A Wholesaler s Optimal Ordering and Quantity Discount Policies for Deteriorating Items Hidefumi Kawakatsu Astract This study analyses the seller s wholesaler s decision to offer quantity discounts to the uyer retailer. The seller purchases products from an upper-leveled supplier manufacturer and then sells them to the uyer who faces her customers demand. The seller attempts to increase her profit y controlling the uyer s order quantity through a quantity discount strategy. The uyer tries to maximize her profit considering the seller s proposal. We formulate the aove prolem for deteriorating items as a Stackelerg game etween the seller and uyer to analyze the existence of the seller s optimal quantity discount pricing policy which maximizes her total profit per unit of time. The same prolem is also formulated as a cooperative game. Numerical examples are presented to illustrate the theoretical underpinnings of the proposed formulation. Index Terms quantity discounts, deteriorating items, total profit, Stackelerg game, cooperative game. I. INTRODUCTION THis paper presents a model for determining optimal allunit quantity discount strategies in a channel of one seller wholesaler and one uyer retailer. Many researchers have developed models to study the effectiveness of quantity discounts. Quantity discounts are widely used y the seller with the ojective of inducing the uyer to order larger quantities in order to reduce their total transaction costs associated with ordering, shipment and inventorying. Monahan1 formulated the transaction etween the seller and the uyer see also 2, 3, and proposed a method for determining an optimal all-unit quantity discount policy with a fixed demand. Lee and Rosenlatt4 generalized Monahan s model to otain the exact discount rate offered y the seller, and to relax the implicit assumption of a lot-forlot policy adopted y the seller. Parlar and Wang5 proposed a model using a game theoretical approach to analyze the quantity discount prolem as a perfect information game. For more work: see also Sarmah et al.6. These models assumed that oth the seller s and the uyer s inventory policies can e descried y classical economic order quantity EOQ models. The classical EOQ model is a cost-minimization inventory model with a constant demand rate. It is one of the most successful models in all the inventory theories due to its simplicity and easiness. In many real-life situations, retailers deal with perishale products such as fresh fruits, food-stuffs and vegetales. The inventory of these products is depleted not only y demand ut also deterioration. Yang7 has developed the model to determine an optimal pricing and a ordering policy for deteriorating items with quantity discount which is offered y the vendor. However, his model assumed that the deterioration Manuscript received Feruary 2, 211; revised Octoer 25, 211. H. Kawakatsu is with Department of Economics & Information Science, Onomichi University, 16 Hisayamadacho, Onomichi 722-856 Japan corresponding author to provide phone: +81-848-22-8312 ex.617; fax: +81-848-22-546; e-mail: kawakatsu@onomichi-u.ac.jp. rate of the vendor s inventory is equal to its rate of the retailer s inventory, and focused on the case where oth the uyer s and the vendor s total profits can e approximated using Taylor series expansion. In this study, we discuss a quantity discount prolem etween a seller wholesaler and a uyer retailer under circumstances where oth the wholesaler s and the retailer s inventory levels of the product are depleted not only y demand ut also y deterioration. The wholesaler purchases products from an upper-leveled supplier manufacturer and then sells them to the retailer who faces her/his customers demand. The shipment cost is characterized y economies of density8. The wholesaler is interested in increasing her/his profit y controlling the retailer s order quantity through the quantity discount strategy. The retailer attempts to maximize her/his profit considering the wholesaler s proposal. Our previous work has formulated the aove prolem as a Stackelerg game etween the wholesaler and the retailer to show the existence of the wholesaler s optimal quantity discount pricing policy which maximizes her/his total profit per unit of time9. In this study, we also formulate the same prolem as a cooperative game. Numerical examples are presented to illustrate the theoretical underpinnings of the proposed model. II. NOTATION AND ASSUMPTIONS The wholesaler uses a quantity discount strategy in order to improve her/his profit. The wholesaler proposes, for the retailer, an order quantity per lot along with the corresponding discounted wholesale price, which induces the retailer to alter her/his replenishment policy. We consider the two options throughout the present study as follows: Option V 1 : The retailer does not adopt the quantity discount proposed y the wholesaler. When the retailer chooses this option, she/he purchases the products from the wholesaler at an initial price in the asence of the discount, and she/he determines her/himself an optimal order quantity which maximizes her/his own total profit per unit of time. Option V 2 : The retailer accepts the quantity discount proposed y the wholesaler. The main notations used in this paper are listed elow: : the retailer s order quantity per lot under Option V i i 1, 2. S i : the wholesaler s order quantity per lot under Option V i i 1, 2. T i : the length of the retailer s order cycle under Option V i i 1, 2. h s, h : the wholesaler s and the retailer s inventory holding costs per item and unit of time, respectively. a s, a : the wholesaler s and the retailer s ordering costs per lot, respectively.
ξt i : the shipment cost per shipment from the wholesaler to the retailer. c s : the wholesaler s unit acquisition cost unit purchasing cost from the upper-leveled manufacturer. p s : the wholesaler s initial unit selling price, i.e., the retailer s unit acquisition cost in the asence of the discount. y: the discount rate for the wholesale price proposed y the wholesaler, i.e., the wholesaler offers a unit discounted price of 1 yp s y < 1. p : the retailer s unit selling price, i.e., unit purchasing price for her/his customers., : the deterioration rates of the wholesaler s inventory and of the retailer s inventory, respectively <. µ: the constant demand rate of the product. The assumptions in this study are as follows: 1 The retailer s inventory level is continuously depleted due to the comined effects of its demand and deterioration. In contrast, the wholesaler s inventory is depleted y deterioration during jt i, j + 1T i j, 1, 2,, ut at time jt i her/his inventory level decreases y ecause of shipment to the retailer. 2 The rate of replenishment is infinite and the delivery is instantaneous. 3 Backlogging and shortage are not allowed. 4 The quantity of the item can e treated as continuous for simplicity. 5 Both the wholesaler and the retailer are rational and use only pure strategies. 6 The shipment cost is characterized y economies of density8, i.e., the shipment cost per shipment decreases as the retailer s lot size increases. We assume, for simplicity, that ξt i β α T i >. 7 The length of the wholesaler s order cycle is given y N i T i under Option V i i 1, 2, where N i is a positive integer. This is ecause the wholesaler can possily improve her/his total profit y increasing the length of her/his order cycle from T i to N i T i. In this case, the wholesaler s lot size can e otained y the sum of N i times of the retailer s lot size and the cumulative quantity of the waste products to e discarded during, N i T i. III. RETAILER S TOTAL PROFIT This section formulates the retailer s total profit per unit of time for the Option V 1 and V 2 availale to the retailer. A. Under Option V 1 If the retailer chooses Option V 1, her/his order quantity per lot and her/his unit acquisition cost are respectively given y Q 1 QT 1 and p s, where p s is the unit initial price in the asence of the discount. In this case, she/he determines her/himself the optimal order quantity Q 1 Q 1 which maximize her/his total profit per unit of time. Since the inventory is depleted due to the comined effect of its demand and deterioration, the inventory level, I t, at time t during, T 1 can e expressed y the following differential equation: di t/dt I t µ. 1 By solving the differential equation in Eq. 1 with a oundary condition I T 1, the retailer s inventory level at time t is given y I t ρ e T 1 t, 2 where ρ µ/. Therefore, the initial inventory level, I Q 1 Q T 1, in the order cycle ecomes QT 1 ρ e T 1. 3 On the other hand, the cumulative inventory, AT 1, held during, T 1 is expressed y e T 1 AT 1 T1 I tdt ρ. 4 Hence, the retailer s total profit per unit of time under Option V 1 is given y π 1 T 1 p T1 µdt p s QT 1 h AT 1 a ρp + h T 1 p s + h QT 1 + a T 1. 5 In the following, the results of analysis are riefly summarized: The proof is given in A. There exists a unique finite T 1 T1 > which maximizes π 1 T 1 in Eq. 5. The optimal order quantity is therefore given y Q 1 ρ e T 1. 6 The total profit per unit of time ecomes π 1 T1 ρ p + h p s + h e T 1. 7 B. Under Option V 2 If the retailer chooses Option V 2, the order quantity and unit discounted wholesale price are respectively given y Q 2 Q 2 T 2 ρ e T 2 and 1 yp s. The retailer s total profit per unit of time can therefore e expressed y π 2 T 2, y ρp + h 1 yp s + h Q 2 T 2 + a. 8 T 2 IV. WHOLESALER S TOTAL PROFIT This section formulates the wholesaler s total profit per unit of time, which depends on the retailer s decision. Figure 1 shows oth the wholesaler s and the retailer s transitions of inventory level in the case of N i 3.
Wholesaler s inventory level The wholesaler s cumulative inventory, held during, N 1 T 1 ecomes S i BN 1, T 1 N 1 1 j1 Q 1T 1 B j T 1 e N 1 T 1 e T 1 N 1. 14 T 2T 3T Retailer s inventory level i i i T 2T 3T i i i Fig. 1. Transition of Inventory Level N i 3 A. Total Profit under Option V 1 If the retailer chooses Option V 1, her/his order quantity per lot and unit acquisition cost are given y Q 1 and p s, respectively. The length of the wholesaler s order cycle can e divided into N 1 shipping cycles N 1 1, 2, 3, as descried in assumption 7, where N 1 is also a decision variale for the wholesaler. The wholesaler s inventory is depleted only due to deterioration during j T 1, jt 1 in jth shipping cycle j 1, 2,, N 1. Therefore, the wholesaler s inventory level, I s t, at time t can e expressed y the following differential equation: di s t/dt I s t, 9 with a oundary condition I s jt 1 z j T 1, where z j T 1 denotes the remaining inventory at the end of the jth shipping cycle. By solving the differential equation in Eq. 9, the wholesaler s inventory level, I s t I s j t, at time t in jth shipment cycle is given y I s j t z j T 1 e jt 1 t. 1 It can easily e confirmed that the inventory level at the end of the N 1 th shipping cycle ecomes Q 1, i.e. z N1 1T 1 Q 1, as also shown in Fig. 1. By induction, we have z j T 1 QT 1 e N 1 j T 1 / e T 1. 11 The wholesaler s order quantity, S 1 SN 1, T 1 z T 1 per lot is then given y SN 1, T 1 QT 1 e N1θsT1 / e θst1. 12 On the other hand, the wholesaler s cumulative inventory, B j T 1, held during jth shipping cycle is expressed y B j T 1 jt1 I s j j 1T 1 tdt QT 1 e N i j T 1 t t. 13 Hence, for a given N 1, the wholesaler s total profit per unit of time under Option V 1 is given y P 1 N 1, T1 1 N 1 T1 p s N 1 QT1 N 1 ξt 1 c s SN 1, T1 h s BN 1, T1 a s p s + h s + α QT1 β T 1 c s + hs SN 1, T1 + a s N 1 T1. 15 B. Total Profit under Option V 2 When the retailer chooses Option V 2, she/he purchases Q 2 QT 2 units of the product at the unit discounted wholesale price 1 yp s. In this case, the wholesaler s order quantity per lot under Option V 2 is expressed as S 2 SN 2, T 2, accordingly the wholesaler s total profit per unit of time under Option V 2 is given y P 2 N 2, T 2, y 1 1 yp s N 2 QT 2 N 2 ξt 2 N 2 T 2 c s SN 2, T 2 h s BN 2, T 2 a s 1 yp s + hs + α QT 2 β where T 2 c s + h s SN 2, T 2 + a s N 2 T 2, 16 QT 2 ρ e T 2, 17 SN 2, T 2 QT 2 e N 2 T 2 / e T 2. 18 V. RETAILER S OPTIMAL RESPONSE This section discusses the retailer s optimal response. The retailer prefers Option V 1 over Option V 2 if π 1 > π 2 T 2, y, ut when π 1 < π 2 T 2, y, she/he prefers V 2 to V 1. The retailer is indifferent etween the two options if π 1 π 2 T 2, y, which is equivalent to y QT2 p s + h ρ T 2 e T 1 + a. 19 p s QT 2 Let us denote, y ψt 2, the right-hand-side of Eq. 19. It can easily e shown from Eq. 19 that ψt 2 is increasing in T 2 T 1.
y where ψt 2 Ω 2 C c s + h s /, 24 HN 2 h s / h / + αn 2. 25 T 1 * Ω 1 T 2 Let us now define LT 2 as follows: Fig. 2. Characterization of retailer s optimal responses VI. WHOLESALER S OPTIMAL POLICY UNDER THE NON-COOPERATIVE GAME The wholesaler s optimal values for T 2 and y can e otained y maximizing her/his total profit per unit of time considering the retailer s optimal response which was discussed in Section V. Henceforth, let Ω i i 1, 2 e defined y Ω 1 {T 2, y y ψt 2 }, Ω 2 {T 2, y y ψt 2 }. Figure 2 depicts the region of Ω i i 1, 2 on the T 2, y plane. A. Under Option V 1 If T 2, y Ω 1 \ Ω 2 in Fig. 2, the retailer will naturally select Option V 1. In this case, the wholesaler can maximize her/his total profit per unit of time independently of T 2 and y on the condition of T 2, y Ω 1 \Ω 2. Hence, the wholesaler s locally maximum total profit per unit of time in Ω 1 \ Ω 2 ecomes P 1 max N 1 N P 1N 1, T 1, 2 where N signifies the set of positive integers. B. Under Option V 2 On the other hand, if T 2, y Ω 2 \Ω 1, the retailer s optimal response is to choose Option V 2. Then the wholesaler s locally maximum total profit per unit of time in Ω 2 \ Ω 1 is given y where P 2 ˆP 2 N 2 max N 2 N ˆP 2 N 2, 21 max P 2 N 2, T 2, y. 22 T 2,y Ω 2\Ω 1 More precisely, we should use sup instead of max in Eq. 22. For a given N 2, we show elow the existence of the wholesaler s optimal quantity discount pricing policy T 2, y T 2, y which attains Eq. 22. It can easily e proven that P 2 N 2, T 2, y in Eq. 16 is strictly decreasing in y, and consequently the wholesaler can attain ˆP 2 N 2 in Eq. 22 y letting y ψt 2 +. By letting y ψt 2 in Eq. 16, the total profit per unit of time on y ψt 2 ecomes P 2 N 2, T 2 ρ p s + h / e T 1 N 2 T 2 C SN 2, T 2 HN 2 QT 2 +a + βn 2 + a s, 23 LT 2 C T 2 QT 2 N 2e N 2 T 2 e T 2 e T 2 e N 2 T 2 e T 2 2 + ρ e T 2 T 2 QT 2 C en2θst2 e θ2t2 HN 2. 26 We here summarize the results of analysis in relation to the optimal quantity discount policy which attains ˆP 2 N 2 in Eq. 22 when N 2 is fixed to a suitale value. The proofs are given in Appendix B. 1 N 2 1: c s + h / α > : In this sucase, there exists a unique finite T 2 > T 1 which maximizes P 2 N 2, T 2 in Eq. 23, and therefore T 2, y is given y T 2, y T 2, ỹ, 27 where ỹ ψ T 2. The wholesaler s total profit then ecomes ˆP 2 N 2 ρ ps + h / e T 1 c s + h / α e θ T 2. 28 c s + h / α : In this sucase, the optimal policy can e expressed y T 2, y ˆT 2, 1, 29 where ˆT 2 > T1 is the unique finite positive solution to ψt 2 1. The wholesaler s total profit is therefore given y ˆP 2 N 2 c 2 αq ˆT 2 ˆT 2 β a s. 3 2 N 2 2: Let us define T 2 T 2 > T 1 as the unique solution if it exists to LT 2 a + βn 2 + a s. 31 In this case, the optimal quantity discount pricing policy is given y Eq. 27. C. Under Option V 1 and V 2 In the case of T 2, y Ω 1 Ω 2, the retailer is indifferent etween Option V 1 and V 2. For this reason, this study confines itself to a situation where the wholesaler does not use a quantity discount policy T 2, y Ω 1 Ω 2.
D. Optimal value for N i For a given T i, we here derive a lower ound and an upper ound for the optimal value of N i Ni Ni 1, 2, 3, which maximizes P 1 N 1, T1 in Eq. 15 and P 2 N 2, T 2, y in Eq. 16. Let KT i e defined y KT i c s + h s / QT i /e T i. 32 In the following, the results of analysis are riefly summarized. The proofs are shown in Appendix C. 1 Lower ound N i N L i T i Ni : e T i 2 a s /KT i : N L i T i 1. e θsti 2 < a s /KT i : There exists a unique finite N L i T i 1 which is the solution to N i e N i T i e T i e N i T i a s /KT i. 33 2 Upper ound N i N U i T i Ni : There exists a unique finite N U i T i N L i T i which is the solution to N i e Ni 1θsTi e θsti e N i T i a s /KT i. 34 The aove results indicate that the optimal N i satisfies 1 N L i T i N i < N U i T i. 35 In the aove, it should e reminded that we can use T 1 T 1 under Option V 1. VII. WHOLESALER S OPTIMAL POLICY UNDER THE COOPERATIVE GAME This section discusses a cooperative game etween the wholesaler and the retailer. We focus on the case where the wholesaler and the retailer maximize their joint profit. We here introduce some more additional notations N 3, T 3 and Q 3, which correspond to N 2, T 2 and Q 2 respectively, under Option V 2 in the previous section. Let JN 3, T 3, y express the joint profit function per unit of time for the wholesaler and the retailer, i.e., let JN 3, T 3, y P 2N 3, T 3, y + π 2 T 3, y, we have JN 3, T 3, y ρp + h N 3 T 3 C SN 3, T 3 HN 3 QT 3 +a + βn 3 + a s. 36 It can easily e proven from Eq. 36 that JN 3, T 3, y is independent of y and we have JN 3, T 3, y P 2 N 3, T 3, ψt 3 + π 1. This signifies that the optimal quantity discount policy T 3, y T 3, y which maximizes JN 3, T 3, y in Eq. 36 is given y T 2, y as shown in Section VI. This is simply ecause, in this study, the inventory holding cost is assumed to e independent of the value of the item. TABLE I SENSITIVITY ANALYSIS a Under Option V 1 a s Q 1 p 1 S1 N 1 P 1 5 47.35 3. 47.351 91.75 1 47.35 3. 99.92 861.24 2 47.35 3. 99.92 84.78 3 47.35 3. 155.613 752.21 Under Option V 2 a s Q 2 p 2 S2 N 2 P 2 5 17.63 285.1 17.631 998.56 1 117.65 281.63 117.651 972.8 2 135.77 275.49 135.771 926.23 3 152.4 27.4 152.41 884.43 VIII. NUMERICAL EXAMPLES Tale I reveals the results of sensitively analysis in reference to Q 1, p 1 p s, S1 SN1, T1, N1, P1, Q 2 QT2, p 2 1 y p s, S2 SN2, T2, N2, P2 for c s, p s, p, a, h s, h,,, µ, α, β 1, 3, 6, 12, 1, 1.1,.1,.15, 5, 2, 1 when a s 5, 1, 2 and 3. In Tale Ia, we can oserve that oth S1 and N1 are nondecreasing in a s. As mentioned in Section II, under Option V 1, the retailer does not adopt the quantity discount offered y the wholesaler, which signifies that the wholesaler cannot control the retailer s ordering schedule. In this case, the wholesaler s cost associated with ordering should e reduced y increasing her/his own length of order cycle and lot size y means of increasing N 1. Tale I shows that, under Option V 2, S2 increases with a s, in contrast, N2 takes a constant value, i.e., we have N2 1. Under Option V 2, the retailer accepts the quantity discount proposed y the wholesaler. The wholesaler s lot size can therefore e increased y stimulating the retailer to alter her/his order quantity per lot through the quantity discount strategy. If the wholesaler increases N 2 one step, her/his lot size also significantly jumps up since N 2 takes a positive integer. Under this option, the wholesaler should increase her/his lot size using the quantity discount rather than increasing N 2 when a s takes larger values. We can also notice in Tale I that we have P1 < P2. This indicates that using the quantity discount strategy can increase the wholesaler s total profit per unit of time. IX. CONCLUSION In this study, we have discussed a quantity discount prolem etween a wholesaler and a retailer under circumstances where oth the wholesaler s and the retailer s inventory levels of the product are depleted not only y demand ut also y deterioration. The wholesaler is interested in increasing her/his profit y controlling the retailer s order quantity through the quantity discount strategy. The retailer attempts to maximize her/his profit considering the wholesaler s proposal. We have formulated the aove prolem as a Stackelerg game etween the wholesaler and the retailer to show the existence of the wholesaler s optimal quantity discount policy that maximizes her/his total profit per unit of time in the same manner as our previous work9. In this study, we have also formulated the same prolem as a
cooperative game. The result of our analysis reveals that the wholesaler is indifferent etween the cooperative and noncooperative options. It should e pointed out that our results are otained under the situation where the inventory holding cost is independent of the value of the item. The relaxation of such a restriction is an interesting extension. APPENDIX A This appendix shows the existence of a unique optimal order quantity which maximizes the retailer s total profit per unit of time under Option V 1. By differentiating π 1 T 1 in Eq. 5 with respect to T 1, we have d π 1 T 1 dt 1 Then ρ p s + h θ T 1 e T 1 e T 1 a. 37 T 2 1 d dt 1 π 1 T 1 agrees with T 1 e T 1 e T 1 a. 38 ρ p s + h Let L T 1 express the left-hand-side of Inequality 38, and we have L T 1 θt 2 1 e T 1 >, 39 L <, 4 ρ p s + h lim L 1T 1 +. 41 T 1 + On the asis of the aove results, we can show that there exists a unique finite T1 >. The retailer s optimal order quantity per lot can therefore e given y Eq. 6. APPENDIX B In this appendix, we discuss the existence of the optimal quantity discount pricing policy which attains ˆP 2 N 2 in Eq. 22 when N 2 is fixed to a suitale value. 1 N 2 1: By differentiating P 2 N 2, T 2 in Eq. 23 with respect to T 2, we have P 2 N 2, T 2 T 2 { ρθ T2 2 e T 2 QT 2 c s + h α θ } a + a s + β. 42 It can easily e shown from Eq. 42 that the sign of T 2 P 2 N 2, T 2 is positive when c s + h / α. In contrast, in the case of c s + h α ><, T 2 P 2 N 2, T 2 agrees with T 2 e T 2 e T 2 a < a + a s + β > ρc s + h / α. 43 Let L 1 T 2 express the left-hand-side of Inequality 43, we have L 1T 2 θt 2 2 e T 2 >, 44 L 1 T1 a ρp s + h /, 45 lim L 1T 2 +. 46 T 2 + From Eqs. 44, 45 and 46, the existence of an optimal quantity discount pricing policy can e discussed for the following two sucases: c s + h / α > : Equiation 45 yields L 1 T 1 < a + a s + β ρc s + h / α. 47 Equiations 44, 46 and 47 indicate that the sign of T 2 P 2 N 2, T 2 changes from positive to negative only once. This signifies that P 2 N 2, T 2 first increases and then decreases as T 2 increases, and thus there exists a unique finite T 2 > T1 which maximizes P 2 N 2, T 2 in Eq. 23. Hence, T2, y is given y Eq. 27. c s + h / α : In this sucase, P 2 N 2, T 2 is increasing in T 2, and consequently the optimal policy can e expressed y Eq. 29. 2 N 2 2: By differentiating P 2 N 2, T 2 in Eq. 23 with respect to T 2, we have T 2 P 2 N 2, T 2 LT 2 a + βn 2 a s N 2 T2 2, 48 where LT 2 is defined y Eq. 26. Then T 2 P 2 N 2, T 2 agrees with LT 2 a + βn 2 + a s. 49 If we assume that there exists a unique solution to Eq. 31, the optimal quantity discount pricing policy can e given y Eq. 27. APPENDIX C For a given T i, this appendix shows the existence of a lower ound and an upper ound for the optimal value of N i N i N i 1, 2, 3, which maximizes P 1 N 1, T 1 in Eq. 15 and P 2 N 2, T 2, y in Eq. 16. Let GT i e defined y GT i p i + h s / + αqt i β T i, 5 then P 1 N 1, T 1 and P 2 N 2, T 2, y can e rewritten as P i N i GT i et i + a s /KT i, 51 N i T i /KT i where p 1 p s, p 2 1 yp s and KT i is defined y Eq. 32. 1 Lower ound N i N L i T i N i :
By calculating the difference etween P i N i + 1 and P i N i, we have P L i P i N 1 + 1 P i N i 1 N i + 1N i T i /KT i γ Ni N i γ Ni γ + a s /KT i,52 where γ e T i. Then P L i agrees with γ Ni N i γ Ni γ a s /KT i. 53 Let us denote, y L L N i, the left-hand-side of Inequality 53, and we have L L L L N i + 1 L L N i N i + 1 γ N i γ 2 <, 54 L L 1 γ 2, 55 lim N LL N i < a s /KT i. 56 i + From Eqs. 54, 55 and 56, we can clarify the conditions where a lower ound N L i T i exists as shown elow. e T i 2 a s /KT i : In this sucase, P i N i is non-increasing in N i, and consequently N L i T i 1. e θsti 2 < a s /KT i : In this sucase, the sign of P L i changes from positive to negative only once, and thus there exists a unique finite N L i T i 1 which is the solution to N i e NiθsTi e θsti e N i T i a s /KT i. 57 2 Upper ound N i N U i T i N i : By calculating the difference etween P i N i and P i N i, we have P U i P i N 1 P i N i 1 N i N i T i /KT i γ Ni N i γ Ni 1 γ +a s /KT i. 58 These oservations can clarify that there exists a unique finite N U i T i N L i T i which is the solution to N i e Ni 1θsTi e θsti e N i T i a s /KT i. 63 REFERENCES 1 J. P. Monahan, A quantity discount pricing model to increase vendor s profit, Management Sci., vol. 3, no. 6, pp. 72 726, 1984. 2 M. Data and K. N. Srikanth, A generalized quantity discount pricing model to increase vendor s profit, Management Sci., vol. 33, no. 1, pp. 1247 1252, 1987. 3 M. J. Rosenlatt and H. L. Lee, Improving pricing profitaility with quantity discounts under fixed demand, IIE Transactions, vol. 17, no. 4, pp. 338 395, 1985. 4 H. L. Lee and M. J. Rosenlatt, A generalized quantity discount pricing model to increase vendor s profit, Management Sci., vol. 32, no. 9, pp. 1177 1185, 1986. 5 M. Parlar and Q. Wang, A game theoretical analysis of the quantity discount prolem with perfect and incomplete information aout the uyer s cost structure, RAIRO/Operations Research, vol. 29, no. 4, pp. 415 439, 1995. 6 S. P. Sarmah, D. Acharya, and S. K. Goyal, Buyer vendor coordination models in supply chain management, European Journal of Operational Research, vol. 175, no. 1, pp. 1 15, 26. 7 P. C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, vol. 157, no. 2, pp. 389 397, 24. 8 K. Behrens, C. Gaigne, G. I. P. Ottaviano, and J. F. Thisse, How density economies in international transportation link the internal geography of trading partners, Journal of Uran Economics, vol. 6, no. 2, pp. 248 263, 26. 9 H. Kawakatsu, A wholesaler s optimal quantity discount policy for deteriorating items, Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 211, WCE 211, 6-8 July, 211, London, U.K., pp. 54 544, 211. Then P U i agrees with γ N i N i γ N i 1 γ a s /KT i.59 Let L U N i express the left-hand-side of Inequality 59, we have L U L U N i L U N i N i γ N i 2 γ 2 <, 6 L U 1 > a s /KT i, 61 lim N i + LU N i < a s /KT i. 62