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Responsive pricing under supply uncertainty
European Journal of Operational Research 182 (2007) 239–255 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Responsive pricing under supply uncertainty Christopher S. Tang *, Rui Yin
*
UCLA Anderson School, UCLA, 110 Westwood Plaza, CA 90095, USA Received 15 January 2006; accepted 17 July 2006 Available online 24 October 2006
Abstract Consider a retailer orders a seasonal product from a supplier and sells the product over a selling season. While the product demand is known to be a linear function of price, the supply yield is uncertain and is distributed according to a general discrete probability distribution. This paper presents a two-stage stochastic model for analyzing two pricing policies: No Responsive Pricing and Responsive Pricing. Under the No Responsive Pricing policy, the retailer would determine the order quantity and the retail price before the supply yield is realized. Under the Responsive Pricing policy, the retailer would specify the order quantity first and then decide on the retail price after observing the realized supply yield. Therefore, the Responsive Pricing policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. Our analysis suggests that the retailer would always obtain a higher expected profit under the Responsive Pricing policy. In addition to examining the impact of yield distribution and system parameters on the optimal order quantities, retail prices, and profits under these two pricing policies, we analyze two issues arising from responsive pricing. The first issue deals with the case in which the retailer can place an emergency order with an alternative source after observing the realized yield, while the second issue deals with a situation in which the retailer has to allocate his order among multiple suppliers. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Responsive pricing; Uncertain supply; Supply management
1. Introduction Due to long supply lead time and short selling season, retailers usually can place their orders only once before the start of the selling season. For example, in the fashion industry, it is common for retailers to place their orders many months before the selling season. The reader is referred to Fisher and Raman (1996) for an excellent description of the ordering process in the fashion industry. Since accurate supply or demand information is rarely available in advance, it is difficult for retailers to determine cost effective order quantities. For instance, due to uncertain supply yields, transportation delays, shrinkage during shipment, retailers usually do not know for sure if they would receive the exact quantity ordered prior to the selling season. Besides uncertain supply, retailers often face uncertain demand as well. As such, many retailers have to struggle with the *
Corresponding authors. Tel.: +1 310 825 4203 (C.S. Tang). E-mail addresses: [email protected] (C.S. Tang), [email protected] (R. Yin).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.07.011
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overstocking and understocking issues. Specifically, overstocking forces a retailer to dispose of unsold items at clearance prices, while understocking creates lost sales. See Tang (2005) for a review of supply chain risk management. To reduce the overstocking and understocking costs, various researchers have examined different approaches for helping retailers to meet uncertain demand. First, Fisher and Raman (1996) consider a situation in which a retailer can place two separate orders prior to the selling season. By allowing the retailer to place the second order in a later period, the retailer can generate a more accurate demand forecast by using the market signals observed during the time between the first order and the second order. Fisher and Raman refer the second order as an ‘accurate response’ for managing uncertain demand. Next, Tang et al. (2004) consider a situation in which a retailer offers each customer two options: pre-commit an order at a reduced price before the selling season, or buy the product at the regular price during the selling season. Clearly, the reduced price serves as an incentive for the customers to pre-commit their orders before the selling season. Tang et al. (2004) show how the retailer can use these pre-committed orders to generate more accurate demand forecasts, which would enable the retailer to place more cost effective orders and manage uncertain demand more efficiently. The ideas articulated in Fisher and Raman (1996) and Tang et al. (2004) would certainly improve supply chain performance; however, some suppliers and retailers may have specific concerns about the requirements associated with these two ideas. For example, to implement the accurate response concept, the supplier needs to have sufficient capacity to handle the second order on short notice. Also, to implement the early commitment program, a retailer need to show his customers a sample of the seasonal products before the selling season, which could make the retailer or the manufacturer more vulnerable to copycats. This paper examines a situation in which the accurate response or the early commitment program is impractical due to the aforementioned concerns. Instead of having the flexibility to place two separate orders or to sell the product at two different retail prices, we consider a situation in which the retailer can place exactly one order and select only one retail price before the selling season. This concept is known as ‘responsive pricing’. Van Mieghem and Dada (1999) is the first to analyze the responsive pricing concept as a mechanism for a retailer to manage uncertain demand. In their paper, they present a two-stage stochastic model in which the retailer places an order in the first period. Then the retailer would determine the retail price after the demand uncertainty is resolved at the end of the first period but before the selling season that starts at the beginning of the second period. They show the benefits of delaying the pricing decision until the demand uncertainty is resolved. Motivated by the product postponement concept examined in Lee and Tang (1997), Chod and Rudi (2005) extend the work of Van Mieghem and Dada to the two-product case. Specifically, they consider the case in which the retailer places an order of a ‘generic’ product in the first period. Then, after the demand uncertainty is resolved, the retailer would customize this order of generic product into two individual products and then determine the retail price for each of these two products. Given the flexibility of delaying the product identity and the pricing decision until demand uncertainty is resolved, Chod and Rudi present a two-period model to illustrate the benefit of product postponement under responsive pricing. While Van Mieghem and Dada (1999) and Chod and Rudi (2005) focus on the issue of demand uncertainty, this paper examines the benefits of responsive pricing under supply uncertainty. As an initial attempt to analyze the issue of responsive pricing under supply uncertainty, we shall consider a situation in which the demand function is known but the supply yield is uncertain. By focusing on the issue of uncertain supply, we develop tractable results which can serve as building blocks for analyzing responsive pricing under uncertain supply and uncertain demand in the near future. In the context of supply uncertainty, various researchers have developed models for determining the optimal order quantity under uncertain supply. The reader is referred to Yano and Lee (1995) for a comprehensive review of lot sizing models with uncertain supply yield. To our knowledge, this paper is the first that examines the joint decisions of order quantity and retail pricing under supply uncertainty. This paper presents a two-stage stochastic model for analyzing two pricing policies: No Responsive Pricing and Responsive Pricing. Under the No Responsive Pricing policy, the retailer would determine the order quantity and the retail price at the beginning of the first period before the supply yield is realized. Under the Responsive Pricing policy, the retailer would determine the order quantity at the beginning of the first period. Then, after the actual supply yield is realized at the end of the first period, the retailer would then determine the retail price at the end of the first period but before the selling season that starts at the beginning of the second period. By delaying the pricing decision after the actual supply yield is realized,
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the Responsive Pricing policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. This paper is divided into two parts. In the first part, it is shown analytically that, under supply uncertainty, the retailer would always obtain a higher expected profit under the Responsive Pricing policy. (This result is consistent with the result obtained by Van Mieghem and Dada (1999) when they analyze the benefits of responsive pricing under demand uncertainty.) We also examine the impact of yield distribution and system parameters on the optimal order quantities, prices, and retailer’s profits under these two pricing policies numerically. The numerical analysis indicates that the Responsive Pricing policy is more beneficial when the supply yield is highly uncertain (low mean or high variance) or when the unit cost is high. The analysis of the Responsive Pricing is extended to examine two separate issues in the second part of this paper. The first issue deals with the case in which the retailer can place an emergency order with an alternative source after observing the actual supply yield, while the second issue deals with a situation in which the retailer has to allocate his order among multiple suppliers. Ramasesh et al. (1991) develop a model for analyzing ways to allocate an order among different suppliers with different lead times, while we examine the issue of order allocation under responsive pricing. This paper is organized as follows. In Section 2, we present the base model for analyzing the retailer’s optimal expected profits under the No Responsive Pricing policy and the Responsive Pricing policy. It is shown analytically that the retailer would always obtain a higher expected profit under the Responsive Pricing policy. We also develop numerical experiments to examine the benefits of the Responsive Pricing policy. Section 3 extends our analysis of the Responsive Pricing policy to the case in which the retailer can place an emergency order after the actual supply yield is realized. Section 4 deals with a situation in which the retailer can order from multiple suppliers with different supply yield distributions. This paper is concluded in Section 5. 2. The base model Consider a situation in which a retailer orders a seasonal product from a supplier and sells the product over a selling season. We assume that the retailer knows that the product demand function is given by: D = a bp, where a > 0 represents the potential market size, b > 0 represents the price sensitivity, and p represents the retail price. While the retailer has perfect information about the demand function, he has to deal with supply uncertainty in the following manner. First, the supply lead time is equal to one period, and hence, the retailer needs to decide on the order quantity Q at the beginning of period 1 so that the retailer will receive the order at the end of period 1 prior to the selling season that starts at the beginning of period 2 and ends at the end of period 2. Second, the supply yield is uncertain in the following sense. For any order quantity Q, the retailer will receive only yQ non-defective units, where y represents the supply yield (essentially, the yield y accounts for the defective rate of the supplier as well as the damage or shrinkage occurred during shipment). It is assumed that y is a random variable that takes on N different discrete values, say, yn for n = 1, 2, . . . , N, where 0 < y1 < y2 < < yN1 < yN 6 1.1 Let kn P be the probability that the actual yield is equal to yn; i.e., ProN b{y = yn} = kn for n = 1, 2, . . . , N. Hence, n¼1 kn ¼ 1. Third, the retailer pays the supplier c per unit at the beginning of period 1, and the retailer disposes of the unsold units at s per unit at the end of period 2. Without loss of generality, it is easy to show that one can transform the cost parameters so that one can set s = 0. To simply our exposition, we shall consider the case when s = 0. This paper examines two pricing policies: No Responsive Pricing (NRP) policy and Responsive Pricing (RP) policy. Under the NRP policy, the retailer specifies the order quantity Q and the retail price p jointly at the beginning of period 1. Under the RP policy, the retailer first decides on the order quantity Q at the beginning of period 1. Then he specifies the retail price p at the beginning of period 2 after he observes the actual supply yield realized at the end of period 1. As such, the RP policy allows the retailer to delay 1 The uncertain supply yield has a general discrete distribution, which allows us to approximate any continuous distribution. Notice that our way of capturing uncertainty is different from the models examined by Van Mieghem and Dada (1999) and Chod and Rudi (2005). First, we examine supply uncertainty, while they consider demand uncertainty. Second, we assume that the supply yield follows a general discrete probability distribution; however, they assume that the demand function is additive (a linear function of price plus an uncertain error term) and that the distribution of the error term is Uniform or Exponential in the former and Bivariate Normal in the latter.
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the pricing decision until the actual supply yield is realized. In other words, the RP policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. Given these two pricing strategies, we are interested in examining the following questions: 1. Would the retailer obtain a higher expected profit under the RP policy? If yes, by how much? 2. How would the yield distribution affect the retailer’s optimal expected profits, order quantities, and prices under these two pricing policies? To answer these questions, we now formulate the retailer’s problem under these two policies. 2.1. Problem formulation First, under the NRP policy, the retailer has to determine the optimal order quantity Q 0 and optimal retail price p 0 at the beginning of period 1 so as to maximize the retailer’s expected profit. In this case, the retailer’s problem P(NRP) can be formulated as follows: P0 ¼ max Ey ðcQ þ p minfyQ; DgÞ: Q;p
ð2:1Þ
Next, under the RP policy, the retailer would first determine the order quantity Q at the beginning of period 1. Then the retailer would determine the retail price p at the beginning of period 2 after observing the actual yield realized at the end of period 1. In order to determine the optimal order quantity Q* and the optimal retail price p* so that the retailer’s expected profit is maximized, we can formulate the retailer’s problem P(RP) as follows: P ¼ max cQ þ Ey ðmaxfp minfyQ; DggÞ: Q
p
ð2:2Þ
Let us compare the retailer’s expected profits under these two policies. Suppose we implement the optimal order quantity Q 0 and the optimal retail price p 0 under the NRP policy. Then it is easy to check from (2.2) and (2.1) that P P cQ0 þ Ey ðp0 minfyQ0 ; ða bp0 ÞgÞ ¼ P0 : This implies that the optimal expected profit under the RP policy is always higher than that under the NRP policy. This result is intuitive because, under the Responsive Pricing policy, the retailer has the flexibility to select a ‘more profitable’ retail price after observing the actual yield. While it is clear that the RP policy enables the retailer to obtain a higher profit, we would like to compare the retailer’s optimal expected profits, order quantities, and retail prices under both pricing policies and to examine the impact of the yield distribution on these quantities. However, such comparison is analytically intractable, but it can be done numerically. In the remainder of this section, we first determine the retailer’s optimal expected profits, order quantities, and prices under these two policies analytically. We then compare these quantities numerically. 2.2. Analysis of the no responsive pricing policy Since D = a bp, the No Responsive Pricing problem P(NRP) given in (2.1) can be rewritten as: P0 ¼ max Ey ðP0 ðQ; pjyÞÞ; where Q;p 8 < cQ þ pyQ ; if y 6 abp Q 0 P ðQ; pjyÞ ¼ abp : cQ þ pða bpÞ if y > : Q Essentially, P 0 (Q, pjy) represents the retailer’s profit associated with order quantity Q and retail price p for any given realization of yield y. The underlying structure of problem P(NRP) resembles the joint pricing and ordering decision for the Newsvendor problem examined by Petruzzi and Dada (1999). While there is no simple closed form solution for this problem, we now present a simple approach for solving problem P(NRP).
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Suppose we set y0 = 0 and yN+1 = Y, where Y is a sufficiently large number. For any given Q and p, there must exist a k, k = 0, 1, . . . , N, so that the ratio abp satisfies y k 6 abp < y kþ1 . For this particular k, we can take Q Q 0 the expectation of P (Q, pjy) to show that the retailer’s expected profit can be rewritten as: Ey fP0 ðQ; pjyÞg ¼
k X
km ðcQ þ py m QÞ þ
m¼1
N X
km ðcQ þ pða bpÞÞ
if y k 6
m¼kþ1
P0
a bp < y kþ1 : Q
PN
By denoting the terms 1 ¼ 0 and N þ1 ¼ 0, we can decompose problem P(NRP) into N + 1 subproblems Pk(NRP), where k = 0, 1, . . . , N and each subproblem Pk(NRP) can be expressed as: ( ) k N X X 0 Pk ¼ max km ðcQ þ py m QÞ þ km ðcQ þ pða bpÞÞ ð2:3Þ Q;p
m¼1
subject to
m¼kþ1
a bp < y kþ1 : yk 6 Q
Let Q0k and p0k be the optimal order quantity and retail price for subproblem Pk(NRP), respectively. Also, let k 0 2 argmaxfP0k : k ¼ 0; 1; 2; . . . ; N g. In this case, one can utilize the solutions of these N + 1 subproblems Pk(NRP) to determine the optimal expected profit and the optimal solutions to the original problem P(NRP) as follows: P0 ¼ P0k0 ;
Q0 ¼ Q0k0 ;
p0 ¼ p0k0 :
ð2:4Þ
However, the uniqueness of the above optimal solution could not be established for the general discrete distribution, which is consistent with the result obtained by Van Mieghem and Dada (1999). It remains to determine the optimal solutions to subproblem Pk(NRP). In preparation, let us define two partial sums that will become useful. Let: uk ¼
k X
km y m
and
N X
vk ¼
m¼1
ð2:5Þ
km :
m¼kþ1
For notational convenience, P let u0 = 0 and vN = 0. In this case, it is easy to check from the definitions that uk is increasing in k and uN ¼ Nn¼1 kn y n ¼ EðyÞ ¼ l. Also, note that vk is decreasing in k and v0 = 1. By considering the terms uk and vk along with subproblem Pk(NRP) given in (2.3), we can establish the following Proposition: Proposition 1. The solutions for subproblem Pk(NRP) can be expressed as follows: 1. Suppose 2. Suppose
p0k ¼
u ðuk þvk y kþ1 Þ bc c < kuk þ2v . Then p0k ¼ 2ðuk þy a k y kþ1 k vk Þ uk ðuk þvk y kþ1 Þ uk ðuk þvk y k Þ bc 6 < . Then a uk þ2vk y kþ1 uk þ2vk y k
a þ 2b ; Q0k ¼ 2ya 2ðuk þvbck y k
2
k Þy k
k þvk y k Þbc and P0k ¼ ½aðu : 4by ðuk þvk y Þ k
8 > > > <
c a þ 2ðuk þ y k vk Þ 2b
if
½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > ; y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ
> > > :
c a þ 2ðuk þ y kþ1 vk Þ 2b
if
½aðuk þ vk y k Þ bc2 ½aðuk þ vk y kþ1 Þ bc2 6 ; y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ
2
2
8 2 2 a bc ½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > > > > ; if < 2y 2ðu þ v y Þy y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ k k k k k Q0k ¼ 2 2 > a bc ½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > > : 6 ; if 2y kþ1 2ðuk þ vk y kþ1 Þy kþ1 y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ 8 ½aðuk þ vk y k Þ bc2 ½aðuk þ vk y k Þ bc2 ½aðuk þ vk y kþ1 Þ bc2 > > > if > ; < 4by k ðuk þ vk y k Þ y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ P0k ¼ 2 2 2 > ½aðuk þ vk y kþ1 Þ bc ½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > > : if 6 : 4by kþ1 ðuk þ vk y kþ1 Þ y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ
k
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uk ðuk þvk y k Þ uk þ2vk y k ½aðuk þvk y kþ1 Þbc2 : 4by kþ1 ðuk þvk y kþ1 Þ
3. Suppose
4. Suppose
bc a
6 bca < uk þ vk y kþ1 .
Then
p0k ¼ 2ðuk þyc
kþ1 vk Þ
a þ 2b ; Q0k ¼ 2ya 2ðuk þvkbc y kþ1
kþ1 Þy kþ1
and P0k ¼
P uk þ vk y kþ1 . Then Q0k ¼ 0; p0k is arbitrary and P0k ¼ 0.
Proof. All proofs are given in the Appendix. Proposition 1 can be interpreted via the following cases. First, consider the case Pk PNwhen the expected Pk yield is bc bc sufficiently low, say, when EðyÞ ¼ l 6 . In this case, P l ¼ k y þ k y P m¼1 m m m¼kþ1 m m m¼1 km y m þ a a PN ½ m¼kþ1 km y kþ1 ¼ uk þ vk y kþ1 for k = 0, 1, 2, . . . , N. By applying the fourth statement in Proposition 1, Q0k ¼ 0 for k = 0, 1, 2, . . . , N. Hence, when the expected yield l is sufficiently low; i.e., when l 6 bca , it is optimal for the retailer to order nothing; i.e., Q 0 = 0. Second, consider the case when the expected yield is suffiu ðuk þvk y kþ1 Þ ciently high, say, when l > bca . In this case, it is easy to check from (2.5) that kuk þ2v ¼ l > bca when k y kþ1 k = N. Hence, the first statement in Proposition 1 implies that it is optimal for the retailer to order something when k = N; i.e., Q0N > 0. Combine this observation with the fact that P0 P P0N , it is easy to show that the retailer’s optimal order quantity Q 0 > 0 when the expected yield l is sufficiently high; i.e., when l > bca . These two cases enable us to establish the threshold for l above which the retailer would order something in the optimal solution. 2.3. Analysis of the responsive pricing policy We now analyze the retailer’s optimal expected profit under the Responsive Pricing policy. Observe from (2.2) that problem P(RP) can be rewritten as: P ¼ max Ey maxfPðQ; pjyÞg ; ð2:6Þ Q
p
where P(Q, pjy) is the retailer’s profit associated with order quantity Q and retail price p for any given realization of y. The term P(Q, pjy) can be expressed as: ( cQ þ pyQ if p 6 ayQ ; b PðQ; pjyÞ ¼ : cQ þ pða bpÞ if p > ayQ b By differentiating P(Q, pjy) with respect to p and by considering the break points, we can determine the optimal retail price p*(Qjy) associated with order quantity Q placed at the beginning of period 1 and the actual yield y realized at the end of period 1 as follows: ( ayQ if Q 6 2ya ; p ðQjyÞ ¼ a b ð2:7Þ if Q > 2ya : 2b Substitute the optimal retail price p*(Qjy) into P(Q, pjy), getting: ( cQ þ ðayQ ÞyQ if Q 6 2ya ; b PðQ; p jyÞ ¼ a2 cQ þ 4b if Q > 2ya :
ð2:8Þ
Before taking the expectation of P(Q, p*jy) with respect to y, let us define an additional partial sum in addition to the terms uk and vk defined in (2.5). Let: k X wk ¼ km y 2m ; ð2:9Þ m¼1
where w0 = 0. Notice that wk is increasing in k. Define the following break points: xN þ1 ¼ 0; xN ¼ 2ya ; . . . ; xk ¼ N a , for k = N, N 1, . . . , 1, and x0 = X, where X is a large number. Since y1 < y2 < < yN1 < yN, xk is 2y k decreasing in k; i.e., xN+1 < xN < xN1 < < x1 < x0. In this case, there must exist a k so that xk+1 < Q 6 xk. By using the terms uk, vk and wk and by using (2.8), the retailer’s expected profit Ey(P(Q, p*jy)) can be expressed as follows:
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Ey ðPðQ; p jyÞÞ ¼
a wk a2 v k uk c Q Q 2 þ b b 4b
245
if xkþ1 < Q 6 xk ;
ð2:10Þ
2
where k = N, N 1, . . . , 0. Since the function ðba uk cÞQ wbk Q2 þ a4bvk is concave in Q, it is easy to see that the objective function of problem P(RP) given in (2.6) is a piece-wise concave function. By evaluating the derivative of this piece-wise concave function at various break points xk ¼ 2ya , the unique optimal solution to the k problem P(RP) can be determined as follows: Proposition 2. Suppose bc a < l. Then wk bc k ¼ argmax uk < : k ¼ N ; N 1; . . . ; 1 : a yk
ð2:11Þ
Also, the optimal order quantity Q* and the retailer’s optimal expected profit P* under the Responsive Pricing policy can be expressed as: Q ¼
auk bc ; 2wk
P ¼
ðauk bcÞ a2 v k : þ 4bwk 4b
and
ð2:12Þ
2
Moreover, suppose
bc a
ð2:13Þ
P l. Then Q* = 0 and P* = 0.
It follows from Propositions 1 and 2 that it is optimal for the retailer to order nothing under both pricing policies when the expected yield l is sufficiently small, say, when l 6 bca . Therefore, it suffices to focus on the case when l > bca throughout the paper. 2.4. Numerical analysis We now utilize Propositions 1 and 2 to construct numerical experiments for comparing the retailer’s optimal expected profits, order quantities, and prices under the NRP and the RP policies. In addition, we shall examine the impact of the yield distribution and the unit cost on these quantities. In the numerical experiments, the supply yield follows a ‘discrete’ Uniform distribution over [a, b], where 0 < a < b 6 1. Specifically, the supply yield takes on N possible values y1 < y2 < < yN, where y k ¼ a þ Nba ðk 1Þ and Pro1 b(y = yk) = 1/N for k = 1, 2, . . . , N.22 Hence, the expected supply yield l ¼ EðyÞ ¼ aþb and the variance of 2 ðN þ1Þ the supply yield r2 ¼ VarðyÞ ¼ ðabÞ . 12ðN 1Þ In our experiments, N = 1000, a = 100, b = 5 and the values of a, b and c are varied so as to examine the impact of the yield distribution and the unit cost. In each set of experiments, the retailer’s optimal expected profits and the optimal order quantities under the NRP and the RP policies are computed. Since the optimal retail price under the Responsive Pricing p*(Q*jy) given in (2.7) is based on the actual realization of the supply yield, we shall compare the optimal retail price under the No Responsive Pricing policy with the expected optimal retail price Ey(p*(Q*jy)) under the Responsive Pricing policy. 2.4.1. The impact of supply yield uncertainty In the first set of experiments, we examine the impact of the standard deviation of the supply yield r on the retailer’s optimal expected profits, order quantities, and prices under the NRP and the RP policies. To do so, we set the unit cost c = 3 and vary the standard deviation r. To isolate the effect of r, the standard deviation of the supply yield r is varied while l is kept constant at l ¼ aþb ¼ 0:5. Specifically, the value of a is increased 2 from 0.1 to 0.48 and b is decreased from 0.9 to 0.52 according to an increment of 0.02. As we vary the values of a and b, the standard deviation of the supply yield r is decreased from 0.23 to 0.012. The impact of r on various quantities are reported in Figs. 1–3. Fig. 1 confirms that the retailer would always obtain a higher 2 Since our analysis is based on the assumption that the supply yield has a general discrete distribution, we can use our general discrete distribution to approximate any continuous probability distribution. For example, our discrete Uniform distribution converges to the (continuous) Uniform distribution over [a, b] when N ! 1.
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300 250 Profit
200 150 100
Profit (RP) Profit (NRP)
50 0 0
0.05
0.1
0.15
0.2
0.25
Sigma
Order quantity
Fig. 1. The impact of sigma on optimal profits.
80 70 60 50 40 30 20 10 0
Q (RP) Q (NRP)
0
0.05
0.1
0.15
0.2
0.25
Sigma Fig. 2. The impact of sigma on optimal order quantities.
Retail price
14.4 14.2
Price (RP)
14 13.8
Price (NRP)
13.6 13.4 13.2 13 12.8 0
0.05
0.1
0.15
0.2
0.25
Sigma Fig. 3. The impact of sigma on optimal prices.
expected profit under the RP policy; i.e., P* > P 0 . Observe from Fig. 1 that the profit gap P* P 0 increases as the supply yield becomes more variable; i.e., when r increases. This observation implies that the RP policy is more beneficial when the supply yield is more variable. Next, notice from Fig. 2 that the retailer would order more under the RP policy. This result is intuitive because the retailer can afford to order more at the beginning of period 1 because he has the flexibility to set the price after the actual yield is realized. Observe from Fig. 2 that the optimal order quantities are decreasing in r under both pricing strategies. This result is consistent with the well-known property of the optimal newsvendor order quantity (i.e., the optimal newsvendor order quantity decreases as the demand uncertainty increases). Finally, Fig. 3 suggests that the optimal retail prices are increasing in r under both pricing strategies. However, as the supply yield becomes more uncertain, the retailer would charge a much higher price under the RP policy. Combining the results displayed in Figs. 2 and 3, we can conclude that, relatively speaking, the retailer would place a larger order and charge a higher price (in expectation) under the RP policy as the standard deviation of the supply yield r increases. This observation explains why the profit gap P* P 0 increases as the supply yield becomes more variable in Fig. 1.
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2.4.2. The impact of the expected supply yield In the second set of experiments, we examine the impact of the expected supply yield l on the retailer’s optimal expected profits, order quantities, and prices. To do so, we set the unit cost c = 3 and vary l. To isolate the effect of l, the value of a is varied from 0.1 to 0.6 according to an increment of 0.02 and the value of b is set at b = a + 0.4. This would allow us to vary the expected supply yield l from 0.3 to 0.8 while keeping r2 conðabÞ2 ðN þ1Þ 2 stant, where r ¼ 12ðN 1Þ ¼ 0:0133. The computational results are summarized in Figs. 4–6. Fig. 4 suggests that, under both pricing policies, the retailer’s optimal expected profits increase as the expected supply yield l increases. This result is intuitive because the retailer should be able to obtain a higher expected profit when the 350 300 Profit
250 200 150
Profit (RP)
100
Profit (NRP)
50 0 0.3
0.4
0.5
0.6
0.7
0.8
μ Fig. 4. The impact of l on profits.
80
Order quantity
70 60 50 40 30
Q (RP)
20
Q (NRP)
10 0 0.3
0.4
0.5
0.6
0.7
0.8
μ Fig. 5. The impact of l on optimal order quantities.
16
Retail price
15.5
Price (RP)
15
Price (NRP)
14.5 14 13.5 13 12.5 12
0.3
0.4
0.5
0.6
0.7
μ Fig. 6. The impact of l on optimal prices.
0.8
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expected supply yield is higher. Notice from Fig. 5 that, under both pricing policies, the optimal order quantities are not monotonic in l. In addition, unlike the result reported in Fig. 2, Fig. 5 suggests that the retailer would order less under the Responsive Pricing policy when the expected supply yield l is sufficiently high, say, when l > 0.56. Observe from Fig. 6 that, under both pricing policies, the optimal retail prices are decreasing in l. This result is intuitive. However, it is interesting to observe from Fig. 6 that, under the RP policy, the retailer would charge a lower price (in expectation) when l is sufficiently large.
Profit
2.4.3. The impact of unit cost In our third set of experiments, we investigate the impact of the unit cost c on the retailer’s optimal expected profits, order quantities, and prices under these two policies. To do so, we set a = 0.1, b = 1, and we vary c from 1 to 10 according to an increment of 0.2. The computational results are summarized in Figs. 7–9. Fig. 7 confirms that the retailer would obtain a higher expected profit under the Responsive Pricing policy; however, the profit gap P* P 0 decreases as the unit cost c increases. Moreover, if one measures the relative 0 benefit of RP policy over the NRP policy according to the relative profit gain D ¼ P PP 0 , then one can check that the relative profit gain D is increasing in the unit cost c. The corresponding figure is omitted here. There400 350 300 250 200 150 100 50 0
Profit (RP) Profit (NRP)
1
3
5 7 Unit Cost
9
Order quantity
Fig. 7. The impact of unit cost on profits.
120 100 80 60 40 20 0
Q (RP) Q (NRP)
1
3
5 7 Unit Cost
9
Fig. 8. The impact of unit cost on optimal order quantities.
Retail price
20 18 16 14
Price (RP)
12
Price (NRP)
10
1
3
5 7 Unit Cost
9
Fig. 9. The impact of unit cost on optimal prices.
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fore, we can conclude that the RP policy becomes more beneficial when the unit cost c is high. Next, Fig. 8 indicates that the retailer would order less under the RP policy when the unit cost c is sufficiently low, say, when c < 2.6. Fig. 9 suggests that the optimal prices are increasing in the unit cost c, which is intuitive. Finally, Fig. 9 implies that the retailer should always set a higher retail price (in expectation) under the RP policy. In this section, we have presented a model for analyzing the retailer’s optimal expected profits, order quantities and prices under the two pricing policies. It is shown analytically and numerically that the Responsive Pricing dominates the No Responsive Pricing policy. In addition, it is demonstrated numerically that the Responsive Pricing policy is even more beneficial to the retailer when the supply yield is more uncertain (low mean or high variance) or when the unit ordering cost is high. Since the Responsive Pricing policy dominates the No Responsive Pricing policy, we shall focus our attention on the Responsive Pricing policy in the remainder of this paper. Specifically, in the next 2 sections, we shall extend our analysis of the Responsive Pricing policy to examine two issues: emergency order and supplier order allocation. 3. Extension 1: Emergency order under responsive pricing Let us consider a situation in which the retailer can place an emergency order from an alternative source as follows: after the retailer receives yQ non-defective units at the end of period 1, he can order (1 y)Q nondefective units from this perfect source and receive this emergency order immediately so as to bring the total non-defective units up to Q at the beginning of period 2. Here we assume that the alternative source can deliver non-defective emergency order. This perfect alternative source can be a superior supplier that commands a higher unit cost ce, where ce > c. By observing the fact that the retailer can always use the emergency order to ensure Q units are available at the beginning of period 2, we can formulate the retailer’s problem with emergency order under the RP policy as problem P(EO), where: Pe ¼ max cQ þ Ey ðce ð1 yÞQ þ maxfp minfQ; DggÞ: Q
p
Rearranging the terms, problem P(EO) can be simplified as: Pe ¼ max maxfPe ðQ; pÞg; where p (Q ; ^ c Q þ pQ if p 6 aQ b Pe ðQ; pÞ ¼ aQ ^cQ þ pða bpÞ if p > b ;
and
^c ¼c þ ce ð1 lÞ: By considering the first order condition, the optimal retail price pe(Q) is given as: ( aQ if Q 6 a2 ; e p ðQÞ ¼ ab if Q > a2 : 2b
ð3:1Þ
Since the retailer can always use the emergency order to bring the total non-defective units up to Q units at the beginning of period 2, it is easy to check from the expression for Pe that the pricing decision in period 2 would depend only on Q and that the ordering decision Q in period 1 would depend only on the expected supply yield. Substitute the optimal retail price pe(Q) into Pe(Q, p), getting: ( ÞQ if Q 6 a2 ; ^cQ þ ðaQ b e e P ðQ; p ðQÞÞ ¼ a2 if Q > a2 : ^cQ þ 4b It remains to determine the optimal order quantity Qe that maximizes the expected profit Pe(Q, pe(Q)). By considering the first order condition at the boundary points and assuming a > b^c, it is easy to show that the optimal order quantity and the retailer’s optimal expected profit can be expressed as: a b^c ; and 2 2 ða b^cÞ Pe ¼ : 4b Qe ¼
ð3:2Þ ð3:3Þ
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By comparing the optimal profits given in (2.13) and (3.3), we can establish the following Proposition. Proposition 3. It is optimal for the retailer to place an emergency order if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða bcÞ ðaukw bcÞ þ a2 v k k e ; c < ð1 lÞb where k* is given by (2.11). Proposition 3 suggests that it is optimal to use the emergency order when the unit cost associated with the emergency order is sufficiently small. Proposition 3 has the following implications. Consider the case when ce is low enough so that the retailer decides to use the emergency order. In this case, the retailer would place an order at the beginning of period 1 c according to the optimal order quantity Qe given in (3.2), i.e., Qe ¼ ab^ . Since Qe < a2, it is easy to check from 2 aQe e e e (3.1) that p ¼ b . This implies that Q = a bp . Combine this result with the fact that the retailer would always place an emergency order to bring the total number of non-defective units up to Qe at the beginning of period 2, we can conclude that the retailer would set the optimal retail price pe so as to sell all Qe units in period 2. 4. Extension 2: Supplier order allocation Consider the case when the retailer can order from multiple suppliers with different yield distributions and different unit costs. To simplify our exposition, we shall consider the 2-supplier case. Consider a situation in which the retailer orders Qi units from supplier i and Qj from supplier j. As such, the retailer will receive yiQi + yjQj non-defective units of the same product at the end of period 1. For s = i, j, it is assumed that the supply yield ys is a discrete random variable that P takes on N different values of ysn, where n = 1, . . . , N. Define the probability Prob{ys = ysn} = ksn so that Nn¼1 ksn ¼ 1 for s = i, j. Under the Responsive Pricing policy, the retailer would first determine the optimal order quantities Qi and Qj at the beginning of period 1. Then he would specify the optimal retail price p after the actual yield of each supplier is realized at the end of period 1. As such, there are three decision variables. To simplify our exposition, we shall transform our decision variables Qi and Qj as follows. Let Q be the total number of units ordered from the suppliers, where Q = Qi + Qj. Let f P 0 be the ratio between the order quantities; i.e., f ¼ QQi . Notice that f = 0 when the retailer orders only from supplier j and f = 1 when the j f 1 retailer orders only from supplier i. By noting that Qi ¼ 1þf Q and Qj ¼ 1þf Q, the retailer’s problem is based on the decision variables Q, f and p. In preparation, let cs be the unit cost of supplier s, s = i, j. For any order quantities Qi and Qj, the total ordering cost can be expressed in terms of caQ, where ca corresponds to the c f þc weighted average unit cost that can be expressed as ca ¼ i1þf j . By observing that the total number of nondefective units received by the retailer at the end of period 1 is equal to yimQi + yjnQj for some m, n = 1, 2, . . . , N, the total number of non-defective units can be expressed in terms of rQ, where r corresponds y f þy to the effective yield with N2 realizations rmn ¼ im1þf jn , and m,n = 1, 2, . . . , N. Notice that the expected effective y f þy l f þl yield la ¼ EðrÞ ¼ Ey i ;y j ð i1þf j Þ ¼ i1þf j . In this case, it is easy to check that, under the RP policy, the retailer’s supplier order allocation problem can be formulated as problem P(SOA), where: a a a P ¼ max c Q þ Er maxfp minfrQ; Dgg ¼ max max Er ðmaxfP ðQ; f ; pjrÞgÞ : ð4:1Þ Q;f
a
p
f
Q
p
Notice that P (Q, f, pjr) is the retailer’s profit associated with order quantity Q, ratio f and retail price p for any given realization of r. To solve problem P(SOA) given in (4.1), we first solve the subproblem for any given ratio f and then determine the optimal ratio fa using a simple search algorithm. For any given value of f, one can check from (4.1) that the subproblem is given by maxQEr(maxp{Pa(Q, f, pjr)}), where: ( a c Q þ prQ if p 6 arQ ; b a P ðQ; f ; pjrÞ ¼ ð4:2Þ arQ a c Q þ pða bpÞ if p > b :
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Suppose we treat the effective yield r as y. Then it is clear that the subproblem has the same structure as problem P(RP) given in (2.2). Therefore, the same approach presented in Section 2.3 can be used to solve this subproblem. Recall from section 2.3 that the solution approach for solving problem P(RP) hinges upon the boundary points 2ya , k = N, N 1, . . . , 1, as well as the terms uk, vk and wk. In order to apply the solution apk proach for solving problem P(RP) to solve our subproblem, we need to develop a new index l, y f þy l = 1, . . . , L = N2 for the effective yield r. To do so, sort the realized effective yield rmn ¼ im1þf jn in ascending order and then assign the new index l so that r1 < r2 < < rl < < rL. After creating the new index l for the effective yield r, define the terms uk, vk and wk as follows: uk ¼
k X
kl r l ;
vk ¼
l¼1
L X
kl ;
l¼kþ1
and
wk ¼
k X
kl r2l ;
ð4:3Þ
l¼1
where u0 = 0, vL = 0, and w0 = 0. Thus one can apply Proposition 2 in Section 2.3 to determine the optimal solution to the problem P(SOA) for any given ratio f as follows: Corollary 1. Suppose ls > bca s for s = i, j. Then, for any given ratio f, the optimal solutions to the retailer’s problem (4.1) can be expressed as follows: auka bca ; where 2wka wk bca k a ¼ argmax uk : k ¼ L; L 1; . . . ; 1 ; < rk a ( arQa a a if Q 6 2r ; pa ¼ a b if Qa > 2ra ; and 2b
Qa ¼
ð4:4Þ ð4:5Þ ð4:6Þ
2
Pa ¼
ðauka bca Þ a2 v k a : þ 4bwka 4b
ð4:7Þ
Given the retailer’s optimal expected profit for any given ratio f, we can determine the optimal ratio fa by using a simple search algorithm. While it is difficult to determine the optimal ratio fa analytically for the general case, we can determine fa for a special case. In this special case, there are two potential suppliers i and j. Supplier i’s yield is constant and it is equal to l. Supplier j’s yield has two possible values yj1 and yj2 with equal probability of 0.5 and supplier j’s expected yield is also equal to l. Hence, we can express yj1 = l r and yj2 = l + r. Given supply yield of both y f þy r suppliers, it is easy to check that the effective yield r ¼ i1þf j takes on two possible values, namely, r1 ¼ l f þ1 r and r2 ¼ l þ f þ1 with equal probability of 0.5. Since supplier i has a constant yield l and since supply j has uncertain yield with mean l, it is reasonable to assume that ci > cj. We now investigate the conditions under which the retailer would order from exactly one supplier or both r r suppliers. First, since r1 ¼ l f þ1 and r2 ¼ l þ f þ1 , it is easy to check from (4.3) that: r u0 ¼ 0; u1 ¼ 0:5 l ; u2 ¼ l; v0 ¼ 1; v1 ¼ 0:5; v2 ¼ 0; f þ1 2 r r2 w0 ¼ 0; w1 ¼ 0:5 l ; and w2 ¼ l2 þ : 2 f þ1 ðf þ 1Þ a
It follows from (4.5) in Corollary 2 that ka = 2 if u2 wr22 bca < 0, and ka = 1, otherwise. By considering the a terms u2, w2, r2 and ca as functions of f, it is easy to show that the function hðf Þ ¼ u2 wr22 bca is a quadratic function of f. Hence, there exists two roots s1 6 s2 that satisfy h(f) = 0. To simplify our exposition, we shall focus on the case when s1 < 0 and s2 > 0.3 By comparing the profit function Pa associated with different values of f, we can establish the following Proposition: 3
One can use the exact same approach to analyze other cases, say, when s1 > 0 and s2 > 0 or when s1 < 0 and s2 < 0 . We omit the details.
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Proposition 4. Suppose bci(l + r) + bcjl arl > 0 and bcj(l + r) + ar2 arl < 0. Then the optimal ka can be expressed as follows: ka ¼
1 2
if 0 6 f 6 s2 ; if s2 < f 6 1:
ð4:8Þ
Moreover, for any given value of f, the retailer’s optimal profit can be expressed as: 8 ½alðf þ1Þar2bðci f þcj Þ2 a2 > þ 8b if 0 6 f 6 s2 ; > < 8b½ðf þ1Þlr2 Pa ðf Þ ¼ > 2 > i f þcj Þ : ½alðf þ1Þbðc if s2 < f 6 1: 2
ð4:9Þ
4b½ðf þ1Þ l2 þr2
By comparing the profit function Pa(f) for different values of f, the optimal ratio fa can be expressed as follows: 2bci Proposition 5. Suppose bci(l + r) + bcjl arl > 0 and bcj(l + r) + ar2 arl < 0 and suppose a > lr . Then:
a
f ¼
8 <0 :
if r 6
lðci cj Þ ; ci
2 ½s2 ; 1Þ if r >
lðci cj Þ : ci lðc c Þ
Proposition 5 can be interpreted as follows. When r is sufficiently small, say, r 6 ici j , it is optimal for the retailer to order only from supplier j (i.e., fa = 0). This result is intuitive because supplier j will dominate supplier i when supply j’s unit cost cj is sufficiently lower or when the standard deviation of supplier j’s yield r is lðc c Þ sufficiently lower. On the other hand, when r is reasonably large so that r > ici j and the suppositions hold, it is optimal for the retailer to order from both suppliers. This implication is consistent with the portfolio theory in which an investor should invest in a stock portfolio instead of a single stock. To elaborate, notice that supplier i has a constant yield with a higher unit cost while supplier j has an uncertain yield with a lower cost. As supplier j’s yield becomes more variable (i.e., when rj is large), Proposition 5 suggests that it is optimal for the retailer to order from both suppliers so as to maintain an optimal tradeoff between lower unit cost and higher yield uncertainty.
5. Conclusion We have developed a two-stage stochastic model for determining the optimal order quantity and optimal retail price under supply uncertainty. Specifically, we show the Responsive Pricing policy dominates the No Responsive Pricing policy in terms of the retailer’s optimal expected profit. By examining the underlying structure of the problems associated with these two pricing policy, we have developed simple approaches for determining the optimal order quantity and retail price for any discrete supply yield distribution. By using these solution approaches, we examine the impact of yield distribution on the optimal order quantity, retail price, and retailer’s expected profit. We have also shown how to extend our analysis of the Responsive Pricing policy to examine two issues including emergency order and order allocation among multiple suppliers. Our model has certain limitations including deterministic demand and linear demand function. We plan to extend our model to examine the issue of responsive pricing under uncertain demand and uncertain supply in our future research. Recently, when the supply yield y is uniformly distributed; i.e., U[a b, a + b] and when the market potential a is normally distributed; i.e., N(l, r2), Tang and Yin (2006) reported that the retailer’s optimal order quantity under responsive pricing is increasing in demand uncertainty r. In addition, they showed that the retailer’s optimal profit under responsive pricing is decreasing in supply uncertainty b but increasing in demand uncertainty r. We plan to generalize these preliminary results in the near future.
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Appendix. Proofs Proof of Proposition 1. For given k = 0, 1, . . ., N, the problem Pk(NRP) in (2.3) can be rewritten as: P0k ¼ maxa maxfðuk p cÞQ þ pða bpÞvk g
ð7:1Þ
06p6b QP0
a bp a bp
subject to
Our solution approach is as follows. We first determine the optimal Q for any given p and then we determine the optimal p. For any given p 2 ½0; ba, the objective function in (7.1) is a linear function of Q with the coefficient ukp c. Let us consider the following cases: Case 1: Suppose ba > uck (or equivalently, bc a < uk ). Depending on the sign of (ukp c), it is easy check from (7.1) that ( abp if ba P p > uck ; yk 0 Qk ¼ abp if p 6 uck : y kþ1
Case 2: Suppose ba 6 uck (or equivalently, bca P uk ). Since p 6 ba and ba 6 uck , we have ukp c 6 0. Therefore, it is easy to check from (7.1) that Q0k ¼ abp . y kþ1 Let us examine further about two subproblems (a) and (b) associated with Case 1: (a) Suppose ba P p > uck . Then we can substitute Q0k ¼ abp into the objective function (7.1), getting a concave yk c a function of p. By considering the first order condition, it is easy to show that p01k ¼ 2ðuk þy þ 2b . Notice k vk Þ uk ðuk þvk y k Þ bc a 0 a c a 0 c P p1k always holds when b > uk . Therefore, the inequality b P p1k > uk holds if and only if a < uk þ2vk y . b k
k þvk y k Þ This implies that when bca < ukuðuk þ2v , the optimal price is given by p01k , the optimal order quantity is 2 k yk bc 0 a k þvk y k Þbc Q1k ¼ 2y 2ðuk þvk y Þy and the optimal expected profit is P01k ¼ ½aðu . However, when 4by k ðuk þvk y k Þ k k k uk ðuk þvk y k Þ bc 0 0 c P uk þ2vk y , the optimal price is given by p3k ¼ uk with expected profit P3k . a k (b) Suppose p 6 uck . Then we can substitute Q0k ¼ abp into the objective function (7.1) and get a concave y kþ1 a function of p. By considering the first order condition, we can show that p02k ¼ 2ðuk þyc vk Þ þ 2b . In this case, kþ1
u ðu þv y
Þ
k k kþ1 it is easy to show that the inequality p02k 6 uck holds if and only if bca P kuk þ2v . Hence, when k y kþ1 uk ðuk þvk y kþ1 Þ bc 0 P , the optimal price is given by p , the optimal order quantity is given by 2k a uk þ2vk y kþ1 ½aðuk þvk y kþ1 Þbc2 bc 0 0 a Q2k ¼ 2y 2ðuk þvk y Þy and the optimal expected profit is given by P2k ¼ 4by ðuk þvk y Þ . However, kþ1 kþ1 kþ1 kþ1 kþ1 u ðuk þvk y kþ1 Þ , the optimal price is given by p03k ¼ uck with expected profit P03k . when bca < kuk þ2v ky kþ1
Notice that the retailer’s optimal expected profit associated with Case 1 is given as u ðuk þvk y kþ1 Þ k þvk y k Þ P0k ¼ maxfP01k ; P02k ; P03k g. Since yk < yk+1, we have kuk þ2v < ukuðuk þ2v . We now use this observation to k y kþ1 k yk determine the optimal solution to problem Pk(NRP) associated with Case 1 as follows: u ðu þv y
Þ
k k kþ1 k þvk y k Þ 1. Suppose bca < kuk þ2v . Our observation implies that bca < ukuðuk þ2v . Hence, we can apply the results k y kþ1 k yk 0 0 0 0 0 0 obtained in (a) to show that Pk ¼ P1k ; pk ¼ p1k ; and Qk ¼ Q1k . This proves the first statement in the Proposition. u ðuk þvk y kþ1 Þ k þvk y k Þ k þvk y k Þ 2. Suppose bca P ukuðuk þ2v . Since kuk þ2v < ukuðuk þ2v , we can apply the results obtained in (b) to show that k yk k y kþ1 k yk 0 0 0 0 0 0 Pk ¼ P2k ; pk ¼ p2k ; and Qk ¼ Q2k . This proves half of the third statement in the Proposition.
u ðu þv y
Þ
k k kþ1 k þvk y k Þ 6 bca < ukuðuk þ2v . Then P0k ¼ maxfP01k ; P02k g and we can determine the optimal price and 3. Suppose kuk þ2v k y kþ1 k yk order quantity accordingly. This proves the second statement in the Proposition.
We now turn our attention to Case 2; i.e., when ba 6 uck . In this case, we have bc a P uk . Let us first substitute Q0k ¼ abp y kþ1 into the objective function (7.1), getting a concave function of p. Then, by examining the first order a condition, we can show that p02k ¼ 2ðuk þyc vk Þ þ 2b . We now determine the optimal solution to problem kþ1 Pk(NRP) associated with Case 2 as follows:
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1. Suppose p02k < ba (or equivalently, bca < uk þ vk y kþ1 Þ. Then the optimal price is p0k ¼ p02k . This proves half of the third statement in the Proposition. 2. Suppose bca P uk þ vk y kþ1 . Then Q0k ¼ 0; p0k is arbitrary, and P0k ¼ 0. This proves the fourth statement in the Proposition. This completes the proof.
h
Proof of Proposition 2. Let us differentiate the objective function given in (2.10) with respect to Q, getting: dEy ðPðQ; p jyÞÞ ¼ dQ
a 2wk uk c Q: b b
ð7:2Þ
First, let us consider the case when l > bca . Notice that the derivative is positive when Q is sufficiently small, say, when Q = xN+1 = 0 and is negative when Q is sufficiently large, say, Q ¼ x1 ¼ 2y1 . Also, by evaluating the 1 derivatives at the boundary points xk+1 and xk, one can use the definitions of uk and wk given in (2.5) and (2.9) to show that: " # k dEy ðPðQ; p jyÞÞ dEy ðPðQ; p jyÞÞ a X 1 1 2 ¼ k y < 0: m m dQ dQ b m¼1 y kþ1 y k Q¼xk Q¼xkþ1 Since xk+1 < xk, we can conclude that the first derivative of the objective function at the boundary points xN+1, xN, . . . , x0 are decreasing. Combine this observation with the fact that the objective function is piece-wise dE ðPðQ;p jyÞÞ dE ðPðQ;p jyÞÞ concave, we can conclude that there exists an k* such that y dQ jQ¼xk þ1 P 0 and y dQ jQ¼xk < 0. This implies that the optimal k* is given by (2.11). By considering the first order condition (7.2), we can show that the optimal order quantity Q* is given by (2.12). Given the optimal order quantity Q*, we can compute the optimal retail price p*(Q*jy) once the actual yield y is observed. In addition, we can substitute Q* into (2.10) to show that the retailer’s optimal expected profit under the responsive pricing policy is given by (2.13). dEy ðPðQ;p jyÞÞ Next, let us consider the case when l 6 bc 6 0 for Q P 0. Therefore dQ a . In this case, we have Q* = 0 is the optimal solution. This completes the proof. h Proof of Proposition 3. The proof is immediate. We omit the details.
h
Proof of Corollary 1. The proof follows directly from Proposition 2. We omit the details.
h
Proof of Proposition 4. The results follow from Corollary 1. Let g1 = bci(l + r) + bcjl arl and Then it is easy p toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi check that the two roots associated with the equation g2 = bcj(l + r) + ar2 parl. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 g21 4bci lg2 g1 þ g21 4bci lg2 and s2 ¼ . Suppose g1 > 0 and g2 < 0. Then we have s1 < 0 h(f) = 0 are: s1 ¼ 2bci l 2bci l and s2 > 0. First, let us consider the case when f > s2. In this case, ka = 2 and Pa(f) = A(f), where: 2
Aðf Þ ¼
½alðf þ 1Þ bðci f þ cj Þ 4b½ðf þ 1Þ2 l2 þ r2
ð7:3Þ
:
Second, consider the case when 0 6 f 6 s2. In this case, ka = 1 and Pa(f) = B(f), where: Bðf Þ ¼
½alðf þ 1Þ ar 2bðci f þ cj Þ2 8b½ðf þ 1Þl r
2
þ
a2 : 8b
ð7:4Þ a
Third, consider the case when f = 1. In this case, w2 = l2, r2 = l and u2 wr22 ¼ 0 < bca holds. Thus, we can conclude that ka = 2 and Pa (f = 1) = A(f = 1). This completes the proof. h
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255
Proof of Proposition 5. It follows from (4.9) that the retailer’s optimal expected profit Pa can be expressed as: Pa ¼ maxfmax06f 6s2 Bðf Þ; maxs2
ð7:5Þ
Suppose a is sufficiently large so that a(l r) > 2bci. Then it is easy to use the fact that ci > cj and l > r to show that a(l r) 2bcj + (al 2bci)f > 0 and that l(f + 1) r > 0. Applying this result to (7.5), we can Þ conclude that sign dBðf ¼ signðci ðr lÞ þ cj lÞ: Let us consider the following two cases: df Þ 6 0. Therefore B(f) is decreasing in 1. When ci(r l) + cjl 6 0. In this case, we have dBðf df f 2 ½0; 1; Bð0Þ ¼ max06f 61 Bðf Þ P maxfmax06f 6s2 Bðf Þ; maxs2 0 and s2 = argmax{B(f) : 0 6 f 6 s2}. It follows from 2. When ci(r l) + cjl > 0. In this case, we have dBðf df (7.3) that:
dAðf Þ ððal bcj Þ þ f ðal bci ÞÞððal bci Þr2 bl2 ð1 þ f Þðci cj ÞÞ ¼ : 2 df 2bðl2 þ r2 þ 2l2 f þ l2 f 2 Þ
ð7:6Þ
Observe that (al bci)r2 bl(1 + f)(ci cj) < 0 when f is sufficiently large. Combine this observation with Þ < 0 when f is sufficiently large. the fact that (al bcj) + f(al bci) > 0, we can check from (7.6) that dAðf df a Therefore, argmax{A(f) : s2 < f 6 1} < 1, which implies that f 2 [s2, 1). We can prove the Proposition by considering the results associated with these two cases.
h
References Chod, J., Rudi, N., 2005. Resource flexibility with responsive pricing. Operations Research 53 (3), 532–548. Fisher, M., Raman, A., 1996. Reducing the cost of demand uncertainty through accurate response. Operations Research 44 (1), 87–99. Lee, H., Tang, C.S., 1997. Modeling the costs and benefits of delayed product differentiation. Management Science 43 (1), 40–53. Petruzzi, N.C., Dada, M., 1999. Pricing and the newsvendor problem: A review with extensions. Operations Research 47 (2), 183–194. Ramasesh, R., Ord, J., Hayya, J., Pan, A., 1991. Sole versus dual sourcing in stochastic lead time (s, Q) inventory models. Management Science 37 (4), 428–443. Tang, C.S., 2005. Perspectives in Supply Chain Risk Management: A Review. working paper UCLA Anderson School, November. Tang, C.S., Yin, R., 2006. Responsive Pricing under Uncertain Supply and Uncertain Demand. INFORMS Hong Kong Conference Presentation. Tang, C.S., Rajaram, K., Alptekinoglu, A., Ou, J.H., 2004. The benefits of advance booking discount programs. Management Science 50 (4), 465–478. Van Mieghem, J., Dada, M., 1999. Price versus production postponement: capacity and competition. Management Science 45 (12), 1631– 1649. Yano, C., Lee, H., 1995. Lot sizing with random yields: A review. Operations Research 43 (2), 311–334.
Production, Manufacturing and Logistics
Responsive pricing under supply uncertainty Christopher S. Tang *, Rui Yin
*
UCLA Anderson School, UCLA, 110 Westwood Plaza, CA 90095, USA Received 15 January 2006; accepted 17 July 2006 Available online 24 October 2006
Abstract Consider a retailer orders a seasonal product from a supplier and sells the product over a selling season. While the product demand is known to be a linear function of price, the supply yield is uncertain and is distributed according to a general discrete probability distribution. This paper presents a two-stage stochastic model for analyzing two pricing policies: No Responsive Pricing and Responsive Pricing. Under the No Responsive Pricing policy, the retailer would determine the order quantity and the retail price before the supply yield is realized. Under the Responsive Pricing policy, the retailer would specify the order quantity first and then decide on the retail price after observing the realized supply yield. Therefore, the Responsive Pricing policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. Our analysis suggests that the retailer would always obtain a higher expected profit under the Responsive Pricing policy. In addition to examining the impact of yield distribution and system parameters on the optimal order quantities, retail prices, and profits under these two pricing policies, we analyze two issues arising from responsive pricing. The first issue deals with the case in which the retailer can place an emergency order with an alternative source after observing the realized yield, while the second issue deals with a situation in which the retailer has to allocate his order among multiple suppliers. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Responsive pricing; Uncertain supply; Supply management
1. Introduction Due to long supply lead time and short selling season, retailers usually can place their orders only once before the start of the selling season. For example, in the fashion industry, it is common for retailers to place their orders many months before the selling season. The reader is referred to Fisher and Raman (1996) for an excellent description of the ordering process in the fashion industry. Since accurate supply or demand information is rarely available in advance, it is difficult for retailers to determine cost effective order quantities. For instance, due to uncertain supply yields, transportation delays, shrinkage during shipment, retailers usually do not know for sure if they would receive the exact quantity ordered prior to the selling season. Besides uncertain supply, retailers often face uncertain demand as well. As such, many retailers have to struggle with the *
Corresponding authors. Tel.: +1 310 825 4203 (C.S. Tang). E-mail addresses: [email protected] (C.S. Tang), [email protected] (R. Yin).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.07.011
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overstocking and understocking issues. Specifically, overstocking forces a retailer to dispose of unsold items at clearance prices, while understocking creates lost sales. See Tang (2005) for a review of supply chain risk management. To reduce the overstocking and understocking costs, various researchers have examined different approaches for helping retailers to meet uncertain demand. First, Fisher and Raman (1996) consider a situation in which a retailer can place two separate orders prior to the selling season. By allowing the retailer to place the second order in a later period, the retailer can generate a more accurate demand forecast by using the market signals observed during the time between the first order and the second order. Fisher and Raman refer the second order as an ‘accurate response’ for managing uncertain demand. Next, Tang et al. (2004) consider a situation in which a retailer offers each customer two options: pre-commit an order at a reduced price before the selling season, or buy the product at the regular price during the selling season. Clearly, the reduced price serves as an incentive for the customers to pre-commit their orders before the selling season. Tang et al. (2004) show how the retailer can use these pre-committed orders to generate more accurate demand forecasts, which would enable the retailer to place more cost effective orders and manage uncertain demand more efficiently. The ideas articulated in Fisher and Raman (1996) and Tang et al. (2004) would certainly improve supply chain performance; however, some suppliers and retailers may have specific concerns about the requirements associated with these two ideas. For example, to implement the accurate response concept, the supplier needs to have sufficient capacity to handle the second order on short notice. Also, to implement the early commitment program, a retailer need to show his customers a sample of the seasonal products before the selling season, which could make the retailer or the manufacturer more vulnerable to copycats. This paper examines a situation in which the accurate response or the early commitment program is impractical due to the aforementioned concerns. Instead of having the flexibility to place two separate orders or to sell the product at two different retail prices, we consider a situation in which the retailer can place exactly one order and select only one retail price before the selling season. This concept is known as ‘responsive pricing’. Van Mieghem and Dada (1999) is the first to analyze the responsive pricing concept as a mechanism for a retailer to manage uncertain demand. In their paper, they present a two-stage stochastic model in which the retailer places an order in the first period. Then the retailer would determine the retail price after the demand uncertainty is resolved at the end of the first period but before the selling season that starts at the beginning of the second period. They show the benefits of delaying the pricing decision until the demand uncertainty is resolved. Motivated by the product postponement concept examined in Lee and Tang (1997), Chod and Rudi (2005) extend the work of Van Mieghem and Dada to the two-product case. Specifically, they consider the case in which the retailer places an order of a ‘generic’ product in the first period. Then, after the demand uncertainty is resolved, the retailer would customize this order of generic product into two individual products and then determine the retail price for each of these two products. Given the flexibility of delaying the product identity and the pricing decision until demand uncertainty is resolved, Chod and Rudi present a two-period model to illustrate the benefit of product postponement under responsive pricing. While Van Mieghem and Dada (1999) and Chod and Rudi (2005) focus on the issue of demand uncertainty, this paper examines the benefits of responsive pricing under supply uncertainty. As an initial attempt to analyze the issue of responsive pricing under supply uncertainty, we shall consider a situation in which the demand function is known but the supply yield is uncertain. By focusing on the issue of uncertain supply, we develop tractable results which can serve as building blocks for analyzing responsive pricing under uncertain supply and uncertain demand in the near future. In the context of supply uncertainty, various researchers have developed models for determining the optimal order quantity under uncertain supply. The reader is referred to Yano and Lee (1995) for a comprehensive review of lot sizing models with uncertain supply yield. To our knowledge, this paper is the first that examines the joint decisions of order quantity and retail pricing under supply uncertainty. This paper presents a two-stage stochastic model for analyzing two pricing policies: No Responsive Pricing and Responsive Pricing. Under the No Responsive Pricing policy, the retailer would determine the order quantity and the retail price at the beginning of the first period before the supply yield is realized. Under the Responsive Pricing policy, the retailer would determine the order quantity at the beginning of the first period. Then, after the actual supply yield is realized at the end of the first period, the retailer would then determine the retail price at the end of the first period but before the selling season that starts at the beginning of the second period. By delaying the pricing decision after the actual supply yield is realized,
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241
the Responsive Pricing policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. This paper is divided into two parts. In the first part, it is shown analytically that, under supply uncertainty, the retailer would always obtain a higher expected profit under the Responsive Pricing policy. (This result is consistent with the result obtained by Van Mieghem and Dada (1999) when they analyze the benefits of responsive pricing under demand uncertainty.) We also examine the impact of yield distribution and system parameters on the optimal order quantities, prices, and retailer’s profits under these two pricing policies numerically. The numerical analysis indicates that the Responsive Pricing policy is more beneficial when the supply yield is highly uncertain (low mean or high variance) or when the unit cost is high. The analysis of the Responsive Pricing is extended to examine two separate issues in the second part of this paper. The first issue deals with the case in which the retailer can place an emergency order with an alternative source after observing the actual supply yield, while the second issue deals with a situation in which the retailer has to allocate his order among multiple suppliers. Ramasesh et al. (1991) develop a model for analyzing ways to allocate an order among different suppliers with different lead times, while we examine the issue of order allocation under responsive pricing. This paper is organized as follows. In Section 2, we present the base model for analyzing the retailer’s optimal expected profits under the No Responsive Pricing policy and the Responsive Pricing policy. It is shown analytically that the retailer would always obtain a higher expected profit under the Responsive Pricing policy. We also develop numerical experiments to examine the benefits of the Responsive Pricing policy. Section 3 extends our analysis of the Responsive Pricing policy to the case in which the retailer can place an emergency order after the actual supply yield is realized. Section 4 deals with a situation in which the retailer can order from multiple suppliers with different supply yield distributions. This paper is concluded in Section 5. 2. The base model Consider a situation in which a retailer orders a seasonal product from a supplier and sells the product over a selling season. We assume that the retailer knows that the product demand function is given by: D = a bp, where a > 0 represents the potential market size, b > 0 represents the price sensitivity, and p represents the retail price. While the retailer has perfect information about the demand function, he has to deal with supply uncertainty in the following manner. First, the supply lead time is equal to one period, and hence, the retailer needs to decide on the order quantity Q at the beginning of period 1 so that the retailer will receive the order at the end of period 1 prior to the selling season that starts at the beginning of period 2 and ends at the end of period 2. Second, the supply yield is uncertain in the following sense. For any order quantity Q, the retailer will receive only yQ non-defective units, where y represents the supply yield (essentially, the yield y accounts for the defective rate of the supplier as well as the damage or shrinkage occurred during shipment). It is assumed that y is a random variable that takes on N different discrete values, say, yn for n = 1, 2, . . . , N, where 0 < y1 < y2 < < yN1 < yN 6 1.1 Let kn P be the probability that the actual yield is equal to yn; i.e., ProN b{y = yn} = kn for n = 1, 2, . . . , N. Hence, n¼1 kn ¼ 1. Third, the retailer pays the supplier c per unit at the beginning of period 1, and the retailer disposes of the unsold units at s per unit at the end of period 2. Without loss of generality, it is easy to show that one can transform the cost parameters so that one can set s = 0. To simply our exposition, we shall consider the case when s = 0. This paper examines two pricing policies: No Responsive Pricing (NRP) policy and Responsive Pricing (RP) policy. Under the NRP policy, the retailer specifies the order quantity Q and the retail price p jointly at the beginning of period 1. Under the RP policy, the retailer first decides on the order quantity Q at the beginning of period 1. Then he specifies the retail price p at the beginning of period 2 after he observes the actual supply yield realized at the end of period 1. As such, the RP policy allows the retailer to delay 1 The uncertain supply yield has a general discrete distribution, which allows us to approximate any continuous distribution. Notice that our way of capturing uncertainty is different from the models examined by Van Mieghem and Dada (1999) and Chod and Rudi (2005). First, we examine supply uncertainty, while they consider demand uncertainty. Second, we assume that the supply yield follows a general discrete probability distribution; however, they assume that the demand function is additive (a linear function of price plus an uncertain error term) and that the distribution of the error term is Uniform or Exponential in the former and Bivariate Normal in the latter.
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the pricing decision until the actual supply yield is realized. In other words, the RP policy enables the retailer to use pricing as a response mechanism for managing uncertain supply. Given these two pricing strategies, we are interested in examining the following questions: 1. Would the retailer obtain a higher expected profit under the RP policy? If yes, by how much? 2. How would the yield distribution affect the retailer’s optimal expected profits, order quantities, and prices under these two pricing policies? To answer these questions, we now formulate the retailer’s problem under these two policies. 2.1. Problem formulation First, under the NRP policy, the retailer has to determine the optimal order quantity Q 0 and optimal retail price p 0 at the beginning of period 1 so as to maximize the retailer’s expected profit. In this case, the retailer’s problem P(NRP) can be formulated as follows: P0 ¼ max Ey ðcQ þ p minfyQ; DgÞ: Q;p
ð2:1Þ
Next, under the RP policy, the retailer would first determine the order quantity Q at the beginning of period 1. Then the retailer would determine the retail price p at the beginning of period 2 after observing the actual yield realized at the end of period 1. In order to determine the optimal order quantity Q* and the optimal retail price p* so that the retailer’s expected profit is maximized, we can formulate the retailer’s problem P(RP) as follows: P ¼ max cQ þ Ey ðmaxfp minfyQ; DggÞ: Q
p
ð2:2Þ
Let us compare the retailer’s expected profits under these two policies. Suppose we implement the optimal order quantity Q 0 and the optimal retail price p 0 under the NRP policy. Then it is easy to check from (2.2) and (2.1) that P P cQ0 þ Ey ðp0 minfyQ0 ; ða bp0 ÞgÞ ¼ P0 : This implies that the optimal expected profit under the RP policy is always higher than that under the NRP policy. This result is intuitive because, under the Responsive Pricing policy, the retailer has the flexibility to select a ‘more profitable’ retail price after observing the actual yield. While it is clear that the RP policy enables the retailer to obtain a higher profit, we would like to compare the retailer’s optimal expected profits, order quantities, and retail prices under both pricing policies and to examine the impact of the yield distribution on these quantities. However, such comparison is analytically intractable, but it can be done numerically. In the remainder of this section, we first determine the retailer’s optimal expected profits, order quantities, and prices under these two policies analytically. We then compare these quantities numerically. 2.2. Analysis of the no responsive pricing policy Since D = a bp, the No Responsive Pricing problem P(NRP) given in (2.1) can be rewritten as: P0 ¼ max Ey ðP0 ðQ; pjyÞÞ; where Q;p 8 < cQ þ pyQ ; if y 6 abp Q 0 P ðQ; pjyÞ ¼ abp : cQ þ pða bpÞ if y > : Q Essentially, P 0 (Q, pjy) represents the retailer’s profit associated with order quantity Q and retail price p for any given realization of yield y. The underlying structure of problem P(NRP) resembles the joint pricing and ordering decision for the Newsvendor problem examined by Petruzzi and Dada (1999). While there is no simple closed form solution for this problem, we now present a simple approach for solving problem P(NRP).
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Suppose we set y0 = 0 and yN+1 = Y, where Y is a sufficiently large number. For any given Q and p, there must exist a k, k = 0, 1, . . . , N, so that the ratio abp satisfies y k 6 abp < y kþ1 . For this particular k, we can take Q Q 0 the expectation of P (Q, pjy) to show that the retailer’s expected profit can be rewritten as: Ey fP0 ðQ; pjyÞg ¼
k X
km ðcQ þ py m QÞ þ
m¼1
N X
km ðcQ þ pða bpÞÞ
if y k 6
m¼kþ1
P0
a bp < y kþ1 : Q
PN
By denoting the terms 1 ¼ 0 and N þ1 ¼ 0, we can decompose problem P(NRP) into N + 1 subproblems Pk(NRP), where k = 0, 1, . . . , N and each subproblem Pk(NRP) can be expressed as: ( ) k N X X 0 Pk ¼ max km ðcQ þ py m QÞ þ km ðcQ þ pða bpÞÞ ð2:3Þ Q;p
m¼1
subject to
m¼kþ1
a bp < y kþ1 : yk 6 Q
Let Q0k and p0k be the optimal order quantity and retail price for subproblem Pk(NRP), respectively. Also, let k 0 2 argmaxfP0k : k ¼ 0; 1; 2; . . . ; N g. In this case, one can utilize the solutions of these N + 1 subproblems Pk(NRP) to determine the optimal expected profit and the optimal solutions to the original problem P(NRP) as follows: P0 ¼ P0k0 ;
Q0 ¼ Q0k0 ;
p0 ¼ p0k0 :
ð2:4Þ
However, the uniqueness of the above optimal solution could not be established for the general discrete distribution, which is consistent with the result obtained by Van Mieghem and Dada (1999). It remains to determine the optimal solutions to subproblem Pk(NRP). In preparation, let us define two partial sums that will become useful. Let: uk ¼
k X
km y m
and
N X
vk ¼
m¼1
ð2:5Þ
km :
m¼kþ1
For notational convenience, P let u0 = 0 and vN = 0. In this case, it is easy to check from the definitions that uk is increasing in k and uN ¼ Nn¼1 kn y n ¼ EðyÞ ¼ l. Also, note that vk is decreasing in k and v0 = 1. By considering the terms uk and vk along with subproblem Pk(NRP) given in (2.3), we can establish the following Proposition: Proposition 1. The solutions for subproblem Pk(NRP) can be expressed as follows: 1. Suppose 2. Suppose
p0k ¼
u ðuk þvk y kþ1 Þ bc c < kuk þ2v . Then p0k ¼ 2ðuk þy a k y kþ1 k vk Þ uk ðuk þvk y kþ1 Þ uk ðuk þvk y k Þ bc 6 < . Then a uk þ2vk y kþ1 uk þ2vk y k
a þ 2b ; Q0k ¼ 2ya 2ðuk þvbck y k
2
k Þy k
k þvk y k Þbc and P0k ¼ ½aðu : 4by ðuk þvk y Þ k
8 > > > <
c a þ 2ðuk þ y k vk Þ 2b
if
½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > ; y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ
> > > :
c a þ 2ðuk þ y kþ1 vk Þ 2b
if
½aðuk þ vk y k Þ bc2 ½aðuk þ vk y kþ1 Þ bc2 6 ; y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ
2
2
8 2 2 a bc ½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > > > > ; if < 2y 2ðu þ v y Þy y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ k k k k k Q0k ¼ 2 2 > a bc ½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > > : 6 ; if 2y kþ1 2ðuk þ vk y kþ1 Þy kþ1 y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ 8 ½aðuk þ vk y k Þ bc2 ½aðuk þ vk y k Þ bc2 ½aðuk þ vk y kþ1 Þ bc2 > > > if > ; < 4by k ðuk þ vk y k Þ y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ P0k ¼ 2 2 2 > ½aðuk þ vk y kþ1 Þ bc ½aðuk þ vk y k Þ bc ½aðuk þ vk y kþ1 Þ bc > > : if 6 : 4by kþ1 ðuk þ vk y kþ1 Þ y k ðuk þ vk y k Þ y kþ1 ðuk þ vk y kþ1 Þ
k
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uk ðuk þvk y k Þ uk þ2vk y k ½aðuk þvk y kþ1 Þbc2 : 4by kþ1 ðuk þvk y kþ1 Þ
3. Suppose
4. Suppose
bc a
6 bca < uk þ vk y kþ1 .
Then
p0k ¼ 2ðuk þyc
kþ1 vk Þ
a þ 2b ; Q0k ¼ 2ya 2ðuk þvkbc y kþ1
kþ1 Þy kþ1
and P0k ¼
P uk þ vk y kþ1 . Then Q0k ¼ 0; p0k is arbitrary and P0k ¼ 0.
Proof. All proofs are given in the Appendix. Proposition 1 can be interpreted via the following cases. First, consider the case Pk PNwhen the expected Pk yield is bc bc sufficiently low, say, when EðyÞ ¼ l 6 . In this case, P l ¼ k y þ k y P m¼1 m m m¼kþ1 m m m¼1 km y m þ a a PN ½ m¼kþ1 km y kþ1 ¼ uk þ vk y kþ1 for k = 0, 1, 2, . . . , N. By applying the fourth statement in Proposition 1, Q0k ¼ 0 for k = 0, 1, 2, . . . , N. Hence, when the expected yield l is sufficiently low; i.e., when l 6 bca , it is optimal for the retailer to order nothing; i.e., Q 0 = 0. Second, consider the case when the expected yield is suffiu ðuk þvk y kþ1 Þ ciently high, say, when l > bca . In this case, it is easy to check from (2.5) that kuk þ2v ¼ l > bca when k y kþ1 k = N. Hence, the first statement in Proposition 1 implies that it is optimal for the retailer to order something when k = N; i.e., Q0N > 0. Combine this observation with the fact that P0 P P0N , it is easy to show that the retailer’s optimal order quantity Q 0 > 0 when the expected yield l is sufficiently high; i.e., when l > bca . These two cases enable us to establish the threshold for l above which the retailer would order something in the optimal solution. 2.3. Analysis of the responsive pricing policy We now analyze the retailer’s optimal expected profit under the Responsive Pricing policy. Observe from (2.2) that problem P(RP) can be rewritten as: P ¼ max Ey maxfPðQ; pjyÞg ; ð2:6Þ Q
p
where P(Q, pjy) is the retailer’s profit associated with order quantity Q and retail price p for any given realization of y. The term P(Q, pjy) can be expressed as: ( cQ þ pyQ if p 6 ayQ ; b PðQ; pjyÞ ¼ : cQ þ pða bpÞ if p > ayQ b By differentiating P(Q, pjy) with respect to p and by considering the break points, we can determine the optimal retail price p*(Qjy) associated with order quantity Q placed at the beginning of period 1 and the actual yield y realized at the end of period 1 as follows: ( ayQ if Q 6 2ya ; p ðQjyÞ ¼ a b ð2:7Þ if Q > 2ya : 2b Substitute the optimal retail price p*(Qjy) into P(Q, pjy), getting: ( cQ þ ðayQ ÞyQ if Q 6 2ya ; b PðQ; p jyÞ ¼ a2 cQ þ 4b if Q > 2ya :
ð2:8Þ
Before taking the expectation of P(Q, p*jy) with respect to y, let us define an additional partial sum in addition to the terms uk and vk defined in (2.5). Let: k X wk ¼ km y 2m ; ð2:9Þ m¼1
where w0 = 0. Notice that wk is increasing in k. Define the following break points: xN þ1 ¼ 0; xN ¼ 2ya ; . . . ; xk ¼ N a , for k = N, N 1, . . . , 1, and x0 = X, where X is a large number. Since y1 < y2 < < yN1 < yN, xk is 2y k decreasing in k; i.e., xN+1 < xN < xN1 < < x1 < x0. In this case, there must exist a k so that xk+1 < Q 6 xk. By using the terms uk, vk and wk and by using (2.8), the retailer’s expected profit Ey(P(Q, p*jy)) can be expressed as follows:
C.S. Tang, R. Yin / European Journal of Operational Research 182 (2007) 239–255
Ey ðPðQ; p jyÞÞ ¼
a wk a2 v k uk c Q Q 2 þ b b 4b
245
if xkþ1 < Q 6 xk ;
ð2:10Þ
2
where k = N, N 1, . . . , 0. Since the function ðba uk cÞQ wbk Q2 þ a4bvk is concave in Q, it is easy to see that the objective function of problem P(RP) given in (2.6) is a piece-wise concave function. By evaluating the derivative of this piece-wise concave function at various break points xk ¼ 2ya , the unique optimal solution to the k problem P(RP) can be determined as follows: Proposition 2. Suppose bc a < l. Then wk bc k ¼ argmax uk < : k ¼ N ; N 1; . . . ; 1 : a yk
ð2:11Þ
Also, the optimal order quantity Q* and the retailer’s optimal expected profit P* under the Responsive Pricing policy can be expressed as: Q ¼
auk bc ; 2wk
P ¼
ðauk bcÞ a2 v k : þ 4bwk 4b
and
ð2:12Þ
2
Moreover, suppose
bc a
ð2:13Þ
P l. Then Q* = 0 and P* = 0.
It follows from Propositions 1 and 2 that it is optimal for the retailer to order nothing under both pricing policies when the expected yield l is sufficiently small, say, when l 6 bca . Therefore, it suffices to focus on the case when l > bca throughout the paper. 2.4. Numerical analysis We now utilize Propositions 1 and 2 to construct numerical experiments for comparing the retailer’s optimal expected profits, order quantities, and prices under the NRP and the RP policies. In addition, we shall examine the impact of the yield distribution and the unit cost on these quantities. In the numerical experiments, the supply yield follows a ‘discrete’ Uniform distribution over [a, b], where 0 < a < b 6 1. Specifically, the supply yield takes on N possible values y1 < y2 < < yN, where y k ¼ a þ Nba ðk 1Þ and Pro1 b(y = yk) = 1/N for k = 1, 2, . . . , N.22 Hence, the expected supply yield l ¼ EðyÞ ¼ aþb and the variance of 2 ðN þ1Þ the supply yield r2 ¼ VarðyÞ ¼ ðabÞ . 12ðN 1Þ In our experiments, N = 1000, a = 100, b = 5 and the values of a, b and c are varied so as to examine the impact of the yield distribution and the unit cost. In each set of experiments, the retailer’s optimal expected profits and the optimal order quantities under the NRP and the RP policies are computed. Since the optimal retail price under the Responsive Pricing p*(Q*jy) given in (2.7) is based on the actual realization of the supply yield, we shall compare the optimal retail price under the No Responsive Pricing policy with the expected optimal retail price Ey(p*(Q*jy)) under the Responsive Pricing policy. 2.4.1. The impact of supply yield uncertainty In the first set of experiments, we examine the impact of the standard deviation of the supply yield r on the retailer’s optimal expected profits, order quantities, and prices under the NRP and the RP policies. To do so, we set the unit cost c = 3 and vary the standard deviation r. To isolate the effect of r, the standard deviation of the supply yield r is varied while l is kept constant at l ¼ aþb ¼ 0:5. Specifically, the value of a is increased 2 from 0.1 to 0.48 and b is decreased from 0.9 to 0.52 according to an increment of 0.02. As we vary the values of a and b, the standard deviation of the supply yield r is decreased from 0.23 to 0.012. The impact of r on various quantities are reported in Figs. 1–3. Fig. 1 confirms that the retailer would always obtain a higher 2 Since our analysis is based on the assumption that the supply yield has a general discrete distribution, we can use our general discrete distribution to approximate any continuous probability distribution. For example, our discrete Uniform distribution converges to the (continuous) Uniform distribution over [a, b] when N ! 1.
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300 250 Profit
200 150 100
Profit (RP) Profit (NRP)
50 0 0
0.05
0.1
0.15
0.2
0.25
Sigma
Order quantity
Fig. 1. The impact of sigma on optimal profits.
80 70 60 50 40 30 20 10 0
Q (RP) Q (NRP)
0
0.05
0.1
0.15
0.2
0.25
Sigma Fig. 2. The impact of sigma on optimal order quantities.
Retail price
14.4 14.2
Price (RP)
14 13.8
Price (NRP)
13.6 13.4 13.2 13 12.8 0
0.05
0.1
0.15
0.2
0.25
Sigma Fig. 3. The impact of sigma on optimal prices.
expected profit under the RP policy; i.e., P* > P 0 . Observe from Fig. 1 that the profit gap P* P 0 increases as the supply yield becomes more variable; i.e., when r increases. This observation implies that the RP policy is more beneficial when the supply yield is more variable. Next, notice from Fig. 2 that the retailer would order more under the RP policy. This result is intuitive because the retailer can afford to order more at the beginning of period 1 because he has the flexibility to set the price after the actual yield is realized. Observe from Fig. 2 that the optimal order quantities are decreasing in r under both pricing strategies. This result is consistent with the well-known property of the optimal newsvendor order quantity (i.e., the optimal newsvendor order quantity decreases as the demand uncertainty increases). Finally, Fig. 3 suggests that the optimal retail prices are increasing in r under both pricing strategies. However, as the supply yield becomes more uncertain, the retailer would charge a much higher price under the RP policy. Combining the results displayed in Figs. 2 and 3, we can conclude that, relatively speaking, the retailer would place a larger order and charge a higher price (in expectation) under the RP policy as the standard deviation of the supply yield r increases. This observation explains why the profit gap P* P 0 increases as the supply yield becomes more variable in Fig. 1.
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247
2.4.2. The impact of the expected supply yield In the second set of experiments, we examine the impact of the expected supply yield l on the retailer’s optimal expected profits, order quantities, and prices. To do so, we set the unit cost c = 3 and vary l. To isolate the effect of l, the value of a is varied from 0.1 to 0.6 according to an increment of 0.02 and the value of b is set at b = a + 0.4. This would allow us to vary the expected supply yield l from 0.3 to 0.8 while keeping r2 conðabÞ2 ðN þ1Þ 2 stant, where r ¼ 12ðN 1Þ ¼ 0:0133. The computational results are summarized in Figs. 4–6. Fig. 4 suggests that, under both pricing policies, the retailer’s optimal expected profits increase as the expected supply yield l increases. This result is intuitive because the retailer should be able to obtain a higher expected profit when the 350 300 Profit
250 200 150
Profit (RP)
100
Profit (NRP)
50 0 0.3
0.4
0.5
0.6
0.7
0.8
μ Fig. 4. The impact of l on profits.
80
Order quantity
70 60 50 40 30
Q (RP)
20
Q (NRP)
10 0 0.3
0.4
0.5
0.6
0.7
0.8
μ Fig. 5. The impact of l on optimal order quantities.
16
Retail price
15.5
Price (RP)
15
Price (NRP)
14.5 14 13.5 13 12.5 12
0.3
0.4
0.5
0.6
0.7
μ Fig. 6. The impact of l on optimal prices.
0.8
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expected supply yield is higher. Notice from Fig. 5 that, under both pricing policies, the optimal order quantities are not monotonic in l. In addition, unlike the result reported in Fig. 2, Fig. 5 suggests that the retailer would order less under the Responsive Pricing policy when the expected supply yield l is sufficiently high, say, when l > 0.56. Observe from Fig. 6 that, under both pricing policies, the optimal retail prices are decreasing in l. This result is intuitive. However, it is interesting to observe from Fig. 6 that, under the RP policy, the retailer would charge a lower price (in expectation) when l is sufficiently large.
Profit
2.4.3. The impact of unit cost In our third set of experiments, we investigate the impact of the unit cost c on the retailer’s optimal expected profits, order quantities, and prices under these two policies. To do so, we set a = 0.1, b = 1, and we vary c from 1 to 10 according to an increment of 0.2. The computational results are summarized in Figs. 7–9. Fig. 7 confirms that the retailer would obtain a higher expected profit under the Responsive Pricing policy; however, the profit gap P* P 0 decreases as the unit cost c increases. Moreover, if one measures the relative 0 benefit of RP policy over the NRP policy according to the relative profit gain D ¼ P PP 0 , then one can check that the relative profit gain D is increasing in the unit cost c. The corresponding figure is omitted here. There400 350 300 250 200 150 100 50 0
Profit (RP) Profit (NRP)
1
3
5 7 Unit Cost
9
Order quantity
Fig. 7. The impact of unit cost on profits.
120 100 80 60 40 20 0
Q (RP) Q (NRP)
1
3
5 7 Unit Cost
9
Fig. 8. The impact of unit cost on optimal order quantities.
Retail price
20 18 16 14
Price (RP)
12
Price (NRP)
10
1
3
5 7 Unit Cost
9
Fig. 9. The impact of unit cost on optimal prices.
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fore, we can conclude that the RP policy becomes more beneficial when the unit cost c is high. Next, Fig. 8 indicates that the retailer would order less under the RP policy when the unit cost c is sufficiently low, say, when c < 2.6. Fig. 9 suggests that the optimal prices are increasing in the unit cost c, which is intuitive. Finally, Fig. 9 implies that the retailer should always set a higher retail price (in expectation) under the RP policy. In this section, we have presented a model for analyzing the retailer’s optimal expected profits, order quantities and prices under the two pricing policies. It is shown analytically and numerically that the Responsive Pricing dominates the No Responsive Pricing policy. In addition, it is demonstrated numerically that the Responsive Pricing policy is even more beneficial to the retailer when the supply yield is more uncertain (low mean or high variance) or when the unit ordering cost is high. Since the Responsive Pricing policy dominates the No Responsive Pricing policy, we shall focus our attention on the Responsive Pricing policy in the remainder of this paper. Specifically, in the next 2 sections, we shall extend our analysis of the Responsive Pricing policy to examine two issues: emergency order and supplier order allocation. 3. Extension 1: Emergency order under responsive pricing Let us consider a situation in which the retailer can place an emergency order from an alternative source as follows: after the retailer receives yQ non-defective units at the end of period 1, he can order (1 y)Q nondefective units from this perfect source and receive this emergency order immediately so as to bring the total non-defective units up to Q at the beginning of period 2. Here we assume that the alternative source can deliver non-defective emergency order. This perfect alternative source can be a superior supplier that commands a higher unit cost ce, where ce > c. By observing the fact that the retailer can always use the emergency order to ensure Q units are available at the beginning of period 2, we can formulate the retailer’s problem with emergency order under the RP policy as problem P(EO), where: Pe ¼ max cQ þ Ey ðce ð1 yÞQ þ maxfp minfQ; DggÞ: Q
p
Rearranging the terms, problem P(EO) can be simplified as: Pe ¼ max maxfPe ðQ; pÞg; where p (Q ; ^ c Q þ pQ if p 6 aQ b Pe ðQ; pÞ ¼ aQ ^cQ þ pða bpÞ if p > b ;
and
^c ¼c þ ce ð1 lÞ: By considering the first order condition, the optimal retail price pe(Q) is given as: ( aQ if Q 6 a2 ; e p ðQÞ ¼ ab if Q > a2 : 2b
ð3:1Þ
Since the retailer can always use the emergency order to bring the total non-defective units up to Q units at the beginning of period 2, it is easy to check from the expression for Pe that the pricing decision in period 2 would depend only on Q and that the ordering decision Q in period 1 would depend only on the expected supply yield. Substitute the optimal retail price pe(Q) into Pe(Q, p), getting: ( ÞQ if Q 6 a2 ; ^cQ þ ðaQ b e e P ðQ; p ðQÞÞ ¼ a2 if Q > a2 : ^cQ þ 4b It remains to determine the optimal order quantity Qe that maximizes the expected profit Pe(Q, pe(Q)). By considering the first order condition at the boundary points and assuming a > b^c, it is easy to show that the optimal order quantity and the retailer’s optimal expected profit can be expressed as: a b^c ; and 2 2 ða b^cÞ Pe ¼ : 4b Qe ¼
ð3:2Þ ð3:3Þ
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By comparing the optimal profits given in (2.13) and (3.3), we can establish the following Proposition. Proposition 3. It is optimal for the retailer to place an emergency order if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða bcÞ ðaukw bcÞ þ a2 v k k e ; c < ð1 lÞb where k* is given by (2.11). Proposition 3 suggests that it is optimal to use the emergency order when the unit cost associated with the emergency order is sufficiently small. Proposition 3 has the following implications. Consider the case when ce is low enough so that the retailer decides to use the emergency order. In this case, the retailer would place an order at the beginning of period 1 c according to the optimal order quantity Qe given in (3.2), i.e., Qe ¼ ab^ . Since Qe < a2, it is easy to check from 2 aQe e e e (3.1) that p ¼ b . This implies that Q = a bp . Combine this result with the fact that the retailer would always place an emergency order to bring the total number of non-defective units up to Qe at the beginning of period 2, we can conclude that the retailer would set the optimal retail price pe so as to sell all Qe units in period 2. 4. Extension 2: Supplier order allocation Consider the case when the retailer can order from multiple suppliers with different yield distributions and different unit costs. To simplify our exposition, we shall consider the 2-supplier case. Consider a situation in which the retailer orders Qi units from supplier i and Qj from supplier j. As such, the retailer will receive yiQi + yjQj non-defective units of the same product at the end of period 1. For s = i, j, it is assumed that the supply yield ys is a discrete random variable that P takes on N different values of ysn, where n = 1, . . . , N. Define the probability Prob{ys = ysn} = ksn so that Nn¼1 ksn ¼ 1 for s = i, j. Under the Responsive Pricing policy, the retailer would first determine the optimal order quantities Qi and Qj at the beginning of period 1. Then he would specify the optimal retail price p after the actual yield of each supplier is realized at the end of period 1. As such, there are three decision variables. To simplify our exposition, we shall transform our decision variables Qi and Qj as follows. Let Q be the total number of units ordered from the suppliers, where Q = Qi + Qj. Let f P 0 be the ratio between the order quantities; i.e., f ¼ QQi . Notice that f = 0 when the retailer orders only from supplier j and f = 1 when the j f 1 retailer orders only from supplier i. By noting that Qi ¼ 1þf Q and Qj ¼ 1þf Q, the retailer’s problem is based on the decision variables Q, f and p. In preparation, let cs be the unit cost of supplier s, s = i, j. For any order quantities Qi and Qj, the total ordering cost can be expressed in terms of caQ, where ca corresponds to the c f þc weighted average unit cost that can be expressed as ca ¼ i1þf j . By observing that the total number of nondefective units received by the retailer at the end of period 1 is equal to yimQi + yjnQj for some m, n = 1, 2, . . . , N, the total number of non-defective units can be expressed in terms of rQ, where r corresponds y f þy to the effective yield with N2 realizations rmn ¼ im1þf jn , and m,n = 1, 2, . . . , N. Notice that the expected effective y f þy l f þl yield la ¼ EðrÞ ¼ Ey i ;y j ð i1þf j Þ ¼ i1þf j . In this case, it is easy to check that, under the RP policy, the retailer’s supplier order allocation problem can be formulated as problem P(SOA), where: a a a P ¼ max c Q þ Er maxfp minfrQ; Dgg ¼ max max Er ðmaxfP ðQ; f ; pjrÞgÞ : ð4:1Þ Q;f
a
p
f
Q
p
Notice that P (Q, f, pjr) is the retailer’s profit associated with order quantity Q, ratio f and retail price p for any given realization of r. To solve problem P(SOA) given in (4.1), we first solve the subproblem for any given ratio f and then determine the optimal ratio fa using a simple search algorithm. For any given value of f, one can check from (4.1) that the subproblem is given by maxQEr(maxp{Pa(Q, f, pjr)}), where: ( a c Q þ prQ if p 6 arQ ; b a P ðQ; f ; pjrÞ ¼ ð4:2Þ arQ a c Q þ pða bpÞ if p > b :
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Suppose we treat the effective yield r as y. Then it is clear that the subproblem has the same structure as problem P(RP) given in (2.2). Therefore, the same approach presented in Section 2.3 can be used to solve this subproblem. Recall from section 2.3 that the solution approach for solving problem P(RP) hinges upon the boundary points 2ya , k = N, N 1, . . . , 1, as well as the terms uk, vk and wk. In order to apply the solution apk proach for solving problem P(RP) to solve our subproblem, we need to develop a new index l, y f þy l = 1, . . . , L = N2 for the effective yield r. To do so, sort the realized effective yield rmn ¼ im1þf jn in ascending order and then assign the new index l so that r1 < r2 < < rl < < rL. After creating the new index l for the effective yield r, define the terms uk, vk and wk as follows: uk ¼
k X
kl r l ;
vk ¼
l¼1
L X
kl ;
l¼kþ1
and
wk ¼
k X
kl r2l ;
ð4:3Þ
l¼1
where u0 = 0, vL = 0, and w0 = 0. Thus one can apply Proposition 2 in Section 2.3 to determine the optimal solution to the problem P(SOA) for any given ratio f as follows: Corollary 1. Suppose ls > bca s for s = i, j. Then, for any given ratio f, the optimal solutions to the retailer’s problem (4.1) can be expressed as follows: auka bca ; where 2wka wk bca k a ¼ argmax uk : k ¼ L; L 1; . . . ; 1 ; < rk a ( arQa a a if Q 6 2r ; pa ¼ a b if Qa > 2ra ; and 2b
Qa ¼
ð4:4Þ ð4:5Þ ð4:6Þ
2
Pa ¼
ðauka bca Þ a2 v k a : þ 4bwka 4b
ð4:7Þ
Given the retailer’s optimal expected profit for any given ratio f, we can determine the optimal ratio fa by using a simple search algorithm. While it is difficult to determine the optimal ratio fa analytically for the general case, we can determine fa for a special case. In this special case, there are two potential suppliers i and j. Supplier i’s yield is constant and it is equal to l. Supplier j’s yield has two possible values yj1 and yj2 with equal probability of 0.5 and supplier j’s expected yield is also equal to l. Hence, we can express yj1 = l r and yj2 = l + r. Given supply yield of both y f þy r suppliers, it is easy to check that the effective yield r ¼ i1þf j takes on two possible values, namely, r1 ¼ l f þ1 r and r2 ¼ l þ f þ1 with equal probability of 0.5. Since supplier i has a constant yield l and since supply j has uncertain yield with mean l, it is reasonable to assume that ci > cj. We now investigate the conditions under which the retailer would order from exactly one supplier or both r r suppliers. First, since r1 ¼ l f þ1 and r2 ¼ l þ f þ1 , it is easy to check from (4.3) that: r u0 ¼ 0; u1 ¼ 0:5 l ; u2 ¼ l; v0 ¼ 1; v1 ¼ 0:5; v2 ¼ 0; f þ1 2 r r2 w0 ¼ 0; w1 ¼ 0:5 l ; and w2 ¼ l2 þ : 2 f þ1 ðf þ 1Þ a
It follows from (4.5) in Corollary 2 that ka = 2 if u2 wr22 bca < 0, and ka = 1, otherwise. By considering the a terms u2, w2, r2 and ca as functions of f, it is easy to show that the function hðf Þ ¼ u2 wr22 bca is a quadratic function of f. Hence, there exists two roots s1 6 s2 that satisfy h(f) = 0. To simplify our exposition, we shall focus on the case when s1 < 0 and s2 > 0.3 By comparing the profit function Pa associated with different values of f, we can establish the following Proposition: 3
One can use the exact same approach to analyze other cases, say, when s1 > 0 and s2 > 0 or when s1 < 0 and s2 < 0 . We omit the details.
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Proposition 4. Suppose bci(l + r) + bcjl arl > 0 and bcj(l + r) + ar2 arl < 0. Then the optimal ka can be expressed as follows: ka ¼
1 2
if 0 6 f 6 s2 ; if s2 < f 6 1:
ð4:8Þ
Moreover, for any given value of f, the retailer’s optimal profit can be expressed as: 8 ½alðf þ1Þar2bðci f þcj Þ2 a2 > þ 8b if 0 6 f 6 s2 ; > < 8b½ðf þ1Þlr2 Pa ðf Þ ¼ > 2 > i f þcj Þ : ½alðf þ1Þbðc if s2 < f 6 1: 2
ð4:9Þ
4b½ðf þ1Þ l2 þr2
By comparing the profit function Pa(f) for different values of f, the optimal ratio fa can be expressed as follows: 2bci Proposition 5. Suppose bci(l + r) + bcjl arl > 0 and bcj(l + r) + ar2 arl < 0 and suppose a > lr . Then:
a
f ¼
8 <0 :
if r 6
lðci cj Þ ; ci
2 ½s2 ; 1Þ if r >
lðci cj Þ : ci lðc c Þ
Proposition 5 can be interpreted as follows. When r is sufficiently small, say, r 6 ici j , it is optimal for the retailer to order only from supplier j (i.e., fa = 0). This result is intuitive because supplier j will dominate supplier i when supply j’s unit cost cj is sufficiently lower or when the standard deviation of supplier j’s yield r is lðc c Þ sufficiently lower. On the other hand, when r is reasonably large so that r > ici j and the suppositions hold, it is optimal for the retailer to order from both suppliers. This implication is consistent with the portfolio theory in which an investor should invest in a stock portfolio instead of a single stock. To elaborate, notice that supplier i has a constant yield with a higher unit cost while supplier j has an uncertain yield with a lower cost. As supplier j’s yield becomes more variable (i.e., when rj is large), Proposition 5 suggests that it is optimal for the retailer to order from both suppliers so as to maintain an optimal tradeoff between lower unit cost and higher yield uncertainty.
5. Conclusion We have developed a two-stage stochastic model for determining the optimal order quantity and optimal retail price under supply uncertainty. Specifically, we show the Responsive Pricing policy dominates the No Responsive Pricing policy in terms of the retailer’s optimal expected profit. By examining the underlying structure of the problems associated with these two pricing policy, we have developed simple approaches for determining the optimal order quantity and retail price for any discrete supply yield distribution. By using these solution approaches, we examine the impact of yield distribution on the optimal order quantity, retail price, and retailer’s expected profit. We have also shown how to extend our analysis of the Responsive Pricing policy to examine two issues including emergency order and order allocation among multiple suppliers. Our model has certain limitations including deterministic demand and linear demand function. We plan to extend our model to examine the issue of responsive pricing under uncertain demand and uncertain supply in our future research. Recently, when the supply yield y is uniformly distributed; i.e., U[a b, a + b] and when the market potential a is normally distributed; i.e., N(l, r2), Tang and Yin (2006) reported that the retailer’s optimal order quantity under responsive pricing is increasing in demand uncertainty r. In addition, they showed that the retailer’s optimal profit under responsive pricing is decreasing in supply uncertainty b but increasing in demand uncertainty r. We plan to generalize these preliminary results in the near future.
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Appendix. Proofs Proof of Proposition 1. For given k = 0, 1, . . ., N, the problem Pk(NRP) in (2.3) can be rewritten as: P0k ¼ maxa maxfðuk p cÞQ þ pða bpÞvk g
ð7:1Þ
06p6b QP0
a bp a bp
subject to
Our solution approach is as follows. We first determine the optimal Q for any given p and then we determine the optimal p. For any given p 2 ½0; ba, the objective function in (7.1) is a linear function of Q with the coefficient ukp c. Let us consider the following cases: Case 1: Suppose ba > uck (or equivalently, bc a < uk ). Depending on the sign of (ukp c), it is easy check from (7.1) that ( abp if ba P p > uck ; yk 0 Qk ¼ abp if p 6 uck : y kþ1
Case 2: Suppose ba 6 uck (or equivalently, bca P uk ). Since p 6 ba and ba 6 uck , we have ukp c 6 0. Therefore, it is easy to check from (7.1) that Q0k ¼ abp . y kþ1 Let us examine further about two subproblems (a) and (b) associated with Case 1: (a) Suppose ba P p > uck . Then we can substitute Q0k ¼ abp into the objective function (7.1), getting a concave yk c a function of p. By considering the first order condition, it is easy to show that p01k ¼ 2ðuk þy þ 2b . Notice k vk Þ uk ðuk þvk y k Þ bc a 0 a c a 0 c P p1k always holds when b > uk . Therefore, the inequality b P p1k > uk holds if and only if a < uk þ2vk y . b k
k þvk y k Þ This implies that when bca < ukuðuk þ2v , the optimal price is given by p01k , the optimal order quantity is 2 k yk bc 0 a k þvk y k Þbc Q1k ¼ 2y 2ðuk þvk y Þy and the optimal expected profit is P01k ¼ ½aðu . However, when 4by k ðuk þvk y k Þ k k k uk ðuk þvk y k Þ bc 0 0 c P uk þ2vk y , the optimal price is given by p3k ¼ uk with expected profit P3k . a k (b) Suppose p 6 uck . Then we can substitute Q0k ¼ abp into the objective function (7.1) and get a concave y kþ1 a function of p. By considering the first order condition, we can show that p02k ¼ 2ðuk þyc vk Þ þ 2b . In this case, kþ1
u ðu þv y
Þ
k k kþ1 it is easy to show that the inequality p02k 6 uck holds if and only if bca P kuk þ2v . Hence, when k y kþ1 uk ðuk þvk y kþ1 Þ bc 0 P , the optimal price is given by p , the optimal order quantity is given by 2k a uk þ2vk y kþ1 ½aðuk þvk y kþ1 Þbc2 bc 0 0 a Q2k ¼ 2y 2ðuk þvk y Þy and the optimal expected profit is given by P2k ¼ 4by ðuk þvk y Þ . However, kþ1 kþ1 kþ1 kþ1 kþ1 u ðuk þvk y kþ1 Þ , the optimal price is given by p03k ¼ uck with expected profit P03k . when bca < kuk þ2v ky kþ1
Notice that the retailer’s optimal expected profit associated with Case 1 is given as u ðuk þvk y kþ1 Þ k þvk y k Þ P0k ¼ maxfP01k ; P02k ; P03k g. Since yk < yk+1, we have kuk þ2v < ukuðuk þ2v . We now use this observation to k y kþ1 k yk determine the optimal solution to problem Pk(NRP) associated with Case 1 as follows: u ðu þv y
Þ
k k kþ1 k þvk y k Þ 1. Suppose bca < kuk þ2v . Our observation implies that bca < ukuðuk þ2v . Hence, we can apply the results k y kþ1 k yk 0 0 0 0 0 0 obtained in (a) to show that Pk ¼ P1k ; pk ¼ p1k ; and Qk ¼ Q1k . This proves the first statement in the Proposition. u ðuk þvk y kþ1 Þ k þvk y k Þ k þvk y k Þ 2. Suppose bca P ukuðuk þ2v . Since kuk þ2v < ukuðuk þ2v , we can apply the results obtained in (b) to show that k yk k y kþ1 k yk 0 0 0 0 0 0 Pk ¼ P2k ; pk ¼ p2k ; and Qk ¼ Q2k . This proves half of the third statement in the Proposition.
u ðu þv y
Þ
k k kþ1 k þvk y k Þ 6 bca < ukuðuk þ2v . Then P0k ¼ maxfP01k ; P02k g and we can determine the optimal price and 3. Suppose kuk þ2v k y kþ1 k yk order quantity accordingly. This proves the second statement in the Proposition.
We now turn our attention to Case 2; i.e., when ba 6 uck . In this case, we have bc a P uk . Let us first substitute Q0k ¼ abp y kþ1 into the objective function (7.1), getting a concave function of p. Then, by examining the first order a condition, we can show that p02k ¼ 2ðuk þyc vk Þ þ 2b . We now determine the optimal solution to problem kþ1 Pk(NRP) associated with Case 2 as follows:
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1. Suppose p02k < ba (or equivalently, bca < uk þ vk y kþ1 Þ. Then the optimal price is p0k ¼ p02k . This proves half of the third statement in the Proposition. 2. Suppose bca P uk þ vk y kþ1 . Then Q0k ¼ 0; p0k is arbitrary, and P0k ¼ 0. This proves the fourth statement in the Proposition. This completes the proof.
h
Proof of Proposition 2. Let us differentiate the objective function given in (2.10) with respect to Q, getting: dEy ðPðQ; p jyÞÞ ¼ dQ
a 2wk uk c Q: b b
ð7:2Þ
First, let us consider the case when l > bca . Notice that the derivative is positive when Q is sufficiently small, say, when Q = xN+1 = 0 and is negative when Q is sufficiently large, say, Q ¼ x1 ¼ 2y1 . Also, by evaluating the 1 derivatives at the boundary points xk+1 and xk, one can use the definitions of uk and wk given in (2.5) and (2.9) to show that: " # k dEy ðPðQ; p jyÞÞ dEy ðPðQ; p jyÞÞ a X 1 1 2 ¼ k y < 0: m m dQ dQ b m¼1 y kþ1 y k Q¼xk Q¼xkþ1 Since xk+1 < xk, we can conclude that the first derivative of the objective function at the boundary points xN+1, xN, . . . , x0 are decreasing. Combine this observation with the fact that the objective function is piece-wise dE ðPðQ;p jyÞÞ dE ðPðQ;p jyÞÞ concave, we can conclude that there exists an k* such that y dQ jQ¼xk þ1 P 0 and y dQ jQ¼xk < 0. This implies that the optimal k* is given by (2.11). By considering the first order condition (7.2), we can show that the optimal order quantity Q* is given by (2.12). Given the optimal order quantity Q*, we can compute the optimal retail price p*(Q*jy) once the actual yield y is observed. In addition, we can substitute Q* into (2.10) to show that the retailer’s optimal expected profit under the responsive pricing policy is given by (2.13). dEy ðPðQ;p jyÞÞ Next, let us consider the case when l 6 bc 6 0 for Q P 0. Therefore dQ a . In this case, we have Q* = 0 is the optimal solution. This completes the proof. h Proof of Proposition 3. The proof is immediate. We omit the details.
h
Proof of Corollary 1. The proof follows directly from Proposition 2. We omit the details.
h
Proof of Proposition 4. The results follow from Corollary 1. Let g1 = bci(l + r) + bcjl arl and Then it is easy p toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi check that the two roots associated with the equation g2 = bcj(l + r) + ar2 parl. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 g21 4bci lg2 g1 þ g21 4bci lg2 and s2 ¼ . Suppose g1 > 0 and g2 < 0. Then we have s1 < 0 h(f) = 0 are: s1 ¼ 2bci l 2bci l and s2 > 0. First, let us consider the case when f > s2. In this case, ka = 2 and Pa(f) = A(f), where: 2
Aðf Þ ¼
½alðf þ 1Þ bðci f þ cj Þ 4b½ðf þ 1Þ2 l2 þ r2
ð7:3Þ
:
Second, consider the case when 0 6 f 6 s2. In this case, ka = 1 and Pa(f) = B(f), where: Bðf Þ ¼
½alðf þ 1Þ ar 2bðci f þ cj Þ2 8b½ðf þ 1Þl r
2
þ
a2 : 8b
ð7:4Þ a
Third, consider the case when f = 1. In this case, w2 = l2, r2 = l and u2 wr22 ¼ 0 < bca holds. Thus, we can conclude that ka = 2 and Pa (f = 1) = A(f = 1). This completes the proof. h
C.S. Tang, R. Yin / European Journal of Operational Research 182 (2007) 239–255
255
Proof of Proposition 5. It follows from (4.9) that the retailer’s optimal expected profit Pa can be expressed as: Pa ¼ maxfmax06f 6s2 Bðf Þ; maxs2
ð7:5Þ
Suppose a is sufficiently large so that a(l r) > 2bci. Then it is easy to use the fact that ci > cj and l > r to show that a(l r) 2bcj + (al 2bci)f > 0 and that l(f + 1) r > 0. Applying this result to (7.5), we can Þ conclude that sign dBðf ¼ signðci ðr lÞ þ cj lÞ: Let us consider the following two cases: df Þ 6 0. Therefore B(f) is decreasing in 1. When ci(r l) + cjl 6 0. In this case, we have dBðf df f 2 ½0; 1; Bð0Þ ¼ max06f 61 Bðf Þ P maxfmax06f 6s2 Bðf Þ; maxs2
dAðf Þ ððal bcj Þ þ f ðal bci ÞÞððal bci Þr2 bl2 ð1 þ f Þðci cj ÞÞ ¼ : 2 df 2bðl2 þ r2 þ 2l2 f þ l2 f 2 Þ
ð7:6Þ
Observe that (al bci)r2 bl(1 + f)(ci cj) < 0 when f is sufficiently large. Combine this observation with Þ < 0 when f is sufficiently large. the fact that (al bcj) + f(al bci) > 0, we can check from (7.6) that dAðf df a Therefore, argmax{A(f) : s2 < f 6 1} < 1, which implies that f 2 [s2, 1). We can prove the Proposition by considering the results associated with these two cases.
h
References Chod, J., Rudi, N., 2005. Resource flexibility with responsive pricing. Operations Research 53 (3), 532–548. Fisher, M., Raman, A., 1996. Reducing the cost of demand uncertainty through accurate response. Operations Research 44 (1), 87–99. Lee, H., Tang, C.S., 1997. Modeling the costs and benefits of delayed product differentiation. Management Science 43 (1), 40–53. Petruzzi, N.C., Dada, M., 1999. Pricing and the newsvendor problem: A review with extensions. Operations Research 47 (2), 183–194. Ramasesh, R., Ord, J., Hayya, J., Pan, A., 1991. Sole versus dual sourcing in stochastic lead time (s, Q) inventory models. Management Science 37 (4), 428–443. Tang, C.S., 2005. Perspectives in Supply Chain Risk Management: A Review. working paper UCLA Anderson School, November. Tang, C.S., Yin, R., 2006. Responsive Pricing under Uncertain Supply and Uncertain Demand. INFORMS Hong Kong Conference Presentation. Tang, C.S., Rajaram, K., Alptekinoglu, A., Ou, J.H., 2004. The benefits of advance booking discount programs. Management Science 50 (4), 465–478. Van Mieghem, J., Dada, M., 1999. Price versus production postponement: capacity and competition. Management Science 45 (12), 1631– 1649. Yano, C., Lee, H., 1995. Lot sizing with random yields: A review. Operations Research 43 (2), 311–334.