Discount pricing decisions in distribution channels with price-sensitive demand

European Journal of Operational Research 149 (2003) 571–587 www.elsevier.com/locate/dsw

Production, Manufacturing and Logistics

Discount pricing decisions in distribution channels with price-sensitive demand S. Viswanathan *, Qinan Wang Nanyang Business School, Nanyang Technological University, Singapore 639798, Singapore Received 19 February 2001; accepted 27 May 2002

Abstract In this paper, we evaluate the effectiveness of quantity discounts and volume discounts as coordination mechanisms in distribution channels with demand that is price-sensitive. We consider a single-vendor, single-retailer, distribution channel. Demand for the product arises only at the retailer. The demand faced by the retailer is assumed to be deterministic, but price elastic. The retailer in turn buys the product from the vendor. The vendor and the retailer act independently and rationally, each maximizing their own respective profit. The equilibrium point that specifies the vendorÕs and buyerÕs inventory and pricing policy is determined by the solution to a Stackelberg game. We consider the cases when quantity discount and volume discount respectively are offered. We develop methods to determine the optimal discount policy for these cases. We then consider the case where both volume and quantity discounts are offered simultaneously, and develop a method to determine the optimal simultaneous discount offer. The relative performance of the alternative discount schemes is then evaluated through a numerical study. The results of the study demonstrate that the effectiveness of volume discount as a coordination mechanism is higher when the sensitivity of demand to price changes is higher. The effectiveness of quantity discount, on the other hand, is higher with lower price sensitivity of demand. Finally, the results also demonstrate that perfect coordination is achieved when volume and quantity discounts are offered simultaneously. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Distribution channels; Vendor–buyer models; Quantity and volume discounts; Inventory

1. Introduction Price discounts have long been used as a strategy for improving the profitability and cost effectiveness of distribution channels. While discounts based on the replenishment batch size or

*

Corresponding author. Tel.: +65-6790-4798; fax: +65-67922313. E-mail address: [email protected] (S. Viswanathan).

order quantity have been in vogue for a long time, in recent decades, discounts based on the total annual volume demanded by the buyer have become popular (Munson and Rosenblatt, 1998; Munson et al., 1999; Sadrian and Yoon, 1992, 1994). Intuitively, the rationale for price discounts is to encourage buyers to buy larger annual volumes and order larger quantities in each replenishment, resulting in lower costs and higher revenue for the vendor. Dolan (1987) provides a survey of the

0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-2217(02)00469-1

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literature from the perspective of both the economics literatureÕs price discrimination motivation for quantity discount and the marketing/management science literatureÕs motivation for quantity discount as a mechanism for improving channel coordination and efficiency. His paper also provides a systematic review of the different types of discount tariff structures and the motivation and rationale for order quantity based pricing mechanisms. The focus of most of the earlier research on price discounts from a channel coordination perspective has been on determining the optimal quantity discount pricing policies or determining coordination mechanisms based on quantity discounts that improve system-wide profits or cost effectiveness. Crowther (1964) demonstrated the use of quantity discounts for improving channel coordination through numerical examples. Subsequent work mostly focused on the single-vendor, single-buyer (or multiple, but identical buyers) model with fixed, price-inelastic demand and price discounts based on the order quantity. Papers that studied this problem include Dolan (1978), Monahan (1984), Rosenblatt and Lee (1985), Lee and Rosenblatt (1986), Dada and Srikanth (1987), Kohli and Park (1989), Chakravarty and Martin (1988), Joglekar (1988), Joglekar and Tharthare (1990), and Goyal (1987a,b). Papers that considered multiple buyers but fixed demand include Lal and Staelin (1984), Kim and Hwang (1988), Drezner and Wesolowsky (1989), and Weng and Wong (1993). All these papers either attempt to determine the optimal quantity discount schedule from the point of view of the vendor, or develop coordination mechanisms that can achieve either an improved solution or the joint optimal solution for the vendor and buyer(s). Parlar and Wang (1994), Weng and Wong (1993) and Weng (1995a) consider price-elastic demand, but only a singlebuyer. Other papers that consider price-elastic demand include Weng (1995b), Abad (1994), and Chakravarty and Martin (1991). With the exception of a recent study by Chen et al. (1997), almost all the papers in the literature study only quantity discount decisions. Under a quantity discount scheme, the vendor offers a lower unit price if the quantity ordered by the retailer in

each replenishment order is larger than a certain minimum order size. Quantity discount schemes help the vendor achieve economies in order processing and inventory costs as it encourages the retailer to order in larger sizes. However, when the demand is price sensitive, there is an opportunity for the vendor to increase annual revenue by getting the retailer to generate a larger volume of annual demand. Quantity discounts, however, are not effective for this purpose. An alternative is to offer volume discounts, wherein price discounts are based on the total volume of annual demand generated by the retailer (Munson et al., 1999). According to a recent field study by Munson and Rosenblatt (1998), 76% of their study participants experienced discount plans that offered price discount based on the total sales volume per year as opposed to individual order sizes. Furthermore, companies do in fact offer alternative discount schemes such as quantity and volume discounts simultaneously. However, neither volume discounts nor a simultaneous offer of multiple discount schemes have been studied in adequate detail in the literature. Also, the focus of many of the papers on quantity discount policies cited earlier have been on developing the algorithm for determining the discount policy rather than on evaluating the impact of discount policies on distribution channel profitability. Recently, Parlar and Wang (1994) and Weng (1995a) have shown that for the model with price-elastic demand, quantity discounts alone cannot achieve perfect channel coordination and the total channel profits with the best quantity discount decision might still be worse off than the total channel profits under joint/combined optimization. This and the industry practice observed by Munson and Rosenblatt (1998) provide the motivation to study other forms of price discount schemes in distribution channels. The objectives of this paper are (i) to develop algorithms for determining the alternative discount policies such as volume discounts and combined volume and quantity discounts and (ii) to generate a better understanding of the benefits and impact of various types of discount policies on channel profits. Towards this end, we consider a setting with a single-vendor supplying a single product in a distribution channel consisting of a

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single-retailer (or buyer). Demand for the product arises only at the retailer. The demand faced by the retailer is assumed to be deterministic, but price elastic. The retailer in turn buys the product from the vendor who is assumed to be a wholesaler. The vendor and the retailer act independently as rational economic agents, each maximizing their own respective profit. This represents a common practice in distribution channels in which the vendor announces his pricing policy in advance and the retailer makes his optimal pricing and ordering decisions independently under the vendorÕs pricing policy. The vendor aims to develop his optimal pricing policy that corresponds to the buyerÕs rational behavior and results in maximum profit. The situation can therefore be modeled as a Stackelberg game with the vendor acting as the leader and the retailer acting as the follower. The equilibrium point that specifies the vendorÕs and buyerÕs inventory and pricing policy is then determined by the solution to the Stackelberg game. The Stackelberg game framework has been adopted by many previous studies on quantity discount such as Monahan (1984), Jueland and Shugan (1983); Rosenblatt and Lee (1985), Lee and Rosenblatt (1986), Lal and Staelin (1984), Kim and Hwang (1988), Drezner and Wesolowsky (1989), Parlar and Wang (1994) and Weng (1995a). Unlike in previous studies, we consider three price discount schemes in this paper, namely, volume discount, quantity discount and simultaneous offer of volume and quantity discount. After developing methods to evaluate the best pricing policy for each of the above price-discount scheme, we then conduct a numerical study to determine the effectiveness of these schemes. The major findings of the numerical study are that a simultaneous offer of volume and quantity discount always achieved better benefits for the vendor than either quantity or volume discounts alone; and for all the problems in the numerical study, a simultaneous offer of volume and quantity discounts achieved perfect coordination (perfect coordination is defined as the situation when total channel profits under the decentralized system is equal to the profits under joint optimal or integrated system).

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The paper is organized as follows. In Section 2, we describe the model assumptions and notation and develop a method for determining the joint optimal profit for the vendor and retailer. In Section 3, we develop a method for determining the Stackelberg equilibrium solution when no discounts are offered by the vendor. In Sections 4 and 5, we consider the cases when volume discount and quantity discount respectively are offered. We develop methods to determine the optimal discount policy for these cases. In Section 6, we consider the case where both volume and quantity discounts are offered simultaneously. The results of a numerical study that compares the three types of discount policies are presented in Section 7. Finally, Section 8 provides a summary and few concluding remarks.

2. Notation, model assumptions and joint optimal profit As stated earlier, the model we consider is a single-vendor, single-retailer, single-product distribution channel where constant, but price-elastic demand at rate D per period arises only at the retailer. We use the term ÔretailerÕ and ÔbuyerÕ interchangeably. The retailer buys the product from the vendor at a certain wholesale price C. Every order placed by the retailer incurs a fixed ordering cost K. The retailer incurs holding cost at rate h per unit per year. The retailer has no variable costs other than the unit purchasing cost and the inventory costs. The vendorÕs holding cost rate is hv per unit per year. For every order placed by the retailer, the vendor incurs a fixed order processing cost A. The vendor in turn places orders for the product with his supplier who charges a unit price of Cv . Each order placed by the vendor with his supplier incurs the vendor a fixed cost Kv . The objective of both the vendor and retailer is to maximize their respective longterm average profit. Note that a majority of the papers in the literature have modeled holding cost rate on a per unit basis rather than as a percentage of the unit cost. Some researchers have argued that this overstates the retailerÕs holding cost when price

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discounts are availed. However, this need not be the case for two reasons. First, the holding cost rate is only a fraction of the unit cost (around 10–20%), and discounts are normally only a small percentage of the original unit price (Monahan, 1988; Lee and Rosenblatt, 1986). Second, discounts, especially volume discounts, are normally credited back to the buyer only on an annual or periodic basis, after the actual quantum of purchases by the retailer are known. Therefore, the retailer effectively incurs the opportunity cost on the original unit price before the discount and this can be approximated very well by a fixed holding cost rate per unit. The demand rate D is a function of the retail price. Let wðDÞ be the price function, which is the inverse of the demand function. In other words, wðDÞ is the price borne by the market at demand rate D. The dollar revenue corresponding to demand rate D is given by RðDÞ ¼ DwðDÞ. We assume that wðDÞ is decreasing in D, and the retailerÕs revenue RðDÞ is concave in D for D > 0. Also, (i) Lt RðDÞ ¼ 0; (ii) RðDÞ, R0 ðDÞ D!0þ and R00 ðDÞ are continuous and differentiable in the range where the price function is defined and (iii) either R000 ðDÞ 6 0 (satisfied for linear demand function) or the demand function is Cobb–Douglas ðD ¼ Du eap Þ or constant pricesensitive ðD ¼ /p2 Þ. The above assumptions on wðDÞ and RðDÞ are satisfied for most demand functions found in practice including the linear demand function. The retailerÕs profit is his sales revenue minus the inventory-related cost. Since the objective of the retailer is to maximize the long-run average profit per period, he will follow a stationary inventory replenishment and pricing policy. For a particular level of demand D, the vendorÕs wholesale price C and replenishment order quantity q, the retailerÕs long-run average profit per period can be written as Zb ðD; qÞ ¼ RðDÞ  DC  ðKD=q þ qh=2Þ:

ð1Þ

For a given demand D, and order quantity q by the retailer, it is optimal for the vendor to place orders with his supplier in quantities that are integer multiples of q, i.e., the vendorÕs order quantity qv ¼ nq. Since q units are delivered to

the retailer as soon as the shipment from the vendorÕs supplier is received, the average inventory held by the vendor will be ðn  1Þq=2 (Rosenblatt and Lee, 1985). Therefore, the longrun average profit per unit time for the vendor can be written as Zv ðC; nÞ ¼ DðC  Cv Þ  ðAD=qÞ  ðKv D=nqÞ  ðqðn  1Þhv =2Þ:

ð2Þ

Note that Zv is a function of only C and n, since for a given C, the retailer would choose the values of D and q that maximizes his profit. The system profit is the sum of the profit for the vendor and the retailer. Adding the expressions (1) and (2), we get the joint system profit for the vendor and buyer as ZJP ðD; q; nÞ ¼ RðDÞ  DCv  ðD=qÞ  ðK þ A þ ðKv =nÞÞ  ðq=2Þ  ðh  hv þ nhv Þ:

ð3Þ

Note that the joint system profit depends only on D, q and n and not on the vendorÕs wholesale price. Since the vendor and the retailer are independent entities striving to maximize their own individual profit, it is unlikely that they will always take the decision that optimizes the joint profit. However, we are interested in determining the optimal joint profit ZJP , since this will serve as a benchmark or upper bound to evaluate the system profits when price discounts are offered. For a particular value of D and n, ZJP is maximized by setting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ q ¼ 2DðK þ A þ ðKv =nÞÞ=ðh  hv þ nhv Þ: Substituting for q from (4) into (3), we get pffiffiffiffiffiffiffiffiffi ZJP ðD; nÞ ¼ RðDÞ  DCv  2LD;

ð5Þ

where L ¼ ðK þ A þ ðKv =nÞÞðh  hv þ nhv Þ. Now, to maximize ZJP ðD; nÞ, we need to minimize L. The value of n that minimizes L is given by the smallest n that satisfies n ðn  1Þ 6 ððh  hv ÞKv Þ=ðhv ðK þ AÞÞ 6 n ðn þ 1Þ: Defining bxc as the largest integer 6 x, a closed form expression for n can be obtained as follows (see Munson and Rosenblatt, 2001)

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n ¼

j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. k 1 þ 1 þ ½ð4ðh  hv ÞKv Þ=ðhv ðK þ AÞÞ 2 : ð6Þ

Since n is independent of q and D, we can now write ZJP in (5) as just a function of D. Now, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dZJP 0 ¼ R0 ðDÞ  Cv  ZJP ðDÞ ¼ L=ð2DÞ ; dD and 00 ðDÞ ¼ R00 ðDÞ þ ZJP

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L=ð8D3 Þ :

Theorem 1. ZJP has its maximum value at either D ¼ 0 or at the largest value of D for which 0 ZJP ðDÞ P 0. Proof. See Appendix A. The above theorem provides an algorithm for determining the optimal joint profit. Since RðDÞ is concave and wðDÞ is decreasing, the largest D for 0 which ZJP ðDÞ P 0 will be the largest positive root of 0 the equation ZJP ðDÞ ¼ 0. One can employ a line search between the range Dmin and Dmax to determine the optimal value of D. Some discussion on how to set the search range is provided in Appendix A. When the demand function is linear, the optimal value of D can be obtained as a closed form expression and this is also provided in Appendix A.

3. Initial market equilibrium The initial market equilibrium price and the corresponding profit for the vendor and retailer serve as a basis for determining the discount pricing policy. Hence, there is a need to derive the initial market equilibrium. In the initial situation, the vendor offers no discounts. That is, the vendor offers a fixed wholesale price to the retailer. The retailerÕs cost parameters are assumed to be known to the vendor. Therefore, for any particular wholesale price, the vendor knows the retailerÕs optimal decision and therefore his own optimal profit corresponding to the retailerÕs decisions. The vendorÕs objective will therefore be to determine

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the wholesale price that will maximize his own profit taking into account the fact that the retailer will optimize his profit for any given wholesale price. For a given wholesale price C, the retailerÕs profit is given by (1). For a given D and C, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of q that maximizes Zb is q ¼ 2KD=h. Substituting for q in (1) we get pffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ Zb ðDÞ ¼ RðDÞ  CD  2KDh: Now the functional form of Zb in (7) is the same as for ZJP in (5), with L ¼ Kh and C ¼ Cv . Therefore, Theorem 1 is also applicable in this case and a method similar to the one used for determining ZJP can be used for determining the optimal retailer profit Zb ðCÞ and the corresponding demand D ¼ D ðCÞ. For a particular wholesale price C, once the retailer takes the optimal decision D ¼ D ðCÞ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 2KD=h the vendorÕs profit can be determined by (2). pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting for q ¼ 2KD=h in (2), we get pffiffiffi pffiffiffi pffiffiffiffi A h Kv h Zv ðC; nÞ ¼ DðC  Cv Þ  D pffiffiffiffiffiffi þ pffiffiffiffiffiffi 2K n 2K pffiffiffiffi pffiffiffiffi ! nhv K hv K þ pffiffiffiffiffi  pffiffiffiffiffi : 2h 2h The value of n that maximizes Zv ðC; nÞ is given by the smallest n that satisfies

 Kv h n ðn  1Þ 6 6 n ðn þ 1Þ or Khv j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k n ¼ 1 þ 1 þ ð4Kv h=Khv Þ =2 : Then the optimal profit for the vendor for the wholesale price C is pffiffiffiffi Zv ðCÞ ¼ DðC  Cv Þ  I D; ð8Þ where I¼

pffiffiffi pffiffiffi pffiffiffiffi pffiffiffiffi ! A h Kv h n hv K hv K pffiffiffiffiffiffi þ pffiffiffiffiffiffi þ pffiffiffiffiffi  pffiffiffiffiffi : 2h 2h 2K n 2K

If the price charged to the retailer C is greater than a certain threshold value, then D ðCÞ ¼ 0; that is, the retailer will stay out of the business.

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Similarly, if the price C is below a certain threshold value, the vendor will make a loss; therefore, the vendor will only consider prices that are larger than this threshold. Clearly, prices smaller than Cv is unattractive to the vendor and prices larger than wð0Þ will force the retailer to stay out of business. As such Zv ðCÞ takes on positive values only in a range Cv < C < wð0Þ. The optimal wholesale price C and the corresponding profit for the vendor can be determined numerically by an exhaustive search within the above range. Though Zv ðCÞ seems to be an increasing (decreasing) function at low (high) values of C, it is not strictly concave. However, local maxima, if any, seem to occur only close to the global maximum. Therefore, we adopt a grid search approach in our computational experiments. We first perform a coarse grid search by dividing the search range into 101 equidistant points. Then the search range is reduced to within 5 grid points on either side of the grid point with the maximum value. The search is then made finer by searching for 101 equidistant points within this narrower search range. The search range is made narrower (and the search finer) in this manner in a iterative fashion until the search range itself is very small. Depending on the problem parameters (for example when Cv is large or the slope of the price function is steep), it is possible that there may be no price C for which both the retailer and the vendor make positive profits. In this case the equilibrium demand D ¼ 0, and the equilibrium price to be charged by the vendor can be any price not less than a minimum threshold price. This threshold price is the price beyond which the retailerÕs optimal demand is zero. Below this threshold price, the retailer will make profits, but the vendor will make a loss. Even if the initial equilibrium has zero demand and zero system profits, it is still possible that a discount price offer will result in positive demand and non-negative profits for the vendor and retailer.

4. Volume discounts The demand faced by the retailer is sensitive to his retail price. Therefore, a discount on the

wholesale price given by the vendor based on the retailerÕs annual volume of demand may motivate the retailer in turn to reduce his retail price, generating higher demand and profits for both the vendor and retailer. In this section we consider such volume discounts. The structure of the volume discount is as follows. Both the vendor and retailer know the equilibrium wholesale price C , and the equilibrium demand D ðC Þ when no price discounts are offered. The vendor offers a wholesale price C VD ¼ C  y (or all-unit discount of y per unit), if the demand rate per period from the retailer is at least DVD ðC VD Þ or more. The retailer will accept the volume discount only if he is better off (or at least not worse off). Note that the retailer will be able to generate this higher level of demand only if he reduces his own retail price. For a wholesale price C and retailer demand D (andpcorresponding replenishment order quantity ffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 2KD=h), the vendorÕs profit is given by (8). Differentiating (8) with respect to D, we get dZv I Zv ðD; CÞ I þ pffiffiffiffi : ¼ C  Cv  pffiffiffiffi ¼ D dD 2 D 2 D

ð9Þ

This implies that if the price C is profitable to the vendor for some level of demand D, then the vendorÕs profit increases as the demand increases, and the marginal profit at a particular level of demand is not less than the average profit per unit at that level. Therefore for a particular price C, the vendor gains by making the required minimum demand DVD ðCÞ as large as possible. Of course, for price C below the threshold value Cv , the vendorÕs profit will remain negative however large the demand. Prices below the threshold value would not be considered for the discount decision. From the proof of Theorem 1 and with further calculus, it is found that as the wholesale price decreases, the retailerÕs optimal demand will increase. Therefore, the objective of the vendor would now be to reduce the wholesale price to C < C (i.e. offer a discount), and entice the retailer to order at a demand rate DVD ðCÞ that is larger than his optimal D ðCÞ at that price. The retailer would be willing to order DVD ðCÞ > D ðCÞ > D ðC Þ provided his profit Zb ðDVD ðCÞ; CÞ P Zb ðC Þ. Therefore, for a particular dis-

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counted price C, the vendor should make the required minimum demand DVD ðCÞ as the largest demand D for which Zb ðD; CÞ ¼ Zb ðC Þ. 1 Recall that according to Theorem 1, the optimal value of demand D ðCÞ for the retailer occurs at the largest D for which Zb0 ðDÞ P 0. Therefore Zb decreases as D increases beyond D ðCÞ. Also, for C < C , Zb ðCÞ > Zb ðC Þ. Therefore, for a particular value of C < C , DVD ðCÞ is found by searching for D > D ðCÞ where Zb ðD; CÞ ¼ Zb ðC Þ. Let ZvVD ðCÞ be the vendorÕs profit corresponding to the discounted wholesale price C and demand volume DVD ðCÞ. As in the initial equilibrium case, an exhaustive numerical search can be performed to determine the wholesale price C ¼ C  y , the corresponding optimal volume discount y and demand DVD ðC  y Þ. The suggested values for the search range are ðCv < C 6 C Þ. Note that if ZvVD ðC  y Þ ¼ Zv ðC Þ, that is, if the search in the range ðCv ; C Þ cannot find a solution that is better for the vendor, then y ¼ 0, and no volume discounts should be offered by the vendor. 5. Quantity discounts The motivation for the vendor to offer volume discount is to increase the demand and thereby increase the dollar revenue and profits. Quantity discounts are primarily offered with the intention of increasing the buyerÕs order quantity in each order, thereby achieving reduction in the vendorÕs order processing cost. However, when the price elasticity of demand is high, the quantity discount offer may also result in a larger demand and thus possibly larger dollar revenues. In this section, we consider the case where only a quantity discount is offered (i.e. no volume discount). The vendor offers a wholesale price C QD ¼ C  z if the order quantity from the re1

Note that, in reality, the vendor should set the minimum demand such that the retailer has strict improvement in profits over the initial equilibrium profit. Otherwise, the retailer will not be motivated to accept the discount offer. One possible way of doing this is to set Zb ðDVD ðCÞ; CÞ ¼ ð1 þ dÞZb ðC Þ, where 100d is the minimum percentage increase in profits required to entice the retailer to take the discount.

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tailer is at least QQD ðC QD Þ. In other words, if the order quantity is larger than or equal to QQD ðC QD Þ, a quantity discount of z per unit is offered on all units bought by the retailer in that particular order. For a particular value of C and q, the optimal D for the retailer is determined by solving oZb ¼ R0 ðDÞ  C  K=q ¼ 0: oD

ð10Þ

Let DðqÞ be the solution to (10) above. Substituting D from the above solution into (1), Zb can be written as just a function of q (for a particular value of C). Zb ðq; CÞ has a functional form similar to Zb ðD; CÞ in (7), therefore q ðCÞ that corresponds to global maximum of Zb ðCÞ will be the largest positive root of the equation oZb =oq ¼ 0. As we state and prove in Theorem 2 in the appendix, for a wholesale price C above a minimum threshold value (by definition, for a wholesale price greater than or equal to the minimum threshold value, there exists a q ¼ q0 and corresponding Dðq0 Þ for which the vendorÕs profit is positive), the vendorÕs profit increases as the order quantity increases. Therefore, the vendor would want to set the order quantity QQD ðCÞ for a particular price C to be as large as possible. At the same time, the retailer would be willing to accept this discount only if he is not worse off than at the initial equilibrium. Therefore, the order quantity QQD ðCÞ is determined by finding the value of q P q ðCÞ for which Zb ðq; CÞ ¼ Zb ðC Þ. For a particular C < C and corresponding q ¼ QQD ðCÞ, the vendorÕs profit is ZvQD ðC; nÞ ¼ DðqÞðC  Cv Þ  ðADðqÞ=qÞ  ðKv DðqÞ=nqÞ  ðqðn  1Þhv =2Þ: ð11Þ The value of n that maximizes ZvQD ðCÞ is given by the smallest n that satisfies

 2Kv DðqÞ n ðn  1Þ 6 ð12Þ 6 n ðn þ 1Þ or q2 hv j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k n ¼ 1 þ 1 þ ð8Kv DðqÞ=q2 hv Þ =2 : ð13Þ The optimal profit for the vendor ZvQD ðCÞ with order quantity QQD ðCÞ at discount z ¼ C  C can

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be determined by substituting for n from (13) into (11). As in the cases for initial equilibrium and volume discounts, a search can be performed in the range ðCv < C 6 C Þ to determine the optimal quantity discount z and corresponding order quantity QQD ðC  z Þ. Note that if ZvQD ðC  z Þ ¼ Zv ðC Þ then, z ¼ 0, and no quantity discounts should be offered by the vendor.

6. Simultaneous offer of volume and quantity discounts We now consider the case where the vendor offers both quantity and volume discounts simultaneously. By offering the discounts simultaneously, the vendorÕs motivation is to obtain larger dollar revenue through larger demand volume and lower per unit inventory costs through larger replenishment order quantities. When offering both the discounts simultaneously, the vendor would naturally want to entice the retailer to accept both. Otherwise, the vendor can offer just the volume or quantity discount based on the analyses carried out in the previous sections. The structure of the simultaneous discount offer is as follows. The vendor offers a wholesale price C QVD ¼ C  x, if the order quantity from the retailer is at least QQVD ðC QVD Þ and the annual demand volume is at least DQVD ðC QVD Þ. As discussed in the previous section, for a given wholesale price C that is profitable at some level of demand, the vendorÕs profit increases with D, and also with q. Therefore, for a particular wholesale price C ¼ C  x, the vendorÕs objective is to specify the combination of demand DQVD ðCÞ, and order quantity QQVD ðCÞ, which if accepted will maximize the vendorÕs profit. For a particular value of C and D, Zb is concave with respect to q. Similarly, for a fixed value of C and q, Zb is concave with respect to D. Therefore, for a particular C < C , there are several combinations of values of ðD; qÞ, where Zb ðD; qÞ ¼ Zb ðC Þ (i.e. the retailer is not worse off compared to the initial equilibrium). Among these several combinations, let the combination that maximizes the vendorÕs profit for the particular C be

ðDQVD ðCÞ; QQVD ðCÞÞ and let the corresponding profit for the vendor be ZvQVD ðCÞ. The optimal C and the corresponding optimal discount x ¼ C  C that maximizes the vendorÕs profit under the simultaneous offer of quantity and volume discount can be determined by a search along C in the range ðCv ; C Þ. For a particular value of C, the combination ðDQVD ðCÞ; QQVD ðCÞÞ is determined as follows. For a given value of C and D, the optimal order quantity p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi for the retailer is the EOQ, which is 2KD=h. Since Zb is concave with respect to q, one can determine the largest value of q for which the retailer is not worse off than at the initial equilibrium, i.e. Zb ðD; qÞ ¼ Zb ðC Þ. For this combination of D and q, the vendorÕs profit can be determined using (2) (after determining the optimal value of n in a manner similar to the previously discussed cases). For the price C, the optimal demand for the retailer without any discount is D ðCÞ. When only volume discounts are offered, the maximum demand at price C at which the retailer is still not worse off than at the initial equilibrium is DVD ðCÞ. Therefore, when volume and quantity discounts are simultaneously offered, DQVD ðCÞ has to be in the range ðD ðCÞ; DVD ðCÞÞ, since the buyer is expected to have a minimum demand per period, at the same time order a minimum quantity that is larger than the EOQ for that demand. Therefore, one has to perform an exhaustive search for DQVD ðCÞ in the interval ðD ðCÞ; DVD ðCÞÞ. For each demand D in the interval, one can find QQVD ðCÞ as the largest order quantity at which the retailerÕs profit is not less than the initial equilibrium profit Zb ðC Þ. Corresponding to this, the vendorÕs profit can be determined. For each C, we determine the combination ðDQVD ðCÞ; QQVD ðCÞÞ that maximizes the vendorÕs profit. Therefore, for a particular value of C, DQVD ðCÞ, QVD Q ðCÞ and the vendorÕs profit ZvQVD ðCÞ can be determined. Now, one can determine the optimal combined discount x and the corresponding price C  x by performing a exhaustive numerical search for ZvQVD ðCÞ along C in the range ðCv ; C Þ. As in the earlier cases, an iterative grid search with decreasing search range and step sizes have to employed for determining the optimal

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ðDQVD ðCÞ; QQVD ðCÞÞ pair for a particular C, as well for determining the optimal x . In practice, implementing the simultaneous discount may require specifying a volume discount y and quantity discount z separately. This can be done as follows. One can determine the minimum price discount y^ required to entice the retailer to accept the volume discount corresponding to demand volume of DQVD ðC QVD Þ. Similarly, one can determine the minimum price discount ^z required to entice the retailer to accept the quantity discount corresponding to order quantity of QQVD ðC QVD Þ. In Theorem 3 (stated and proved in the appendix), we show that y^ þ ^z > x . Therefore, when the total discount under the simultaneous offer is x , the vendor can offer a volume discount y < y^ corresponding to volumes not less than DQVD ðC QVD Þ and quantity discount z < ^z corresponding to order quantity not less than QQVD ðC QVD Þ, such that y þ z ¼ x . With such an offer, the retailer will always find it more attractive to accept both offers together rather than either one separately.

7. Effectiveness of the alternative discount schemes The methods developed earlier for determining the initial equilibrium, volume discount, quantity discount and simultaneous offer of quantity and volume discounts were used to evaluate the performance of the alternative discount schemes on a large number of test problems. For all the problems, the linear price function wðDÞ ¼ P0  ‘D was used. The values used for the different model parameters in the performance evaluation are given in Table 1. For each problem, the system profit under joint optimization was determined first. The joint optimal system profit serves as a benchmark for evaluating the various discount schemes, since the system profit under the various discount schemes cannot be better than the joint optimal profit. For all the performance results discussed in this paper, the system profit for various discount schemes or scenarios are reported as a ratio of the system profit for the particular scenario to the joint optimal system profit. The profit ratio indicates the

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Table 1 Value of the various model parameters used for the performance evaluation Model parameters

Base value

Values used in the test

P0 ‘

10 0.005

Cv K A Kv hv h

5 100 100 100 0.40 1:30hv ¼ 0:52

10, 15, 20, 25, 50, 100, 400 0.02, 0.005, 0.002, 0.0005, 0.0002, 0.00002 5 20, 50, 100, 200, 500 20, 50, 100, 200, 500 20, 50, 100, 200, 500 0.40, 0.60, 0.80, 1.00 1:30hv

effectiveness of the particular discount scheme. Detailed results for all the problems may be obtained from the corresponding author. We now discuss the performance of the alternative discount schemes with respect to the various problem parameters. To analyze the impact of a particular problem parameter, we fix the values of all the other parameters at their base value, and vary only the value of the parameter under study. The relationship between the system profit and the slope of the price function ‘ is reported in Fig. 1. Note that in this figure the value of ‘ was varied from 0.00002 to 0.02, while the values of the other parameters were kept fixed at their base values. When the slope of the price function ‘ is small, it implies a high price sensitivity of demand. As can be seen from Fig. 1, the system profit under initial equilibrium is lower under low price sensitivity and increases as the price sensitivity of demand increases. Overall, the system profit under initial equilibrium was 65% of the joint optimal system profit. When ‘ ¼ 0:00002, the arms length Stackelberg relationship with no price discounts was able to achieve 75% of the joint optimal system profit. Fig. 1 also shows that quantity discount becomes a more effective channel coordination mechanism for lower price sensitivity of demand or higher value of the slope ‘. The performance under the volume discount behaves in an exact opposite fashion as would be expected. For low price sensitivity ð‘ ¼ 0:02Þ, volume discount scheme achieves system profit that is 78% of the joint optimal. But as the price sensitivity of

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Fig. 1. Relationship between total profit and I (slope of the demand curve).

demand increases, this percentage increases and for ‘ ¼ 0:00002, the volume discount scheme achieves almost perfect coordination, i.e. system profit almost close to the joint optimal system profit. Overall, volume discount achieved 93% of the joint optimal system profit, whereas quantity discount achieved only 80%. This seems to justify the industry practice observed in the Munson and Rosenblatt (1998) study. The most remarkable result noticeable from Fig. 1 is that under the simultaneous offer of quantity and volume discounts (indicated as ‘‘QV discount’’ in the figures), the system profit is always equal to the joint optimal profit. That is, perfect channel coordination can be achieved with a simultaneous offer of quantity and volume discounts. This was noted for all the test problems. This result has been formally proved in another paper by Viswanathan (2000). The relationship between the system profit and retail price of the product is shown in Fig. 2. The surrogate used for retail price in Fig. 2 is P0 . For a particular value of demand volume D, a higher P0 implies a higher retail price. The system profit under initial equilibrium and volume discount improves dramatically as P0 increases from 10 to 15, but as P0 increases further, there is no appreciable change in the system profit for any of the scenarios. The system profit under quantity discount alone decreases as P0 increases from 10 to

15, but thereafter remains steady. Fig. 2 also shows that for P0 P 15, volume discount can achieve almost perfect coordination. As mentioned before, simultaneous offer of volume and quantity discounts always achieves perfect coordination. In Fig. 3, the system profit for the different schemes is plotted against the retailerÕs ordering cost K. When the retailerÕs ordering cost increases, he is motivated to order in larger quantities, which in turn benefits the vendor. Therefore, the system profit under initial equilibrium increases as K increases, albeit in a concave fashion. As the retailer is anyway motivated to increase his order quantity as K increases, the impact of K on the system profit under quantity discount is relatively flat. The effectiveness of volume discount increases as K increases since the vendor gains from the larger demand volume as well as larger order quantities (due to large K) from the retailer. For larger values of K, volume discount is almost as effective in achieving perfect coordination as simultaneous offer of volume and quantity discounts. The effectiveness of the alternative discount schemes for various values of the vendorÕs order processing cost A is studied in Fig. 4. Fig. 5 plots the system profit against the vendorÕs ordering cost Kv for the various discount schemes. Both these graphs behave in a somewhat similar fashion. With larger ordering or order processing cost, the vendorÕs cost increases, whereas it does not have

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Fig. 2. Plot of total profit for various values of P0 .

Fig. 3. Plot of total profit for various values of retailerÕs order cost (K).

any effect on the retailerÕs cost. Therefore, this affects the system profit adversely. The system profit under initial equilibrium and under volume discount decreases sharply as A or Kv increases. However, this is mitigated under quantity discounts since the retailer is encouraged to order less often and in larger quantities. The effectiveness of quantity discounts therefore increases mildly as A or Kv increases. Fig. 6 plots the system profit against the holding cost for the vendor. While the effectiveness of

volume discount as a coordination mechanism decreases as the holding cost increases, for quantity discount, it is the reverse. One plausible explanation for this is that volume discounts are primarily intended to achieve larger demand volumes rather than decrease the inventory-related costs. In all the results reported so far, the optimal discount policies for the vendor were determined by setting the retailer profit under the scenario as equal to the one at initial equilibrium. The retailer

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Fig. 4. Plot of total profit for various values of order processing cost (A).

Fig. 5. Plot of total profit for various values of vendorÕs ordering cost (Kv ).

would accept a discount offer only if he is not worse off compared to the initial equilibrium. The vendorÕs own objective is to maximize his profit. Mathematically, this implies that the profit offered to the retailer would be exactly the same as in the initial equilibrium. In reality, this might not be sufficient to entice the retailer to accept the discount offer. Therefore, further tests were conducted to examine the effect on the system profit if discount policies are determined such that the retailer

gains a fixed additional percentage of profits over the initial equilibrium. Of course, if the additional profits required for the retailer are set too large, then the vendor would be worse off with the discount offer and would therefore not offer any discounts. For the model parameters used in the tests, this did not happen so long as the additional profits were less than 100%. Therefore, for the problem with parameters set at their base values, the system profit under alternative schemes were determined for different values of additional

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Fig. 6. Relationship between total profit and holding cost h.

Fig. 7. Relationship between total system profit and additional profit offered to buyer.

profits for the retailer (20%, 40%, 60%, and 80%). The plot of the system profit against the additional profit offered for the retailer is reported in Fig. 7. As can be seen from Fig. 7, the additional profit offered to the retailer does not have any impact on the system profit except in the case of quantity discount. Of course, the vendorÕs profit would decrease as more profits are offered to the retailer, but the system profit remains the same. One plausible explanation for the different behavior for quantity discount is as follows. For all

the other scenarios, the retailer replenishment order quantity is the EOQ corresponding to the particular annual demand volume. Under the quantity discount scheme, the demand volume is optimized for a particular order quantity but not vice-versa. Essentially, what Fig. 7 implies is that for discount schemes under a Stackelberg game framework, the retailerÕs benefit from improved coordination can be adjusted by just changing the wholesale price (or the price discount) without

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affecting the total system profit. Of course, the vendorÕs profit would decrease in this process and therefore, the vendor would be willing to offer a higher discount only if the retailer does not bite and take the initial discount offered. The vendor by observing the past ordering patterns of the retailer would have better information about the retailerÕs cost parameters rather than vice-versa. Hence, the retailer would theoretically not be in as good a position to evaluate a discount offer (other than by comparing with his own initial profit). In this case, a very small additional percentage increase in profits would be sufficient to entice the retailer to take the discount. In reality, the actual discount price offered might depend on the channel power exerted by the vendor and retailer.

8. Summary and conclusions In this paper, we evaluate the effectiveness of alternative discount schemes in a simplified setting of a single-vendor, single-retailer, distribution channel. Methods are developed to determine the optimal volume discount, quantity discount, and combined quantity and volume discount policies. Numerical evaluation of the alternative discount schemes indicate that volume discount achieves almost perfect coordination (93% of joint optimal profit) and is very effective when price sensitivity of demand is high. On the other hand, quantity discount is effective when price sensitivity of demand is low. Finally, for all the test problems, a simultaneous offer of quantity and volume discount is able to achieve perfect coordination!

Appendix A. Proof of Theorem 1 To prove the theorem, we prove that either (i) 0 0 ZJP ðDÞ < 0, for D P 0 or (ii) ZJP P 0 in only one contiguous range of D P 0. In the former case, D ¼ 0 maximizes ZJP . In the latter case, ZJP has its maximum value at either D ¼ 0, or at the largest 0 value of D for which ZJP ðDÞ P 0. First we prove the result for the case where R000 ðDÞ 6 0. Since RðDÞ, R0 ðDÞ and R00 ðDÞ are con-

0 00 and ZJP are tinuous and differentiable, ZJP , ZJP also continuous and differentiable. The third derivative of the first term in (5) R000 ðDÞ 6 0 by the assumption and clearly the third derivative of the last two terms are also non-positive. This im000 00 plies that ZJP ðDÞ 6 0, for D P 0. Therefore, ZJP ðDÞ is non-increasing for D P 0. This implies that if 00 ~ , then Z 00 ðDÞ < 0 for all ZJP ðDÞ < 0 for a D ¼ D JP 0 ~ D P D. This in turn means that ZJP is decreasing 0 ~ for D P D, or in other words, once the slope ZJP stops increasing, it can never increase again for larger values of D. 0 Therefore, if ZJP ðDÞ P 0 for some D, then it can only be in one contiguous range of D P 0. In this case, clearly, ZJP is increasing (or nondecreasing) in this contiguous range of D, and ZJP has its only local maxima at the largest value of D in this range. If ZJP > 0 at this local maxima, then the corresponding D is the optimal demand, otherwise D ¼ 0 is the optimal. It is pos0 sible that ZJP ðDÞ < 0, for all D P 0, in which case ZJP ðDÞ < 0 for all D > 0; therefore D ¼ 0 will maximize ZJP . This completes the proof for the case where R000 ðDÞ 6 0. For the Cobb–Douglas demand function, the assumption R000 ðDÞ 6 0 is not true. However, as we show in Lemma 1 below, for this demand function 000 00 ZJP ðDÞ 6 0, for D 6 DA ¼ 0:282La2 and ZJP ðDÞ 6 0, 2 for D P DB ¼ 0:125La . Since DB < DA , this im00 plies that ZJP ðDÞ either non-increasing or nonpositive for all D P 0, and therefore the proof given above for the case where R000 ðDÞ 6 0 is still applicable. For the constant price-sensitive demand function D ¼ /p2 considered by Weng (1995a), it can be shown that ZJP ðDÞ is either concave or nonpositive. When ZJP ðDÞ is concave, clearly there can only be one contiguous range of D P 0 for which 0 ZJP ðDÞ P 0. This completes the proof of the theorem. 

Lemma 1. For the Cobb–Douglas demand function, D ¼ Du eap , 000 ðDÞ 6 0; ZJP

for D 6 DA ¼ 0:282La2

and 00 ðDÞ 6 0 ZJP

for D P DB ¼ 0:125La2 :

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Proof. Inverting the above demand function, the price p ¼ wðDÞ ¼ ð1=aÞ logðD=Du Þ. Therefore pffiffiffiffiffiffiffiffiffi ZJP ðDÞ ¼ ðD=aÞ logðD=Du Þ  Cv D  2LD;

for any particular wholesale price C above a minimum threshold value, the vendor’s profit increases as the buyer’s order quantity q increases.

 rffiffiffiffiffiffiffiffi 1 L and ¼ þ aD 8D3

 rffiffiffiffiffiffiffiffiffiffiffi 1 9L 000 ZJP ðDÞ ¼ :  aD2 32D5

Proof. For a particular value of C, the vendorÕs profit is given by (11). Since the price C is above a minimum threshold value, by definition, there exists a q ¼ q0 and corresponding Dðq0 Þ for which ZvQD ðq0 Þ > 0. The approach for the proof is as follows. We first show that for a fixed q ¼ q0 , ZvQD increases as D increases. We then show that DðqÞ is increasing with q. Finally, we show that for a fixed D, ZvQD increases as q increases. These three results together prove the proposition. First, we show that for a fixed q ¼ q0 , ZvQD increases as D increases. By definition ZvQD ðq0 Þ > 0, therefore from (11) we have

00 ðDÞ ZJP



The result is then obtained by algebraic manipulation of the above expressions.  Using Theorem 1, an algorithm based on line search can be developed for determining the optimal joint profit for vendor and retailer as well as for determining the optimal profit for the retailer (for a given wholesale price C). One can employ a line search between the range Dmin and 0 Dmax to determine the largest D where ZJP ðDÞ P 0. For both the linear and Cobb–Douglas functions this value of D (if it exists) will be larger than the 00 value of D where ZJP ðDÞ ¼ 0. Therefore, one good value for Dmin would be the root of the 00 equation ZJP ðDÞ ¼ 0. Dmax can be set as a value 0 that is large enough. Of course, if ZJP ðDmin Þ < 0, then this implies that the optimal demand D ¼ 0. For, the constant price-sensitive demand function, it can be shown that ZJP ðDÞ is either concave or non-positive. Therefore, a simple search algorithm can again be used to determine the optimal demand. When the demand function is linear ðwðDÞ ¼ P0  ‘DÞ, the optimal D can be obtained as a closed form expression. In this case, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ðDÞ ¼ P0  2‘D  Cv  L=ð2DÞ ¼ 0: ZJP Solving this we obtain the optimal demand

ðP C Þ 2 v 0 D ¼ 2 3‘ ðcos hÞ ; if 3‘LðP0  Cv Þ P 0; 0; otherwise; pffiffiffiffiffiffiffiffi 3=2 where h ¼ ð1=3Þ arccosð3 3‘L=2ðP0  Cv Þ Þ. The corresponding n and q can be determined using (6) and (4). Theorem 2. Consider the case of quantity discount offer, wherein the retailer sets the order quantity q according to the discount offer, and chooses DðqÞ such that his profits are maximized for that q. Then,

Dðq0 ÞðC  Cv  A=q0  Kv =nq0 Þ P q0 ðn  1Þhv =2: ðA:1Þ Therefore, as D increases, the left-hand side of (A.1) increases (so long as n remains at the same value), which implies that ZvQD increases. When D increases, the value of n will remain the same as long as (12) is satisfied. At the transition point for D, where the value of n changes to n þ 1, (12) is satisfied as equality on the right-hand side. Therefore, the sum of the last two terms of ZvQD in (11) will not change in value at this transition point. Thereafter, the value of n will remain unchanged at its new value until the next transition point for D, and as argued above, ZvQD increases as D increases. The same argument holds for every transition point of D. This proves that ZvQD is increasing in D, for fixed a q ¼ q0 . We now show that DðqÞ determined by solving (10) is non-decreasing in q. Differentiating (10) with respect to q, we obtain R00 ðDÞ

dD K þ ¼ 0: dq q2

Therefore, dD=dq ¼ K=q2 R00 ðDÞ P 0, since RðDÞ is concave. This proves that DðqÞ is increasing. Finally, we prove that for a fixed D, ZvQD increases as q increases. For a fixed D, the value of n determined by (12) is optimal only for a particular range of q, and as q increases beyond this range, n

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will decrease. Let n~ be the optimal value of n for q in the range ql 6 q < qu . At the transition point qu , (12) will be satisfied for n~  1 with equality on the right-hand side and both n~ and n~  1 will result in same value of ZvQD . That is n~ð~ n  1Þ ¼ 2Kv D=q2u hv . Therefore, nq2u ¼ ð~ n  1Þhv =2: Kv D=~

ðA:2Þ

Now for ql 6 q < qu , dZvQD ¼ AD=q2 þ Kv D=~ nq2  ð~ n  1Þhv =2: dq

ðA:3Þ

From (A.2), the second and thirds terms in (A.3) are equal at q ¼ qu . Since q < qu , the second term is larger than the third term in (A.3) and therefore dZvQD =dq > 0. The same result can be proved in a similar manner for every range of q, corresponding to different values of n. This implies that for fixed D, ZvQD increases as q increases, and this completes the proof of the proposition.  Theorem 3. Let the initial equilibrium price, demand and replenishment order quantity be C , D , and Q respectively. Consider the case of simultaneous offer of volume and quantity discount. Let the optimal combined discount be x (as determined in Section 6) if the retailer orders an annual volume of at least DDVD ðC  x Þ > D and a replenishment quantity in each order of at least QDVD ðC  x Þ > Q . Let y^ be the minimum pure volume discount required to entice the retailer to order an annual volume of at least DDVD ðC  x Þ. Let ^z be the minimum pure quantity discount required to entice the retailer to order a replenishment quantity in each order of at least QDVD ðC  x Þ. Then, y^ þ ^z > x . Proof. Let Qv be the retailerÕs optimal replenishment order quantity corresponding to annual demand volume of DQVD ðC  x Þ. Let Dq be the optimal annual demand for the retailer corresponding to a replenishment order quantity of QDVD ðC  x Þ. For notational convenience, we use DQVD to indicate DQVD ðC  x Þ and QQVD to indicate QQVD ðC  x Þ for the rest of the proof. Since the retailerÕs profits cannot be worse off than at the initial equilibrium for any of the three offers x , y^ and ^z, we have

Zb ðDQVD ; QQVD ; C  x Þ ¼ Zb ðD ; Q ; C Þ; Zb ðDQVD ; Qv ; C  y^Þ ¼ Zb ðD ; Q ; C Þ and Zb ðDq ; QQVD ; C  ^zÞ ¼ Zb ðD ; Q ; C Þ: Expanding and simplifying each of the above using (1), we get x DQVD ¼ RðD Þ  RðDQVD Þ þ C ðDQVD  D Þ

QVD  D D h þK  þ ðQQVD  Q Þ; 2 QQVD Q ðA:4Þ y^DQVD ¼ RðD Þ  RðDQVD Þ þ C ðDQVD  D Þ

QVD  D D h þK  þ ðQv  Q Þ; 2 Qv Q ðA:5Þ ^zDq ¼ RðD Þ  RðDq Þ þ C ðDq  D Þ

q  D D h þK  þ ðQQVD  Q Þ: 2 QQVD Q ðA:6Þ Using (A.4)–(A.6) and some algebra, we can get

 1 KD hQ y^ þ ^z  x ¼ q RðD Þ  C D   D Q 2

q 1 KD  q RðDq Þ  C Dq  v D Q

  hQQVD h Qv  QQVD  þ 2 2 DQVD 1 ¼ q ðZb ðC ; D ; Q Þ  Zb ðC ; Dq ; Qv ÞÞ D

 h QVD 1 1 v þ ðQ Q Þ  : 2 Dq DQVD By the nature in which the simultaneous discount offer is determined, QQVD > Qv , and DQVD > Dq . Also, ðD ; Q Þ maximizes Zb for C ¼ C . Therefore, y^ þ ^z  x > 0. 

References Abad, P.L., 1994. Supplier pricing and lot-sizing when demand is price sensitive. European Journal of Operational Research 78, 334–354.

S. Viswanathan, Q. Wang / European Journal of Operational Research 149 (2003) 571–587 Chakravarty, A.K., Martin, G.E., 1988. An optimal joint buyer-seller discount pricing model. Computers and Operations Research 15, 271–281. Chakravarty, A.K., Martin, G.E., 1991. Operational economies of a process positioning determinant. Computers and Operations Research 18, 515–530. Chen, F.A., Federgruen, Zheng, Y.S., 1997. Coordination mechanisms for decentralized distribution systems, Working Paper, Graduate School of Business, Columbia University, New York, NY 100027. Crowther, J., 1964. Rationale for quantity discounts. Harvard Business Review March–April, 121–127. Dada, M., Srikanth, K.N., 1987. Pricing policies for quantity discounts. Management Science 33, 1247–1252. Dolan, R.J., 1978. A normative model of industrial buyer response to quantity discounts. In: Jain, S.C. (Ed.), Research Frontiers in Marketing: Dialogues and Directions. American Marketing Association, Chicago, IL, pp. 121– 125. Dolan, R.J., 1987. Quantity discounts: Managerial issues and research opportunities. Marketing Science 6, 1–22. Drezner, Z., Wesolowsky, G.O., 1989. Multi-buyer discount pricing. European Journal of Operational Research 40, 38– 42. Goyal, S.K., 1987a. Comment on a generalized quantity discount pricing model to increase supplierÕs profits. Management Science 222, 1635–1636. Goyal, S.K., 1987b. Determination of a supplierÕs economic ordering policy. Journal of the Operational Research Society 28, 853–857. Joglekar, P., 1988. Comments on a quantity discount pricing model to increase vendor profits. Management Science 34, 1391–1398. Joglekar, P., Tharthare, S., 1990. The individually responsible and rationale decision approach to economic lot sizes for one vendor and many purchases. Decision Sciences 21, 492– 506. Jueland, A., Shugan, S., 1983. Managing channel profits. Marketing Science 2, 239–272. Kim, K.H., Hwang, H., 1988. An incremental discount pricing schedule with multiple customers and single price break. European Journal of Operational Research 35, 71– 79. Kohli, R., Park, H., 1989. A cooperative game theory model of quantity discounts. Management Science 35, 693–707.

587

Lal, R., Staelin, R., 1984. An approach for developing an optimal discount pricing policy. Management Science 30, 1524–1539. Lee, H.L., Rosenblatt, M.J., 1986. A generalized quantity discount pricing model to increase supplierÕs profits. Management Science 32, 1177–1185. Monahan, J.P., 1984. A quantity pricing model to increase vendor profits. Management Science 34, 1398–1400. Monahan, J.P., 1988. On comments on a quantity discount pricing model to increase vendor profits. Management Science 34, 1398–1400. Munson, C.L., Rosenblatt, M.J., 1998. Theories and realities of quantity discounts: An exploratory study. Production and Operations Management 7, 352–369. Munson, C.L., Rosenblatt, M.J., 2001. Coordinating a threelevel supply chain with quantity discounts. IIE Transactions 33, 371–384. Munson, C.L., Rosenblatt, M.J., Rosenblatt, Z., 1999. The use and abuse of power in supply chains. Business Horizons 42 (1), 55–65. Parlar, M., Wang, Q., 1994. Discounting decisions in a supplier-buyer relationship with a linear buyerÕs demand. IIE Transactions 26, 34–41. Rosenblatt, M.J., Lee, H.L., 1985. Improving profitability with quantity discounts under fixed demand. IIE Transactions 17, 388–395. Sadrian, A.A., Yoon, Y.S., 1992. Business volume discount: A new perspective on discount pricing strategy. International Journal of Purchasing and Materials Management (Spring), 43–46. Sadrian, A.A., Yoon, Y.S., 1994. A procurement decision support system in business volume discount environments. Operations Research 42, 14–23. Viswanathan, S., 2000. Coordination in supply chains: On price discounts, stackelberg game and joint optimization, Working Paper, Nanyang Business School, Nanyang Technological University, Singapore. Weng, Z.K., Wong, R.T., 1993. General models for the supplierÕs all-unit quantity discount policy. Naval Research Logistics 40, 971–991. Weng, Z.K., 1995a. Channel coordination and quantity discounts. Management Science 41, 1509–1522. Weng, Z.K., 1995b. Modeling quantity discounts under general price-sensitive demand functions: Optimal policies and relationships. European Journal of Operational Research 86, 300–314.