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Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy
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European Journal of Operational Research 195 (2009) 358–363 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy Jinn-Tsair Teng a, Chun-Tao Chang b,* a
Department of Marketing and Management Sciences, The William Paterson University of New Jersey, Wayne, NJ 07470-2103, USA b Department of Statistics, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC Received 18 January 2007; accepted 2 February 2008 Available online 9 February 2008
Abstract In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Inventory; EPQ; Trade Credits; Permissible delay in payments
1. Introduction Goyal (1985) first developed an economic order quantity (EOQ) model under the conditions of permissible delay in payments. Aggarwal and Jaggi (1995) extended Goyal’s model to the case of deterioration. Jamal et al. (1997) analyzed Aggarwal and Jaggi’s model to allow for shortages. Teng (2002) amended Goyal’s model by considering the difference between unit price and unit cost and established an easy analytical closed-form solution to the problem. Chung and Huang (2003) proposed an economic production quantity (EPQ) inventory model for a retailer when the supplier offers a permissible delay in payments by assuming that the selling price is the same as the purchase cost. Huang (2003) extended Goyal’s model to develop an EOQ model in which supplier offers the retailer the permissible delay period M (i.e., the supplier trade credit), and the retailer in turn provides the trade credit period N (with N 6 M) to its customers (i.e., the retailer trade credit). *
Corresponding author. E-mail address: [email protected] (C.-T. Chang).
0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.02.001
Huang (2007) incorporated both Chung and Huang (2003) and Huang (2003) to investigate the optimal retailer’s replenishment decisions with two levels of trade credit policy in the EPQ framework. Jaggi et al. (2008) incorporated the concept of credit-linked demand and developed an inventory model under two levels of trade credit policy to determine the optimal credit as well as replenishment policy jointly for the retailer. Ho et al. (2008) formulated an integrated supplier–buyer inventory model with the assumption that the market demand is sensitive to the retail price and the supplier adopts a trade credit policy to determine the optimal pricing, shipment and payment policy. Lately, Chang et al. (in press) reviewed the contributions on the literature in modeling of inventory lot-sizing under trade credits. Huang (2007) proposed the optimal retailers replenishment decisions in the EPQ model under two levels of trade credit policy. He then developed the theoretical results. However, he ignored the fact that the retailer (or manufacturer) offers its customers a permissible delay period N, hence, the retailer receives its revenue from N to T + N, not from 0 to T as shown in Huang’s model formulations.
J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
In this paper, we not only extend his EPQ model to complement the above mentioned shortcoming but also relax some dispensable assumptions of N < M and others. In our view the permissible delay period offered by the retailer (N) is independent of the permissible delay period offered by the supplier (M) to the retailer. The retailer based on the prevalent market conditions must choose the appropriate value of N. In many situations retailers are forced to offer a permissible delay period to their customers while receiving no permissible delay period (M = 0) from their suppliers. We then propose the generalized formulation to the problem, and establish the theoretical results to obtain the optimal solution. Finally, a numerical example is given to illustrate the proposed model and its optimal solution. 2. Mathematical formulation The following notation will be adopted are the similar as those in Huang’s EPQ model under trade credit. D P
the demand rate per year the replenishment rate (i.e., production rate) per year, P P D A the ordering (or set-up) cost per order (lot) q 1 DP P 0, the fraction of no production c the unit purchasing price s the unit selling price, s P c h the unit stock holding cost per item per year excluding interest charges the interest earned per dollar per year Ie the interest charged per dollar in stocks per year Ik by the supplier M the manufacturer’s trade credit period offered by supplier in years N the customer’s trade credit period offered by manufacturer in years T the cycle time in years TVC(T) the annual total relevant cost, which is a function of T the optimal cycle time of TVC(T) T* the optimal lot size of TVC(T) Q*
359
Case 1. N 6 M The manufacturer buys all parts at time zero and must pay the purchasing cost at time M. Based on the values of M (i.e., the time at which the manufacturer must pay the supplier to avoid interest charge) and T + N (i.e., the time at which the manufacturer receives the payment from the last customer), we have two possible sub-cases. Sub-case 11: T + N P M and Sub-case 1-2: T + N < M. Now, let us discuss the detailed formulation in each sub-case. Sub-case 1-1. M 6 T + N In this sub-case, the manufacturer pays off all units sold by M N at time M, keeps the profits, and starts paying for the interest charges on the items sold after M N. The graphical representation of this sub-case is shown in Fig. 1. However, the manufacturer can not payoff the supplier by M because the supplier credit period M is shorter than the customer last payment time T + N. Hence, the manufacturer must finance all items sold after time M N at an interest charged Ik per dollar per year. The interest charged per cycle is cIk times the area of the triangle BCD shown in Fig. 1. Therefore, the interest charged per year is given by ( ) cI k D½T þ N M2 : ð1Þ T 2 Notice that Huang (2007) did not recognize that the last customer buys the product at time T, and pays the manufacturer at time T + N due to its customer trade credit period N. Consequently, he obtained the interest charged per year as " # 2 cI k qðDT 2 PM 2 Þ cI k DðT MÞ or ; T T 2 2 which is different from ours in (1). On the other hand, the manufacturer starts selling products at time 0, but getting the money at time N. Consequently, the manufacturer accumulates revenue in an account that earns Ie per dollar per year starting from N
Inventory level
Cumulative revenue C
Huang (2007) assumed that Ik P Ie and M P N. In this note, we will relax these two dispensable assumptions. The other assumptions are the same as those in Huang (2007). The annual total relevant cost consists of the following elements: 1. Annual ordering cost = TA 2. Annual stock holding cost = hDT2 q 3. According to assumptions (6) and (7) in Huang (2007), as well as the values of N and M, there are two cases (Case 1: N 6 M and Case 2: N P M) to occur in interest charged and interest earned per year.
B
D
0
N
M
T
Fig. 1. M 6 T + N.
T+N
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J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
through M. Therefore, the interest earned per cycle is sIe multiplied by the area of the triangle NMD as shown is Fig. 1. Hence, the interest earned per year is given by " # sI e DðM N Þ2 : ð2Þ T 2
Inventory level
Cumulative revenue
D
A
Imax
Notice that Huang (2007) ignored the fact that the manufacturer starts getting the revenue at time N, not at time 0 as shown in his model. As a result, he obtained the following annual interest earned as sI e DðM 2 N 2 Þ : 2T
M
Using the Eqs. (1) and (2), we yield the annual total relevant cost for the manufacturer as ( ) 2 A hDT q cI k D½T þ N M þ TVC1–1 ðT Þ ¼ þ T 2 T 2 " # 2 sI e DðM N Þ : ð3Þ T 2
N
T
T+N
Fig. 3. N P M.
Using the Eqs. (3) and (4), we obtain the annual total relevant cost for the manufacturer as TVC1–2 ðT Þ ¼
A hDT q þ T 2 sI e DT 2 þ DT ðM T N Þ : T 2
ð5Þ
Sub-case 1-2. M > T + N
Case 2. N P M
In this sub-case, the manufacturer receives the total revenue at time T + N, and is able to pay the supplier the total purchase cost at time M. Since the customer last payment time T + N is shorter than the supplier credit period M, the manufacturer faces no interest charged. The interest earned per cycle is sIe multiplied by the area of the trapezoid on the interval [N, M] as shown in Fig. 2. As a result, the interest earned per year is given by
In this case, the customer’s trade credit period N is equal to or larger than the supplier credit period M. Consequently, there is no interest earned for the manufacturer. In addition, the manufacturer must finance all items ordered at time M at an interest charged Ik per dollar per year, and start to payoff the loan after time N. Hence, the interest charged per cycle is cIk multiplied by the area of the trapezoid on the interval [M, T + N], as shown in Fig. 3. Therefore, the interest charged per year is given by
sI e DT 2 þ DT ðM T N Þ : T 2
ð4Þ
Notice that Huang (2007) did not recognize the same fact as Sub-case 1-1. Hence, he obtained the following two possible annual interests earned as sI e Dð2MT N 2 T 2 Þ 2T
and
sI e DðM N Þ:
I nventory level
cI k DT 2 ðN MÞDT þ : T 2
ð6Þ
Therefore, we obtain the annual total relevant cost for the manufacturer as A hDT q cI k DT 2 ðN MÞDT þ þ TVC2 ðT Þ ¼ þ : ð7Þ T 2 T 2
Cumulative revenue
C
D
3. Optimal solution To minimize the annual total relevant cost, taking the first-order and the second-order derivatives of TVC1–1(T), TVC1–2(T), and TVC2(T) with respect to T, we obtain " # 2 oTVC1–1 ðT Þ 1 DðM N Þ ðsI e cI k Þ ¼ 2 A oT 2 T Dðqh þ cI k Þ ; 2 2 o TVC1–1 ðT Þ 1 2 ¼ 3 ½2A DðM N Þ ðsI e cI k Þ; oT 2 T þ
N
T
Fig. 2. M > T + N.
T+N
M
ð8Þ ð9Þ
J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
oTVC1–2 ðT Þ A Dðqh þ sI e Þ ¼ 2 þ ; oT 2 T o2 TVC1–2 ðT Þ 2A ¼ 3 > 0; oT 2 T oTVC2 ðT Þ A Dðqh þ cI k Þ ¼ 2 þ ; oT 2 T
ð10Þ ð11Þ ð12Þ
and o2 TVC2 ðT Þ 2A ¼ 3 > 0: oT 2 T
ð13Þ
It is clear from (11) that TVC1–2(T) is a strictly convex function in T whenN 6 M. Consequently, we obtain the corresponding unique optimal cycle time T 1–2 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ T 1–2 ¼ 2A=½Dðqh þ sI e Þ: Therefore, the optimal lot size Q1–2 is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q1–2 ¼ T 1–2 D ¼ 2AD=ðqh þ sI e Þ:
ð15Þ
To ensure T 1–2 þ N 6 M, we substitute (14) into inequality T + N 6 M, and obtain that if and only if d1 2A ðqh þ sI e ÞDðM N Þ
2
6 0; then T 1–2 þ N 6 M:
ð16Þ
From (16), we know that if T is not in Sub-case 1-2, then M 6 T + N and d1 P 0. Thus, we know that if T is not in Sub-case 1-2, then 2
2
2A þ DðM N Þ ðcI k sI e Þ > 2A þ DðM N Þ ð0 sI e Þ 2
¼ 2A DðM N Þ ð0 þ sI e Þ > 2A DðM N Þ
2
ðsI e þ qhÞ ¼ d1 P 0;
ð17Þ
which implies o2 TVC1–1 ðT Þ 1 ¼ 3 ½2A þ DðM N Þ2 ðcI k sI e Þ > 0: 2 oT T
ð18Þ
Consequently, we know that TVC1–1(T) is also a strictly convex function in T. Likewise, we can easily obtain the unique optimal replenishment cycle time T 1–1 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T 1–1 ¼ ½2A þ DðM N Þ ðcI k sI e Þ=½Dðqh þ cI k Þ: ð19Þ Hence the optimal lot size Q1–1 is Q11 ¼ DT 11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ ½2AD þ D2 ðM N Þ ðcI k sI e Þ=ðqh þ cI k Þ:
361
Proof. It immediately follows from (14)–(17), (19) and (20). Note that Theorem 1 is not only an extension of Theorem 1 in Teng (2002), in which it assumes N = 0 and q = 1 (i.e.,P ? 1), but also a generalization of Theorem 1 in Teng and Goyal (2007), in which q = 1. In the classical EPQ model, both the manufacturer and the customer are assumed to pay for the parts or products as soon as they receive them. Hence, it is a special case of Case 1 with M = N = 0. Therefore, the classical optimal EPQ is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þ Q0 ¼ 2AD=ðqh þ cI k Þ: As a result, we can easily obtain the following theoretical result. Theorem 2. When N 6 M and 2A D(M N)2(sIe cIk) P 0, (A) if sIe < c Ik, then both Q1–1 and Q1–2 are larger than Q0 . (B) if sIe > c Ik, then both Q1–1 and Q1–2 are smaller than Q0 . (C) if sIe = c Ik, then Q1–1 ¼ Q1–2 ¼ Q0 . Proof. It immediately follows from (15), (20), and (21). Note that if 2A D(M N)2(sIe cIk) < 0, then from (20) we know that Q1–1 does not exist, and the rest of Theorem 2 is still true. Theorem 2 is not only a generalization of Theorem 2 in Teng (2002) but also an extension of Theorem 2 in Teng and Goyal (2007). Next, let us discuss the case in which N P M. When N P M, we know from (13) that TVC2(T) is a strictly convex function in T. Consequently we obtain the unique optimal cycle time T 2 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22Þ T 2 ¼ 2A=½Dðqh þ cI k Þ: Therefore, the optimal lot size Q2 is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q2 ¼ T 2 D ¼ 2AD=ðqh þ cI k Þ ¼ Q0 :
ð23Þ
Hence, the manufacturer’s optimal lot size is exactly the same as the classical economic production quantity when N P M. 4. Comparisons between Huang’s model and the proposed model
ð20Þ
Notice that if 2A D(M N)2(s Ie cIk) < 0, then both T 1–1 and Q1–1 do not exist. From the above arguments, we obtain the following results. Theorem 1. When N 6 M, (A) if d1 P 0, then T ¼ T 1–1 , and Q ¼ Q1–1 . (B) if d1 6 0, then T ¼ T 1–2 , and Q ¼ Q1–2 . (C) if d1 = 0, then T ¼ T 1–1 ¼ T 1–2 , and Q* = D(M N).
Both models are to investigate the optimal replenishment policies in the EPQ model under two levels of trade credit financing. Both models assume that the manufacturer buys and receives all parts at time zero and must pay the purchasing cost at time M, which is the time the manufacturer must pay the supplier in full to avoid interest charge. Since the manufacturer offers its customers the permissible delay of N periods, the manufacturer starts receiving its revenue at time N (i.e., the time at which the manufacturer receives the payment from the first customer). Based on the values
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J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
of N and M, there are two possible cases (Case 1: N 6 M and Case 2: N P M). In fact, Huang (2007) assumed that M is not shorter than N. Consequently, the comparison between these two models is only in the Case 1: N 6 M. From the values of M and T + N (i.e., the last payment time when the manufacturer receives the payment from the last customer), we have two possible sub-cases related to the interest charged and the interest earned per year. Sub-case 1-1: T + N P M and Sub-case 1-2: T + N < M. Sub-case 1-1: T + N P M. In this sub-case, we obtain the annual total relevant cost for the manufacturer as shown in the following equation: ( ) 2 A hDT q cI k D½T þ N M þ TVC1–1 ðT Þ ¼ þ T 2 T 2 " # 2 sI e DðM N Þ : ð3Þ T 2
Similarly, Huang’s model did not recognize the same fact as Sub-case 1-1. Hence, he obtained the annual interest earned as
Huang’s model did not recognize that the last customer buys the product at time T, and pays the manufacturer at time T + N due to its customer trade credit period N. Consequently, he obtained the interest charged per year as " # 2 cI k qðDT 2 PM 2 Þ cI k DðT MÞ or : T T 2 2
We consider the data as the same as Huang (2007): A = $150/order, D = 2500 units/year, c = $50/unit, s = $75/unit, h = $15/unit/year, Ik = 0.15/$/year, P = 3000 units/year, M = 0.1 year and N = 0.05 year. Since d1 2A (qh + sIe)D(M N)2 = 300 [(15/6) + 75Ie] (2500) (0.1 0.05)2 > 0 for Ie={0.08, 0.1, 0.15, 0.18}, we know from Theorem 1 that the optimal replenishment interval is T ¼ T 1–1 . Substituting the numerical values into (20), we obtain the optimal production quantity Q11 as shown in Table 1. From (21), we have the classical optimal economic production quantity (i.e., M = N = 0) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ Q0 ¼ 2AD=ðqh þ cI k Þ ¼ 273:8613:
In addition, Huang’s model ignored the fact that the manufacturer starts getting the revenue at time N, not at time 0 as shown in his model. As a result, he obtained the following annual interest earned as sI e DðM 2 N 2 Þ : 2T
sI e Dð2MT N 2 T 2 Þ or sI e DðM N Þ: 2T Therefore, his annual total relevant cost for the retailer is TVC3 ðT Þ ¼ or TVC4 ðT Þ ¼
5. Numerical example
Using Table 1 and (24), we obtain the following results: (1) If Ie=0.10, then sIe = cIk and Q1–1 ¼ 273:8613 ¼ Q0 . It means that the manufacturer’s optimal production quantity is exactly the same as the classical EPQ when the interest earned per unit sIe is equal to the interest charged per unit cIk. (2) If Ie = 0.08, then sIe < cIk and Q1–1 ¼ 278:1074 > 273:8613 ¼ Q0 . It implies that the manufacturer should product more than the classical EPQ when the interest earned per unit sIe is smaller than the interest charged per unit cIk. (3) If Ie = 0.15 (or 0.18), then sIe > cIk and Q1–1 ¼ 262:9460ðor256:1738Þ < 273:8613 ¼ Q0 . It reveals that if the interest earned per unit sIe is larger than the interest charged per unit cIk, then the manufac-
sI e DðM 2 N 2 Þ ; 2T
or " # A hDT q cI k DðT MÞ2 TVC2 ðT Þ ¼ þ þ T 2 T 2
sI e DðM 2 N 2 Þ : 2T
Notice that if N = 0, then TVC1–1(T) = TVC2(T). That is, his annual total relevant cost is the same as ours if N = 0. Sub-case 1-2: T + N < M In this sub-case, we know the annual total relevant cost for the manufacturer as shown in the following equation: A hDT q TVC1–2 ðT Þ ¼ þ T 2 sI e DT 2 þ DT ðM T N Þ : T 2
A hDT q þ sI e DðM N Þ: T 2
Notice that if N = 0, then TVC1–2(T) = TVC3(T). That is, his annual total relevant cost is the same as ours if N = 0.
Therefore, his annual total relevant cost for the retailer is A hDT q cI k qðDT 2 PM 2 Þ þ TVC1 ðT Þ ¼ þ T T 2 2
A hDT q sI e Dð2MT N 2 T 2 Þ þ ; T 2 2T
ð5Þ
Table 1 Computational results with respect to different values of Ie Ie
T*
0.08 0.10 0.15 0.18
T 1–1 T 1–1 T 1–1 T 1–1
Q* ¼ 0:111243 ¼ 0:109545 ¼ 0:105178 ¼ 0:102470
Q1–1 Q1–1 Q1–1 Q1–1
¼ 278:1074 ¼ 273:8613 ð¼ Q0 Þ ¼ 262:9460 ¼ 256:1738
J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
turer should product less quantity than the classical EPQ in order to take the benefits of the permissible delay more frequently. Acknowledgements The authors would like to thank three anonymous referees for their detailed and constructive comments. The principal author’s research was supported by the ART for Research and a Summer Research Funding from the William Paterson University of New Jersey. References Aggarwal, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payment. Journal of the Operational Research Society 46, 658–662. Chang, C.T., Teng, J.T., Goyal, S.K., in press. Inventory lot-size models under trade credits: A review. Asia Pacific Journal of Operational Research.. Chung, K.J., Huang, Y.F., 2003. The optimal cycle time for EPQ inventory model under permissible delay in payments. International Journal of Production Economics 84, 307–318.
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Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338. Ho, C.H., Ouyang, L.Y., Su, C.H., 2008. Optimal pricing, shipment and payment policy for an integrated supplier–buyer inventory model with two-part trade credit. European Journal of Operational Research 187, 496–510. Huang, Y.F., 2003. Optimal retailer’s ordering policies in the EOQ model under trade credit financing. Journal of the Operational Research Society 54, 1011–1015. Huang, Y.F., 2007. Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy. European Journal of Operational Research 176, 1577–1591. Jaggi, C.K., Goyal, S.K., Goel, S.K., 2008. Retailer’s optimal replenishment decisions with credit-linked demand under permissible delay in payments. European Journal of Operational Research 190, 130–135. Jamal, A.M.M., Sarker, B.R., Wang, S., 1997. An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. Journal of the Operational Research Society 48, 826–833. Teng, J.T., 2002. On the economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 53, 915–918. Teng, J.T., Goyal, S.K., 2007. Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits. Journal of the Operational Research Society 58, 1252–1255.
European Journal of Operational Research 195 (2009) 358–363 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy Jinn-Tsair Teng a, Chun-Tao Chang b,* a
Department of Marketing and Management Sciences, The William Paterson University of New Jersey, Wayne, NJ 07470-2103, USA b Department of Statistics, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC Received 18 January 2007; accepted 2 February 2008 Available online 9 February 2008
Abstract In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Inventory; EPQ; Trade Credits; Permissible delay in payments
1. Introduction Goyal (1985) first developed an economic order quantity (EOQ) model under the conditions of permissible delay in payments. Aggarwal and Jaggi (1995) extended Goyal’s model to the case of deterioration. Jamal et al. (1997) analyzed Aggarwal and Jaggi’s model to allow for shortages. Teng (2002) amended Goyal’s model by considering the difference between unit price and unit cost and established an easy analytical closed-form solution to the problem. Chung and Huang (2003) proposed an economic production quantity (EPQ) inventory model for a retailer when the supplier offers a permissible delay in payments by assuming that the selling price is the same as the purchase cost. Huang (2003) extended Goyal’s model to develop an EOQ model in which supplier offers the retailer the permissible delay period M (i.e., the supplier trade credit), and the retailer in turn provides the trade credit period N (with N 6 M) to its customers (i.e., the retailer trade credit). *
Corresponding author. E-mail address: [email protected] (C.-T. Chang).
0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.02.001
Huang (2007) incorporated both Chung and Huang (2003) and Huang (2003) to investigate the optimal retailer’s replenishment decisions with two levels of trade credit policy in the EPQ framework. Jaggi et al. (2008) incorporated the concept of credit-linked demand and developed an inventory model under two levels of trade credit policy to determine the optimal credit as well as replenishment policy jointly for the retailer. Ho et al. (2008) formulated an integrated supplier–buyer inventory model with the assumption that the market demand is sensitive to the retail price and the supplier adopts a trade credit policy to determine the optimal pricing, shipment and payment policy. Lately, Chang et al. (in press) reviewed the contributions on the literature in modeling of inventory lot-sizing under trade credits. Huang (2007) proposed the optimal retailers replenishment decisions in the EPQ model under two levels of trade credit policy. He then developed the theoretical results. However, he ignored the fact that the retailer (or manufacturer) offers its customers a permissible delay period N, hence, the retailer receives its revenue from N to T + N, not from 0 to T as shown in Huang’s model formulations.
J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
In this paper, we not only extend his EPQ model to complement the above mentioned shortcoming but also relax some dispensable assumptions of N < M and others. In our view the permissible delay period offered by the retailer (N) is independent of the permissible delay period offered by the supplier (M) to the retailer. The retailer based on the prevalent market conditions must choose the appropriate value of N. In many situations retailers are forced to offer a permissible delay period to their customers while receiving no permissible delay period (M = 0) from their suppliers. We then propose the generalized formulation to the problem, and establish the theoretical results to obtain the optimal solution. Finally, a numerical example is given to illustrate the proposed model and its optimal solution. 2. Mathematical formulation The following notation will be adopted are the similar as those in Huang’s EPQ model under trade credit. D P
the demand rate per year the replenishment rate (i.e., production rate) per year, P P D A the ordering (or set-up) cost per order (lot) q 1 DP P 0, the fraction of no production c the unit purchasing price s the unit selling price, s P c h the unit stock holding cost per item per year excluding interest charges the interest earned per dollar per year Ie the interest charged per dollar in stocks per year Ik by the supplier M the manufacturer’s trade credit period offered by supplier in years N the customer’s trade credit period offered by manufacturer in years T the cycle time in years TVC(T) the annual total relevant cost, which is a function of T the optimal cycle time of TVC(T) T* the optimal lot size of TVC(T) Q*
359
Case 1. N 6 M The manufacturer buys all parts at time zero and must pay the purchasing cost at time M. Based on the values of M (i.e., the time at which the manufacturer must pay the supplier to avoid interest charge) and T + N (i.e., the time at which the manufacturer receives the payment from the last customer), we have two possible sub-cases. Sub-case 11: T + N P M and Sub-case 1-2: T + N < M. Now, let us discuss the detailed formulation in each sub-case. Sub-case 1-1. M 6 T + N In this sub-case, the manufacturer pays off all units sold by M N at time M, keeps the profits, and starts paying for the interest charges on the items sold after M N. The graphical representation of this sub-case is shown in Fig. 1. However, the manufacturer can not payoff the supplier by M because the supplier credit period M is shorter than the customer last payment time T + N. Hence, the manufacturer must finance all items sold after time M N at an interest charged Ik per dollar per year. The interest charged per cycle is cIk times the area of the triangle BCD shown in Fig. 1. Therefore, the interest charged per year is given by ( ) cI k D½T þ N M2 : ð1Þ T 2 Notice that Huang (2007) did not recognize that the last customer buys the product at time T, and pays the manufacturer at time T + N due to its customer trade credit period N. Consequently, he obtained the interest charged per year as " # 2 cI k qðDT 2 PM 2 Þ cI k DðT MÞ or ; T T 2 2 which is different from ours in (1). On the other hand, the manufacturer starts selling products at time 0, but getting the money at time N. Consequently, the manufacturer accumulates revenue in an account that earns Ie per dollar per year starting from N
Inventory level
Cumulative revenue C
Huang (2007) assumed that Ik P Ie and M P N. In this note, we will relax these two dispensable assumptions. The other assumptions are the same as those in Huang (2007). The annual total relevant cost consists of the following elements: 1. Annual ordering cost = TA 2. Annual stock holding cost = hDT2 q 3. According to assumptions (6) and (7) in Huang (2007), as well as the values of N and M, there are two cases (Case 1: N 6 M and Case 2: N P M) to occur in interest charged and interest earned per year.
B
D
0
N
M
T
Fig. 1. M 6 T + N.
T+N
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through M. Therefore, the interest earned per cycle is sIe multiplied by the area of the triangle NMD as shown is Fig. 1. Hence, the interest earned per year is given by " # sI e DðM N Þ2 : ð2Þ T 2
Inventory level
Cumulative revenue
D
A
Imax
Notice that Huang (2007) ignored the fact that the manufacturer starts getting the revenue at time N, not at time 0 as shown in his model. As a result, he obtained the following annual interest earned as sI e DðM 2 N 2 Þ : 2T
M
Using the Eqs. (1) and (2), we yield the annual total relevant cost for the manufacturer as ( ) 2 A hDT q cI k D½T þ N M þ TVC1–1 ðT Þ ¼ þ T 2 T 2 " # 2 sI e DðM N Þ : ð3Þ T 2
N
T
T+N
Fig. 3. N P M.
Using the Eqs. (3) and (4), we obtain the annual total relevant cost for the manufacturer as TVC1–2 ðT Þ ¼
A hDT q þ T 2 sI e DT 2 þ DT ðM T N Þ : T 2
ð5Þ
Sub-case 1-2. M > T + N
Case 2. N P M
In this sub-case, the manufacturer receives the total revenue at time T + N, and is able to pay the supplier the total purchase cost at time M. Since the customer last payment time T + N is shorter than the supplier credit period M, the manufacturer faces no interest charged. The interest earned per cycle is sIe multiplied by the area of the trapezoid on the interval [N, M] as shown in Fig. 2. As a result, the interest earned per year is given by
In this case, the customer’s trade credit period N is equal to or larger than the supplier credit period M. Consequently, there is no interest earned for the manufacturer. In addition, the manufacturer must finance all items ordered at time M at an interest charged Ik per dollar per year, and start to payoff the loan after time N. Hence, the interest charged per cycle is cIk multiplied by the area of the trapezoid on the interval [M, T + N], as shown in Fig. 3. Therefore, the interest charged per year is given by
sI e DT 2 þ DT ðM T N Þ : T 2
ð4Þ
Notice that Huang (2007) did not recognize the same fact as Sub-case 1-1. Hence, he obtained the following two possible annual interests earned as sI e Dð2MT N 2 T 2 Þ 2T
and
sI e DðM N Þ:
I nventory level
cI k DT 2 ðN MÞDT þ : T 2
ð6Þ
Therefore, we obtain the annual total relevant cost for the manufacturer as A hDT q cI k DT 2 ðN MÞDT þ þ TVC2 ðT Þ ¼ þ : ð7Þ T 2 T 2
Cumulative revenue
C
D
3. Optimal solution To minimize the annual total relevant cost, taking the first-order and the second-order derivatives of TVC1–1(T), TVC1–2(T), and TVC2(T) with respect to T, we obtain " # 2 oTVC1–1 ðT Þ 1 DðM N Þ ðsI e cI k Þ ¼ 2 A oT 2 T Dðqh þ cI k Þ ; 2 2 o TVC1–1 ðT Þ 1 2 ¼ 3 ½2A DðM N Þ ðsI e cI k Þ; oT 2 T þ
N
T
Fig. 2. M > T + N.
T+N
M
ð8Þ ð9Þ
J.-T. Teng, C.-T. Chang / European Journal of Operational Research 195 (2009) 358–363
oTVC1–2 ðT Þ A Dðqh þ sI e Þ ¼ 2 þ ; oT 2 T o2 TVC1–2 ðT Þ 2A ¼ 3 > 0; oT 2 T oTVC2 ðT Þ A Dðqh þ cI k Þ ¼ 2 þ ; oT 2 T
ð10Þ ð11Þ ð12Þ
and o2 TVC2 ðT Þ 2A ¼ 3 > 0: oT 2 T
ð13Þ
It is clear from (11) that TVC1–2(T) is a strictly convex function in T whenN 6 M. Consequently, we obtain the corresponding unique optimal cycle time T 1–2 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ T 1–2 ¼ 2A=½Dðqh þ sI e Þ: Therefore, the optimal lot size Q1–2 is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q1–2 ¼ T 1–2 D ¼ 2AD=ðqh þ sI e Þ:
ð15Þ
To ensure T 1–2 þ N 6 M, we substitute (14) into inequality T + N 6 M, and obtain that if and only if d1 2A ðqh þ sI e ÞDðM N Þ
2
6 0; then T 1–2 þ N 6 M:
ð16Þ
From (16), we know that if T is not in Sub-case 1-2, then M 6 T + N and d1 P 0. Thus, we know that if T is not in Sub-case 1-2, then 2
2
2A þ DðM N Þ ðcI k sI e Þ > 2A þ DðM N Þ ð0 sI e Þ 2
¼ 2A DðM N Þ ð0 þ sI e Þ > 2A DðM N Þ
2
ðsI e þ qhÞ ¼ d1 P 0;
ð17Þ
which implies o2 TVC1–1 ðT Þ 1 ¼ 3 ½2A þ DðM N Þ2 ðcI k sI e Þ > 0: 2 oT T
ð18Þ
Consequently, we know that TVC1–1(T) is also a strictly convex function in T. Likewise, we can easily obtain the unique optimal replenishment cycle time T 1–1 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T 1–1 ¼ ½2A þ DðM N Þ ðcI k sI e Þ=½Dðqh þ cI k Þ: ð19Þ Hence the optimal lot size Q1–1 is Q11 ¼ DT 11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ ½2AD þ D2 ðM N Þ ðcI k sI e Þ=ðqh þ cI k Þ:
361
Proof. It immediately follows from (14)–(17), (19) and (20). Note that Theorem 1 is not only an extension of Theorem 1 in Teng (2002), in which it assumes N = 0 and q = 1 (i.e.,P ? 1), but also a generalization of Theorem 1 in Teng and Goyal (2007), in which q = 1. In the classical EPQ model, both the manufacturer and the customer are assumed to pay for the parts or products as soon as they receive them. Hence, it is a special case of Case 1 with M = N = 0. Therefore, the classical optimal EPQ is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þ Q0 ¼ 2AD=ðqh þ cI k Þ: As a result, we can easily obtain the following theoretical result. Theorem 2. When N 6 M and 2A D(M N)2(sIe cIk) P 0, (A) if sIe < c Ik, then both Q1–1 and Q1–2 are larger than Q0 . (B) if sIe > c Ik, then both Q1–1 and Q1–2 are smaller than Q0 . (C) if sIe = c Ik, then Q1–1 ¼ Q1–2 ¼ Q0 . Proof. It immediately follows from (15), (20), and (21). Note that if 2A D(M N)2(sIe cIk) < 0, then from (20) we know that Q1–1 does not exist, and the rest of Theorem 2 is still true. Theorem 2 is not only a generalization of Theorem 2 in Teng (2002) but also an extension of Theorem 2 in Teng and Goyal (2007). Next, let us discuss the case in which N P M. When N P M, we know from (13) that TVC2(T) is a strictly convex function in T. Consequently we obtain the unique optimal cycle time T 2 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22Þ T 2 ¼ 2A=½Dðqh þ cI k Þ: Therefore, the optimal lot size Q2 is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q2 ¼ T 2 D ¼ 2AD=ðqh þ cI k Þ ¼ Q0 :
ð23Þ
Hence, the manufacturer’s optimal lot size is exactly the same as the classical economic production quantity when N P M. 4. Comparisons between Huang’s model and the proposed model
ð20Þ
Notice that if 2A D(M N)2(s Ie cIk) < 0, then both T 1–1 and Q1–1 do not exist. From the above arguments, we obtain the following results. Theorem 1. When N 6 M, (A) if d1 P 0, then T ¼ T 1–1 , and Q ¼ Q1–1 . (B) if d1 6 0, then T ¼ T 1–2 , and Q ¼ Q1–2 . (C) if d1 = 0, then T ¼ T 1–1 ¼ T 1–2 , and Q* = D(M N).
Both models are to investigate the optimal replenishment policies in the EPQ model under two levels of trade credit financing. Both models assume that the manufacturer buys and receives all parts at time zero and must pay the purchasing cost at time M, which is the time the manufacturer must pay the supplier in full to avoid interest charge. Since the manufacturer offers its customers the permissible delay of N periods, the manufacturer starts receiving its revenue at time N (i.e., the time at which the manufacturer receives the payment from the first customer). Based on the values
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of N and M, there are two possible cases (Case 1: N 6 M and Case 2: N P M). In fact, Huang (2007) assumed that M is not shorter than N. Consequently, the comparison between these two models is only in the Case 1: N 6 M. From the values of M and T + N (i.e., the last payment time when the manufacturer receives the payment from the last customer), we have two possible sub-cases related to the interest charged and the interest earned per year. Sub-case 1-1: T + N P M and Sub-case 1-2: T + N < M. Sub-case 1-1: T + N P M. In this sub-case, we obtain the annual total relevant cost for the manufacturer as shown in the following equation: ( ) 2 A hDT q cI k D½T þ N M þ TVC1–1 ðT Þ ¼ þ T 2 T 2 " # 2 sI e DðM N Þ : ð3Þ T 2
Similarly, Huang’s model did not recognize the same fact as Sub-case 1-1. Hence, he obtained the annual interest earned as
Huang’s model did not recognize that the last customer buys the product at time T, and pays the manufacturer at time T + N due to its customer trade credit period N. Consequently, he obtained the interest charged per year as " # 2 cI k qðDT 2 PM 2 Þ cI k DðT MÞ or : T T 2 2
We consider the data as the same as Huang (2007): A = $150/order, D = 2500 units/year, c = $50/unit, s = $75/unit, h = $15/unit/year, Ik = 0.15/$/year, P = 3000 units/year, M = 0.1 year and N = 0.05 year. Since d1 2A (qh + sIe)D(M N)2 = 300 [(15/6) + 75Ie] (2500) (0.1 0.05)2 > 0 for Ie={0.08, 0.1, 0.15, 0.18}, we know from Theorem 1 that the optimal replenishment interval is T ¼ T 1–1 . Substituting the numerical values into (20), we obtain the optimal production quantity Q11 as shown in Table 1. From (21), we have the classical optimal economic production quantity (i.e., M = N = 0) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ Q0 ¼ 2AD=ðqh þ cI k Þ ¼ 273:8613:
In addition, Huang’s model ignored the fact that the manufacturer starts getting the revenue at time N, not at time 0 as shown in his model. As a result, he obtained the following annual interest earned as sI e DðM 2 N 2 Þ : 2T
sI e Dð2MT N 2 T 2 Þ or sI e DðM N Þ: 2T Therefore, his annual total relevant cost for the retailer is TVC3 ðT Þ ¼ or TVC4 ðT Þ ¼
5. Numerical example
Using Table 1 and (24), we obtain the following results: (1) If Ie=0.10, then sIe = cIk and Q1–1 ¼ 273:8613 ¼ Q0 . It means that the manufacturer’s optimal production quantity is exactly the same as the classical EPQ when the interest earned per unit sIe is equal to the interest charged per unit cIk. (2) If Ie = 0.08, then sIe < cIk and Q1–1 ¼ 278:1074 > 273:8613 ¼ Q0 . It implies that the manufacturer should product more than the classical EPQ when the interest earned per unit sIe is smaller than the interest charged per unit cIk. (3) If Ie = 0.15 (or 0.18), then sIe > cIk and Q1–1 ¼ 262:9460ðor256:1738Þ < 273:8613 ¼ Q0 . It reveals that if the interest earned per unit sIe is larger than the interest charged per unit cIk, then the manufac-
sI e DðM 2 N 2 Þ ; 2T
or " # A hDT q cI k DðT MÞ2 TVC2 ðT Þ ¼ þ þ T 2 T 2
sI e DðM 2 N 2 Þ : 2T
Notice that if N = 0, then TVC1–1(T) = TVC2(T). That is, his annual total relevant cost is the same as ours if N = 0. Sub-case 1-2: T + N < M In this sub-case, we know the annual total relevant cost for the manufacturer as shown in the following equation: A hDT q TVC1–2 ðT Þ ¼ þ T 2 sI e DT 2 þ DT ðM T N Þ : T 2
A hDT q þ sI e DðM N Þ: T 2
Notice that if N = 0, then TVC1–2(T) = TVC3(T). That is, his annual total relevant cost is the same as ours if N = 0.
Therefore, his annual total relevant cost for the retailer is A hDT q cI k qðDT 2 PM 2 Þ þ TVC1 ðT Þ ¼ þ T T 2 2
A hDT q sI e Dð2MT N 2 T 2 Þ þ ; T 2 2T
ð5Þ
Table 1 Computational results with respect to different values of Ie Ie
T*
0.08 0.10 0.15 0.18
T 1–1 T 1–1 T 1–1 T 1–1
Q* ¼ 0:111243 ¼ 0:109545 ¼ 0:105178 ¼ 0:102470
Q1–1 Q1–1 Q1–1 Q1–1
¼ 278:1074 ¼ 273:8613 ð¼ Q0 Þ ¼ 262:9460 ¼ 256:1738
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turer should product less quantity than the classical EPQ in order to take the benefits of the permissible delay more frequently. Acknowledgements The authors would like to thank three anonymous referees for their detailed and constructive comments. The principal author’s research was supported by the ART for Research and a Summer Research Funding from the William Paterson University of New Jersey. References Aggarwal, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payment. Journal of the Operational Research Society 46, 658–662. Chang, C.T., Teng, J.T., Goyal, S.K., in press. Inventory lot-size models under trade credits: A review. Asia Pacific Journal of Operational Research.. Chung, K.J., Huang, Y.F., 2003. The optimal cycle time for EPQ inventory model under permissible delay in payments. International Journal of Production Economics 84, 307–318.
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Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338. Ho, C.H., Ouyang, L.Y., Su, C.H., 2008. Optimal pricing, shipment and payment policy for an integrated supplier–buyer inventory model with two-part trade credit. European Journal of Operational Research 187, 496–510. Huang, Y.F., 2003. Optimal retailer’s ordering policies in the EOQ model under trade credit financing. Journal of the Operational Research Society 54, 1011–1015. Huang, Y.F., 2007. Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy. European Journal of Operational Research 176, 1577–1591. Jaggi, C.K., Goyal, S.K., Goel, S.K., 2008. Retailer’s optimal replenishment decisions with credit-linked demand under permissible delay in payments. European Journal of Operational Research 190, 130–135. Jamal, A.M.M., Sarker, B.R., Wang, S., 1997. An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. Journal of the Operational Research Society 48, 826–833. Teng, J.T., 2002. On the economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 53, 915–918. Teng, J.T., Goyal, S.K., 2007. Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits. Journal of the Operational Research Society 58, 1252–1255.