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Role of forecast effort on supply chain profitability under various information sharing scenarios
Int. J. Production Economics 129 (2011) 284–291
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Role of forecast effort on supply chain profitability under various information sharing scenarios Xiaowei Zhu a,1, Samar K. Mukhopadhyay b,n, Xiaohang Yue c,2 a
College of Business and Public Affairs, West Chester University of Pennsylvania, West Chester, PA 19383, USA Sungkyunkwan University-GSB, 53 Myeongnyung-dong 3-Ga, Jongno-gu, Seoul 110-745, Republic of Korea c School of Business Administration, University of Wisconsin-Milwaukee, P.O. Box 742, Milwaukee, WI 53201, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 May 2009 Accepted 21 October 2010 Available online 27 October 2010
In this paper, we consider a manufacturer who sells a product to a retailer in a single selling season. Each party obtains a forecast of the market demand independent of each other. We study three different forecast scenarios: Non-Information Sharing, Information Sharing, and Retailer Forecasting cases. In the first scenario, both parties make their forecasts, but do not share the information with the other firm. In the second scenario, they share the information with each other, while in the last scenario, only the retailer makes the forecast, and shares it with the manufacturer. Noting that the forecast accuracy comes at a cost, we derive the optimal price and forecast accuracy level for each of the three cases. We then compare the optimal policies of the three cases and derive conditions under which the two parties should share information with each other. Results of extensive numerical experimentation are also presented. & 2010 Elsevier B.V. All rights reserved.
Keywords: Forecast cost Forecast variance Information sharing Combining forecasts
1. Introduction Supply chain participants are becoming increasingly more coupled in the sense that decisions taken by one firm in a supply chain directly affect the performance of other firms. Firms, therefore, have an incentive to devise methods to gain information about other firms’ decisions and other operating parameters. Recent advancements in information technology have been helpful in this respect. Today, different types of information like point-of-sales data, inventory, forecast data, and sales trends are being shared among supply chain partners quickly and inexpensively. Examples of well-known information sharing implementations include Wal-Mart’s Retail Link program, providing point-of-sales data to suppliers like Johnson and Johnson and Lever Brothers. Many other recent supply chain initiatives like Quick Response (QR), Efficient Consumer Response (ECR), Vendor-Managed Inventory (VMI), Continuous Replenishment Program (CRP), Radio-frequency identification (RFID), and Collaborative Planning, Forecasting and Replenishment (CPFR) are built on the principles of information sharing. Examples of major successes in the benefits of information sharing are Campbell Soup and Barilla SpA as narrated by Lee et al. (2000).
n
Corresponding author. Tel.: + 822 740 1522; fax: + 822 740 1503. E-mail addresses: [email protected] (X. Yue), [email protected] (X. Zhu), [email protected] (S.K. Mukhopadhyay). 1 Tel.: +1 610 738 0588. 2 Tel.: +1 414 229 4657. 0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2010.10.021
These developments in the real world have motivated the academic community to explore the benefits of information sharing. Particularly, many recent papers study the effects of information sharing in supply chain management. Mukhopadhyay et al., (2009) examined information sharing for complimentary goods, while that in a multi-generational product setting was studied by Sohn and Lim ˚ (2008). Smaros (2007) presents the results of a case study examining four collaboration projects involving a total of four suppliers and one retailer operating in the European grocery sector. He finds strong support for collaborative forecasting as a means of improving supply chain efficiency and advocate using valid assumptions to benefit from information sharing. Another case study was presented by Byrne and Heavey (2006) examining the effect of information sharing and forecasting on performance parameters of an industrial supply chain consisting of small-to-medium sized enterprises. They have shown that potential cost savings of up to 9.7% can be achieved through the use of improved forecasting techniques. The research cited above underscores the importance of information sharing. One very important aspect of information sharing in the context of supply chain is the forecast developed by various parties in the supply chain. Generally speaking, forecast is used for production and inventory planning, and the demand–supply matching is vastly improved, at any stage of a supply chain, when the forecast error is reduced or in other words, forecast accuracy is improved. While all the channel partners develop their own forecasts using their own methods, it has been shown by researchers that sharing of information among them does help in forecast improvement albeit in different degrees. Vives (1984) and Gal-Or (1985) study incentives for a firm to share private information
X. Zhu et al. / Int. J. Production Economics 129 (2011) 284–291
horizontally with competitors in an oligopolistic market. Impact of forecast improvement on optimal order quantity and selling price determination for a second cycle order is shown by Serel (2009). Hosoda and Disney (2009) show that if the supply chain attempts to minimize the demand forecast errors, then the impact of the ¨ demand mis-specification on cost is minor. Kerkkanen et al. (2009) study the role of forecast errors for defining a realistic target of forecast accuracy. Reiner and Fichtinger (2009) develop a dynamic model to evaluate different forecasting methods. Sarvary and Parker (1997) show that information from different sources could be substitutes or complements depending on characteristics such as variance and correlation. While the above researchers have established the value of information sharing in supply chain management, there is a body of research that study the incentives for information sharing are examined by Corbett and de Groote (2000), Cachon and Lariviere (2001), Ha (2001), and Mukhopadhyay et al. (2008). Agrawal et al. (2009) study the impact of information sharing and lead time on bullwhip effect and on-hand inventory of a supply chain. They show that while information sharing is beneficial for reduction in bullwhip effect, lead time reduction is even more beneficial. Information sharing in decentralized supply chains where one manufacturer supplies to multiple retailers competing on price or quality has been the focus of several papers like Li and Zhang (2008), Bernstein and Federgruen (2005), and Gaughety and Reinganum (2007). Raju and Roy (2000) use forecast in their pricing decision in both Stackelberg and Bertrand-Nash models. They found that information is more valuable for products with higher substitutability, and for larger firms. Roy (2000) studied channel pricing when there are two competing manufacturer– retailer channels with both channels facing stochastic downwardsloping demand functions. Their paper discusses how the forecast variance affects profit, but did not obtain the optimal level of forecast variance. Qi and Zhu (2009) study an endogenous determination of efforts put into information acquisition and its impact on supply chain management. They measure the effort level of a number of experts (hired by the buyer) in information acquisition. The forecast cost is measured by a commission fee. We see from current research cited above that sharing of information, especially the forecast information, could be beneficial for the supply chain partners who are not privy to the information. Nevertheless, the fact remains that the forecast information obtained by one party remains with that party and is seldom shared with others in the supply chain. We study this practical scenario in this paper. We analyze several forecast system designs from the information sharing point of view, and their impact on factors like forecast accuracy, forecast cost, and profit. We focus on the impact of forecast improvement/information sharing on pricing through mathematical modeling. Chen (1998), Cachon and Fisher (2000), Donohue (2000), and Yue et al., (2006) use similar approach in their papers. We use a more general model to measure the forecast cost. We include costs of obtaining data, and any forecast related costs like cost of purchasing software, hardware, labor hour, and overhead. In the specific scenario considered in our paper, a manufacturer sells a product to a retailer in a single selling season. Three different forecast systems studied in this paper are: Non-Information Sharing, Information Sharing, and Retailer Forecasting. The first case is where the manufacturer and the retailer both forecast separately but do not share the forecast information with each other. In the second case, the manufacturer and the retailer both forecast separately and share the forecast information with each other. The third is the case where only one side (the retailer) forecasts, and shares the information with the other side (the manufacturer). For each of these cases, we derive optimal forecast accuracy level and price, and discuss the forecast variance and the related forecast cost. Our study complements the
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literature by explicitly characterizing the value of forecasting cost under a Stackelberg game mode and by deriving optimal policies regarding retail price, wholesale price, and forecast accuracy (as given by forecast variance). In the next section, we introduce our notation and model. In Section 3, we first present the analytical results under the three scenarios (Non-Information Sharing, Information Sharing, and Retailer Forecasting), and then compare the results of these three cases. In Section 4, we present the results of our numerical experiments and the insights obtained from them. Section 5 concludes the paper with some future research directions.
2. Model framework In this supply chain, the manufacturer first decides the wholesale price w. The retailer then sets the retail price p, after knowing w. We consider a one period model. The consumer demand d, is a linear function of retail price and is given by d ¼ abp
ð1Þ
where a is the primary demand, i.e., the demand determined by all factors except price, and b is the slope of the demand function. All parameters are assumed to be non-negative. To capture uncertainty in demand, we assume that a is a random variable defined as follows: a ¼ aþe
ð2Þ
where a is the deterministic part of the demand (mean) and e is the random part. Note that Eq. (2) is the commonly used forecast equation. e is normally distributed with a mean of zero and a variance of U. a is common knowledge for both the manufacturer and the retailer and is commonly derived from historical data. We assume that each party obtains a forecast about the unknown primary demand a using the market-information-gathering techniques at its disposal. We consider the general case that the forecasts obtained by the manufacturer and the retailer are also random variables. We denote manufacturer’s and retailer’s forecasts of primary demand as fm and fr, respectively. Therefore, fm ¼ a þ em
ð3Þ
fr ¼ a þ er
ð4Þ
where em and er are the forecast errors associated with the two parties, respectively. We also assume that em and er are independent of a and are normally distributed with a mean of zero and variances of Vm and Vr, respectively. Note from Eq. (3) that in case the forecast is very accurate, (i.e., em ¼0, say), then the forecast is equal to the actual primary demand, as it should be. A higher variance for the forecast error implies a less precise forecast. The forecast errors em and er can be correlated. The extent of correlation depends on the data and methodology used by the retailer and the manufacturer in their forecasting processes. If they use data and methodology very similar to each other, then it will result in a higher correlation between forecasts. The covariance matrix of forecast errors is represented by ! X Vr Vrm ¼ ð5Þ Vrm Vm The covariance between the two forecasts is Vrm and the pffiffiffiffiffipffiffiffiffiffiffiffi coefficient of correlation is r ¼ Vrm = Vr Vm . In this paper, we consider the case where r ¼ 0 i.e., the forecasts of the manufacturer and the retailer are not correlated. We also assume that the forecasts are obtained at a cost of Ci =Vi (i¼r, m). This says that to obtain a more accurate forecast (i.e., a lower forecast variance), there will be a need to invest more. Ferbar et al. (2009) show that
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that the consensus forecast E a9fr ,fm is more accurate than one sided forecast, like fr or fm . The equilibrium variance of the forecast can be obtained by maximizing the unconditional expected profit.
the use of a particular forecasting method affects the costs of a supply chain. Based on the model in Winkler (1981), we have the following parameters of the consensus distribution. When the manufacturer obtains his forecast fm, he computes the expected value of the market primary demand a, i.e.,E½a9fm as the weighted average of a (which is common knowledge) and fm, his own forecast. i.e.,
We derive the equilibrium price, variance, and profit of each firm in this scenario in Proposition 1. Proofs of all results are given in the Appendix.
E½a9fm ¼ a þ tm ðfm aÞ Am
Proposition 1. For the non-information sharing (N) case:
ð6Þ
For clarity, we denote this quantity as Am. Following the same path, the other similar parameters are derived as E½a9fr ¼ a þ tr ðfr aÞ Ar , say, E½a9fr ,fm ¼ a þ Jðfm aÞ þ Kðfr aÞ A, say, E½fr 9fm ¼ a þ dm ðfm aÞ, and E½fm 9fr ¼ a þ dr ðfr aÞ The respective variances of these parameters are given by Var a9fm ¼ tm Vm Var a9fr ¼ tr Vr ðVr Vm ÞVð1r2 Þ pffiffiffiffiffiffiffipffiffiffiffiffi Var a9fr ,fm ¼ ð1r2 ÞVr Vm þ VðVr þVm 2r Vm Vr Þ It can be shown that U U U þ Vrm U þVrm , tm ¼ , dr ¼ , dm ¼ U þ Vr U þVm U þVr U þ Vm tm ðdr 1Þ tr ðdm 1Þ ,K ¼ J¼ dr dm 1 dr dm 1 Because of our assumption of r ¼ 0, we have Vrm ¼ 0, hence the above parameters are reduced to tr ¼
tr ¼ dr ¼
U U tm ðtr 1Þ tr ðtm 1Þ ,K¼ , tm ¼ dm ¼ , J¼ U þVr U þ Vm tr tm 1 tr tm 1
The profit functions of the retailer and the manufacturer are, respectively Cr Vr C pm ¼ wðabpÞ m Vm
pr ¼ ðpwÞðabpÞ
3. Model analysis We assume, as stated earlier, that the manufacturer makes the first decision (on wholesale price) and therefore, acts as a leader. A Stackelberg type game analysis is appropriate in this situation. We analyze three different scenarios of the Stackelberg game as given below. 3.1. Stackelberg game with no information sharing (N) In this form of Stackelberg game, information about manufacturer’s forecast is not shared with the other firm. We use the superscript N for this case. The sequence of moves is as follows: 1. The manufacturer obtains and announces his optimal price w based on his forecast fm and his belief about retailer’s forecast (i.e.,Eðfr 9fm Þ in Eq. (6)), and retailer’s best response function (retailer’s optimal action as a function of manufacturer’s action). 2. Next, the retailer uses manufacturer’s announced wholesale price w, and the consensus forecast of primary demand a9fr ,fm , and obtains and announces her optimal price p. We will see in Proposition 1 that w ¼ Am =2b. Where Am ¼ Am ¼ a þ tm ðfm aÞ (from Eq. (6)). This is a linear function of fm. Hence, once w is known, the retailer can infer manufacturer’s forecast fm . This observation is consistent with Raju and Roy (2000). We assume
a. The Stackelberg equilibrium prices and forecasting variances for the manufacturer and the retailer are AN AN AN m , pN ¼ þ m, 2b 2b 4b pffiffiffiffiffiffiffiffiffiffiffiffi 2Uð4Cm b þ 2Cm bUÞ N Vm ¼ , with U 2 4 8bCm U 2 8Cmb wN ¼
Vr N satisfies the following equation: Cr ðVrN Þ2
N N N 2 N 2 U 2 Vm ðVrN U 2 þ3U 2 Vm þ UðVm Þ þ Vr N ðVm Þ Þ N þV N U þ V N V N Þ3 4bðUVm r r m
¼0
b. The unconditional expected profits of the retailer and the manufacturer are: 1 U2 Cm N EðpN a2 þ N m Þ ¼ EðEðpm 9fm ÞÞ ¼ N 8b U þVm Vm ! 1 a2 ð2Jtm Þ2 U K 2 U Cr N N þ Eðpr Þ ¼ EðEðpr 9fr ,fm ÞÞ ¼ þ N 16tm 4tr b 16 Vr
3.2. Stackelberg game with information sharing (I) In this subsection, we consider that the manufacturer and the retailer share information about each other’s forecast. We use the superscript I for the Stackelberg Information sharing case. Intuitively, this will increase the quality of their decisions. The sequence of moves is as follows: 1. The manufacturer obtains and announces his optimal wholesale price w, based on his own forecast fm , retailer’s forecast fr , which is now a shared information, and retailer’s best response function. 2. Next, the retailer obtains and announces her optimal price p, based on manufacturer’s action w, her own forecast fr , and manufacturer’s forecast fm . The manufacturer and the retailer maximize their own expected profits based on both fr and fm (note: each firm would come up with a consensus forecast about the primary demanda using his/her own forecast and the shared forecast from the other firm based on Winkler’s consensus model). The optimum equilibrium prices, forecast variance, and corresponding profits for this case are derived in the next proposition. Proposition 2. For information sharing (I) case (a) The Stackelberg equilibrium prices and forecasting variances for the manufacturer and the retailer are: a 3a I wI ¼ , pI ¼ , V ¼ VrI ¼ 0 2b 4b m (b) The unconditional expected profits of the retailer and the manufacturer are given below. 1 U Cm I a2 þ 8b V m 2 1 U Cr I EðpIr Þ ¼ EðEðpr 9fr ,fm ÞÞ ¼ a2 þ 16b 2 Vr
EðpIm Þ ¼ EðEðpm 9fm ÞÞ ¼
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The optimal variance for both the forecasts came out as zero denoting that both should have a very high degree of accuracy. Obviously, the investment for such a forecasting system will be prohibitively high. Thus, practically, it would be hard to follow the optimal policies using the information sharing mode, from the implementation point of view. 3.3. Stackelberg game with retailer forecasting (R) There are times when the two firms would enter into a strategic alliance to benefit from each other’s core competency and unique advantages. It is very common for the retailer to make more accurate forecast than the manufacturer because she is closer to customers and has better information about the market demand. In this section, we study the third scenario where only the retailer forecasts the market demand, denoted by fr , and then shares her forecasting information to be used by the manufacturer. The manufacturer uses retailer’s forecast informationfr , to decide his wholesale price w. The forecasting cost here is Cr =Vr R for the retailer. Because of the strategic alliance, we assume that the manufacturer shares a fraction of this forecast cost. Retailer’s portion of cost is Cr =VrR X, and manufacturer’s portion is Cr =VrR ð1XÞ (with 0 r X r 1). The fraction of the forecast cost X, will be decided by the manufacturer and is one of his decision variables. We use the superscript R for this case. The sequence of moves is as follows: 1. The manufacturer obtains and announces his optimal wholesale price w, based on retailer’s forecast fr , known to him, and retailer’s best response function. 2. Next, the retailer obtains and announces her optimal price p, based on manufacturer’s action w, and her own forecast fr . The optimal forecast variance VrR is derived by the retailer through maximizing her unconditional expected profit function. The optimal level of forecast cost sharing percentage X is derived by manufacturer through maximizing his unconditional expected profit and using the best response function of VrR . These results are shown in the next proposition. Proposition 3. For retailer forecasting (R) case (a) The Stackelberg equilibrium prices and forecasting variances for the manufacturer and the retailer are AR 3ARr wR ¼ r , pR ¼ 2b 4b VrR ¼
pffiffiffiffiffiffiffiffiffiffiffi 4Uð4Cr bX þU Cr bX Þ where U 2 Z 16Cr bX,X ¼ 0:7435412222 U 2 16Cr bX
(b) The unconditional expected profits of the retailer and the manufacturer are given below 1 U2 Cr ð1XÞ EðpRm Þ ¼ EðEðpm 9fr ÞÞ ¼ a2 þ R 8b VrR U þ Vr 2 1 U Cr X R EðpRr Þ ¼ EðEðpr 9fr ÞÞ ¼ a2 þ 16b Vr U þVrR
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We next introduce the following two propositions wherein we derive some important results regarding the profits and optimum pricing strategies of the manufacturer and retailer under the three different cases. Recall that the manufacturer is the leader firm and also the firm which suffers from information asymmetry because it does not know the forecast of the other firm at the time it makes the pricing decision. The retailer, as the follower firm, has the privilege of knowing both the forecasts and manufacturer’s wholesale pricing decision while making her own retail pricing decision. Proposition 4. The optimum manufacturer wholesale price and retail price compared as follows: (a) wI owN and pI o pN if and only if fr oEðfr 9fm Þ. (b) wI owR and pI o pR if and only if fm oEðfm 9fr Þ. This proposition gives the insight that when manufacturer’s estimate of retailer’s forecast is higher than the true value, the manufacturer sets a high wholesale price in the case of no information sharing compared with the information sharing case. At the same time, the retailer sets a higher retail price. Similarly, when retailer’s estimate of manufacturer’s forecast is higher than the true value, the manufacturer sets a lower wholesale price in information sharing case, compared with the case of retailer forecasts. From Proposition 4(a) and (b), we see that the information sharing case is always the first-best case among the three scenarios if the manufacturer fears an overestimation of retailer’s true forecast value. A similar analysis can be applied to the retail price setting. The managerial insight we get here is as follows. In case the retailer shares her forecast information with the manufacturer (i.e., go from case (N) into case (I)), the manufacturer reduces the wholesale price. The retailer can take this as a signal from manufacturer’s action. From Proposition 4(a), the retailer can conclude that manufacturer’s estimate of retailer’s forecast is higher than her true value, i.e., fr oEðfr 9fm Þ, and retailer should also reduce the retail price correspondingly. In the other case, the manufacturer makes his forecast and shares his forecast information with the retailer (rather than just use retailer’s forecast, i.e. go from case (R) to case (I)). The retailer will persuade the manufacturer to reduce the wholesale price if the retailer finds that she overestimates manufacturer’s forecast, fm oEðfm 9fr Þ, as seen in Proposition 4(b). The retailer should also reduce the retail price after the manufacturer reduces the wholesale price. In the next proposition, we will compare the manufacturer and retailer’s unconditional expected profits under certain assumptions. Proposition 5. The comparisons of unconditional expected profits for the manufacturer and retailer under the assumption of Cm ¼ Cr ¼ 0 N I R R (a) EðpIm Þ Z EðpN m Þ when U r Vm and Eðpm Þ Z Eðpm Þ when U rVr (b) EðpIr Þ ZEðpRr Þ when U rVrR I 2 N 2 N U2 U (c) EðpN r Þ ZEðpr Þ, if 4J ðU þ Vm Þ þ 4K ðU þVr Þ4JU þ ðU þ V N Þ Z 2 m
3.4. Analysis of the three cases In the previous three subsections, we derived the optimal pricing decisions and profits for the three scenarios. Note that each scenario represents a strategic business model for the firms. In this section, we focus on analyzing how the sharing of forecast information impacts firm’s performance in different scenarios. All the results obtained earlier for the three cases are summarized in Table 1.
This proposition is derived under the special case of no forecasting cost, i.e., Cm ¼ Cr ¼ 0. We see that, in case the true market demand variance is less than manufacturer’s forecast variance, then manufacturer earns more profit with information sharing than without. The manufacturer reaches his profit upper bound of 1=8b ða2 þU=2Þ. We also see that when the market demand variance is lower than retailer’s forecast variance, both manufacturer and retailer will earn higher profit with information sharing
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Table 1 Equilibrium prices, forecast variances and profits under three cases. Non-information sharing (N)
Information sharing (I)
Retailer Forecasting (R)
Manufacturer information Retailer information Manufacturer wholesale price (w)
fm fr and fm
fr and fm fr and fm
fr fr
A 2b
Retailer Retailing price (p)
A 2b
3A 4b
Ar 2b 3Ar 4b
Vm I ¼ 0
N/A
VrI ¼ 0
VrI ¼
Manufacturer forecast variance (Vm)
Am 2b
þ A4mb
N Vm ¼
Retailer forecast variance (Vr)
Cr ðVrN Þ2
Manufacturer’s expected profits
A2m 8b
Retailer’s expected profits Manufacturer’s unconditional expected profits Retailer’s unconditional expected profits
pffiffiffiffiffiffiffiffiffi
2Uð4Cm b þ 2Cm bUÞ U 2 8bCm
N N N 2 N 2 U 2 Vm ðVrN U 2 þ 3U 2 Vm þ UðVm Þ þ Vr N ðVm Þ Þ N þ V N U þ V N V N Þ3 4bðUVm r r m
¼0
A2 VCmI 8b m
VCmN m 2 Am 1 A VCNr 2 4b r 2 1 U a2 þ U þ VCmN 8b VN m
1 4b
2
a 4
A2r VCmR 8b m
A2
VCrI r 2 Cm 1 U a þ 2 VI 8b 16b
m
m
2
N þ J 2 ðU þ Vm Þþ K 2 ðU þ VrN ÞJU þ 4ðUUþ V N Þ VCNr m
r
1 16b
pffiffiffiffiffiffiffiffi
4Uð4Cr bX þ U Cr bX Þ U 2 16Cr bX
a2 þ U2 VCrI r
A2r VCrR 16b r 1 8b
2 a2 þ U þU V R VCrR ð1XÞ r
r
X ¼ 0:7435412222
1 U2 a2 þ U þ VCrR X ¼ 0:7435412222 16b VR r
r
than with using retailer’s forecast. From Propositions 5(a) and (b), we state that the Information Sharing (I) is always the first-best solution for the manufacturer when there is no forecast cost and the market variance is smaller than forecast variance. The comparison of retailer’s profits under Non-Information Sharing (N) and Information Sharing (I) is shown in Propositions 5(c). There is no dominant scenario for the retailer. In the next section, we will continue to discuss and compare the manufacturer and retailer’s profits including costs of forecasting.
4. Numerical experiments In Section 3, we have been able to generate a number of propositions giving useful managerial guidelines. In this section, we take the analysis further using extensive numerical experimentation and gain more insights into the optimal policies and profits of the two parties. The numerical values used in this experiment are as follows: Parameters A
b U Cm Cr
Base Value 100 1 20 1 1
Fig. 1 shows the manufacturer and retailer’s forecast variances under Non-Information Sharing (N). From Proposition 1, the N ¼ optimal forecast variance for the manufacturer is Vm pffiffiffiffiffiffiffiffiffiffiffiffi 2 2Uð4Cm b þ 2Cm bUÞ=U 8bCm . Retailer’s forecast variance satisN N N 2 the equation Cr =ðVrN Þ2 U 2 Vm ðVrN U 2 þ 3U 2 Vm þ UðVm Þ N N 2 N N N N 3 þVr ðVm Þ Þ=4bðUVm þ Vr U þVr Vm Þ ¼ 0. Fig. 1 shows a positive
fies
relationship between manufacturer’s variance and retailer’s variance. We see that the manufacturer and retailer’s forecast variance both need to be relatively small or one of the variances (manufacturer’s or retailer’s) need to be relatively small if the other has relatively large forecast variance. In Fig. 2, we draw manufacturer’s profit under non-information sharing with or without including the cost of forecasting. We observe that manufacturer’s profit line excluding forecast cost is decreasing with forecast variance. This is an expected result. Manufacturer’s profit will decrease as the forecast inaccuracy increases. On the other hand, we observe that manufacturer’s profit including forecast cost is concave in his forecast variance and he gets his highest profit at Vm N ¼ 3:29, which is consistent with our analytical result. Hence, we see that there is an optimal forecasting
Fig. 1. Manufacturer and retailer’s forecast variance under Non-Information Sharing (N).
accuracy. The difference between the two lines is the forecast cost. We also see the forecast cost is significantly higher when forecast variance is low. Fig. 3 shows retailer’s profit under Retailer Forecast (R) case. The trend is similar to that seen in Fig. 2. Retailer’s profit line excluding forecast cost is decreasing with forecast variance. The retailer reaches her highest profit at Vr R ¼ 4:17 which is the optimal forecast variance and is consistent with the analytical result. In the next set of experiments, we compare manufacturer’s profit under Retailer Only Forecast (R) case and under NonInformation Sharing (N) case with the assumption of V m N ¼ V r R (manufacturer’s forecast variance under Non-Information Sharing (N) equals to his forecast variance shared by the retailer under Retailer Forecasting (R) case) and Cm ¼ Cr . Fig. 4 shows that manufacturer’s profit under (R), is always higher than that under (N), when the manufacturer and the retailer have the same level of forecast accuracy and same effective forecast cost. This result can be mathematically proved. This is an interesting finding. Even if the manufacturer can get the same level of forecast accuracy as the
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5. Conclusion
Fig. 2. Manufacturer’s profit vs. various forecast variances under Non-Information Sharing (N).
Fig. 3. Retailer’s profit vs. various her forecast variance under Retailer Sharing Forecast Information (R) case.
The market scenario studied in this paper is increasingly prevalent in reality. Getting more information for decision making is appealing and, at the same time, possible with new technology. The effort of gathering information may turn out to be a mixed blessing, though. In this paper, we study the scenario where a manufacturer and a retailer invest on their respective forecasting system, incurring costs of hardware, software, data, and labor hours, in order to improve the forecast accuracy and thereby profit. We first derived the equilibrium prices, forecast variance, and profits under three cases, Non-Information Sharing, Information Sharing, and Retailer Forecasting. We studied how the forecast variance and its related forecasting cost would impact firms’ expected profit. We also investigated which case should be preferred by the manufacturer and the retailer and under what conditions. Our numerical experiments illustrated and verified our analytical findings and provided further managerial insights and interpretations. One limitation of our model is that the demand is modeled to be only price-sensitive. Some other marketing variables such as promotion, advertising, etc., are excluded in the model. Also, our linear demand functions, combined with the linear decision rules that follow our assumptions about the forecast errors, make the analysis tractable, but may somehow limit the generalizability of our results. Further research can extend the model to include nonlinear demand functions, non-normal distributions of errors, and uncertainty about the slope coefficients. Future study could also address the forecasting cost’s impact on both the pricing decisions and inventory management simultaneously. Our findings would be relevant for helping firms understand how the forecast system design, like who forecasts, whether to share forecast or not and related forecast cost, will affect the profit. The insights from our research may also be useful for corporations who consider developing partnership mechanisms and quantify direct financial impact of investing on the forecasting systems. Despite the limitations mentioned above, we believe that our model addresses an important issue and hope that the proposed approach can prompt more research in this area.
Appendix Proof of Proposition 1. The retailer and the manufacturer maximize her and his profit, respectively, Eðpr 9fr ,fm Þ ¼ ðpwÞðEða9fr ,fm ÞbpÞ
Fig. 4. Manufacturer’s profits vs. various forecast variance.
retailer does, the manufacturer would still prefer to let retailer forecast the demand and then make her share it with him, instead of forecasting it himself.
Cr Vr
@Eðpr 9fr ,fm Þ ¼ Eða9fr ,fm Þ2bp þ wb ¼ 0 @p Eða9f ,f r m Þ þ wb pBR ¼ 2b Cm Eðpm 9fm Þ ¼ w Eða9fm ÞbEðpBR 9fm Þ Vm
Eða9fr ,fm Þ þ wb Cm ¼ w Eða9fm ÞbEð 9fm Þ 2b
Vm Eða9fm Þ þ wb Cm ¼ w Eða9fm Þ 2
Vm Eða9fm Þ wb Cm ¼ w Eða9fm Þ 2 Vm 2 EðaÞ w2 b ¼ wEða9fm Þw 2 2 @Eðpm 9fm Þ Eða9fm Þ ¼ Eða9fm Þ bw ¼ 0 @w 2 Eða9fm Þ a þtm ðfm aÞ ¼ w ¼ 2b 2b Please note that the retailer can infer manufacturer’s forecast, fm, after the manufacturer announce w from the above
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equation (solving fm by given w). p ¼
Eða9fr ,fm Þ þ bw 2b
a þJðfm aÞ þ Kðfr aÞ a þtm ðfm aÞ þ 4b 2b 3 K 2J þ tm A Am ¼ aþ ðfm aÞ ¼ þ ðfr aÞ þ 4b 2b 2b 4b 4b Cm Eðpm 9fm Þ ¼ w Eðabp 9fm Þ Vm
3 K 2J þ tm Cm ¼ w Eða9fm Þb aþ ðfm aÞ Eððfr aÞ9fm Þ þ 4b Vm 4b 2b ¼
3 K 2J þ tm Cm ðfm aÞ ¼ w a þ tm ðfm aÞ a ðEðfr 9fm ÞaÞ 4 Vm 4 2 3 K 2J þ tm Cm ðfm aÞ ¼ w a þ tm ðfm aÞ a tm ðfm aÞ 4 Vm 4 2 a þ tm ðfm aÞ 1 3tm 2Ktm 2J Cm ¼ aþ ðfm aÞ Vm 2b 4 4 2 a tm ð3tm 2Ktm 2JÞ að3tm 2Ktm 2JÞ ¼ þ ðfm aÞ2 þ ðfm aÞ 8b 8b 8b
atm ðfm aÞ Cm Am 2 Cm þ ¼ Since Eðfi aÞ ¼ 0 and 8b Vm 8b Vm 2 Eðfi aÞ ¼ U þVi ¼ U=ti EðEðpm 9fm ÞÞ ¼
a2 tm ð3tm 2Ktm 2JÞ Cm þ Eðfm aÞ2 8b 8b Vm
a2 ð3tm 2Ktm 2JÞ Cm þ U 8b 8b Vm a2 tm Cm 1 U2 Cm ¼ þ U ¼ a2 þ 8b 8b 8b Vm U þVm Vm ¼
Optimal Vm can be got by solving @EðEðpm 9fm ÞÞ=@Vm ¼ 0. We get N ¼ Vm
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2Uð4Cm bU 2Cm bÞ 2Uð4Cm b þ U 2Cm bÞ or U 2 8bCm U 2 8bCm
Only
pffiffiffiffiffiffiffiffiffiffiffiffi 2Uð4Cm b þ U 2Cm bÞ have valid solution and with U 2 4 8bCm : U 2 8bCm
Cr E pr 9fr ,fm ¼ Eððp w Þðabp Þ9fr ,fm Þ Vr Cr ¼ ðp w ÞðEða9fr ,fm Þbp Þ Vr Eða9fr ,fm Þw b Eða9fr ,fm Þbw Cr ¼ 2b 2 V 0 12 r a þ tm ðfm aÞ 1 a þJðfm aÞ þ Kðfr aÞb 2b A Cr ¼ @ b 2 Vr ¼
1 a
b 4
þ
2 ð2Jtm Þðfm aÞ Kðfr aÞ Cr 1 A m 2 Cr or A þ Vr 2 Vr 4 2 4b
Since Eðfi aÞ ¼ 0, Eðfi aÞ2 ¼ U þVi ¼ U=ti , Eðfm aÞðfr aÞ ¼ 0 ! 1 a2 ð2Jtm Þ2 Eðfm aÞ2 K 2 Eðfr aÞ2 Cr þ þ E E pr 9fr ,fm ¼ b 16 16 4 Vr ! 1 a2 ð2Jtm Þ2 U K 2 U Cr þ ¼ þ 16tm 4tr b 16 Vr or can be written as ! 1 a2 U2 Cr 2 2 þ J ðU þVm Þ þ K ðU þ Vr ÞJU þ 4b 4 4ðU þ Vm Þ Vr Optimal Vr can be got by solving @EðEðpr 9fr ,fm ÞÞ=@Vr ¼ 0. We get Vr 2 2 Cr U 2 Vm ðVr U 2 þ 3U 2 Vm þ UVm þVr Vm Þ ¼ 0: & 2 3 þ V U þV V Þ Vr 4bðUVm r r m
Proof of Propositions 2 and 3. Similar to Proposition 1 and is omitted. & Proof of Proposition 4. Part (a), Similar to Proposition 4(b) and is omitted.
Part (b) wI wR ¼
ðAAr Þ o0 2b
if
A o Ar
or
fm o Eðfm 9fr Þ
We next show that A oAr is equivalent to fm o Eðfm 9fr Þ. Proof of E A9fr ¼ Ar EðA9fr Þ ¼ E½ð1JKÞa þ Jfm 9fr þ Kfr ¼ ð1JKÞa þJEðfm 9fr Þ þKfr ¼ a þ ðfr aÞðJtr þKÞ ¼ a þ ðfr aÞtr ¼ Ar Using above conclusion, Ar ¼ EðA9fr Þ ¼ ð1JKÞa þ JEðfm 9fr Þ þ Kfr ComparingAr with the A from the Eq. (6), A E½a9fr ,fm ¼ a þ Jðfm aÞ þKðfr aÞ ¼ ð1JKÞa þ Jfm þ Kfr We can get A o Ar is equivalent to fm o Eðfm 9fr Þ.
&
Proof of Proposition 5. Straightforward and is omitted.
&
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Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Role of forecast effort on supply chain profitability under various information sharing scenarios Xiaowei Zhu a,1, Samar K. Mukhopadhyay b,n, Xiaohang Yue c,2 a
College of Business and Public Affairs, West Chester University of Pennsylvania, West Chester, PA 19383, USA Sungkyunkwan University-GSB, 53 Myeongnyung-dong 3-Ga, Jongno-gu, Seoul 110-745, Republic of Korea c School of Business Administration, University of Wisconsin-Milwaukee, P.O. Box 742, Milwaukee, WI 53201, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 May 2009 Accepted 21 October 2010 Available online 27 October 2010
In this paper, we consider a manufacturer who sells a product to a retailer in a single selling season. Each party obtains a forecast of the market demand independent of each other. We study three different forecast scenarios: Non-Information Sharing, Information Sharing, and Retailer Forecasting cases. In the first scenario, both parties make their forecasts, but do not share the information with the other firm. In the second scenario, they share the information with each other, while in the last scenario, only the retailer makes the forecast, and shares it with the manufacturer. Noting that the forecast accuracy comes at a cost, we derive the optimal price and forecast accuracy level for each of the three cases. We then compare the optimal policies of the three cases and derive conditions under which the two parties should share information with each other. Results of extensive numerical experimentation are also presented. & 2010 Elsevier B.V. All rights reserved.
Keywords: Forecast cost Forecast variance Information sharing Combining forecasts
1. Introduction Supply chain participants are becoming increasingly more coupled in the sense that decisions taken by one firm in a supply chain directly affect the performance of other firms. Firms, therefore, have an incentive to devise methods to gain information about other firms’ decisions and other operating parameters. Recent advancements in information technology have been helpful in this respect. Today, different types of information like point-of-sales data, inventory, forecast data, and sales trends are being shared among supply chain partners quickly and inexpensively. Examples of well-known information sharing implementations include Wal-Mart’s Retail Link program, providing point-of-sales data to suppliers like Johnson and Johnson and Lever Brothers. Many other recent supply chain initiatives like Quick Response (QR), Efficient Consumer Response (ECR), Vendor-Managed Inventory (VMI), Continuous Replenishment Program (CRP), Radio-frequency identification (RFID), and Collaborative Planning, Forecasting and Replenishment (CPFR) are built on the principles of information sharing. Examples of major successes in the benefits of information sharing are Campbell Soup and Barilla SpA as narrated by Lee et al. (2000).
n
Corresponding author. Tel.: + 822 740 1522; fax: + 822 740 1503. E-mail addresses: [email protected] (X. Yue), [email protected] (X. Zhu), [email protected] (S.K. Mukhopadhyay). 1 Tel.: +1 610 738 0588. 2 Tel.: +1 414 229 4657. 0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2010.10.021
These developments in the real world have motivated the academic community to explore the benefits of information sharing. Particularly, many recent papers study the effects of information sharing in supply chain management. Mukhopadhyay et al., (2009) examined information sharing for complimentary goods, while that in a multi-generational product setting was studied by Sohn and Lim ˚ (2008). Smaros (2007) presents the results of a case study examining four collaboration projects involving a total of four suppliers and one retailer operating in the European grocery sector. He finds strong support for collaborative forecasting as a means of improving supply chain efficiency and advocate using valid assumptions to benefit from information sharing. Another case study was presented by Byrne and Heavey (2006) examining the effect of information sharing and forecasting on performance parameters of an industrial supply chain consisting of small-to-medium sized enterprises. They have shown that potential cost savings of up to 9.7% can be achieved through the use of improved forecasting techniques. The research cited above underscores the importance of information sharing. One very important aspect of information sharing in the context of supply chain is the forecast developed by various parties in the supply chain. Generally speaking, forecast is used for production and inventory planning, and the demand–supply matching is vastly improved, at any stage of a supply chain, when the forecast error is reduced or in other words, forecast accuracy is improved. While all the channel partners develop their own forecasts using their own methods, it has been shown by researchers that sharing of information among them does help in forecast improvement albeit in different degrees. Vives (1984) and Gal-Or (1985) study incentives for a firm to share private information
X. Zhu et al. / Int. J. Production Economics 129 (2011) 284–291
horizontally with competitors in an oligopolistic market. Impact of forecast improvement on optimal order quantity and selling price determination for a second cycle order is shown by Serel (2009). Hosoda and Disney (2009) show that if the supply chain attempts to minimize the demand forecast errors, then the impact of the ¨ demand mis-specification on cost is minor. Kerkkanen et al. (2009) study the role of forecast errors for defining a realistic target of forecast accuracy. Reiner and Fichtinger (2009) develop a dynamic model to evaluate different forecasting methods. Sarvary and Parker (1997) show that information from different sources could be substitutes or complements depending on characteristics such as variance and correlation. While the above researchers have established the value of information sharing in supply chain management, there is a body of research that study the incentives for information sharing are examined by Corbett and de Groote (2000), Cachon and Lariviere (2001), Ha (2001), and Mukhopadhyay et al. (2008). Agrawal et al. (2009) study the impact of information sharing and lead time on bullwhip effect and on-hand inventory of a supply chain. They show that while information sharing is beneficial for reduction in bullwhip effect, lead time reduction is even more beneficial. Information sharing in decentralized supply chains where one manufacturer supplies to multiple retailers competing on price or quality has been the focus of several papers like Li and Zhang (2008), Bernstein and Federgruen (2005), and Gaughety and Reinganum (2007). Raju and Roy (2000) use forecast in their pricing decision in both Stackelberg and Bertrand-Nash models. They found that information is more valuable for products with higher substitutability, and for larger firms. Roy (2000) studied channel pricing when there are two competing manufacturer– retailer channels with both channels facing stochastic downwardsloping demand functions. Their paper discusses how the forecast variance affects profit, but did not obtain the optimal level of forecast variance. Qi and Zhu (2009) study an endogenous determination of efforts put into information acquisition and its impact on supply chain management. They measure the effort level of a number of experts (hired by the buyer) in information acquisition. The forecast cost is measured by a commission fee. We see from current research cited above that sharing of information, especially the forecast information, could be beneficial for the supply chain partners who are not privy to the information. Nevertheless, the fact remains that the forecast information obtained by one party remains with that party and is seldom shared with others in the supply chain. We study this practical scenario in this paper. We analyze several forecast system designs from the information sharing point of view, and their impact on factors like forecast accuracy, forecast cost, and profit. We focus on the impact of forecast improvement/information sharing on pricing through mathematical modeling. Chen (1998), Cachon and Fisher (2000), Donohue (2000), and Yue et al., (2006) use similar approach in their papers. We use a more general model to measure the forecast cost. We include costs of obtaining data, and any forecast related costs like cost of purchasing software, hardware, labor hour, and overhead. In the specific scenario considered in our paper, a manufacturer sells a product to a retailer in a single selling season. Three different forecast systems studied in this paper are: Non-Information Sharing, Information Sharing, and Retailer Forecasting. The first case is where the manufacturer and the retailer both forecast separately but do not share the forecast information with each other. In the second case, the manufacturer and the retailer both forecast separately and share the forecast information with each other. The third is the case where only one side (the retailer) forecasts, and shares the information with the other side (the manufacturer). For each of these cases, we derive optimal forecast accuracy level and price, and discuss the forecast variance and the related forecast cost. Our study complements the
285
literature by explicitly characterizing the value of forecasting cost under a Stackelberg game mode and by deriving optimal policies regarding retail price, wholesale price, and forecast accuracy (as given by forecast variance). In the next section, we introduce our notation and model. In Section 3, we first present the analytical results under the three scenarios (Non-Information Sharing, Information Sharing, and Retailer Forecasting), and then compare the results of these three cases. In Section 4, we present the results of our numerical experiments and the insights obtained from them. Section 5 concludes the paper with some future research directions.
2. Model framework In this supply chain, the manufacturer first decides the wholesale price w. The retailer then sets the retail price p, after knowing w. We consider a one period model. The consumer demand d, is a linear function of retail price and is given by d ¼ abp
ð1Þ
where a is the primary demand, i.e., the demand determined by all factors except price, and b is the slope of the demand function. All parameters are assumed to be non-negative. To capture uncertainty in demand, we assume that a is a random variable defined as follows: a ¼ aþe
ð2Þ
where a is the deterministic part of the demand (mean) and e is the random part. Note that Eq. (2) is the commonly used forecast equation. e is normally distributed with a mean of zero and a variance of U. a is common knowledge for both the manufacturer and the retailer and is commonly derived from historical data. We assume that each party obtains a forecast about the unknown primary demand a using the market-information-gathering techniques at its disposal. We consider the general case that the forecasts obtained by the manufacturer and the retailer are also random variables. We denote manufacturer’s and retailer’s forecasts of primary demand as fm and fr, respectively. Therefore, fm ¼ a þ em
ð3Þ
fr ¼ a þ er
ð4Þ
where em and er are the forecast errors associated with the two parties, respectively. We also assume that em and er are independent of a and are normally distributed with a mean of zero and variances of Vm and Vr, respectively. Note from Eq. (3) that in case the forecast is very accurate, (i.e., em ¼0, say), then the forecast is equal to the actual primary demand, as it should be. A higher variance for the forecast error implies a less precise forecast. The forecast errors em and er can be correlated. The extent of correlation depends on the data and methodology used by the retailer and the manufacturer in their forecasting processes. If they use data and methodology very similar to each other, then it will result in a higher correlation between forecasts. The covariance matrix of forecast errors is represented by ! X Vr Vrm ¼ ð5Þ Vrm Vm The covariance between the two forecasts is Vrm and the pffiffiffiffiffipffiffiffiffiffiffiffi coefficient of correlation is r ¼ Vrm = Vr Vm . In this paper, we consider the case where r ¼ 0 i.e., the forecasts of the manufacturer and the retailer are not correlated. We also assume that the forecasts are obtained at a cost of Ci =Vi (i¼r, m). This says that to obtain a more accurate forecast (i.e., a lower forecast variance), there will be a need to invest more. Ferbar et al. (2009) show that
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that the consensus forecast E a9fr ,fm is more accurate than one sided forecast, like fr or fm . The equilibrium variance of the forecast can be obtained by maximizing the unconditional expected profit.
the use of a particular forecasting method affects the costs of a supply chain. Based on the model in Winkler (1981), we have the following parameters of the consensus distribution. When the manufacturer obtains his forecast fm, he computes the expected value of the market primary demand a, i.e.,E½a9fm as the weighted average of a (which is common knowledge) and fm, his own forecast. i.e.,
We derive the equilibrium price, variance, and profit of each firm in this scenario in Proposition 1. Proofs of all results are given in the Appendix.
E½a9fm ¼ a þ tm ðfm aÞ Am
Proposition 1. For the non-information sharing (N) case:
ð6Þ
For clarity, we denote this quantity as Am. Following the same path, the other similar parameters are derived as E½a9fr ¼ a þ tr ðfr aÞ Ar , say, E½a9fr ,fm ¼ a þ Jðfm aÞ þ Kðfr aÞ A, say, E½fr 9fm ¼ a þ dm ðfm aÞ, and E½fm 9fr ¼ a þ dr ðfr aÞ The respective variances of these parameters are given by Var a9fm ¼ tm Vm Var a9fr ¼ tr Vr ðVr Vm ÞVð1r2 Þ pffiffiffiffiffiffiffipffiffiffiffiffi Var a9fr ,fm ¼ ð1r2 ÞVr Vm þ VðVr þVm 2r Vm Vr Þ It can be shown that U U U þ Vrm U þVrm , tm ¼ , dr ¼ , dm ¼ U þ Vr U þVm U þVr U þ Vm tm ðdr 1Þ tr ðdm 1Þ ,K ¼ J¼ dr dm 1 dr dm 1 Because of our assumption of r ¼ 0, we have Vrm ¼ 0, hence the above parameters are reduced to tr ¼
tr ¼ dr ¼
U U tm ðtr 1Þ tr ðtm 1Þ ,K¼ , tm ¼ dm ¼ , J¼ U þVr U þ Vm tr tm 1 tr tm 1
The profit functions of the retailer and the manufacturer are, respectively Cr Vr C pm ¼ wðabpÞ m Vm
pr ¼ ðpwÞðabpÞ
3. Model analysis We assume, as stated earlier, that the manufacturer makes the first decision (on wholesale price) and therefore, acts as a leader. A Stackelberg type game analysis is appropriate in this situation. We analyze three different scenarios of the Stackelberg game as given below. 3.1. Stackelberg game with no information sharing (N) In this form of Stackelberg game, information about manufacturer’s forecast is not shared with the other firm. We use the superscript N for this case. The sequence of moves is as follows: 1. The manufacturer obtains and announces his optimal price w based on his forecast fm and his belief about retailer’s forecast (i.e.,Eðfr 9fm Þ in Eq. (6)), and retailer’s best response function (retailer’s optimal action as a function of manufacturer’s action). 2. Next, the retailer uses manufacturer’s announced wholesale price w, and the consensus forecast of primary demand a9fr ,fm , and obtains and announces her optimal price p. We will see in Proposition 1 that w ¼ Am =2b. Where Am ¼ Am ¼ a þ tm ðfm aÞ (from Eq. (6)). This is a linear function of fm. Hence, once w is known, the retailer can infer manufacturer’s forecast fm . This observation is consistent with Raju and Roy (2000). We assume
a. The Stackelberg equilibrium prices and forecasting variances for the manufacturer and the retailer are AN AN AN m , pN ¼ þ m, 2b 2b 4b pffiffiffiffiffiffiffiffiffiffiffiffi 2Uð4Cm b þ 2Cm bUÞ N Vm ¼ , with U 2 4 8bCm U 2 8Cmb wN ¼
Vr N satisfies the following equation: Cr ðVrN Þ2
N N N 2 N 2 U 2 Vm ðVrN U 2 þ3U 2 Vm þ UðVm Þ þ Vr N ðVm Þ Þ N þV N U þ V N V N Þ3 4bðUVm r r m
¼0
b. The unconditional expected profits of the retailer and the manufacturer are: 1 U2 Cm N EðpN a2 þ N m Þ ¼ EðEðpm 9fm ÞÞ ¼ N 8b U þVm Vm ! 1 a2 ð2Jtm Þ2 U K 2 U Cr N N þ Eðpr Þ ¼ EðEðpr 9fr ,fm ÞÞ ¼ þ N 16tm 4tr b 16 Vr
3.2. Stackelberg game with information sharing (I) In this subsection, we consider that the manufacturer and the retailer share information about each other’s forecast. We use the superscript I for the Stackelberg Information sharing case. Intuitively, this will increase the quality of their decisions. The sequence of moves is as follows: 1. The manufacturer obtains and announces his optimal wholesale price w, based on his own forecast fm , retailer’s forecast fr , which is now a shared information, and retailer’s best response function. 2. Next, the retailer obtains and announces her optimal price p, based on manufacturer’s action w, her own forecast fr , and manufacturer’s forecast fm . The manufacturer and the retailer maximize their own expected profits based on both fr and fm (note: each firm would come up with a consensus forecast about the primary demanda using his/her own forecast and the shared forecast from the other firm based on Winkler’s consensus model). The optimum equilibrium prices, forecast variance, and corresponding profits for this case are derived in the next proposition. Proposition 2. For information sharing (I) case (a) The Stackelberg equilibrium prices and forecasting variances for the manufacturer and the retailer are: a 3a I wI ¼ , pI ¼ , V ¼ VrI ¼ 0 2b 4b m (b) The unconditional expected profits of the retailer and the manufacturer are given below. 1 U Cm I a2 þ 8b V m 2 1 U Cr I EðpIr Þ ¼ EðEðpr 9fr ,fm ÞÞ ¼ a2 þ 16b 2 Vr
EðpIm Þ ¼ EðEðpm 9fm ÞÞ ¼
X. Zhu et al. / Int. J. Production Economics 129 (2011) 284–291
The optimal variance for both the forecasts came out as zero denoting that both should have a very high degree of accuracy. Obviously, the investment for such a forecasting system will be prohibitively high. Thus, practically, it would be hard to follow the optimal policies using the information sharing mode, from the implementation point of view. 3.3. Stackelberg game with retailer forecasting (R) There are times when the two firms would enter into a strategic alliance to benefit from each other’s core competency and unique advantages. It is very common for the retailer to make more accurate forecast than the manufacturer because she is closer to customers and has better information about the market demand. In this section, we study the third scenario where only the retailer forecasts the market demand, denoted by fr , and then shares her forecasting information to be used by the manufacturer. The manufacturer uses retailer’s forecast informationfr , to decide his wholesale price w. The forecasting cost here is Cr =Vr R for the retailer. Because of the strategic alliance, we assume that the manufacturer shares a fraction of this forecast cost. Retailer’s portion of cost is Cr =VrR X, and manufacturer’s portion is Cr =VrR ð1XÞ (with 0 r X r 1). The fraction of the forecast cost X, will be decided by the manufacturer and is one of his decision variables. We use the superscript R for this case. The sequence of moves is as follows: 1. The manufacturer obtains and announces his optimal wholesale price w, based on retailer’s forecast fr , known to him, and retailer’s best response function. 2. Next, the retailer obtains and announces her optimal price p, based on manufacturer’s action w, and her own forecast fr . The optimal forecast variance VrR is derived by the retailer through maximizing her unconditional expected profit function. The optimal level of forecast cost sharing percentage X is derived by manufacturer through maximizing his unconditional expected profit and using the best response function of VrR . These results are shown in the next proposition. Proposition 3. For retailer forecasting (R) case (a) The Stackelberg equilibrium prices and forecasting variances for the manufacturer and the retailer are AR 3ARr wR ¼ r , pR ¼ 2b 4b VrR ¼
pffiffiffiffiffiffiffiffiffiffiffi 4Uð4Cr bX þU Cr bX Þ where U 2 Z 16Cr bX,X ¼ 0:7435412222 U 2 16Cr bX
(b) The unconditional expected profits of the retailer and the manufacturer are given below 1 U2 Cr ð1XÞ EðpRm Þ ¼ EðEðpm 9fr ÞÞ ¼ a2 þ R 8b VrR U þ Vr 2 1 U Cr X R EðpRr Þ ¼ EðEðpr 9fr ÞÞ ¼ a2 þ 16b Vr U þVrR
287
We next introduce the following two propositions wherein we derive some important results regarding the profits and optimum pricing strategies of the manufacturer and retailer under the three different cases. Recall that the manufacturer is the leader firm and also the firm which suffers from information asymmetry because it does not know the forecast of the other firm at the time it makes the pricing decision. The retailer, as the follower firm, has the privilege of knowing both the forecasts and manufacturer’s wholesale pricing decision while making her own retail pricing decision. Proposition 4. The optimum manufacturer wholesale price and retail price compared as follows: (a) wI owN and pI o pN if and only if fr oEðfr 9fm Þ. (b) wI owR and pI o pR if and only if fm oEðfm 9fr Þ. This proposition gives the insight that when manufacturer’s estimate of retailer’s forecast is higher than the true value, the manufacturer sets a high wholesale price in the case of no information sharing compared with the information sharing case. At the same time, the retailer sets a higher retail price. Similarly, when retailer’s estimate of manufacturer’s forecast is higher than the true value, the manufacturer sets a lower wholesale price in information sharing case, compared with the case of retailer forecasts. From Proposition 4(a) and (b), we see that the information sharing case is always the first-best case among the three scenarios if the manufacturer fears an overestimation of retailer’s true forecast value. A similar analysis can be applied to the retail price setting. The managerial insight we get here is as follows. In case the retailer shares her forecast information with the manufacturer (i.e., go from case (N) into case (I)), the manufacturer reduces the wholesale price. The retailer can take this as a signal from manufacturer’s action. From Proposition 4(a), the retailer can conclude that manufacturer’s estimate of retailer’s forecast is higher than her true value, i.e., fr oEðfr 9fm Þ, and retailer should also reduce the retail price correspondingly. In the other case, the manufacturer makes his forecast and shares his forecast information with the retailer (rather than just use retailer’s forecast, i.e. go from case (R) to case (I)). The retailer will persuade the manufacturer to reduce the wholesale price if the retailer finds that she overestimates manufacturer’s forecast, fm oEðfm 9fr Þ, as seen in Proposition 4(b). The retailer should also reduce the retail price after the manufacturer reduces the wholesale price. In the next proposition, we will compare the manufacturer and retailer’s unconditional expected profits under certain assumptions. Proposition 5. The comparisons of unconditional expected profits for the manufacturer and retailer under the assumption of Cm ¼ Cr ¼ 0 N I R R (a) EðpIm Þ Z EðpN m Þ when U r Vm and Eðpm Þ Z Eðpm Þ when U rVr (b) EðpIr Þ ZEðpRr Þ when U rVrR I 2 N 2 N U2 U (c) EðpN r Þ ZEðpr Þ, if 4J ðU þ Vm Þ þ 4K ðU þVr Þ4JU þ ðU þ V N Þ Z 2 m
3.4. Analysis of the three cases In the previous three subsections, we derived the optimal pricing decisions and profits for the three scenarios. Note that each scenario represents a strategic business model for the firms. In this section, we focus on analyzing how the sharing of forecast information impacts firm’s performance in different scenarios. All the results obtained earlier for the three cases are summarized in Table 1.
This proposition is derived under the special case of no forecasting cost, i.e., Cm ¼ Cr ¼ 0. We see that, in case the true market demand variance is less than manufacturer’s forecast variance, then manufacturer earns more profit with information sharing than without. The manufacturer reaches his profit upper bound of 1=8b ða2 þU=2Þ. We also see that when the market demand variance is lower than retailer’s forecast variance, both manufacturer and retailer will earn higher profit with information sharing
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Table 1 Equilibrium prices, forecast variances and profits under three cases. Non-information sharing (N)
Information sharing (I)
Retailer Forecasting (R)
Manufacturer information Retailer information Manufacturer wholesale price (w)
fm fr and fm
fr and fm fr and fm
fr fr
A 2b
Retailer Retailing price (p)
A 2b
3A 4b
Ar 2b 3Ar 4b
Vm I ¼ 0
N/A
VrI ¼ 0
VrI ¼
Manufacturer forecast variance (Vm)
Am 2b
þ A4mb
N Vm ¼
Retailer forecast variance (Vr)
Cr ðVrN Þ2
Manufacturer’s expected profits
A2m 8b
Retailer’s expected profits Manufacturer’s unconditional expected profits Retailer’s unconditional expected profits
pffiffiffiffiffiffiffiffiffi
2Uð4Cm b þ 2Cm bUÞ U 2 8bCm
N N N 2 N 2 U 2 Vm ðVrN U 2 þ 3U 2 Vm þ UðVm Þ þ Vr N ðVm Þ Þ N þ V N U þ V N V N Þ3 4bðUVm r r m
¼0
A2 VCmI 8b m
VCmN m 2 Am 1 A VCNr 2 4b r 2 1 U a2 þ U þ VCmN 8b VN m
1 4b
2
a 4
A2r VCmR 8b m
A2
VCrI r 2 Cm 1 U a þ 2 VI 8b 16b
m
m
2
N þ J 2 ðU þ Vm Þþ K 2 ðU þ VrN ÞJU þ 4ðUUþ V N Þ VCNr m
r
1 16b
pffiffiffiffiffiffiffiffi
4Uð4Cr bX þ U Cr bX Þ U 2 16Cr bX
a2 þ U2 VCrI r
A2r VCrR 16b r 1 8b
2 a2 þ U þU V R VCrR ð1XÞ r
r
X ¼ 0:7435412222
1 U2 a2 þ U þ VCrR X ¼ 0:7435412222 16b VR r
r
than with using retailer’s forecast. From Propositions 5(a) and (b), we state that the Information Sharing (I) is always the first-best solution for the manufacturer when there is no forecast cost and the market variance is smaller than forecast variance. The comparison of retailer’s profits under Non-Information Sharing (N) and Information Sharing (I) is shown in Propositions 5(c). There is no dominant scenario for the retailer. In the next section, we will continue to discuss and compare the manufacturer and retailer’s profits including costs of forecasting.
4. Numerical experiments In Section 3, we have been able to generate a number of propositions giving useful managerial guidelines. In this section, we take the analysis further using extensive numerical experimentation and gain more insights into the optimal policies and profits of the two parties. The numerical values used in this experiment are as follows: Parameters A
b U Cm Cr
Base Value 100 1 20 1 1
Fig. 1 shows the manufacturer and retailer’s forecast variances under Non-Information Sharing (N). From Proposition 1, the N ¼ optimal forecast variance for the manufacturer is Vm pffiffiffiffiffiffiffiffiffiffiffiffi 2 2Uð4Cm b þ 2Cm bUÞ=U 8bCm . Retailer’s forecast variance satisN N N 2 the equation Cr =ðVrN Þ2 U 2 Vm ðVrN U 2 þ 3U 2 Vm þ UðVm Þ N N 2 N N N N 3 þVr ðVm Þ Þ=4bðUVm þ Vr U þVr Vm Þ ¼ 0. Fig. 1 shows a positive
fies
relationship between manufacturer’s variance and retailer’s variance. We see that the manufacturer and retailer’s forecast variance both need to be relatively small or one of the variances (manufacturer’s or retailer’s) need to be relatively small if the other has relatively large forecast variance. In Fig. 2, we draw manufacturer’s profit under non-information sharing with or without including the cost of forecasting. We observe that manufacturer’s profit line excluding forecast cost is decreasing with forecast variance. This is an expected result. Manufacturer’s profit will decrease as the forecast inaccuracy increases. On the other hand, we observe that manufacturer’s profit including forecast cost is concave in his forecast variance and he gets his highest profit at Vm N ¼ 3:29, which is consistent with our analytical result. Hence, we see that there is an optimal forecasting
Fig. 1. Manufacturer and retailer’s forecast variance under Non-Information Sharing (N).
accuracy. The difference between the two lines is the forecast cost. We also see the forecast cost is significantly higher when forecast variance is low. Fig. 3 shows retailer’s profit under Retailer Forecast (R) case. The trend is similar to that seen in Fig. 2. Retailer’s profit line excluding forecast cost is decreasing with forecast variance. The retailer reaches her highest profit at Vr R ¼ 4:17 which is the optimal forecast variance and is consistent with the analytical result. In the next set of experiments, we compare manufacturer’s profit under Retailer Only Forecast (R) case and under NonInformation Sharing (N) case with the assumption of V m N ¼ V r R (manufacturer’s forecast variance under Non-Information Sharing (N) equals to his forecast variance shared by the retailer under Retailer Forecasting (R) case) and Cm ¼ Cr . Fig. 4 shows that manufacturer’s profit under (R), is always higher than that under (N), when the manufacturer and the retailer have the same level of forecast accuracy and same effective forecast cost. This result can be mathematically proved. This is an interesting finding. Even if the manufacturer can get the same level of forecast accuracy as the
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289
5. Conclusion
Fig. 2. Manufacturer’s profit vs. various forecast variances under Non-Information Sharing (N).
Fig. 3. Retailer’s profit vs. various her forecast variance under Retailer Sharing Forecast Information (R) case.
The market scenario studied in this paper is increasingly prevalent in reality. Getting more information for decision making is appealing and, at the same time, possible with new technology. The effort of gathering information may turn out to be a mixed blessing, though. In this paper, we study the scenario where a manufacturer and a retailer invest on their respective forecasting system, incurring costs of hardware, software, data, and labor hours, in order to improve the forecast accuracy and thereby profit. We first derived the equilibrium prices, forecast variance, and profits under three cases, Non-Information Sharing, Information Sharing, and Retailer Forecasting. We studied how the forecast variance and its related forecasting cost would impact firms’ expected profit. We also investigated which case should be preferred by the manufacturer and the retailer and under what conditions. Our numerical experiments illustrated and verified our analytical findings and provided further managerial insights and interpretations. One limitation of our model is that the demand is modeled to be only price-sensitive. Some other marketing variables such as promotion, advertising, etc., are excluded in the model. Also, our linear demand functions, combined with the linear decision rules that follow our assumptions about the forecast errors, make the analysis tractable, but may somehow limit the generalizability of our results. Further research can extend the model to include nonlinear demand functions, non-normal distributions of errors, and uncertainty about the slope coefficients. Future study could also address the forecasting cost’s impact on both the pricing decisions and inventory management simultaneously. Our findings would be relevant for helping firms understand how the forecast system design, like who forecasts, whether to share forecast or not and related forecast cost, will affect the profit. The insights from our research may also be useful for corporations who consider developing partnership mechanisms and quantify direct financial impact of investing on the forecasting systems. Despite the limitations mentioned above, we believe that our model addresses an important issue and hope that the proposed approach can prompt more research in this area.
Appendix Proof of Proposition 1. The retailer and the manufacturer maximize her and his profit, respectively, Eðpr 9fr ,fm Þ ¼ ðpwÞðEða9fr ,fm ÞbpÞ
Fig. 4. Manufacturer’s profits vs. various forecast variance.
retailer does, the manufacturer would still prefer to let retailer forecast the demand and then make her share it with him, instead of forecasting it himself.
Cr Vr
@Eðpr 9fr ,fm Þ ¼ Eða9fr ,fm Þ2bp þ wb ¼ 0 @p Eða9f ,f r m Þ þ wb pBR ¼ 2b Cm Eðpm 9fm Þ ¼ w Eða9fm ÞbEðpBR 9fm Þ Vm
Eða9fr ,fm Þ þ wb Cm ¼ w Eða9fm ÞbEð 9fm Þ 2b
Vm Eða9fm Þ þ wb Cm ¼ w Eða9fm Þ 2
Vm Eða9fm Þ wb Cm ¼ w Eða9fm Þ 2 Vm 2 EðaÞ w2 b ¼ wEða9fm Þw 2 2 @Eðpm 9fm Þ Eða9fm Þ ¼ Eða9fm Þ bw ¼ 0 @w 2 Eða9fm Þ a þtm ðfm aÞ ¼ w ¼ 2b 2b Please note that the retailer can infer manufacturer’s forecast, fm, after the manufacturer announce w from the above
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equation (solving fm by given w). p ¼
Eða9fr ,fm Þ þ bw 2b
a þJðfm aÞ þ Kðfr aÞ a þtm ðfm aÞ þ 4b 2b 3 K 2J þ tm A Am ¼ aþ ðfm aÞ ¼ þ ðfr aÞ þ 4b 2b 2b 4b 4b Cm Eðpm 9fm Þ ¼ w Eðabp 9fm Þ Vm
3 K 2J þ tm Cm ¼ w Eða9fm Þb aþ ðfm aÞ Eððfr aÞ9fm Þ þ 4b Vm 4b 2b ¼
3 K 2J þ tm Cm ðfm aÞ ¼ w a þ tm ðfm aÞ a ðEðfr 9fm ÞaÞ 4 Vm 4 2 3 K 2J þ tm Cm ðfm aÞ ¼ w a þ tm ðfm aÞ a tm ðfm aÞ 4 Vm 4 2 a þ tm ðfm aÞ 1 3tm 2Ktm 2J Cm ¼ aþ ðfm aÞ Vm 2b 4 4 2 a tm ð3tm 2Ktm 2JÞ að3tm 2Ktm 2JÞ ¼ þ ðfm aÞ2 þ ðfm aÞ 8b 8b 8b
atm ðfm aÞ Cm Am 2 Cm þ ¼ Since Eðfi aÞ ¼ 0 and 8b Vm 8b Vm 2 Eðfi aÞ ¼ U þVi ¼ U=ti EðEðpm 9fm ÞÞ ¼
a2 tm ð3tm 2Ktm 2JÞ Cm þ Eðfm aÞ2 8b 8b Vm
a2 ð3tm 2Ktm 2JÞ Cm þ U 8b 8b Vm a2 tm Cm 1 U2 Cm ¼ þ U ¼ a2 þ 8b 8b 8b Vm U þVm Vm ¼
Optimal Vm can be got by solving @EðEðpm 9fm ÞÞ=@Vm ¼ 0. We get N ¼ Vm
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2Uð4Cm bU 2Cm bÞ 2Uð4Cm b þ U 2Cm bÞ or U 2 8bCm U 2 8bCm
Only
pffiffiffiffiffiffiffiffiffiffiffiffi 2Uð4Cm b þ U 2Cm bÞ have valid solution and with U 2 4 8bCm : U 2 8bCm
Cr E pr 9fr ,fm ¼ Eððp w Þðabp Þ9fr ,fm Þ Vr Cr ¼ ðp w ÞðEða9fr ,fm Þbp Þ Vr Eða9fr ,fm Þw b Eða9fr ,fm Þbw Cr ¼ 2b 2 V 0 12 r a þ tm ðfm aÞ 1 a þJðfm aÞ þ Kðfr aÞb 2b A Cr ¼ @ b 2 Vr ¼
1 a
b 4
þ
2 ð2Jtm Þðfm aÞ Kðfr aÞ Cr 1 A m 2 Cr or A þ Vr 2 Vr 4 2 4b
Since Eðfi aÞ ¼ 0, Eðfi aÞ2 ¼ U þVi ¼ U=ti , Eðfm aÞðfr aÞ ¼ 0 ! 1 a2 ð2Jtm Þ2 Eðfm aÞ2 K 2 Eðfr aÞ2 Cr þ þ E E pr 9fr ,fm ¼ b 16 16 4 Vr ! 1 a2 ð2Jtm Þ2 U K 2 U Cr þ ¼ þ 16tm 4tr b 16 Vr or can be written as ! 1 a2 U2 Cr 2 2 þ J ðU þVm Þ þ K ðU þ Vr ÞJU þ 4b 4 4ðU þ Vm Þ Vr Optimal Vr can be got by solving @EðEðpr 9fr ,fm ÞÞ=@Vr ¼ 0. We get Vr 2 2 Cr U 2 Vm ðVr U 2 þ 3U 2 Vm þ UVm þVr Vm Þ ¼ 0: & 2 3 þ V U þV V Þ Vr 4bðUVm r r m
Proof of Propositions 2 and 3. Similar to Proposition 1 and is omitted. & Proof of Proposition 4. Part (a), Similar to Proposition 4(b) and is omitted.
Part (b) wI wR ¼
ðAAr Þ o0 2b
if
A o Ar
or
fm o Eðfm 9fr Þ
We next show that A oAr is equivalent to fm o Eðfm 9fr Þ. Proof of E A9fr ¼ Ar EðA9fr Þ ¼ E½ð1JKÞa þ Jfm 9fr þ Kfr ¼ ð1JKÞa þJEðfm 9fr Þ þKfr ¼ a þ ðfr aÞðJtr þKÞ ¼ a þ ðfr aÞtr ¼ Ar Using above conclusion, Ar ¼ EðA9fr Þ ¼ ð1JKÞa þ JEðfm 9fr Þ þ Kfr ComparingAr with the A from the Eq. (6), A E½a9fr ,fm ¼ a þ Jðfm aÞ þKðfr aÞ ¼ ð1JKÞa þ Jfm þ Kfr We can get A o Ar is equivalent to fm o Eðfm 9fr Þ.
&
Proof of Proposition 5. Straightforward and is omitted.
&
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