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Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers
ARTICLE IN PRESS Int. J. Production Economics 119 (2009) 415–423
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Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers Jinn-Tsair Teng Department of Marketing and Management Sciences, The William Paterson University of New Jersey, Wayne, NJ 07470-2103, USA
a r t i c l e in fo
abstract
Article history: Received 23 May 2008 Accepted 6 March 2009 Available online 18 April 2009
In practice, to reduce default risks, a retailer frequently offers its bad credit customers a partial trade credit, in which the retailer requests its customers to pay a portion of the purchase amount at the time of placing an order as a collateral deposit, and then grants a permissible delay on the rest of the purchase amount. By contrast, the retailer usually provides a full trade credit to its good credit customers without the collateral deposits. For generality, in this paper, I establish an economic order quantity (EOQ) model for a retailer who receives a full trade credit by its supplier, and offers either a partial or a full trade credit to its customers. The proposed model is in a general framework that includes numerous previous models as special cases. I then analyze the characteristics of the optimal solution, and provide an easy-to-use closed-form optimal solution. Finally, I use a real-world inventory problem to illustrate the proposed model and its optimal solution. Published by Elsevier B.V.
Keywords: Inventory EOQ Partial trade credit Permissible delay in payments
1. Introduction In the classical inventory economic order quantity (or EOQ) model, it was assumed that the retailer must pay for the items as soon as receiving them. In practice, a supplier frequently offers a retailer a delay of payment (i.e., an upstream trade credit) for settling the amount owed to him. Usually, there is no interest charge if the outstanding amount is paid within the permissible delay period. However, if the payment is not paid in full by the end of the permissible delay period, then interest is charged on the outstanding amount. Therefore, it is clear that a retailer will delay the payment up to the last moment of the permissible period allowed by the supplier. The permissible delay in payments produces two benefits to the supplier: (1) it attracts new buyers who consider it to be a type of price reduction, and (2) it may be applied as an alternative to price discount because it does not
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provoke competitors to reduce their prices and thus introduce lasting price reductions. On the other hand, the policy of granting credit terms adds not only an additional cost but also an additional dimension of default risk to the supplier. Similarly, a retailer may offer its customers a permissible delay (i.e., a down-stream trade credit) to settle the outstanding balance. Goyal (1985) developed an EOQ model under conditions of permissible delay in payments. He ignored the difference between the selling price and the purchase cost. Although Dave (1985) corrected Goyal’s model by addressing the fact that the selling price is necessarily higher than its purchase cost, his viewpoint did not draw much attention to the recent researchers. Aggarwal and Jaggi (1995) extended Goyal’s model to consider the deteriorating items. Jamal et al. (1997) further generalized Aggarwal and Jaggi’s model to allow for shortages. Hwang and Shinn (1997) added the pricing strategy to the model, and developed the optimal price and lot sizing for a retailer under the condition of permissible delay in payments. In contrast to all above models, Jamal et al. (2000) and Sarker et al. (2000a) obtained the optimal
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cycle and payment times for a retailer when a supplier offers a specified credit period for payment without penalty. Teng (2002) amended Goyal’s model by considering the difference between unit price and unit cost, and found that it makes economic sense for a well-established buyer to order less quantity and take the benefits of the permissible delay more frequently. Chang et al. (2003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Chung and Huang (2003) developed an EPQ inventory model for a retailer when the supplier offers a permissible delay in payments. Huang (2003) extended Goyal’s model to develop an EOQ model in which the supplier offers the retailer the permissible delay period M, and the retailer in turn provides the trade credit period N (with NpM) to his/her customers. Recently, Teng and Goyal (2007) complemented the shortcoming of Huang’s model and proposed a generalized formulation. Many related articles can be found in Chang and Teng (2004), Chang and Dye (2001), Chung and Liao (2004), Goyal et al. (2007), Huang (2004, 2007), Huang and Hsu (2008), Liao et al. (2000), Khouja and Mehrez (1996), Ouyang et al. (2005, 2006), Sarker et al. (2000b), Shinn and Hwang (2003), Teng and Chang (2009), Teng et al. (2005, 2006, 2007), and their references. To reduce default risks, in practice, a retailer frequently offers a partial down-stream trade credit to its credit risk customers who must pay a portion of the purchase amount at the time of placing an order as a collateral deposit, and then receive a permissible delay on the rest of the outstanding amount. In contrast, the retailer usually provides its good credit customers a full trade credit without the collateral deposit. In this paper, I first establish an EOQ model for a retailer who receives a full trade credit by its supplier, and offers either a partial trade credit to its bad credit customers or a full trade credit to its good credit customers. My proposed model is in a general framework that includes numerous previous models such as Goyal (1985), Teng (2002), Huang (2003), Teng and Goyal (2007), and others as special cases. In addition, the proposed model provides the optimal ordering policies for a retailer not only to its bad credit customers but also to its good credit customers. I then derive the theoretical results of the optimal solution, and provide an easy-to-use closed-form optimal solution. Finally, I use a real-world inventory problem to illustrate the proposed model and its optimal solution. 2. Notation and assumptions The following notation is used throughout the entire paper: D the annual demand rate A the ordering cost per order c the purchasing cost per unit p the selling price per unit, with p4c h the unit holding cost per year excluding interest charge the interest earned per dollar per year Ie Ic the interest charged per dollar per year a the fraction of the purchase cost in which the customer must pay the retailer at the time of placing an order, with 0pap1 the portion of the purchase cost in which the retailer offers its 1a customer a permissible delay of N periods. M the retailer’s trade credit period in years offered by the supplier
N T Q TRCðTÞ T Q
the customer’s trade credit period in years offered by the retailer the replenishment cycle time in years the order quantity the annual total relevant cost, which is a function of T the optimal replenishment cycle time of TRC(T) the optimal order quantity ¼ DT*
In addition, the mathematical models proposed in this paper are based on the following assumptions: (1) At the maturity phase of a product life cycle, the annual demand rate is known and constant. In this paper, I deal with good and bad credit customers separately. To good credit customers, set a ¼ 0, and D to be the annual demand for good credit customers. For credit risk customers, set a40, and D to be the annual demand for bad credit customers. (2) In today’s time-based competition, we assume WLOG that shortages are not allowed. (3) Time horizon is infinite, and replenishments are instantaneous. (4) The supplier offers the retailer a full trade credit of M periods to settle the entire purchase cost. Meanwhile, the retailer also provides a partial trade credit to its credit-risk customers who must pay a portion of the purchase amount at the time of placing an order as a collateral deposit, and then receive a permissible delay of N periods on the outstanding amount. (5) If MXN, then the retailer deposits the sales revenue into an interest bearing account. If MXT+N (i.e., the permissible delay period is longer than the time at which the retailer receives the last payment from its customers), then the retailer receives all revenue and pays off the entire purchase cost at the end of the permissible delay M. Otherwise (if MpT+N), the retailer pays the supplier the sum of all units sold by MN and the collateral deposit received from N to M, keeps the profit for the use of the other activities, and starts paying for the interest charges on the items sold after MN. (6) If NXM, then the retailer finances and pays the supplier the entire amount of the delayed payment (1a) pDT at the end of the trade credit M, and then pays down the load after time N at which the retailer starts to receive sales revenue from its customers. For the collateral deposit, the retailer deposits the sales revenue into an interest bearing account until the end of the permissible delay M. If TXM, then the retailer pays the supplier all units sold by M, keeps the profit for the use of the other activities, and starts paying for the interest charges on the items sold after M.
3. Mathematical models From the values of N and M, we have two potential cases: (1)NpM and (2) NXM. Case 1. NpM Based on the values of M (i.e., the time at which the retailer must pay the supplier to avoid interest charge), T (i.e., the replenishment cycle time), and T+N (i.e., the time
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at which the retailer receives the payment from the last customer), we have the following three possible subcases: (i) MpT, (ii) TpMpT+N, and (iii) TpT+NpM. Now, let us discuss the detailed formulation in each sub-case. Sub-case 1.1. MpT By the time M, the retailer has two resources to accumulate revenue in an account that earns Ie per dollar per year: (1) from the portion of immediate payment (starting 0 through M), and (2) from the portion of delayed payment (starting N through M). Therefore, the interest earned per cycle is Ie multiplied by the total area of the triangle OMA and the triangle NMA’ as shown in Fig. 1. Hence, the interest earned per year is pIe D ½aM 2 þ ð1 aÞðM NÞ2 . 2T
(1)
If MpT, then the retailer pays acDM+(1a)cD(MN) to the supplier at time M, keeps its profits, and starts paying for the interest charged after M. The retailer must finance (1) all items sold after M for the portion of immediate payment and (2) all items sold after MN for the portion of delayed payment at an interest charged Ic per dollar per year. As a result, the interest charged per cycle is (c/p)Ic times the total area of the triangle ABC and the triangle A0 B0 C0 as shown in Fig. 1. Therefore, the interest charged per year is given by cIc D ½aðT MÞ2 þ ð1 aÞðT þ N MÞ2 . 2T
(2)
417
Since the annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2, we obtain the annual total relevant cost for the retailer as TRC 1 ðTÞ ¼
A hDT cIc D þ þ ½aðT MÞ2 T 2 2T þ ð1 aÞðT þ N MÞ2 pIe D ½aM2 þ ð1 aÞðM NÞ2 . 2T
(3)
Sub-case 1.2. TpMpT+N Again, before settling the account with the supplier at time M, the retailer has two resources to accumulate revenue in an account that earns Ie per dollar per year: (1) from the portion of immediate payment (starting 0 through M) and (2) from the portion of delayed payment (starting N through M). Therefore, the interest earned per cycle is Ie multiplied by the total area of the trapezoid on the interval [0, M] and the triangle NMA0 as shown in Fig. 2. As a result, the interest earned per year is pIe D ½aT 2 þ 2aTðM TÞ þ ð1 aÞðM NÞ2 . 2T
(4)
Since TpM, the retailer has acDT to pay the supplier for the portion of immediate payment at time M. However, if MpT+N, then the retailer cannot payoff the supplier by M because M is shorter than the customer last payment time t ¼ T+N. Consequently, the retailer must finance all items sold after MN at an interest charged Ic per dollar per year.
Fig. 1. MXN and MpT.
Fig. 2. MXN and TpMpT+N.
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As a result, the interest charged per cycle is (c/p)Ic times the area of the triangle A0 B0 C0 as shown in Fig. 2. Therefore, the interest charged per year is
Since the annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2, we obtain the annual total relevant cost for the retailer as
cIc D ð1 aÞðT þ N MÞ2 . 2T
TRC 3 ðTÞ ¼
(5)
As we know, the annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2. Therefore, we obtain the annual total relevant cost for the retailer as TRC 2 ðTÞ ¼
A hDT cIc D þ þ ð1 aÞðT þ N MÞ2 T 2 2T pIe D ½aT 2 þ 2aTðM TÞ þ ð1 aÞðM NÞ2 . (6) 2T
Sub-case 1.3. TpT+NpM In this sub-case, the retailer receives the total revenue at time T+N, and is able to pay the supplier the total purchase cost at time M. Consequently, there are no interest charges while the interest earned per cycle is Ie multiplied by the total area of the trapezoid on the interval [0, M] and the trapezoid on the interval [N, M] as shown in Fig. 3. As a result, the annual interest earned is pIe D ½aT 2 þ 2aTðM TÞ þ ð1 aÞT 2 2T þ 2ð1 aÞTðM T NÞ pIe D ¼ ½T þ 2aðM TÞ þ 2ð1 aÞðM T NÞ. 2
(7)
A hDT þ T 2 pIe D ½T þ 2aðM TÞ 2 þ 2ð1 aÞðM T NÞ.
(8)
It is obvious from (3), (6), and (8) that TRC1(M) ¼ TRC2(M) and TRC2(MN) ¼ TRC3(MN). Case 2. NXM Based on the values of M and T, we have the following two possible sub-cases: (i) MpT and (ii) TpM. Next, let us discuss the detailed formulation in each sub-case. Sub-case 2.1. MpT By settling the account with the supplier at time M, the retailer accumulates the collateral deposit in an account that earns Ie per dollar per year. Therefore, the interest earned per cycle is Ie multiplied by the area of the triangle OMA as shown in Fig. 4. As a result, the interest earned per year is pIe D aM2 . 2T
(9)
Since NXM, the retailer must finance and pay the supplier acD(TM) for the items sold after M plus the entire amount of the delayed payment (1a)cDT at the end of the trade credit M, and then pay off the load
Fig. 3. MXN and TpT+NpM.
Fig. 4. NXM and MpT.
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after time N at which the retailer starts to receive sales revenue from its customers. Consequently, the interest charged per cycle is (c/p)Ic times the area of the trapezoid on the interval [M, T+N] as shown in Fig. 4. Therefore, the interest charged per year is cIc D faðT MÞ2 þ ð1 aÞT½T þ 2ðN MÞg. 2T
(10)
The annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2. Therefore, we obtain the annual total relevant cost for the retailer as TRC 4 ðTÞ ¼
A hDT cIc D aðT MÞ2 þ þ T 2 2T cIc D pIe D þ aM 2 . ð1 aÞ½T þ 2ðN MÞ 2 2T
4. Optimal order quantity Let us discuss the first case of NpM, and then the case of NXM. To minimize the annual total relevant cost, taking the first-order and the second-order derivatives of TRC1(T), TRC2(T), and TRC3(T) with respect to T, we obtain ( ) dTRC 1 ðTÞ 1 D½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe Þ ¼ 2 Aþ dT 2 T þ 2
d TRC 1 ðTÞ dT 2
(11)
¼
Dðh þ cIc Þ , 2
1 f2A þ D½aM2 T3 þ ð1 aÞðM NÞ2 ðcIc pIe Þg,
Sub-case 2.2. TpM During [0, M], the retailer accumulates the collateral deposit in an account that earns Ie per dollar per year. Therefore, the interest earned per cycle is Ie multiplied by the area of the trapezoid on the interval [0, M] as shown in Fig. 5. Hence, the interest earned per year is
" # dTRC 2 ðTÞ 1 Dð1 aÞðM NÞ2 ðcIc pIe Þ ¼ 2 Aþ dT 2 T
apIe D
d TRC 2 ðTÞ
2
þ 2
½T þ 2ðM TÞ.
(12)
Similar to sub-case 2.1, if NXM, then the retailer must finance and pay the supplier the entire amount of the delayed payment (1a)cDT at the end of the trade credit M, and then pay off the load after time N at which the retailer starts to receive sales revenue from its customers. Consequently, the interest charged per cycle is (c/p)Ic times the area of the trapezoid on the interval [M, T+N] as shown in Fig. 4. Therefore, the interest charged per year is cIc D ð1 aÞ½T þ 2ðN MÞ. 2
(13)
Therefore, we obtain the annual total relevant cost for the retailer as TRC 5 ðTÞ ¼
A hDT cIc D þ þ ð1 aÞ½T þ 2ðN MÞ T 2 2 apIe D ½T þ 2ðM TÞ. 2
From (11) and (14), we obtain TRC5(M) ¼ TRC4(M).
(14)
419
dT 2
¼
D½h þ ð1 aÞcIc þ apIe , 2
1 T3
½2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ,
dTRC 3 ðTÞ A Dðh þ pIe Þ ¼ 2 þ , dT 2 T
(15)
(16)
(17)
(18)
(19)
and 2
d TRC 3 ðTÞ dT 2
¼
2A T3
40.
(20)
It is clear from (20) that TRC 3 ðTÞ is a strictly convex function in T. Consequently we obtain the corresponding unique optimal cycle time T 3 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 ¼ 2A=½Dðh þ pIe Þ. (21) Therefore, the optimal order quantity Q 3 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 3 ¼ T 3 D ¼ 2AD=ðh þ pIe Þ.
(22)
T 3 +NpM,
To ensure we substitute (21) into inequality T+NpM, and obtain that if and only if
D1 2A ðh þ pIe ÞDðM NÞ2 p0; then T 3 þ NpM.
Fig. 5. NXM and TpM.
(23)
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If 2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ40, then we get 2
d TRC 2 ðTÞ dT 2
¼
1 T3
Theorem 1. For NpM,
½2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ40. (24)
Hence, we know that TRC 2 ðTÞ is also a strictly convex function in T. Likewise, we can easily obtain the unique optimal replenishment cycle time T 2 as T 2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ=fD½h þ ð1 aÞcIc þ apIe g.
(A) if D2 X0, then T ¼ T 1 , and Q ¼ Q 1 . (B) if D1 X0, and D2 p0, then T ¼ T 2 , and Q ¼ Q 2 . (C) if D1 p0, then T ¼ T 3 , and Q ¼ Q 3 . Proof. If D2 40, then D1 40, ( ) dTRC 2 ðTÞ 1 Dð1 aÞðM NÞ2 ðcIc pIe Þ ¼ 2 Aþ dT 2 T D½h þ ð1 aÞcIc þ apIe ! 2 M2 D½h þ ð1 aÞcIc þ apIe o 1 2 2 T
(25) Hence the optimal order quantity
Q 2
þ
is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 2 ¼ DT 2 ¼ ½2AD þ D2 ð1 aÞðM NÞ2 ðcIc pIe Þ=½h þ ð1 aÞcIc þ apIe Þ.
o0; if ToM;
(26) and
It is obvious from (25) and (26) that if 2A þ Dð1 aÞðM NÞ2 ðcIc pIe ÞX0,
(27)
Q 2
T 2
Q 2
T 2
and exist. Otherwise, both and do then both not exist. To ensure T 2 pMpT 2 þ N, we substitute (25) into inequality TpMpT þ N, and obtain that if and only if D1 X0 and D2 2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ D½h þ ð1 aÞcIc þ apIe M 2 p0, then T 2 pMpT 2 þ N.
(28)
Note that 2
2
If 2A þ D½aM þ ð1 aÞðM NÞ ðcIc pIe Þ40, then we obtain d TRC 1 ðTÞ dT 2
¼
1 f2A þ D½aM 2 T3 þ ð1 aÞðM NÞ2 ðcIc pIe Þg40.
(30)
(31) Hence the optimal order quantity Q 1 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f2AD þ D2 ½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe Þg=ðh þ cIc Þ.
(32) It is obvious from (31) and (32) that if 2A þ D½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe ÞX0,
(33)
then both T 1 and Q 1 exist. Otherwise, both T 1 and Q 1 do not exist. To ensure MpT 1 , we substitute (31) into inequality MpT, and obtain that if and only if D2 X0; then MpT 1 .
(36)
Consequently, if D2 40, then both TRC 2 ðTÞ and TRC 3 ðTÞ are strictly decreasing functions for all ToM or MN, respectively. From (34), we know that if D2 40, then T 1 is the optimal solution of TRC 1 ðTÞ. Therefore, we have
From TRC 1 ðT 1 ÞoTRC 2 ðM NÞ as shown as the above, and (36), we yield TRC 1 ðT 1 ÞoTRC 2 ðM NÞ ¼ TRC 3 ðM NÞ oTRC 3 ðTÞ; for all ToM N.
As a result, we know that TRC 1 ðTÞ is also a strictly convex function in T. Similarly, we can easily obtain the unique optimal replenishment cycle time T 1 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 ¼ f2A þ D½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe Þg=½Dðh þ cIc Þ.
Q 1 ¼ DT 1 ¼
o0; if ToM N.
oTRC 2 ðTÞ; for all ToM. (29)
2
dTRC 3 ðTÞ A Dðh þ pIe Þ ¼ 2 þ dT 2 "T # ðM NÞ2 Dðh þ pIe Þ o 1 2 T2
TRC 1 ðT 1 ÞpTRC 1 ðMÞ ¼ TRC 2 ðMÞ
2
D1 D2 ¼ D½h þ ð1 aÞcIc þ apIe ½M ðM NÞ 40. 2
(35)
(34)
From the above arguments and the fact of D1 4D2 , we obtain the following results.
This completes the proof that if D2 40, then T ¼ T 1 , and Q ¼ Q 1 . By using the similar arguments, the reader can prove the rest of Theorem 1. & Judging from (34), (28), and (23), we know that a higher value of A leads to larger values of D1 and D2 which in turn imply a longer replenishment cycle time. Hence, Theorem 1 tells us that the larger the ordering cost, the longer the replenishment cycle. In today’s internet purchasing (e-procurement) systems, many companies have reduced their ordering costs significantly. The theoretical result obtained in Theorem 1 confirms the fact that the replenishment cycle time for a modern company with e-procurement system is getting shorter everyday. Note that Theorem 1 is a general form of the corresponding theoretical result in Chung (1998), in which it requires a ¼ 0; Ic XIe and p ¼ c. In addition, it is also an extension of Theorem 1 in Teng (2002) with a ¼ 0 ¼ N, and Teng and Goyal (2007) with a ¼ 0. In the classical EOQ model, both the retailer and the customer are assumed to pay for the items as soon as they receive them. Hence, it is a special case of Sub-case 1.1 with M ¼ N ¼ a ¼ 0. Therefore, the classical optimal
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EOQ is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q e ¼ 2AD=ðh þ cIc Þ.
(37)
As a result, we can easily obtain the following theoretical result.
421
To ensure T 5 pM, we substitute (42) into inequality TpM, and obtain that if and only if D4 2A D½h þ ð1 aÞcIc þ apIe M 2 p0; then T 5 pM.
(44)
2
Theorem 2. When NpM and the conditions in (27) and (33) hold, (A) if (B) if (C) if
pIe ocIc , then all Q 1 , Q 2 , and pIe 4cIc , then all Q 1 , Q 2 , and pIe ¼ cIc , then Q 1 ¼ Q 2 ¼ Q 3
Q 3 Q 3
are are ¼ Q e .
larger than Q e . smaller than Q e .
Proof. It immediately follows from (37), (32), (26), and (22). A simply economical interpretation of Theorem 2 is as follows. If NpM, then a well-established retailer with its selling price reasonably higher than the purchasing cost (i.e., pIe 4cIc ) should order less quantity and take the upstream trade credit more frequently, and vice versa. The theoretical results from (23), (28), and (34) also show the following managerial phenomena: (1) for a well-established retailer, a higher value of MN causes smaller value of T and Q , (2) conversely, for an ill-established retailer with pIe ocIc , a higher value of M–N causes larger value of T and Q , and (3) if pIe ¼ cIc , then the optimal solution of T or Q does not change with MN. Notice that if the conditions in (33) and (27) do not hold, then both Q 1 and Q 2 do not exist, but the rest of Theorem 2 still holds. In fact, Theorem 2 is a generalization of Theorem 2 in Teng (2002) with a ¼ 0 ¼ N, and in Teng and Goyal (2007) with a ¼ 0. Next, let us discuss the last case in which NXM. To minimize the annual total relevant cost, taking the firstorder and the second-order derivatives of TRC4(T) and TRC5(T) with respect to T, we obtain " # dTRC 4 ðTÞ 1 DaM 2 ðcIc pIe Þ D½h þ cIc ¼ 2 Aþ þ , (38) dT 2 2 T 2
d TRC 4 ðTÞ dT 2
¼
1 T3
½2A þ DaM 2 ðcIc pIe Þ,
dTRC 5 ðTÞ A D½h þ ð1 aÞcIc þ apIe ¼ 2 þ , dT 2 T
(39)
(40)
and 2
d TRC 5 ðTÞ dT 2
¼
2A T3
If 2A þ DaM ðcIc pIe Þ40, then we have 2
d TRC 4 ðTÞ dT 2
¼
1 T3
½2A þ DaM 2 ðcIc pIe Þ40.
As a result, we know that TRC 4 ðTÞ is also a strictly convex function in T. Likewise, we can easily obtain the unique optimal replenishment cycle time T 4 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 4 ¼ ½2A þ DaM 2 ðcIc pIe Þ=½Dðh þ cIc . (46) Hence the optimal order quantity Q 4 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 4 ¼ DT 4 ¼ ½2AD þ D2 aM 2 ðcIc pIe Þ=½h þ cIc .
(41)
It is obvious from (41) that TRC 5 ðTÞ is a strictly convex function in T. Therefore, we can obtain the corresponding unique optimal cycle time T 5 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (42) T 5 ¼ 2A=fD½h þ ð1 aÞcIc þ apIe g. Therefore, the optimal order quantity Q 5 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 5 ¼ T 5 D ¼ 2AD=½h þ ð1 aÞcIc þ apIe .
(43)
(47)
Notice that if 2A þ DaM 2 ðcIc pIe ÞX0,
(48)
and Q 4 then we know from (45) and (46) that both exist. Otherwise, neither T 4 nor Q 4 exists. To ensure MpT 4 , we substitute (46) into MpT, and obtain that T 4
if and only if D4 X0; then MpT 4 .
(49)
From the above arguments, we obtain the following results. & Theorem 3. For MpN, (A) if D4 X0, then T ¼ T 4 , and Q ¼ Q 4 . (B) if D4 p0, then T ¼ T 5 , and Q ¼ Q 5 . Proof. If D4 2A D½h þ ð1 aÞcIc þ apIe M 2 40, then dTRC 5 ðTÞ A D½h þ ð1 aÞcIc þ ape ¼ 2 þ dT 2 T ! 2 M D½h þ ð1 aÞcIc þ apIe o 1 2 2 T o0; if ToM.
(50)
Consequently, if D4 40, then TRC 5 ðTÞ is a strictly decreasing function for all ToM. From (49), we know that if D4 40, then T 4 is the optimal solution of TRC 4 ðTÞ. Therefore, we have TRC 4 ðT 4 ÞpTRC 4 ðMÞ ¼ TRC 5 ðMÞoTRC 5 ðTÞ;
40.
(45)
for all TpM.
This completes the proof that if D4 40, then T ¼ T 4 , and Q ¼ Q 4 . & Similarly, if D4 2A D½h þ ð1 aÞcIc þ apIe M 2 o0, then we have " # dTRC 4 ðTÞ 1 DaM 2 ðcIc pIe Þ D½h þ cIc ¼ 2 Aþ þ dT 2 2 T 2 M Dðh þ cIc Þ Dðh þ cIc Þ 4 2 þ 2 2 T 40; if MpT. (51)
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Hence, if D4 o0, then TRC 4 ðTÞ is an increasing function for all TXM. From (44), we know that if D4 o0, then T 5 is the optimal solution of TRC 5 ðTÞ. Therefore, we have TRC 5 ðT 5 Þ TRC 5 ðMÞ ¼ TRC 4 ðMÞoTRC 4 ðTÞ; for all TXM. This proves that if D4 o0, then T ¼ T 5 , and Q ¼ Q 5 . Similar to Theorem 1, Theorem 3 reveals that the smaller the ordering cost, the shorter the replenishment time. In addition, we know from (42), (43), (46), and (47) that if MpN, then the optimal solution does not affect by the changes in the down-stream trade credit N. Theorem 4. For MpN and 2A þ DaM 2 ðcIc pIe ÞX0, (A) if pIe ocIc , then both Q 4 and Q 5 are larger than Q e . (B) if pIe 4cIc , then both Q 4 and Q 5 are smaller than Q e . (C) if pIe ¼ cIc , then Q 4 ¼ Q 5 ¼ Q e . Proof. It immediately follows from (37), (43), and (47). The result in Theorem 4 reveals the same phenomena as shown in Theorem 2 that a well-established retailer (i.e., pIe 4cIc ) should order less quantity and take the upstream trade credit more often, and vice versa. Notice that if 2A þ DaM 2 ðcIc pIe Þo0, then Q 4 does not exist, but the rest of Theorem 2 is still true. & 5. A numerical example In order to illustrate the previous results, let us apply the theoretical results to solve the following real-world example. Example 1. A one-dollar store (i.e., p ¼ $1) buys nail cutters from a supplier at c ¼ $0.50 a piece. The supplier offers a permissible delay if the payment is made within 60 days (i.e., M ¼ 2/12 ¼ 1/6). This credit term in finance management is usually denoted as ‘‘net 60’’ (e.g., see Brigham, 1995). However, if the payment is not paid in full by the end of 60 days, then 8% interest (i.e., Ic ¼ 0.08) is charged on the outstanding amount. To avoid default risks, the store owner (or the retailer) offers a partial trade credit (e.g., a ¼ 0.5 and N ¼ 1/12) to those credit-risk customers without credit cards. We assume that D ¼ 3,600 units, h ¼ $0.2/unit/year, A ¼ $12 per order, and Ie ¼ 2% if the store deposits its revenue into a moneymarket account; or Ie ¼ 10% if it invests its revenue into a mutual fund account. In this example NpM, and D1 2A ðh þ pIe ÞDðM NÞ2 ¼ 12ð12Þ ð0:2 þ Ie Þð3600Þð1=12Þ2 40; for Ie ¼ 2% or 10%. However,
3600ð0:2 þ 0:02 þ 0:5Ie Þð1=6Þ2 ( 1:25 if Ie ¼ 2%; 3:75
if Ie ¼ 10%:
The numerical result in (53) and (54) verifies the theoretical result in Theorem 2 that if pIe4cIc, then the store owner should order less quantity than the classical EOQ, and take the benefits of the permissible delay more frequently. 6. Conclusions In this paper, I have established an inventory lot-sizing model for a retailer who receives a full trade credit from its supplier, and offers either a partial trade credit to its bad credit customers or a full trade credit to its good credit customers. Then I have derived the theoretical results, and provided an easy-to-use closed-form optimal solution. Finally, I have used a real-world inventory problem to illustrate the proposed model and its optimal solution. In this paper, we assume that the partners across a supply chain are different firms, and make their decision independently. Consequently, locally optimal decision can cause operational inefficiency and globally suboptimal decision for the entire supply chain. To improve operational efficiency and/or supply chain coordination, there is a growing research interest in supply chain contract analysis recently, such as wholesale price contracts, buy back contracts, quantity-based contracts, etc. Please see Tang (2006) for details. To find a more useful solution to today’s coordinated supply chain, we can extend this paper in several ways. For instance, we may add a price contract or a buy-back agreement between the supplier and the retailer to obtain a win-win solution for both partners. Also, we could consider a more general model to reflect the demand risk and the environment risk in the supply chain. Finally, we could generalize the model to allow for shortages, quantity discounts, inflation rates, and others.
2
¼ 12ð12Þ þ 3600ð1 0:5Þð1=12Þ2 ð0:04 Ie Þ
¼
From (34), we have the classical optimal economic order quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q e ¼ 2AD=ðh þ cIc Þ ¼ 600. (54)
Acknowledgments
D2 2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ D½h þ ð1 aÞcIc þ apIe M
Consequently, we know from Theorem 1 that the optimal replenishment interval is T ¼ T 1 if Ie ¼ 2%. However, T ¼ T 2 if Ie ¼ 10%. Substituting the numerical values into (32) and (26), we obtain the optimal economic order quantity Q as follows: ( 600:00 if Ie ¼ 2%; (53) Q ¼ 565:68 if Ie ¼ 10%:
(52)
The author deeply appreciates two anonymous referees for their detailed and constructive comments. He also likes to thank Mr. Jenner Chen and Professor Chun-Tao Chang for their five graphical figures. The research was supported by the ART for research from the William Paterson University of New Jersey.
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers Jinn-Tsair Teng Department of Marketing and Management Sciences, The William Paterson University of New Jersey, Wayne, NJ 07470-2103, USA
a r t i c l e in fo
abstract
Article history: Received 23 May 2008 Accepted 6 March 2009 Available online 18 April 2009
In practice, to reduce default risks, a retailer frequently offers its bad credit customers a partial trade credit, in which the retailer requests its customers to pay a portion of the purchase amount at the time of placing an order as a collateral deposit, and then grants a permissible delay on the rest of the purchase amount. By contrast, the retailer usually provides a full trade credit to its good credit customers without the collateral deposits. For generality, in this paper, I establish an economic order quantity (EOQ) model for a retailer who receives a full trade credit by its supplier, and offers either a partial or a full trade credit to its customers. The proposed model is in a general framework that includes numerous previous models as special cases. I then analyze the characteristics of the optimal solution, and provide an easy-to-use closed-form optimal solution. Finally, I use a real-world inventory problem to illustrate the proposed model and its optimal solution. Published by Elsevier B.V.
Keywords: Inventory EOQ Partial trade credit Permissible delay in payments
1. Introduction In the classical inventory economic order quantity (or EOQ) model, it was assumed that the retailer must pay for the items as soon as receiving them. In practice, a supplier frequently offers a retailer a delay of payment (i.e., an upstream trade credit) for settling the amount owed to him. Usually, there is no interest charge if the outstanding amount is paid within the permissible delay period. However, if the payment is not paid in full by the end of the permissible delay period, then interest is charged on the outstanding amount. Therefore, it is clear that a retailer will delay the payment up to the last moment of the permissible period allowed by the supplier. The permissible delay in payments produces two benefits to the supplier: (1) it attracts new buyers who consider it to be a type of price reduction, and (2) it may be applied as an alternative to price discount because it does not
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provoke competitors to reduce their prices and thus introduce lasting price reductions. On the other hand, the policy of granting credit terms adds not only an additional cost but also an additional dimension of default risk to the supplier. Similarly, a retailer may offer its customers a permissible delay (i.e., a down-stream trade credit) to settle the outstanding balance. Goyal (1985) developed an EOQ model under conditions of permissible delay in payments. He ignored the difference between the selling price and the purchase cost. Although Dave (1985) corrected Goyal’s model by addressing the fact that the selling price is necessarily higher than its purchase cost, his viewpoint did not draw much attention to the recent researchers. Aggarwal and Jaggi (1995) extended Goyal’s model to consider the deteriorating items. Jamal et al. (1997) further generalized Aggarwal and Jaggi’s model to allow for shortages. Hwang and Shinn (1997) added the pricing strategy to the model, and developed the optimal price and lot sizing for a retailer under the condition of permissible delay in payments. In contrast to all above models, Jamal et al. (2000) and Sarker et al. (2000a) obtained the optimal
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cycle and payment times for a retailer when a supplier offers a specified credit period for payment without penalty. Teng (2002) amended Goyal’s model by considering the difference between unit price and unit cost, and found that it makes economic sense for a well-established buyer to order less quantity and take the benefits of the permissible delay more frequently. Chang et al. (2003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Chung and Huang (2003) developed an EPQ inventory model for a retailer when the supplier offers a permissible delay in payments. Huang (2003) extended Goyal’s model to develop an EOQ model in which the supplier offers the retailer the permissible delay period M, and the retailer in turn provides the trade credit period N (with NpM) to his/her customers. Recently, Teng and Goyal (2007) complemented the shortcoming of Huang’s model and proposed a generalized formulation. Many related articles can be found in Chang and Teng (2004), Chang and Dye (2001), Chung and Liao (2004), Goyal et al. (2007), Huang (2004, 2007), Huang and Hsu (2008), Liao et al. (2000), Khouja and Mehrez (1996), Ouyang et al. (2005, 2006), Sarker et al. (2000b), Shinn and Hwang (2003), Teng and Chang (2009), Teng et al. (2005, 2006, 2007), and their references. To reduce default risks, in practice, a retailer frequently offers a partial down-stream trade credit to its credit risk customers who must pay a portion of the purchase amount at the time of placing an order as a collateral deposit, and then receive a permissible delay on the rest of the outstanding amount. In contrast, the retailer usually provides its good credit customers a full trade credit without the collateral deposit. In this paper, I first establish an EOQ model for a retailer who receives a full trade credit by its supplier, and offers either a partial trade credit to its bad credit customers or a full trade credit to its good credit customers. My proposed model is in a general framework that includes numerous previous models such as Goyal (1985), Teng (2002), Huang (2003), Teng and Goyal (2007), and others as special cases. In addition, the proposed model provides the optimal ordering policies for a retailer not only to its bad credit customers but also to its good credit customers. I then derive the theoretical results of the optimal solution, and provide an easy-to-use closed-form optimal solution. Finally, I use a real-world inventory problem to illustrate the proposed model and its optimal solution. 2. Notation and assumptions The following notation is used throughout the entire paper: D the annual demand rate A the ordering cost per order c the purchasing cost per unit p the selling price per unit, with p4c h the unit holding cost per year excluding interest charge the interest earned per dollar per year Ie Ic the interest charged per dollar per year a the fraction of the purchase cost in which the customer must pay the retailer at the time of placing an order, with 0pap1 the portion of the purchase cost in which the retailer offers its 1a customer a permissible delay of N periods. M the retailer’s trade credit period in years offered by the supplier
N T Q TRCðTÞ T Q
the customer’s trade credit period in years offered by the retailer the replenishment cycle time in years the order quantity the annual total relevant cost, which is a function of T the optimal replenishment cycle time of TRC(T) the optimal order quantity ¼ DT*
In addition, the mathematical models proposed in this paper are based on the following assumptions: (1) At the maturity phase of a product life cycle, the annual demand rate is known and constant. In this paper, I deal with good and bad credit customers separately. To good credit customers, set a ¼ 0, and D to be the annual demand for good credit customers. For credit risk customers, set a40, and D to be the annual demand for bad credit customers. (2) In today’s time-based competition, we assume WLOG that shortages are not allowed. (3) Time horizon is infinite, and replenishments are instantaneous. (4) The supplier offers the retailer a full trade credit of M periods to settle the entire purchase cost. Meanwhile, the retailer also provides a partial trade credit to its credit-risk customers who must pay a portion of the purchase amount at the time of placing an order as a collateral deposit, and then receive a permissible delay of N periods on the outstanding amount. (5) If MXN, then the retailer deposits the sales revenue into an interest bearing account. If MXT+N (i.e., the permissible delay period is longer than the time at which the retailer receives the last payment from its customers), then the retailer receives all revenue and pays off the entire purchase cost at the end of the permissible delay M. Otherwise (if MpT+N), the retailer pays the supplier the sum of all units sold by MN and the collateral deposit received from N to M, keeps the profit for the use of the other activities, and starts paying for the interest charges on the items sold after MN. (6) If NXM, then the retailer finances and pays the supplier the entire amount of the delayed payment (1a) pDT at the end of the trade credit M, and then pays down the load after time N at which the retailer starts to receive sales revenue from its customers. For the collateral deposit, the retailer deposits the sales revenue into an interest bearing account until the end of the permissible delay M. If TXM, then the retailer pays the supplier all units sold by M, keeps the profit for the use of the other activities, and starts paying for the interest charges on the items sold after M.
3. Mathematical models From the values of N and M, we have two potential cases: (1)NpM and (2) NXM. Case 1. NpM Based on the values of M (i.e., the time at which the retailer must pay the supplier to avoid interest charge), T (i.e., the replenishment cycle time), and T+N (i.e., the time
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at which the retailer receives the payment from the last customer), we have the following three possible subcases: (i) MpT, (ii) TpMpT+N, and (iii) TpT+NpM. Now, let us discuss the detailed formulation in each sub-case. Sub-case 1.1. MpT By the time M, the retailer has two resources to accumulate revenue in an account that earns Ie per dollar per year: (1) from the portion of immediate payment (starting 0 through M), and (2) from the portion of delayed payment (starting N through M). Therefore, the interest earned per cycle is Ie multiplied by the total area of the triangle OMA and the triangle NMA’ as shown in Fig. 1. Hence, the interest earned per year is pIe D ½aM 2 þ ð1 aÞðM NÞ2 . 2T
(1)
If MpT, then the retailer pays acDM+(1a)cD(MN) to the supplier at time M, keeps its profits, and starts paying for the interest charged after M. The retailer must finance (1) all items sold after M for the portion of immediate payment and (2) all items sold after MN for the portion of delayed payment at an interest charged Ic per dollar per year. As a result, the interest charged per cycle is (c/p)Ic times the total area of the triangle ABC and the triangle A0 B0 C0 as shown in Fig. 1. Therefore, the interest charged per year is given by cIc D ½aðT MÞ2 þ ð1 aÞðT þ N MÞ2 . 2T
(2)
417
Since the annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2, we obtain the annual total relevant cost for the retailer as TRC 1 ðTÞ ¼
A hDT cIc D þ þ ½aðT MÞ2 T 2 2T þ ð1 aÞðT þ N MÞ2 pIe D ½aM2 þ ð1 aÞðM NÞ2 . 2T
(3)
Sub-case 1.2. TpMpT+N Again, before settling the account with the supplier at time M, the retailer has two resources to accumulate revenue in an account that earns Ie per dollar per year: (1) from the portion of immediate payment (starting 0 through M) and (2) from the portion of delayed payment (starting N through M). Therefore, the interest earned per cycle is Ie multiplied by the total area of the trapezoid on the interval [0, M] and the triangle NMA0 as shown in Fig. 2. As a result, the interest earned per year is pIe D ½aT 2 þ 2aTðM TÞ þ ð1 aÞðM NÞ2 . 2T
(4)
Since TpM, the retailer has acDT to pay the supplier for the portion of immediate payment at time M. However, if MpT+N, then the retailer cannot payoff the supplier by M because M is shorter than the customer last payment time t ¼ T+N. Consequently, the retailer must finance all items sold after MN at an interest charged Ic per dollar per year.
Fig. 1. MXN and MpT.
Fig. 2. MXN and TpMpT+N.
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As a result, the interest charged per cycle is (c/p)Ic times the area of the triangle A0 B0 C0 as shown in Fig. 2. Therefore, the interest charged per year is
Since the annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2, we obtain the annual total relevant cost for the retailer as
cIc D ð1 aÞðT þ N MÞ2 . 2T
TRC 3 ðTÞ ¼
(5)
As we know, the annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2. Therefore, we obtain the annual total relevant cost for the retailer as TRC 2 ðTÞ ¼
A hDT cIc D þ þ ð1 aÞðT þ N MÞ2 T 2 2T pIe D ½aT 2 þ 2aTðM TÞ þ ð1 aÞðM NÞ2 . (6) 2T
Sub-case 1.3. TpT+NpM In this sub-case, the retailer receives the total revenue at time T+N, and is able to pay the supplier the total purchase cost at time M. Consequently, there are no interest charges while the interest earned per cycle is Ie multiplied by the total area of the trapezoid on the interval [0, M] and the trapezoid on the interval [N, M] as shown in Fig. 3. As a result, the annual interest earned is pIe D ½aT 2 þ 2aTðM TÞ þ ð1 aÞT 2 2T þ 2ð1 aÞTðM T NÞ pIe D ¼ ½T þ 2aðM TÞ þ 2ð1 aÞðM T NÞ. 2
(7)
A hDT þ T 2 pIe D ½T þ 2aðM TÞ 2 þ 2ð1 aÞðM T NÞ.
(8)
It is obvious from (3), (6), and (8) that TRC1(M) ¼ TRC2(M) and TRC2(MN) ¼ TRC3(MN). Case 2. NXM Based on the values of M and T, we have the following two possible sub-cases: (i) MpT and (ii) TpM. Next, let us discuss the detailed formulation in each sub-case. Sub-case 2.1. MpT By settling the account with the supplier at time M, the retailer accumulates the collateral deposit in an account that earns Ie per dollar per year. Therefore, the interest earned per cycle is Ie multiplied by the area of the triangle OMA as shown in Fig. 4. As a result, the interest earned per year is pIe D aM2 . 2T
(9)
Since NXM, the retailer must finance and pay the supplier acD(TM) for the items sold after M plus the entire amount of the delayed payment (1a)cDT at the end of the trade credit M, and then pay off the load
Fig. 3. MXN and TpT+NpM.
Fig. 4. NXM and MpT.
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after time N at which the retailer starts to receive sales revenue from its customers. Consequently, the interest charged per cycle is (c/p)Ic times the area of the trapezoid on the interval [M, T+N] as shown in Fig. 4. Therefore, the interest charged per year is cIc D faðT MÞ2 þ ð1 aÞT½T þ 2ðN MÞg. 2T
(10)
The annual ordering cost is A/T, and the annual holding cost excluding interest charges is hDT/2. Therefore, we obtain the annual total relevant cost for the retailer as TRC 4 ðTÞ ¼
A hDT cIc D aðT MÞ2 þ þ T 2 2T cIc D pIe D þ aM 2 . ð1 aÞ½T þ 2ðN MÞ 2 2T
4. Optimal order quantity Let us discuss the first case of NpM, and then the case of NXM. To minimize the annual total relevant cost, taking the first-order and the second-order derivatives of TRC1(T), TRC2(T), and TRC3(T) with respect to T, we obtain ( ) dTRC 1 ðTÞ 1 D½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe Þ ¼ 2 Aþ dT 2 T þ 2
d TRC 1 ðTÞ dT 2
(11)
¼
Dðh þ cIc Þ , 2
1 f2A þ D½aM2 T3 þ ð1 aÞðM NÞ2 ðcIc pIe Þg,
Sub-case 2.2. TpM During [0, M], the retailer accumulates the collateral deposit in an account that earns Ie per dollar per year. Therefore, the interest earned per cycle is Ie multiplied by the area of the trapezoid on the interval [0, M] as shown in Fig. 5. Hence, the interest earned per year is
" # dTRC 2 ðTÞ 1 Dð1 aÞðM NÞ2 ðcIc pIe Þ ¼ 2 Aþ dT 2 T
apIe D
d TRC 2 ðTÞ
2
þ 2
½T þ 2ðM TÞ.
(12)
Similar to sub-case 2.1, if NXM, then the retailer must finance and pay the supplier the entire amount of the delayed payment (1a)cDT at the end of the trade credit M, and then pay off the load after time N at which the retailer starts to receive sales revenue from its customers. Consequently, the interest charged per cycle is (c/p)Ic times the area of the trapezoid on the interval [M, T+N] as shown in Fig. 4. Therefore, the interest charged per year is cIc D ð1 aÞ½T þ 2ðN MÞ. 2
(13)
Therefore, we obtain the annual total relevant cost for the retailer as TRC 5 ðTÞ ¼
A hDT cIc D þ þ ð1 aÞ½T þ 2ðN MÞ T 2 2 apIe D ½T þ 2ðM TÞ. 2
From (11) and (14), we obtain TRC5(M) ¼ TRC4(M).
(14)
419
dT 2
¼
D½h þ ð1 aÞcIc þ apIe , 2
1 T3
½2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ,
dTRC 3 ðTÞ A Dðh þ pIe Þ ¼ 2 þ , dT 2 T
(15)
(16)
(17)
(18)
(19)
and 2
d TRC 3 ðTÞ dT 2
¼
2A T3
40.
(20)
It is clear from (20) that TRC 3 ðTÞ is a strictly convex function in T. Consequently we obtain the corresponding unique optimal cycle time T 3 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 ¼ 2A=½Dðh þ pIe Þ. (21) Therefore, the optimal order quantity Q 3 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 3 ¼ T 3 D ¼ 2AD=ðh þ pIe Þ.
(22)
T 3 +NpM,
To ensure we substitute (21) into inequality T+NpM, and obtain that if and only if
D1 2A ðh þ pIe ÞDðM NÞ2 p0; then T 3 þ NpM.
Fig. 5. NXM and TpM.
(23)
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If 2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ40, then we get 2
d TRC 2 ðTÞ dT 2
¼
1 T3
Theorem 1. For NpM,
½2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ40. (24)
Hence, we know that TRC 2 ðTÞ is also a strictly convex function in T. Likewise, we can easily obtain the unique optimal replenishment cycle time T 2 as T 2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ=fD½h þ ð1 aÞcIc þ apIe g.
(A) if D2 X0, then T ¼ T 1 , and Q ¼ Q 1 . (B) if D1 X0, and D2 p0, then T ¼ T 2 , and Q ¼ Q 2 . (C) if D1 p0, then T ¼ T 3 , and Q ¼ Q 3 . Proof. If D2 40, then D1 40, ( ) dTRC 2 ðTÞ 1 Dð1 aÞðM NÞ2 ðcIc pIe Þ ¼ 2 Aþ dT 2 T D½h þ ð1 aÞcIc þ apIe ! 2 M2 D½h þ ð1 aÞcIc þ apIe o 1 2 2 T
(25) Hence the optimal order quantity
Q 2
þ
is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 2 ¼ DT 2 ¼ ½2AD þ D2 ð1 aÞðM NÞ2 ðcIc pIe Þ=½h þ ð1 aÞcIc þ apIe Þ.
o0; if ToM;
(26) and
It is obvious from (25) and (26) that if 2A þ Dð1 aÞðM NÞ2 ðcIc pIe ÞX0,
(27)
Q 2
T 2
Q 2
T 2
and exist. Otherwise, both and do then both not exist. To ensure T 2 pMpT 2 þ N, we substitute (25) into inequality TpMpT þ N, and obtain that if and only if D1 X0 and D2 2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ D½h þ ð1 aÞcIc þ apIe M 2 p0, then T 2 pMpT 2 þ N.
(28)
Note that 2
2
If 2A þ D½aM þ ð1 aÞðM NÞ ðcIc pIe Þ40, then we obtain d TRC 1 ðTÞ dT 2
¼
1 f2A þ D½aM 2 T3 þ ð1 aÞðM NÞ2 ðcIc pIe Þg40.
(30)
(31) Hence the optimal order quantity Q 1 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f2AD þ D2 ½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe Þg=ðh þ cIc Þ.
(32) It is obvious from (31) and (32) that if 2A þ D½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe ÞX0,
(33)
then both T 1 and Q 1 exist. Otherwise, both T 1 and Q 1 do not exist. To ensure MpT 1 , we substitute (31) into inequality MpT, and obtain that if and only if D2 X0; then MpT 1 .
(36)
Consequently, if D2 40, then both TRC 2 ðTÞ and TRC 3 ðTÞ are strictly decreasing functions for all ToM or MN, respectively. From (34), we know that if D2 40, then T 1 is the optimal solution of TRC 1 ðTÞ. Therefore, we have
From TRC 1 ðT 1 ÞoTRC 2 ðM NÞ as shown as the above, and (36), we yield TRC 1 ðT 1 ÞoTRC 2 ðM NÞ ¼ TRC 3 ðM NÞ oTRC 3 ðTÞ; for all ToM N.
As a result, we know that TRC 1 ðTÞ is also a strictly convex function in T. Similarly, we can easily obtain the unique optimal replenishment cycle time T 1 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 ¼ f2A þ D½aM 2 þ ð1 aÞðM NÞ2 ðcIc pIe Þg=½Dðh þ cIc Þ.
Q 1 ¼ DT 1 ¼
o0; if ToM N.
oTRC 2 ðTÞ; for all ToM. (29)
2
dTRC 3 ðTÞ A Dðh þ pIe Þ ¼ 2 þ dT 2 "T # ðM NÞ2 Dðh þ pIe Þ o 1 2 T2
TRC 1 ðT 1 ÞpTRC 1 ðMÞ ¼ TRC 2 ðMÞ
2
D1 D2 ¼ D½h þ ð1 aÞcIc þ apIe ½M ðM NÞ 40. 2
(35)
(34)
From the above arguments and the fact of D1 4D2 , we obtain the following results.
This completes the proof that if D2 40, then T ¼ T 1 , and Q ¼ Q 1 . By using the similar arguments, the reader can prove the rest of Theorem 1. & Judging from (34), (28), and (23), we know that a higher value of A leads to larger values of D1 and D2 which in turn imply a longer replenishment cycle time. Hence, Theorem 1 tells us that the larger the ordering cost, the longer the replenishment cycle. In today’s internet purchasing (e-procurement) systems, many companies have reduced their ordering costs significantly. The theoretical result obtained in Theorem 1 confirms the fact that the replenishment cycle time for a modern company with e-procurement system is getting shorter everyday. Note that Theorem 1 is a general form of the corresponding theoretical result in Chung (1998), in which it requires a ¼ 0; Ic XIe and p ¼ c. In addition, it is also an extension of Theorem 1 in Teng (2002) with a ¼ 0 ¼ N, and Teng and Goyal (2007) with a ¼ 0. In the classical EOQ model, both the retailer and the customer are assumed to pay for the items as soon as they receive them. Hence, it is a special case of Sub-case 1.1 with M ¼ N ¼ a ¼ 0. Therefore, the classical optimal
ARTICLE IN PRESS J.-T. Teng / Int. J. Production Economics 119 (2009) 415–423
EOQ is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q e ¼ 2AD=ðh þ cIc Þ.
(37)
As a result, we can easily obtain the following theoretical result.
421
To ensure T 5 pM, we substitute (42) into inequality TpM, and obtain that if and only if D4 2A D½h þ ð1 aÞcIc þ apIe M 2 p0; then T 5 pM.
(44)
2
Theorem 2. When NpM and the conditions in (27) and (33) hold, (A) if (B) if (C) if
pIe ocIc , then all Q 1 , Q 2 , and pIe 4cIc , then all Q 1 , Q 2 , and pIe ¼ cIc , then Q 1 ¼ Q 2 ¼ Q 3
Q 3 Q 3
are are ¼ Q e .
larger than Q e . smaller than Q e .
Proof. It immediately follows from (37), (32), (26), and (22). A simply economical interpretation of Theorem 2 is as follows. If NpM, then a well-established retailer with its selling price reasonably higher than the purchasing cost (i.e., pIe 4cIc ) should order less quantity and take the upstream trade credit more frequently, and vice versa. The theoretical results from (23), (28), and (34) also show the following managerial phenomena: (1) for a well-established retailer, a higher value of MN causes smaller value of T and Q , (2) conversely, for an ill-established retailer with pIe ocIc , a higher value of M–N causes larger value of T and Q , and (3) if pIe ¼ cIc , then the optimal solution of T or Q does not change with MN. Notice that if the conditions in (33) and (27) do not hold, then both Q 1 and Q 2 do not exist, but the rest of Theorem 2 still holds. In fact, Theorem 2 is a generalization of Theorem 2 in Teng (2002) with a ¼ 0 ¼ N, and in Teng and Goyal (2007) with a ¼ 0. Next, let us discuss the last case in which NXM. To minimize the annual total relevant cost, taking the firstorder and the second-order derivatives of TRC4(T) and TRC5(T) with respect to T, we obtain " # dTRC 4 ðTÞ 1 DaM 2 ðcIc pIe Þ D½h þ cIc ¼ 2 Aþ þ , (38) dT 2 2 T 2
d TRC 4 ðTÞ dT 2
¼
1 T3
½2A þ DaM 2 ðcIc pIe Þ,
dTRC 5 ðTÞ A D½h þ ð1 aÞcIc þ apIe ¼ 2 þ , dT 2 T
(39)
(40)
and 2
d TRC 5 ðTÞ dT 2
¼
2A T3
If 2A þ DaM ðcIc pIe Þ40, then we have 2
d TRC 4 ðTÞ dT 2
¼
1 T3
½2A þ DaM 2 ðcIc pIe Þ40.
As a result, we know that TRC 4 ðTÞ is also a strictly convex function in T. Likewise, we can easily obtain the unique optimal replenishment cycle time T 4 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 4 ¼ ½2A þ DaM 2 ðcIc pIe Þ=½Dðh þ cIc . (46) Hence the optimal order quantity Q 4 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 4 ¼ DT 4 ¼ ½2AD þ D2 aM 2 ðcIc pIe Þ=½h þ cIc .
(41)
It is obvious from (41) that TRC 5 ðTÞ is a strictly convex function in T. Therefore, we can obtain the corresponding unique optimal cycle time T 5 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (42) T 5 ¼ 2A=fD½h þ ð1 aÞcIc þ apIe g. Therefore, the optimal order quantity Q 5 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 5 ¼ T 5 D ¼ 2AD=½h þ ð1 aÞcIc þ apIe .
(43)
(47)
Notice that if 2A þ DaM 2 ðcIc pIe ÞX0,
(48)
and Q 4 then we know from (45) and (46) that both exist. Otherwise, neither T 4 nor Q 4 exists. To ensure MpT 4 , we substitute (46) into MpT, and obtain that T 4
if and only if D4 X0; then MpT 4 .
(49)
From the above arguments, we obtain the following results. & Theorem 3. For MpN, (A) if D4 X0, then T ¼ T 4 , and Q ¼ Q 4 . (B) if D4 p0, then T ¼ T 5 , and Q ¼ Q 5 . Proof. If D4 2A D½h þ ð1 aÞcIc þ apIe M 2 40, then dTRC 5 ðTÞ A D½h þ ð1 aÞcIc þ ape ¼ 2 þ dT 2 T ! 2 M D½h þ ð1 aÞcIc þ apIe o 1 2 2 T o0; if ToM.
(50)
Consequently, if D4 40, then TRC 5 ðTÞ is a strictly decreasing function for all ToM. From (49), we know that if D4 40, then T 4 is the optimal solution of TRC 4 ðTÞ. Therefore, we have TRC 4 ðT 4 ÞpTRC 4 ðMÞ ¼ TRC 5 ðMÞoTRC 5 ðTÞ;
40.
(45)
for all TpM.
This completes the proof that if D4 40, then T ¼ T 4 , and Q ¼ Q 4 . & Similarly, if D4 2A D½h þ ð1 aÞcIc þ apIe M 2 o0, then we have " # dTRC 4 ðTÞ 1 DaM 2 ðcIc pIe Þ D½h þ cIc ¼ 2 Aþ þ dT 2 2 T 2 M Dðh þ cIc Þ Dðh þ cIc Þ 4 2 þ 2 2 T 40; if MpT. (51)
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Hence, if D4 o0, then TRC 4 ðTÞ is an increasing function for all TXM. From (44), we know that if D4 o0, then T 5 is the optimal solution of TRC 5 ðTÞ. Therefore, we have TRC 5 ðT 5 Þ TRC 5 ðMÞ ¼ TRC 4 ðMÞoTRC 4 ðTÞ; for all TXM. This proves that if D4 o0, then T ¼ T 5 , and Q ¼ Q 5 . Similar to Theorem 1, Theorem 3 reveals that the smaller the ordering cost, the shorter the replenishment time. In addition, we know from (42), (43), (46), and (47) that if MpN, then the optimal solution does not affect by the changes in the down-stream trade credit N. Theorem 4. For MpN and 2A þ DaM 2 ðcIc pIe ÞX0, (A) if pIe ocIc , then both Q 4 and Q 5 are larger than Q e . (B) if pIe 4cIc , then both Q 4 and Q 5 are smaller than Q e . (C) if pIe ¼ cIc , then Q 4 ¼ Q 5 ¼ Q e . Proof. It immediately follows from (37), (43), and (47). The result in Theorem 4 reveals the same phenomena as shown in Theorem 2 that a well-established retailer (i.e., pIe 4cIc ) should order less quantity and take the upstream trade credit more often, and vice versa. Notice that if 2A þ DaM 2 ðcIc pIe Þo0, then Q 4 does not exist, but the rest of Theorem 2 is still true. & 5. A numerical example In order to illustrate the previous results, let us apply the theoretical results to solve the following real-world example. Example 1. A one-dollar store (i.e., p ¼ $1) buys nail cutters from a supplier at c ¼ $0.50 a piece. The supplier offers a permissible delay if the payment is made within 60 days (i.e., M ¼ 2/12 ¼ 1/6). This credit term in finance management is usually denoted as ‘‘net 60’’ (e.g., see Brigham, 1995). However, if the payment is not paid in full by the end of 60 days, then 8% interest (i.e., Ic ¼ 0.08) is charged on the outstanding amount. To avoid default risks, the store owner (or the retailer) offers a partial trade credit (e.g., a ¼ 0.5 and N ¼ 1/12) to those credit-risk customers without credit cards. We assume that D ¼ 3,600 units, h ¼ $0.2/unit/year, A ¼ $12 per order, and Ie ¼ 2% if the store deposits its revenue into a moneymarket account; or Ie ¼ 10% if it invests its revenue into a mutual fund account. In this example NpM, and D1 2A ðh þ pIe ÞDðM NÞ2 ¼ 12ð12Þ ð0:2 þ Ie Þð3600Þð1=12Þ2 40; for Ie ¼ 2% or 10%. However,
3600ð0:2 þ 0:02 þ 0:5Ie Þð1=6Þ2 ( 1:25 if Ie ¼ 2%; 3:75
if Ie ¼ 10%:
The numerical result in (53) and (54) verifies the theoretical result in Theorem 2 that if pIe4cIc, then the store owner should order less quantity than the classical EOQ, and take the benefits of the permissible delay more frequently. 6. Conclusions In this paper, I have established an inventory lot-sizing model for a retailer who receives a full trade credit from its supplier, and offers either a partial trade credit to its bad credit customers or a full trade credit to its good credit customers. Then I have derived the theoretical results, and provided an easy-to-use closed-form optimal solution. Finally, I have used a real-world inventory problem to illustrate the proposed model and its optimal solution. In this paper, we assume that the partners across a supply chain are different firms, and make their decision independently. Consequently, locally optimal decision can cause operational inefficiency and globally suboptimal decision for the entire supply chain. To improve operational efficiency and/or supply chain coordination, there is a growing research interest in supply chain contract analysis recently, such as wholesale price contracts, buy back contracts, quantity-based contracts, etc. Please see Tang (2006) for details. To find a more useful solution to today’s coordinated supply chain, we can extend this paper in several ways. For instance, we may add a price contract or a buy-back agreement between the supplier and the retailer to obtain a win-win solution for both partners. Also, we could consider a more general model to reflect the demand risk and the environment risk in the supply chain. Finally, we could generalize the model to allow for shortages, quantity discounts, inflation rates, and others.
2
¼ 12ð12Þ þ 3600ð1 0:5Þð1=12Þ2 ð0:04 Ie Þ
¼
From (34), we have the classical optimal economic order quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q e ¼ 2AD=ðh þ cIc Þ ¼ 600. (54)
Acknowledgments
D2 2A þ Dð1 aÞðM NÞ2 ðcIc pIe Þ D½h þ ð1 aÞcIc þ apIe M
Consequently, we know from Theorem 1 that the optimal replenishment interval is T ¼ T 1 if Ie ¼ 2%. However, T ¼ T 2 if Ie ¼ 10%. Substituting the numerical values into (32) and (26), we obtain the optimal economic order quantity Q as follows: ( 600:00 if Ie ¼ 2%; (53) Q ¼ 565:68 if Ie ¼ 10%:
(52)
The author deeply appreciates two anonymous referees for their detailed and constructive comments. He also likes to thank Mr. Jenner Chen and Professor Chun-Tao Chang for their five graphical figures. The research was supported by the ART for research from the William Paterson University of New Jersey.
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