An EOQ model under retailer partial trade credit policy in supply chain

Transcription

1 ARTICLE IN PRESS Int J Production Economics 11 (008) wwwelseviercom/locate/ijpe An EOQ model under retailer partial trade credit policy in supply chain Yung-Fu Huang a, Kuang-Hua Hsu b, a Department of Marketing Logistics Management, Chaoyang University of Technology, Taichung, Taiwan, ROC b Department of Finance, Chaoyang University of Technology, Taichung, Taiwan, ROC Received 13 January 006; accepted 5 May 007 Available online 10 July 007 Abstract The main purpose of this paper is to investigate the retailer s inventory policy under two levels of trade credit to reflect the supply chain management situation In this paper, we assume that the retailer has the powerful decision-making right So, we extend the assumption that the retailer can obtain the full trade credit offered by the supplier the retailer just offers the partial trade credit to his/her customer Then, we investigate the retailer s inventory system as a cost minimization problem to determine the retailer s optimal inventory policy under the supply chain management Two easyto-use theorems are developed to efficiently determine the optimal inventory policy for the retailer We deduce some previously published results of other researchers as special cases Finally, numerical examples are given to illustrate the theorems obtain a lot of managerial phenomena r 007 Elsevier BV All rights reserved Keywords: EOQ; Inventory; Partial trade credit; Supply chain 1 Introduction The traditional economic order quantity (EOQ) model assumes that the retailer s capitals are unrestricting must be paid for the items as soon as they are received However, this may not be true In practice, the supplier will offer the retailer a delay period, that is the trade credit period, in paying for the amount of purchasing cost Before the end of the trade credit period, the retailer can sell the goods accumulate revenue earn Corresponding author No 168, Jifong E Road, Wufong Township, Taichung County 41349, Taiwan, ROC Tel: ; fax: addresses: huf@cyutedutw (Y-F Huang), khhsu@cyutedutw (K-H Hsu) interest A higher interest is charged if the payment is not settled by the end of the trade credit period In the real world, the supplier often makes use of this policy to promote his/her commodities Goyal (1985) established a single-item inventory model under trade credit Chung (1998) developed an alternative approach to determine the EOQ under the condition of trade credit Aggarwal Jaggi (1995) considered the inventory model with an exponential deterioration rate under the condition of trade credit Chang et al (00) extended this issue to the varying rate of deterioration Liao et al (000) Sarker et al (000a) investigated this topic with inflation Jamal et al (1997) Chang Dye (001) extended this issue with allowable shortage Chang et al (001) extended this issue /$ - see front matter r 007 Elsevier BV All rights reserved doi:101016/jijpe

2 656 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) with linear trend dem Chen Chuang (1999) investigated a light buyer s inventory policy under trade credit by the concept of discounted cash flow Hwang Shinn (1997) modeled an inventory system for a retailer s pricing lot-sizing policy for exponentially deteriorating products under the condition of permissible delay in payment Jamal et al (000) Sarker et al (000b) addressed the optimal payment time under permissible delay in payment with deterioration Teng (00) assumed that the selling price is not equal to the purchasing price to modify Goyal s model (1985) Chung et al (00) discussed this issue under the selling price not equal to the purchasing price different payment rule Shinn Hwang (003) determined the retailer s 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 retailer s order size, also the dem rate is a function of the selling price Chung Huang (003) extended this problem within the economic production quantity (EPQ) framework developed an efficient procedure to determine the retailer s optimal ordering policy Huang Chung (003) extended Goyal s model (1985) to cash discount policy for early payment Salameh et al (003) extended this issue to the continuous review inventory model Chang et al (003) Chung Liao (004) dealt with the problem of determining the EOQ for exponentially deteriorating items under permissible delay in payments depending on the ordering quantity Chang (004) extended this issue to inflation finite time horizon Huang (004) investigated that the unit selling price the unit purchasing price are not necessarily equal within the EPQ framework under a supplier s trade credit policy There are several interesting relevant papers related to trade credit such as Chung et al (005), Chung Liao (006), Huang (007) their references All the above articles assumed that the supplier would offer the retailer a delay period the retailer could sell the goods accumulate revenue earn interest within the trade credit period They implicitly assumed that the customer would pay for the items as soon as the items are received from the retailer That is, they assumed that the supplier would offer the retailer a delay period but the retailer would not offer the trade credit period to his/her customer in previously published results That is one level of trade credit In most business transactions, this assumption is unrealistic Recently, Huang (003) modified this assumption to assume that the retailer will adopt the trade credit policy to stimulate his/her customer dem to develop the retailer s replenishment model That is two levels of trade credit This new viewpoint is more matched to real-life situations in the supply chain model Therefore, we want to extend Huang s model (003) to investigate the situation under which the retailer has the powerful decision-making right That is, we want to assume that the retailer can obtain the full trade credit offered by the supplier the retailer just offers the partial trade credit to his/her customer In practice, this circumstance is very realistic For example, the Toyota Company can require his supplier to offer the full trade credit to him just offer partial trade credit to his dealership That is, the Toyota Company can delay the full amount of purchasing cost until the end of the delay period offered by his supplier But the Toyota Company only offers partial delay payment to his dealership on the permissible credit period the rest of the total amount is payable at the time the dealership places a replenishment order In addition, we want to relax three assumptions in Huang s model (003) that unit purchasing price equals unit selling price, c ¼ s, interest charge rate is larger than interest earned rate, I k XI e, the retailer s trade credit period offered by the supplier is longer than the customer s trade credit period offered by the retailer, MXN Under these conditions, we model the retailer s inventory system as a cost minimization problem to determine the retailer s optimal ordering policies Model formulation convexity The following notation assumptions will be used throughout: Notation D A c s h a I e dem rate per year ordering cost per order unit purchasing price unit selling price, sxc unit stock-holding cost per year excluding interest charges customer s fraction of the total amount owed payable at the time of placing an order offered by the retailer, 0pap1 interest earned per $ per year

3 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) I k interest charged per $ in stocks per year by the supplier M retailer s trade credit period offered by the supplier in years N customer s trade credit period offered by the retailer in years T cycle time in years TRC(T) annual total relevant cost, which is a function of T T* optimal cycle time of TRC(T) Q* optimal order quantity ¼ DT* Assumptions (1) Dem rate, D, is known constant () Shortages are not allowed (3) Time horizon is infinite (4) Replenishments are instantaneous (5) The supplier offers the full trade credit to the retailer When TXM, the account is settled at T ¼ M, the retailer pays off all units sold keeps his/her profits, starts paying for the interest charges on the items in stock with rate I k When TpM, the account is settled at T ¼ M the retailer does not need to pay any interest charge (6) The retailer just offers the partial trade credit to his/her customer Hence, his/her customer must make a partial payment to the retailer when the item is sold Then his/her customer must pay off the remaining balance at the end of the trade credit period offered by the retailer That is, the retailer can accumulate interest from his/her customer payment with rate I e In this case, annual interest payable ¼ 0 (4) According to assumption (6), there are three cases that occur in interest earned per year (i) MpT, as shown in Fig 1 Annual interest earned adn ðdn þ DMÞðM NÞ ¼ si e þ T ¼ si e D½M ð1 aþn Š=T (ii) NpTpM, as shown in Fig Annual interest earned adn ¼ ðdn þ DTÞðT NÞ si e þ þ DTðM TÞ T ¼ si e D½MT ð1 aþn T Š=T (iii) TpN, as shown in Fig 3 Annual interest earned adt ¼ si e þ adtðn TÞþDTðM NÞ T ¼ si e D M ð1 aþn at From the above arguments, the annual total relevant cost for the retailer can be expressed as $ sdt The annual total relevant cost consists of the following elements Two situations may arise: (I) MXN (II) MoN Case I: Suppose that MXN (1) Annual ordering cost ¼ A=T () Annual stock holding cost (excluding interest charges) ¼ DTh= (3) According to assumption (5), there are three cases that occur in costs of interest charges for the items kept in stock per year (i) MpT Annual interest payable ¼ ci k D(T M) /T (ii) NpTpM In this case, annual interest payable ¼ 0 (iii) TpN αsdn N M T Time Fig 1 Total amount of interest earned when MpT $ sdt αsdn N T M Time Fig Total amount of interest earned when NpTpM

4 658 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) TRC(T) ¼ ordering cost+stock-holding cost+ interest payable interest earned 8 >< TRC 1 ðtþ if TXM; TRCðTÞ ¼ TRC ðtþ if NpTpM; (1ac) >: TRC 3 ðtþ if 0oTpN; where $ sdt αsdt TRC 1 ðtþ ¼ A T þ DTh þ ci kdðt MÞ =T si e D½M ð1 aþn Š=T, TRC ðtþ ¼ A T þ DTh si ed½mt ð1 aþn T Š=T, ð3þ TRC 3 ðtþ ¼ A T þ DTh si ed M ð1 aþn at (4) Since TRC 1 (M) ¼ TRC (M) TRC (N) ¼ TRC 3 (N), TRC(T) is continuous well-defined All TRC 1 (T), TRC (T), TRC 3 (T), TRC(T) are defined on T40 Eqs () (4) yield TRC 0 1 ðtþ ¼ A þ cdm I k sdi e ðm ð1 aþn Þ þ D h þ ci k T N M Time Fig 3 Total amount of interest earned when TpN ðþ T, ð5þ TRC 00 1 ðtþ ¼A þ cdm I k sdi e ðm ð1 aþn Þ T 3, (6) TRC 0 ðtþ ¼ A þ sdð1 aþn I e T þ D h þ si e, ð7þ TRC 00 ðtþ ¼A þ sdð1 aþn I e T 3 40, (8) TRC 0 3ðTÞ ¼ A T þ D h þ sai e, (9) TRC 00 3 ðtþ ¼A 40 (10) 3 T Eqs (8) (10) imply that TRC (T) TRC 3 (T) are convex on T40 Eq (6) implies that TRC 1 (T) is convex on T40 when A+cDM I k s- DI e (M (1 a)n )40 Furthermore, we have TRC 0 1 ðmþ ¼TRC 0 ðmþ TRC0 ðnþ ¼TRC0 3 ðnþ Therefore, Eqs (1a c) imply that TRC(T) is convex on T40 when A+cDM I k sdi e (M (1 a) N )40 Case II: Suppose that MoN (1) Annual ordering cost ¼ A=T () Annual stock holding cost (excluding interest charges) ¼ DTh= (3) According to assumption (5), there are two cases that occur in costs of interest charges for the items kept in stock per year (i) MpT Annual interest payable ¼ ci k D(T M) /T (ii) MXT In this case, annual interest payable ¼ 0 (4) According to assumption (6), there are two cases that occur in interest earned per year (i) MpT, as shown in Fig 4 Annual interest earned ¼ si e DaM /T (ii) MXT, as shown in Fig 5 Annual interest earned adt ¼ si e þ adtðm TÞ T ¼ si e D am at From the above arguments, the annual total relevant cost for the retailer can be expressed as TRC(T) ¼ ordering cost+stock-holding cost+ interest payable interest earned ( TRCðTÞ ¼ TRC 4ðTÞ if TXM; (11ab) TRC 5 ðtþ if 0oTpM;

5 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) $ sdt αsdm M N T Time TRC 00 5 ðtþ ¼A 40 (17) 3 T Eq (17) implies that TRC 5 (T) is convex on T4 0 Eq (15) implies that TRC 4 (T) is convex on T40 when A+DM (ci k sai e )40 Furthermore, we have TRC 0 4 ðmþ ¼TRC0 5ðMÞ Therefore, Eqs (11a,b) imply that TRC(T) is convex on T40 when A+DM (ci k sai e )40 $ sdt αsdt where Fig 4 Total amount of interest earned when MpT T M N Fig 5 Total amount of interest earned when TpM TRC 4 ðtþ ¼ A T þ DTh þ ci k DðT MÞ =T si e DaM =T ð1þ TRC 5 ðtþ ¼ A T þ DTh si ed am at (13) Since TRC 4 (M) ¼ TRC 5 (M), TRC(T) is continuous well-defined All TRC 4 (T), TRC 5 (T), TRC(T) are defined on T40 Eqs (1) (13) yield TRC 0 4 ðtþ ¼ A þ DM ðci k sai e Þ T Time þ D h þ ci k, ð14þ TRC 00 4 ðtþ ¼A þ DM ðci k sai e Þ T 3, (15) TRC 0 5ðTÞ ¼ A T þ D h þ sai e, (16) 3 Determination of the optimal cycle time T* Case I: Suppose that MXN Let TRC 0 i ðt i Þ¼0 for all i ¼ 1,, 3 We can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 ¼ A þ cdm I k sdi e ½M ð1 aþn ÞŠ Dðh þ ci k Þ if A þ cdm I k sdi e ½M ð1 aþn ÞŠ40, (18) T ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ sdð1 aþn I e, Dðh þ si e Þ (19) T 3 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Dðh þ sai e Þ (0) Eq (18) gives the optimal value of T* for the case when TXM so that T 1XM We substitute Eq (18) into T 1XM; then we obtain that T 1XM if only if A þ DM h þ sdi e ½M ð1 aþn Šp0 Similarly, Eq (19) gives the optimal value of T* for the case when NpTpM so that NpT pm We substitute Eq (19) into NpT pm; then we obtain that T pm if only if A þ DM h þ sdi e ½M ð1 aþn ŠX0 NpT if only if A þ DN ðh þ sai e Þp0 Finally, Eq (0) gives the optimal value of T* for the case when TpN so that T 3pN We substitute Eq (0) into T 3pN; then we obtain that T 3pN if only if A þ DN ðh þ sai e ÞX0

6 660 Furthermore, we let ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) Special cases D 1 ¼ AþDM h þ sdi e ½M ð1 aþn Š (1) D ¼ AþDN ðh þ sai e Þ () Eqs (1) () imply that D 1 XD From the above arguments, we obtain the following results: Theorem 1 (A) If D X0, then TRCðT Þ¼TRCðT 3 Þ T ¼ T 3 (B) If D 1 40 D o0, then TRCðT Þ¼TRCðT Þ T ¼ T (C) If D 1 p0, then TRCðT Þ¼TRCðT 1 Þ T ¼ T 1 Case II: Suppose that MoN Let TRC 0 i ðt i Þ¼0 for all i ¼ 4, 5 We obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 4 ¼ A þ DM ðci k sai e Þ Dðh þ ci k Þ if A þ DM ðci k sai e Þ40 ð3þ T 5 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Dðh þ sai e Þ (4) Eq (3) gives the optimal value of T* for the case when TXM so that T 4XM We substitute Eq (3) into T 4XM; then we obtain that T 4XM if only if A þ DM ðh þ sai e Þp0 Similarly, Eq (4) gives the optimal value of T * for the case when TpM so that T 5pM We substitute Eq (4) into T 5pM; then we obtain that T 5pM if only if A þ DM ðh þ sai e ÞX0 Furthermore, we let D 3 ¼ AþDM ðh þ sai e Þ (5) From the above arguments, we can obtain the following results Theorem (A) If D 3 X0, then TRCðT Þ¼TRCðT 5 Þ T ¼ T 5 (B) If D 3 o0, then TRCðT Þ¼TRCðT 4 Þ T ¼ T 4 41 Huang s model When MXN, s ¼ c, a ¼ 0 (it means that the retailer also offers the full trade credit to his/her customer), let TRC 6 ðtþ ¼ A T þ DTh þ ci kdðt MÞ =T ci e DðM N Þ=T, ð6þ TRC 7 ðtþ ¼ A T þ DTh ci edðmt N T Þ=T, (7) TRC 8 ðtþ ¼ A T þ DTh ci edðm NÞ, (8) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 6 ¼ A þ cd½m ði k I e ÞþN I e Š, Dðh þ ci k Þ (9) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 7 ¼ A þ cdn I e ; Dðh þ ci e Þ (30) rffiffiffiffiffiffi T 8 ¼ A (31) Dh Then TRC 0 i ðt i Þ¼0 for i ¼ 6, 7, 8 Eqs 1(a c) will be modified as follows: 8 >< TRC 6 ðtþ if TXM; TRCðTÞ ¼ TRC 7 ðtþ if NpTpM; >: TRC 8 ðtþ if 0oTpN; (3ac) Eqs 3(a c) will be consistent with Eqs 1(a c), in Huang (003), respectively Eqs (1) () can be modified as D 1 ¼ A+DM (h+ci e ) cdn I e D ¼ A+DN h, respectively If we let D 1 ¼ A þ DM ðh þ ci e Þ cdn I e D ¼ Aþ DN h, Theorem 1 can be modified as follows: Theorem 3 (A) If D 1 40 D X0, then TRCðT Þ¼TRCðT 8 Þ T ¼ T 8 (B) If D 1 40 D o0 then TRCðT Þ¼TRCðT 7 Þ T ¼ T 7

7 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) (C) If D 1 p0 D o0, then TRCðT Þ¼TRCðT 6 Þ T ¼ T 6 Theorem 3 has been discussed in Theorem 1 of Huang (003) Hence, Huang (003) will be a special case of this paper 4 Goyal s model When N ¼ 0, it means that the supplier would offer the retailer a delay period but the retailer would not offer the delay period to his/her customer That is one level of trade credit Therefore, when s ¼ c, a ¼ 0, N ¼ 0, let TRC 9 ðtþ ¼ A T þ DTh DðT MÞ DM þ ci k T ci e T, ð33þ TRC 10 ðtþ ¼ A T þ DTh DT ci e þ DTðM TÞ T, ð34þ T 9 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ DM cði k I e Þ, Dðh þ ci k Þ (35) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 10 ¼ A Dðh þ ci e Þ (36) Then TRC 0 i ðt i Þ¼0for i ¼ 9, 10 Eqs (1a c) will be reduced as follows: ( TRCðTÞ ¼ TRC 9ðTÞ if MpT; TRC 10 ðtþ if 0oTpM: (37ab) Eqs (37a,b) will be consistent with Eqs (1) (4) in Goyal (1985), respectively Eq (1) can be modified as D 1 ¼ A+DM (h+ci e ) If we let Table 1 Optimal solutions when MXN a N s D 1 D Theorem T* Q* TRC(T*) Let A ¼ $80/order, D ¼ 000 units/year, c ¼ $10/unit, h ¼ $7/unit/year, I k ¼ $015/$/year, I e ¼ $013/$/year, M ¼ 01 year o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 o0 1-(A) T 1 ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 o0 1-(A) T 1 ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: o0 1-(B) T ¼ 0: (C) T 3 ¼ 0:

8 66 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) D ¼ A+DM (h+ci e ), Theorem 1 can be modified as follows: Theorem 4 (A) If D40, then T ¼ T 10 (B) If Do0, then T ¼ T 9 (C) If D ¼ 0, then T ¼ T 9 ¼ T 10 ¼ M Theorem 4 has been discussed in Theorem 1 of Chung (1998) Hence, Goyal (1985) will be a special case of this paper 5 Numerical examples To illustrate the results developed in this paper, let us apply the proposed method to solve the following numerical examples For convenience, the values of the parameters are selected romly The optimal solutions for different parameters of a, N, s when MXN MoN are shown in Tables 1, respectively The following inferences can be made based on Tables 1 (1) For fixed N s, the larger the value of a, the smaller the value of the optimal cycle time the lower the value of the annual total relevant cost () For fixed a s, the larger the value of N, the larger the value of the optimal cycle time the higher the value of the annual total relevant cost when MXN; the optimal cycle time the annual total relevant cost will be independent of N when MoN (3) Finally, for fixed a N, the larger the value of s, the smaller value of the optimal cycle time the smaller the value of the annual total relevant cost Table Optimal solutions when MoN a N s D 3 Theorem T* Q* TRC(T*) Let A ¼ $80/order, D ¼ 5000 units/year, c ¼ $10/unit, h ¼ $10/unit/year, I k ¼ $01/$/year, I e ¼ $0/$/year, M ¼ 005 year o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: o0 -(B) T 4 ¼ 0: (A) T 5 ¼ 0: (A) T 5 ¼ 0: o0 -(B) T 4 ¼ 0: (A) T 5 ¼ 0: (A) T 5 ¼ 0: o0 -(B) T 4 ¼ 0: (A) T 5 ¼ 0: (A) T 5 ¼ 0: o0 -(B) T 4 ¼ 0: (A) T 5 ¼ 0: (A) T 5 ¼ 0: o0 -(B) T 4 ¼ 0: (A) T 5 ¼ 0: (A) T 5 ¼ 0: o0 -(B) T 4 ¼ 0: (A) T 5 ¼ 0: (A) T 5 ¼ 0:

9 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) Conclusions This paper extends the assumption of the two levels of trade credit policy in the previously published result to investigate that the inventory problem of the retailer has powerful decisionmaking right Theorems 1 help the retailer accurately speedily to determine the optimal ordering policy after computing the numbers D 1, D, D 3 Huang s model (003) Goyal s model (1985) are the special cases of this extended model discussed in this paper Finally, numerical examples are given to illustrate the results developed in this paper There are some managerial phenomena as follows: (1) When the customer s fraction of the total amount owed payable at the time of placing an order offered by the retailer is increasing, the retailer will order less quantity increase order frequency The retailer can accumulate more interest under higher order frequency higher customer s fraction of the total amount owed payable at the time of placing an order offered by the retailer () When the customer s trade credit period offered by the retailer is increasing, the retailer will order more quantity to accumulate more interest to compensate the loss of interest earned when longer trade credit period is offered to his/her customer under the condition of MXN (3) When the unit selling price is increasing, the retailer will order less quantity to take the benefits of the trade credit more frequently A future study will further incorporate the proposed model into more realistic assumptions, such as probabilistic dem, allowable shortages, deteriorating items, or finite replenishment rate Acknowledgments The authors would like to thank anonymous referees for their valuable constructive comments suggestions that have led to significant improvement on an earlier version of this paper The NSC in Taiwan supported this research, the project no was NSC H References Aggarwal, SP, Jaggi, CK, 1995 Ordering policies of deteriorating items under permissible delay in payments Journal of the Operational Research Society 46, Chang, CT, 004 An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity International Journal of Production Economics 88, Chang, HJ, Dye, CY, 001 An inventory model for deteriorating items with partial backlogging permissible delay in payments International Journal of Systems Science 3, Chang, HJ, Hung, CH, Dye, CY, 001 An inventory model for deteriorating items with linear trend dem under the condition of permissible delay in payments Production Planning & Control 1, 74 8 Chang, HJ, Hung, CH, Dye, CY, 00 A finite time horizon inventory model with deterioration time-value of money under the conditions of permissible delay in payments International Journal of Systems Science 33, Chang, CT, Ouyang, LY, Teng, JT, 003 An EOQ model for deteriorating items under supplier credits linked to ordering quantity Applied Mathematical Modelling 7, Chen, MS, Chuang, CC, 1999 An analysis of light buyer s economic order model under trade credit Asia-Pacific Journal of Operational Research 16, 3 34 Chung, KJ, 1998 A theorem on the determination of economic order quantity under conditions of permissible delay in payments Computers Operations Research 5, 49 5 Chung, KJ, Huang, YF, 003 The optimal cycle time for EPQ inventory model under permissible delay in payments International Journal of Production Economics 84, Chung, KJ, Liao, JJ, 004 Lot-sizing decisions under trade credit depending on the ordering quantity Computers Operations Research 31, Chung, KJ, Liao, JJ, 006 The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity International Journal of Production Economics 100, Chung, KJ, Huang, YF, Huang, CK, 00 The replenishment decision for EOQ inventory model under permissible delay in payments Opsearch 39, Chung, KJ, Goyal, SK, Huang, YF, 005 The optimal inventory policies under permissible delay in payments depending on the ordering quantity International Journal of Production Economics 95, Goyal, SK, 1985 Economic order quantity under conditions of permissible delay in payments Journal of the Operational Research Society 36, Huang, YF, 003 Optimal retailer s ordering policies in the EOQ model under trade credit financing Journal of the Operational Research Society 54, Huang, YF, 004 Optimal retailer s replenishment policy for the EPQ model under supplier s trade credit policy Production Planning & Control 15, 7 33 Huang, YF, 007 Economic order quantity under conditionally permissible delay in payments European Journal of Operational Research 176, Huang, YF, Chung, KJ, 003 Optimal replenishment payment policies in the EOQ model under cash discount

10 664 ARTICLE IN PRESS Y-F Huang, K-H Hsu / Int J Production Economics 11 (008) trade credit Asia-Pacific Journal of Operational Research 0, Hwang, H, Shinn, SW, 1997 Retailer s pricing lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments Computers Operations Research 4, Jamal, AMM, Sarker, BR, Wang, S, 1997 An ordering policy for deteriorating items with allowable shortages permissible delay in payment Journal of the Operational Research Society 48, Jamal, AMM, Sarker, BR, Wang, S, 000 Optimal payment time for a retailer under permitted delay of payment by the wholesaler International Journal of Production Economics 66, Liao, HC, Tsai, CH, Su, CT, 000 An inventory model with deteriorating items under inflation when a delay in payment is permissible International Journal of Production Economics 63, Salameh, MK, Abboud, NE, El-Kassar, AN, Ghattas, RE, 003 Continuous review inventory model with delay in payments International Journal of Production Economics 85, Sarker, BR, Jamal, AMM, Wang, S, 000a Supply chain model for perishable products under inflation permissible delay in payment Computers Operations Research 7, Sarker, BR, Jamal, AMM, Wang, S, 000b Optimal payment time under permissible delay in payment for products with deterioration Production Planning & Control 11, Shinn, SW, Hwang, H, 003 Optimal pricing ordering policies for retailers under order-size-dependent delay in payments Computers Operations Research 30, Teng, JT, 00 On the economic order quantity under conditions of permissible delay in payments Journal of the Operational Research Society 53,