (Policy 1) pt ¼
pR t ¼ kðaH bp t Þp t þ ð1 kÞðaL bp t Þp t ðaH bp t Þwt ; Taking FOC and setting it to zero, we get pt ¼
kaH þ ð1 kÞaL þ bwt : 2b
Let aH ¼ kaH þ ð1 kÞaL : Thus, 2aH aH bwt : 2 Case 2: Suppose at1 = aL. If qt = aL bpt (Policies 1 and 2) qt ¼
ðB:1Þ
aL þ bw ; 2b aL bw : qt ¼ 2 pt ¼
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