Economic order quantity under conditionally permissible delay in payments

Transcription

1 European Journal of Operational Research 176 (007) Production, Manufacturing Logistics Economic order quantity under conditionally permissible delay in payments Yung-Fu Huang * Department of Business Administration, Chaoyang University of Technology, No. 168, Jifong E. Road, Wufong Township, Taichung 41349, Taiwan, ROC Received 13 August 004; accepted 9 August 005 Available online 16 November Abstract Within the economic order quantity (EOQ) framework, the main purpose of this paper is to investigate the retailerõs optimal replenishment policy under permissible delay in payments. All previously published articles dealing with optimal order quantity with permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this paper, we want to extend this extreme case by assuming that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity. Under this condition, we model the retailerõs inventory system as a cost minimization problem to determine the retailerõs optimal inventory cycle time optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer. Some previously published results of other researchers can be deduced as special cases. Finally, numerical examples are given to illustrate all these theorems to draw managerial insights. Ó 005 Elsevier B.V. All rights reserved. Keywords: Inventory; EOQ; Conditionally permissible delay in payments; Trade credit 1. Introduction The traditional economic order quantity (EOQ) model assumes that the retailer must be paid for the items as soon as the items were received. In practice, the supplier hopes to stimulate his products so he will offer the retailer a delay period, namely, the trade credit period: Before the end of the trade credit period, the retailer can sell the goods accumulate revenue earn interest. On the other h, a higher * Tel.: ; fax: address: huf@mail.cyut.edu.tw /$ - see front matter Ó 005 Elsevier B.V. All rights reserved. doi: /j.ejor

2 91 Y.-F. Huang / European Journal of Operational Research 176 (007) interest is charged if the payment is not settled by the end of the trade credit period. Therefore, it makes economic sense for the retailer to delay the settlement of the replenishment account up to the last moment of the permissible period allowed by the supplier. Several papers discussing this topic have appeared in the literatures that investigate inventory problems under varying conditions. Some of the prominent papers are discussed below. Goyal [8] established a singleitem inventory model for determining the economic ordering quantity in the case that the supplier offers the retailer the opportunity to delay his payment within a fixed time period. Chung [5] simplified the search of the optimal solution for the problem explored by Goyal [8]. Aggarwal Jaggi [] considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Jamal et al. [13] then further generalized the model to allow for shortages. Hwang Shinn [1] developed the model for determining the retailerõs optimal price lot-size simultaneously when the supplier permits delay in payments for an order of a product whose dem rate is a function of constant price elasticity. Jamal et al. [14] formulated a model where the retailer can pay the wholesaler either at the end of the credit period or later, incurring interest charges on the unpaid balances for the overdue period. They developed a retailerõs policy for the optimal cycle payment times for a retailer in a deteriorating-item inventory scenario, in which a wholesaler allows a specified credit period for payment without penalty. Teng [17] assumed that the selling price is not equal to the purchasing price to modify GoyalÕs model [8]. The important finding from TengÕs study [17] is that it makes economic sense for a well-established retailer to order small lot sizes so take more frequently the benefits of the permissible delay in payments. Chung Huang [6] extended Goyal [8] to consider the case that the units are replenished at a finite rate under permissible delay in payments developed an efficient solution-finding procedure to determine the retailerõs optimal ordering policy. Huang [9] extended one-level trade credit into two-level trade credit to develop the retailerõs replenishment model from the viewpoint of the supply chain. He assumed that not only the supplier offers the retailer trade credit but also the retailer offers the trade credit to his/her customer. This viewpoint reflected more real-life situations in the supply chain model. Khouja [15] showed that for many supply chain configurations, complete synchronization would result in some members of the chain being ÔlosersÕ in terms of cost. He used the economic delivery scheduling problem model analyzed supply chains dealing with single multiple components in developing his model. Huang Chung [11] extended GoyalÕs model [8] to discuss the replenishment payment policies to minimize the annual total average cost under cash discount payment delay from the retailerõs point of view. They assumed that the supplier could adopt a cash discount policy to attract retailer to pay the full payment of the amount of purchasing at an earlier time as a means to shorten the collection period. Arcelus et al. [3] modeled the retailerõs profit-maximizing retail promotion strategy, when confronted with a vendorõs trade promotion offer of credit /or price discount on the purchase of regular or perishable merchise. Abad Jaggi [1] formulated models of seller-buyer relationship, they provided procedures for finding the sellerõs buyerõs best policies under non-cooperative cooperative relationship respectively. Huang [10] extended Chung HuangÕs model [6], in allowing the retailer adopts different payment policy finding differences between unit purchase selling price, developed an efficient solution-finding procedure to determine the retailerõs optimal cycle time optimal order quantity. All above published papers assumed that the supplier offer the retailer fully permissible delay in payments independent of the order quantity. Recently, Shinn Hwang [16] determined the retailerõs optimal price order size simultaneously under the condition of order-size-dependent delay in payments. They assumed that the length of the credit period is a function of the retailerõs order size, also the dem rate is a function of the selling price. Chung Liao [7] dealt with the problem of determining the economic order quantity for exponentially deteriorating items under permissible delay in payments depending on the ordering quantity developed an efficient solution-finding procedure to determine the retailerõs optimal ordering policy. In this regard, Chang [4] extended Chung Liao [7] by taking into account inflation finite time horizon. However, all above published papers dealing with economic order quantity in the presence of

3 Y.-F. Huang / European Journal of Operational Research 176 (007) permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer orders a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. We know that this policy of the supplier to stimulate the dems from the retailer is very practical. But this is just an extreme case. That is, the retailer would obtain 100% permissible delay in payments if the retailer ordered a large enough quantity. Otherwise, 0% permissible delay in payments would happen. In reality, the supplier can relax this extreme case to offer the retailer partially permissible delay in payments rather than 0% permissible delay in payments when the order quantity is smaller than a predetermined quantity. That is, the retailer must make a partial payment to the supplier when the order is received to enjoy some portion of the trade credit. Then, the retailer must pay off the remaining balances at the end of the permissible delay period. For example, the supplier provides 100% delay payment permitted if the retailer ordered a sufficient quantity, otherwise only a% (0 6 a 6 100) delay payment permitted. From the viewpoint of supplierõs marketing policy, the supplier can use the fraction of the permissible delay in payments to agilely control the effects of stimulating the dems from the retailer. This viewpoint is a realistic novel one in this research field, hence, forms the focus of the present study. Therefore, we ignore the effect of deteriorating item; inflation finite time horizon similar to most previously published articles. Under these conditions, we model the retailerõs inventory system as a cost minimization problem to determine the retailerõs optimal inventory cycle time optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer under the more general framework. Some previously published results of other researchers can be viewed as special cases. Finally, numerical examples are given to illustrate all these theorems to draw managerial insights.. Model formulation the convexity In this section, the present study develops a retailerõs inventory model under conditionally permissible delay in payments. The following notation assumptions are used throughout this paper. Notation D dem rate per year A ordering cost per order W quantity at which the fully delay payments permitted per order c unit purchasing price h unit stock holding cost per year excluding interest charges I e interest earned per $ per year I k interest charged per $ in stocks per year M the length of the trade credit period, in years a the fraction of the delay payments permitted by the supplier per order, 0 6 a 6 1 T the length of the cycle time, in years Q the order quantity TRC(T) the annual total relevant cost, which is a function of T T * the optimal cycle time of TRC(T) Q * the optimal order quantity = DT * Assumptions: (1) Replenishments are instantaneous. () Dem rate, D, is known constant. (3) Shortages are not allowed.

4 914 Y.-F. Huang / European Journal of Operational Research 176 (007) (4) The inventory system involves only one type of inventory. (5) Time horizon is infinite. (6) If Q < W, i.e. T < W/D, the partially delayed payment is permitted. Otherwise, fully delayed payment is permitted. Hence, if Q P W, pay cq after Mtime periods from the time the order is filled. Otherwise, as the order is filled, the retailer must make a partial payment, (1 a)cdt, to the supplier. Then the retailer must pay off the remaining balances, acdt, at the end of the trade credit period. This assumption constitutes the major difference of the proposed model from previous ones. (7) During the time period that the account is not settled, generated sales revenue is deposited in an interest-bearing account. (8) I k P I e. The model: The annual total relevant cost consists of the following elements. There are three cases to occur: (1) M P W/D; () M < W/D 6 M/(1 a); (3) M/(1 a) <W/D..1. Case I: Suppose that M P W/D (1) Annual ordering cost ¼ A. T () Annual stock holding costðexcluding interest chargesþ ¼ DTh. (3) From assumptions (6) (7), there are three sub-cases in terms of annual opportunity cost of the capital. (i) M 6 T.. The annual opportunity cost of capital ¼ ci kdðt MÞ DM ci T e T. (ii) W/D 6 T 6 M. h i. DT The annual opportunity cost of capital ¼ ci e þ DT ðm T Þ T ¼ ci e DT M T T. (iii) 0 < T < W/D, as shown in Fig. 1. h i. ð1 aþ The annual opportunity cost of capital ¼ ci DT k T ci e DT M T T. From the above conditions, the annual total relevant cost for the retailer can be expressed as TRC(T) = ordering cost + stock-holding cost + opportunity cost of capital. 8 < TRC 1 ðt Þ; if M 6 T ; ðaþ TRCðT Þ¼ TRC ðt Þ; if W 6 T 6 M; ðbþ D ð1þ : TRC 3 ðt Þ; if 0 < T < W ; ðcþ D Inventory level DT αdt (1 α)t T M Fig. 1. The inventory level the total saved amount of interest payable when 0 < T 6 M. Time

5 where Y.-F. Huang / European Journal of Operational Research 176 (007) TRC 1 ðt Þ¼ A T þ DTh þ ci kdðt MÞ T TRC ðt Þ¼ A T þ DTh ci edt M T DM ci e ; ðþ T T ð3þ TRC 3 ðt Þ¼ A T þ DTh þð1 aþ ci k DT =T ci e DT M T T. ð4þ Since TRC 1 (M) = TRC (M) TRC (W/D) 6 TRC 3 (W/D), TRC(T) is continuous except at T = W/D. Furthermore, we have TRC 3 (T) P TRC (T) for all T > 0 TRC 3 (T) will reduce to TRC (T) when a = 1. Eqs. () (4) yield TRC 0 1 ðt Þ¼ ½A þ cdm ði k I e ÞŠ þ Dðh þ ci kþ ; ð5þ T TRC 00 1 ðt Þ¼A þ cdm ði k I e Þ > 0; ð6þ T 3 TRC 0 ðt Þ¼ A T þ Dðh þ ci eþ ; ð7þ TRC 00 ðt Þ¼A T > 0; ð8þ 3 TRC 0 3 ðt Þ¼ A T þ Dfh þ c½ð1 aþ I k þ I e Šg ð9þ TRC 00 3ðT Þ¼A T > 0. ð10þ 3 Eqs. (6), (8) (10) imply that TRC 1 (T), TRC (T) TRC 3 (T) are convex on T > 0. Moreover, we have TRC 0 1 ðmþ ¼TRC0 ðmþ TRC0 ðw =DÞ 6¼ TRC0 3ðW =DÞ except when a =1... Case II: Suppose that M < W/D 6 M/(1 a) If M < W/D 6 M/(1 a), Eqs. (1(a) (c)) will be modified as 8 >< TRC 1 ðt Þ; if W 6 T ; ðaþ D TRCðT Þ¼ TRC 4 ðt Þ; if M 6 T < W ; ðbþ D >: TRC 3 ðt Þ; if 0 < T 6 M; ðcþ ð11þ when M < T < W/D 6 M/(1 a), the annual total relevant cost, TRC 4 (T), consists of the following elements: (1) Annual ordering cost ¼ A. T () Annual stock holding cost ¼ DTh. (3) According to assumption (6), the annual opportunity cost of capital (as shown in Fig. )= h i... ð1 aþ ci DT DðT MÞ DM k þ T ci e T ¼ cdi k ½ð1 aþ T þðt MÞ DM Š=T ci e T.

6 916 Y.-F. Huang / European Journal of Operational Research 176 (007) Inventory level DT αdt (1 α)t M T Fig.. The inventory level the total saved amount of interest payable when M 6 T 6 M/(1 a). Time Combining the above conditions, we get TRC 4 ðt Þ¼ A T þ DTh þ ci kd½ð1 aþ T þðt MÞ Š=T ci e DM =T. ð1þ Since TRC 1 (W/D) 6 TRC 4 (W/D) TRC 4 (M) = TRC 3 (M), TRC(T) is continuous except at T = W/D. Furthermore, we have TRC 4 (T) P TRC 1 (T) for all T > 0 TRC 4 (T) will reduce to TRC 1 (T) when a = 1. Eq. (1) yields TRC 0 4 ðt Þ¼ A þ cdm ði k I e Þ þ Dfh þ ci k½1 þð1 aþ Šg ð13þ T TRC 00 4 ðt Þ¼A þ cdm ði k I e Þ > 0. ð14þ T 3 Eq. (14) implies that TRC 4 (T) is convex on T > 0. Moreover, we have TRC 0 1 ðw =DÞ 6¼ TRC0 4ðW =DÞ except when a = 1 TRC 0 4 ðmþ ¼TRC0 3 ðmþ..3. Case III: Suppose that M/(1 a) <W/D If M/(1 a) <W/D, Eqs. (1(a) (c)) (11(a) (c)) will be modified as 8 TRC 1 ðt Þ; if W D >< 6 T ; ðaþ M TRC 5 ðt Þ; if 1 a TRCðT Þ¼ 6 T < W ; ðbþ D TRC 4 ðt Þ; if M 6 T 6 M ; ðcþ 1 a >: TRC 3 ðt Þ; if 0 < T 6 M; ðdþ when M/(1 a) 6 T < W/D, the annual total relevant cost, TRC 5 (T), consists of the following elements: (1) Annual ordering cost ¼ A. T () Annual stock holding cost ¼ DTh. (3) According to assumption. (6),. the annual opportunity cost of capital (as shown in Fig. 3)¼ DT ci DM k adtm T ci e T. ð15þ

7 Y.-F. Huang / European Journal of Operational Research 176 (007) Inventory level DT αdt Time M (1 α)t T Fig. 3. The inventory level the total saved amount of interest payable when M/(1 a) 6 T. Combining the above conditions, we get TRC 5 ðt Þ¼ A T þ DTh þ ci kdt T am T ci e DM =T. ð16þ M Since TRC 1 (W/D) 5 TRC 5 (W/D), TRC 5 1 a ¼ M TRC4 1 a TRC4 (M) = TRC 3 (M), TRC(T) is continuous except at T = W/D. Furthermore, we have TRC 5 (T) P TRC 1 (T) for T P M/(1 a). Eq. (16) yields TRC 0 5 ðt Þ¼ A cdm I e þ D h þ ci k ð17þ T TRC 00 5 ðt Þ¼A cdm I e T 3. ð18þ Eq. (18) implies that TRC 5 (T) is convex on T > 0 if A cdm I e > 0. Moreover, we have TRC 0 1ðW =DÞ 6¼ TRC 0 5 ðw =DÞ; TRC0 5 ¼ TRC 0 4 TRC 0 4 ðmþ ¼TRC0 3 ðmþ. M 1 a M 1 a 3. Decision rules for the optimal cycle time T * In this section, the present study demonstrates the determination of the optimal cycle time for the above three cases, under the condition of minimizing annual total relevant costs Case I: Suppose that M P W/D From Eqs. (5), (7) (9), find T i* such that TRC 0 i ðt i Þ¼0 for each i = 1,, 3. Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 ¼ A þ cdm ði k I e Þ ; ð19þ Dðh þ ci k Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ A ð0þ Dðh þ ci e Þ

8 918 Y.-F. Huang / European Journal of Operational Research 176 (007) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 ¼ A. Dfh þ c½ð1 aþ I k þ I e Šg ð1þ Eq. (19) gives that the optimal value T * for the case when M 6 T so that M 6 T 1. Substituting Eq. (19) into M 6 T 1, then we can obtain that T 1 P M; if only if A þ DM ðh þ ci e Þ 6 0. Likewise, Eq. (0) gives that the optimal value T * for the case when W/D 6 T 6 M so that W =D 6 T 6 M. Substituting Eq. (0) into W =D 6 T 6 M, then we can obtain that T 6 M; if only if A þ DM ðh þ ci e Þ P 0 T W P W =D; if only if A þ D ðh þ ci eþ 6 0. Finally, Eq. (1) gives that the optimal value T * for the case when T < W/D so that T 3 < W =D. Substituting Eq. (1) into T 3 < W =D, then we can obtain that T 3 W < W =D; if only if A þ D fh þ c½ð1 aþ I k þ I e Šg > 0. Furthermore, to simplify, we let D 1 ¼ AþDM ðh þ ci e Þ; D ¼ Aþ W D ðh þ ci eþ D 3 ¼ Aþ W D fh þ c½ð1 aþ I k þ I e Šg. ðþ ð3þ ð4þ Eqs. () (4) imply that D 1 P D D 3 P D. In addition, we know TRC 3 (T) P TRC (T) for all T >0 from Eqs. (3) (4). From above arguments, we can summarize the above results in Theorem 1. Theorem 1. Suppose that M P W/D, then (A) IfD 1 > 0, D > 0 D 3 > 0, then TRCðT Þ¼TRC 3 ðt 3 Þ T ¼ T 3. (B) IfD 1 > 0, D 6 0 D 3 > 0, then TRCðT Þ¼TRC ðt Þ T ¼ T. (C) IfD 1 > 0, D 6 0 D 3 6 0, then TRCðT Þ¼TRC ðt Þ T ¼ T. (D) IfD 1 6 0, D 6 0 D 3 > 0, then TRCðT Þ¼minfTRC 1 ðt 1 Þ; TRC 3ðT 3 Þg. Hence, T* is T 1 or T 3 whichever has the least cost. (E) IfD 1 6 0, D 6 0 D 3 6 0, then TRCðT Þ¼TRC 1 ðt 1 Þ T ¼ T 1.

9 3.. Case II: Suppose that M < W/D 6 M/(1 a) If M < W/D 6 M/(1 a), from Eqs. (11(a) (c)), we know that 8 TRC 1 ðt Þ; if >< W 6 T ; D TRCðT Þ¼ TRC 4 ðt Þ; if M 6 T < W ; D >: TRC 3 ðt Þ; if 0 < T 6 M. From Eq. (13), find T 4 such that TRC0 4 ðt 4Þ¼0. Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 4 ¼ A þ cdm ði k I e Þ. ð5þ Dfh þ ci k ½1 þð1 aþ Šg In a similar fashion, we can obtain following results: T 1 P W =D; if only if ½A þ cdm ði k I e ÞŠ þ W T 4 < W =D; if only if ½A þ cdm ði k I e ÞŠ þ W D ðh þ ci kþ 6 0; T 4 P M; if only if A þ DM fh þ c½ð1 aþ I k þ I e Šg 6 0; T 3 6 M; if only if A þ DM fh þ c½ð1 aþ I k þ I e Šg P 0. Furthermore, to simplify, we let Y.-F. Huang / European Journal of Operational Research 176 (007) D fh þ ci k½1 þð1 aþ Šg > 0 D 4 ¼ ½AþcDM ði k I e ÞŠ þ W D ðh þ ci kþ; D 5 ¼ ½AþcDM ði k I e ÞŠ þ W D fh þ ci k½1 þð1 aþ Šg ð6þ ð7þ D 6 ¼ AþDM fh þ c½ð1 aþ I k þ I e Šg. ð8þ Eqs. (6) (8) imply that D 5 P D 4 D 5 > D 6. In addition, we know TRC 4 (T) P TRC 1 (T) for all T >0 from Eqs. () (1). From above arguments, we can summarize the above results in Theorem. Theorem. Suppose that M < W/D 6 M/(1 a), then (A) IfD 4 > 0, D 5 > 0 D 6 P 0, then TRCðT Þ¼TRC 3 ðt 3 Þ T ¼ T 3. (B) IfD 4 > 0, D 5 > 0 D 6 < 0, then TRCðT Þ¼TRC 4 ðt 4 Þ T ¼ T 4. (C) IfD 4 6 0, D 5 > 0 D 6 P 0, then TRCðT Þ¼minfTRC 1 ðt 1 Þ; TRC 3ðT 3 Þg. Hence, T* is T 1 or T 3 whichever has the least cost. (D) IfD 4 6 0, D 5 > 0 D 6 < 0, then TRCðT Þ¼TRC 1 ðt 1 Þ T ¼ T 1. (E) IfD 4 6 0, D 5 6 0D 6 < 0, then TRCðT Þ¼TRC 1 ðt 1 Þ T ¼ T 1.

10 90 Y.-F. Huang / European Journal of Operational Research 176 (007) Case III: Suppose that M/(1 a) <W/D If M/(1 a) <W/D, from Eqs. (15(a) (d)), we know that 8 TRC 1 ðt Þ; if W D >< 6 T ; M TRC 5 ðt Þ; if 1 a TRCðT Þ¼ D TRC 4 ðt Þ; if M 6 T 6 M 1 a >: TRC 3 ðt Þ; if 0 < T 6 M. From Eq. (17), find T 5 such that TRC0 5 ðt 5Þ¼0. Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 5 ¼ A cdm I e ; Dðh þ ci k Þ if A cdm I e > 0. ð9þ In a similar fashion, we can obtain following results: T 1 P W =D; if only if ½AþcDM ði k I e ÞŠ þ W D ðh þ ci kþ 6 0. T 5 < W =D; if only if ½A cdm I e Šþ W D ðh þ ci kþ > 0 T 5 P M=ð1 aþ; if only if ½A cdm I e ŠþD M ðh þ ci k Þ a T 4 6 M=ð1 aþ; if only if ½A cdm I e ŠþD M ðh þ ci k Þ P 0 1 a T 4 P M; if only if A þ DM fh þ c½ð1 aþ I k þ I e Šg 6 0; T 3 6 M; if only if A þ DM fh þ c½ð1 aþ I k þ I e Šg P 0. Furthermore, to simplify, we let D 7 ¼ ½A cdm I e Šþ W D ðh þ ci kþ ð30þ D 8 ¼ ½A cdm I e ŠþD M ðh þ ci k Þ. 1 a ð31þ Eqs. (30), (31), (6) (8) imply that D 7 > D 8 P D 6 D 7 > D 4. In addition, we know TRC 4 (T) P TRC 1 (T) for all T > 0 from Eqs. () (1) TRC 5 (T) P TRC 1 (T) for T P M/(1 a) from Eqs. () (16). From above arguments, we can summarize the above results in Theorem 3. Theorem 3. Suppose that M/(1 a) < W/D, then (A) IfD 4 > 0, D 7 > 0, D 8 P 0 D 6 P 0, then TRCðT Þ¼TRC 3 ðt 3 Þ T ¼ T 3. (B) IfD 4 > 0, D 7 > 0, D 8 P 0 D 6 < 0, then TRCðT Þ¼TRC 4 ðt 4 Þ T ¼ T 4. (C) IfD 4 > 0, D 7 > 0, D 8 < 0 D 6 < 0, then TRCðT Þ¼TRC 5 ðt 5 Þ T ¼ T 5.

11 Y.-F. Huang / European Journal of Operational Research 176 (007) (D) IfD 4 6 0, D 7 > 0, D 8 P 0 D 6 P 0, then TRCðT Þ¼minfTRC 1 ðt 1 Þ; TRC 3ðT 3 Þg. Hence, T* is T 1 or T 3 whichever has the least cost. (E) IfD 4 6 0, D 7 > 0, D 8 P 0 D 6 < 0, then TRCðT Þ¼TRC 1 ðt 1 Þ T ¼ T 1. (F) IfD 4 6 0, D 7 > 0, D 8 < 0 D 6 < 0, then TRCðT Þ¼TRC 1 ðt 1 Þ T ¼ T 1. (G) IfD 4 < 0, D 7 6 0, D 8 < 0 D 6 < 0, then TRCðT Þ¼TRC 1 ðt 1 Þ T ¼ T A special case The value a = 1 means that the supplier offers the retailer fully permissible delay in payments. The value W = 0 means that the supplier offers the retailer permissible delay in payments independent of the order quantity. Therefore, when a = 1 W = 0, Eqs. (1(a) (c)) will reduced to TRCðT Þ¼ TRC 1ðT Þ; if M 6 T ; ðaþ ð3þ TRC ðt Þ; if 0 < T 6 M; ðbþ Eqs. (3(a) (b)) will be consistent with Eqs. (1) (4) in Goyal [8], respectively. Hence, Goyal [8] will be a special case of this paper. From Eq. (), we know D 1 = A + DM (h + ci e ). If we let D = A + DM (h + ci e ), Theorem 1 can be modified as follows: Theorem 4 (A) IfD > 0, then T ¼ T. (B) IfD < 0, then T ¼ T 1. (C) IfD = 0, then T ¼ T ¼ T 1 ¼ M. Theorem 4 has been discussed in Chung [5]. Hence, Theorem 1 in Chung [5] is a special case of Theorem 1 of the present study. 5. Numerical examples In this section, the present study provides the following numerical examples to illustrate all the theoretical results as reported in Section 3. For convenience, the values of the parameters are selected romly. The optimal cycle time optimal order quantity for different parameters of a (0., 0.5, 0.8), W (100, 00, 300) c (10, 30, 50) are shown in Table 1. The following inferences can be made based on Table 1. (1) For fixed W c, increasing the value of a will result in a significant increase in the value of the optimal order quantity a significant decrease in the value of the annual total relevant costs as the retailerõs order quantity is smaller only the partially delayed payment is permitted. For example, when W = 300, c = 50 a increases from 0. to 0.5, the optimal order quantity will increase 9.51% (( )/9.5) the annual total relevant costs will decrease 14.4% (( )/ ). However, if the fully delayed payment is permitted, the optimal order quantity the annual total relevant cost are independent of the value of a. It implies that the retailer will order a larger quantity since the retailer can enjoy greater benefits when the fraction of the delay payments permitted is increasing. So the supplier can use the policy of increasing a to stimulate the dems from the retailer. Consequently, the supplierõs marketing policy under partially permissible delay in payments will be more agile than fully permissible delay in payments. () For fixed a c, increasing the value of W will result in a significant decrease in the value of the optimal order quantity a significant increase in the value of the annual total relevant costs. For example, when a = 0., c = 50 W increases from 100 to 00, the optimal order quantity will

12 9 Y.-F. Huang / European Journal of Operational Research 176 (007) Table 1 Optimal solutions under different parametric values a W c W/D M/(1 a) Judgments of D i (i = 1 8) T * Q * TRC(T * ) Theorem D 1 <0,D <0,D 3 <0 T 1 ¼ 0: (F) D 1 >0,D <0,D 3 <0 T ¼ 0: (C) D 1 >0,D <0,D 3 >0 T ¼ 0: (B) D 4 >0,D 7 >0,D 8 >0,D 6 <0 T 4 ¼ 0: (B) D 4 >0,D 7 >0,D 8 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 7 >0,D 8 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 7 >0,D 8 >0,D 6 <0 T 4 ¼ 0: (B) D 4 >0,D 7 >0,D 8 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 7 >0,D 8 >0,D 6 >0 T 3 ¼ 0: (A) D 1 <0,D <0,D 3 <0 T 1 ¼ 0: (F) D 1 >0,D <0,D 3 <0 T ¼ 0: (C) D 1 >0,D <0,D 3 <0 T ¼ 0: (C) D 4 >0,D 5 >0,D 6 <0 T 4 ¼ 0: (B) D 4 >0,D 5 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 5 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 7 >0,D 8 >0,D 6 <0 T 4 ¼ 0: (B) D 4 >0,D 7 >0,D 8 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 7 >0,D 8 >0,D 6 >0 T 3 ¼ 0: (A) D 1 <0,D <0,D 3 <0 T 1 ¼ 0: (F) D 1 >0,D <0,D 3 <0 T ¼ 0: (C) D 1 >0,D <0,D 3 <0 T ¼ 0: (C) D 4 >0,D 5 >0,D 6 <0 T 4 ¼ 0: (B) D 4 >0,D 5 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 5 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 5 >0,D 6 <0 T 4 ¼ 0: (B) D 4 >0,D 5 >0,D 6 >0 T 3 ¼ 0: (A) D 4 >0,D 5 >0,D 6 >0 T 3 ¼ 0: (A) Let A = $50/order, D = 1000 units/year, h = $5/unit/year, I k = $0.1/$/year, I e = $0.07/$/year M = 0.1 year. decrease 14.75% (( )/108.5) the annual total relevant costs will increase 31.8% (( )/501.95). It implies that the retailer will not order a quantity as large as the minimum order quantity as required to obtain fully permissible delay in payments. Hence, the effect of stimulating the dems from the retailer turns negative when the supplier adopts a policy to increase the value of W. (3) Last, for fixed a W, increasing the value of c will result in a significant decrease in the value of the optimal order quantity a significant decrease in the value of the annual total relevant cost. For example, when a = 0., W = 100 c increases from 10 to 50, the optimal order quantity will decrease 17.74% (( )/131.9) the annual total relevant costs will decrease 5.1% (( )/671.15). This result implies that the retailer will order a smaller quantity to enjoy the benefits of either the fully or partially the permissible delay in payments more frequently in the presence of an increased unit purchasing price. 6. Summary conclusions The supplier offers the permissible delay in payments to the retailer in order to stimulate the dem. Hence, the assumption in previously published results that the fully permissible delay in payments is permitted under a sufficient quantity is practical. On the other h, the permissible delay in payments will not be permitted when the order quantity is smaller than a predetermined quantity obviously is an extreme case.

13 Y.-F. Huang / European Journal of Operational Research 176 (007) In this paper, the proposed model allows the supplier to offer an alternative policy, i.e., partially permissible delay in payments, when the retailerõs order quantity is not large enough to get the fully permissible delay in payments. Viewed from such perspective, we model the inventory system to take care of the following states: The retailer under fully permissible delay in payments if the retailer orders a large quantity; Otherwise, the retailer will just obtain partially permissible delay in payments. In addition, we establish three effective easy-to-use theorems to help the retailer to find the optimal replenishment policy. Finally, some numerical examples are provided to illustrate all the theorems, to obtain the following managerial insights: (1) a higher value of the fraction of the delay payments permitted brings about a larger order quantity smaller annual total relevant costs; () a higher value of the minimum order quantity as required to obtain fully permissible delay in payments brings about a smaller order quantity larger annual total relevant costs; (3) a higher value of unit purchasing price brings about a smaller order quantity smaller annual total relevant costs. From the viewpoint of supplierõs marketing policy, the supplier can use the fraction of the delay payments permitted to control more agilely the effects of stimulating the dem from the retailer. For example, the supplier can offer the larger fraction of the delay payments permitted to stimulate the larger order quantity from the retailer. On the other h, the supplier can use the smaller fraction of the delay payments permitted to decrease the order quantity of the retailer. As such, the more realistic flexible marketing policy is valuable to the supplier. The proposed model can be extended in several ways. For instance, we may generalize the model to allow for shortages, deteriorating item, probabilistic dem, time value of money, finite time horizon, finite replenishment rate worthy of future research. Acknowledgments The author would like to express his heartfelt thanks to anonymous referees professor Gili Yen at CYUT for their most valuable comments suggestions that have led to a significant improvement on an earlier version of this paper. References [1] P.L. Abad, C.K. Jaggi, A joint approach for setting unit price the length of the credit period for a seller when end dem is price sensitive, International Journal of Production Economics 83 (003) [] S.P. Aggarwal, C.K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society 46 (1995) [3] F.J. Arcelus, N.H. Shah, G. Srinivasan, RetailerÕs pricing, credit inventory policies for deteriorating items in response to temporary price/credit incentives, International Journal of Production Economics 81 8 (003) [4] C.T. Chang, An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity, International Journal of Production Economics 88 (004) [5] K.J. Chung, A theorem on the determination of economic order quantity under conditions of permissible delay in payments, Computers Operations Research 5 (1998) [6] K.J. Chung, Y.F. Huang, The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics 84 (003) [7] K.J. Chung, J.J. Liao, Lot-sizing decisions under trade credit depending on the ordering quantity, Computers Operations Research 31 (004) [8] S.K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society 36 (1985) [9] Y.F. Huang, Optimal retailerõs ordering policies in the EOQ model under trade credit financing, Journal of the Operational Research Society 54 (003) [10] Y.F. Huang, Optimal retailerõs replenishment policy for the EPQ model under supplierõs trade credit policy, Production Planning Control 15 (004) 7 33.

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