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Retailer’s optimal ordering and discounting policies under advance sales discount and trade credits
Computers & Industrial Engineering 56 (2009) 208–215
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Retailer’s optimal ordering and discounting policies under advance sales discount and trade credits Yu-Chung Tsao * Department of Business Management, Tatung University, No. 300, Jhongda Road, Taipei 104, Taiwan
a r t i c l e
i n f o
Article history: Received 8 September 2007 Received in revised form 13 April 2008 Accepted 21 May 2008 Available online 28 May 2008 Keywords: EOQ Ordering Discounting Trade credits Advance sales discount Two-echelon supply chain
a b s t r a c t Providing advance sales discount program to attract consumers or eliminate demand uncertainty is commonly in today’s business, such as music disk, apparel, books, etc. Also, trade credit is a popular payment behavior in B2B and B2C transactions. In this research, we use EOQ to model the decisions under advance sales discount and two-echelon trade credits, which involve how much to order and how much to discount to minimize the total related cost. Several examples are given to illustrate the solution procedures and discuss the impact of various system parameters. We conclude with a computational analysis that leads to variety of managerial insights. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction To date, companies have provided several activities to boost consumer demand. For examples, retailers Wall-Mart and Costco often try to stimulate demand for specific types of electric equipment by offering price discounts; clothiers A&F and American Eagle make shelf space for specific clothes items available for longer periods; fast-food restaurants McDonald’s and Burger King often use coupons to attract consumers. In this paper, we consider such a strategy in which the retailer develops a program called an ‘‘advance sales discount” (ASD) program that entices customers to commit to their orders at a discount price prior to the selling season. This means the retailer offers his/her customers a price discount if they can commit their orders prior to the sales period. Due to the progress of information technique, many companies provide on-line transaction to their customers on internet. This accelerates the use of ASD program. Now customers can commit their orders on-line before the sales period to get the price discount. For companies, the ASD strategy is common and useful in reality to decrease the estimation error in demand and to increase the market share. Also, customers can save their money if they make sure that they need the product earlier. In addition, traditional EOQ model tacitly assumes that the payment must be made to the supplier for the items immediately after the retailer receive the products. However, such an assumption is not quite practical in the real world. In practice, the supplier allows * Tel.: +886 2 25925252 2435 19. E-mail addresses: [email protected], [email protected] 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.05.006
the retailer a credit period for settling the amount owed and does not charge any interest on the amount owed during this period. Over years, a number of researches have been published which dealt with the inventory model under permissible delay in payments. Haley and Higgins (1973), Chapman, Ward, Cooper, and Page (1984), Goyal (1985), Daellenbach (1986) and Rachamadugu (1989) examined the effect of the credit period on the optimal inventory policy. Recently, Chung (1998) simplified the search of the optimal solution for the problem explored by Goyal (1985). Jamal, Sarker, and Wang (2000) developed a model for optimal cycle and payment time with deteriorating products. Teng (2002) then amended Goyal (1985) model by considering the difference between unit price and unit cost. For deteriorating items, Aggarwal and Jaggi (1995) and Chu et al. (1998) researched the ordering policy for deteriorating items under permissible delay in payment. Jamal, Sarker, and Wang (1997) generalized Aggarwal and Jaggi (1995) model to allow for shortage. Chang and Dye (2001) extended the model of Jamal et al. (1997) to allow for a varying deteriorating rate of time and partial backlogging rate. Chang, Hung, and Dye (2001) considered the linear demand for deteriorating items under credit period. Then Chang, Hung, and Dye (2002) developed a finite time horizon inventory model with deterioration and time-value of money when payment periods are offered. Chung and Huang (2003) considered the case that the units are replenished at a finite rate under permissible delay in payments. Recently, Chang, Ouyang, and Teng (2003), Shinn and Hwang (2003), Chung and Liao (2004) and Chung et al. (2005) considered the inventory model under the condition of order-size-dependent credit period. All these above researches assumed the supplier pro-
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vided a credit period to the retailer, but the customer must pay for the goods as soon as the goods are received. In most business transactions, the retailer may adopt the credit period policy to stimulate his customer demand. Huang (2003) considered not only the supplier offers a credit period to the retailer, but also the retailer offers a credit period to his customers. In this note, this paper both considers the condition of advance sales discount and two-echelon trade credits in a one supplier-one retailer supply chain. In this model, the retailer offers his/her customers a price discount if they can commit their orders prior to the sales period. The supplier offers a credit period to the retailer and the retailer offers a credit period to his customers. To the best of our knowledge, this research is the first to incorporate the conditions of permissible delay in payments and advance sales discount into EOQ model. This research not only deals with the inventory replenishment decisions but also determines the discounting policy in the ASD program. The objective is to determine the optimal replenishment cycle time and the optimal price discount while minimizing the total related cost. We conclude with a computational analysis that leads to variety of managerial insights. The remainder of this paper is given as follows. First, we describe the assumptions and notations. We formulate the model in Section 3. In Section 4, we develop theorems based on lemmas to determine the optimal replenishment cycle time when price discount is exogenous. Then we develop an algorithm based on properties to determine ordering and discounting decisions simultaneously in Section 5. In Section 6, some computational analyses with respect to various system parameters are also conducted. Finally, some conclusions are made. 2. Notations and assumptions The following notation is used in this study: unit retail price unit purchase cost ordering cost per order unit inventory holding cost per unit time retailer’s credit period provided by supplier customer’s credit period provided by retailer, where t2 < t1 price discount, 0 6 r < 1 the interest paid per dollar per unit time, Ip > 0 the interest earned per dollar per unit time, Ie > 0 replenishment cycle time annual demand for the retailer, we call the retailer in this paper retailer 1 annual demand for other retailers, i.e., not retailer 1 D2 fraction of retailer 1’s customers who use advance sales Y1 discount program, where Y1 is a function of r, i.e., Y1 (r) fraction of other retailers’ customers who switch to retailer Y2 1 under advance sales discount program, where Y2 is a function of r, i.e., Y2 (r). Y1D1 + Y2D2 annual demand of customers who use advance sales discount program, where Y2D2 is the increasing demand when provide ASD program (Fig. 1) (1 Y1)D1 annual demand of customers who are not use advance sales discount program (Fig. 1) D total annual demand, D = Y1D1 + Y2D2 + (1 Y1)D1 TCi total related cost, i = 1 when T P t1, i = 2 when t2 6 T 6 t1, i = 3 when T 6 t2.
P C A H t1 t2 r Ip Ie T D1
The mathematical model in this paper is developed on the following assumptions: 1. We consider the problem of single product. 2. The retailer offers a price discount r to his customers if they can commit their orders prior to the sales period. 3. Y1 percentage of the retailer’s (retailer 1’s) customers use advance sales discount program and Y2 percentage of other retailers’ customers use advance sales discount program.
D1 1 − Y1
D2 Y1
Y2
(1 − Y1 ) D1
Y1 D1 + Y2 D2
Not use ASD program
Use ASD program
Fig. 1. The impact of ASD program on end customer demand.
4. All customers (include who use ASD program) should settle his/ her payment at t2. 5. The unit retail price of the products sold during the period t2 to t1 is deposited in an interest bearing account with rate Ie. At the end of t1, the credit is settled and the retailer starts paying for the interest charges for the items in stocks with rate Ip. 6. Since the retailer cannot earn any interest in this situation t2 > t1, it is reasonable to assume that the retailer’s credit period provided by supplier t1 is longer than the customer’s credit period provided by retailer t2. (Huang (2003) made the same assumption.) 3. Model formulation The total related cost consists of the following elements: (1) (2) (3) Case 1:
Annual ordering cost = A/T. Annual inventory holding cost = (1 Y1)D1TH/2. There are three cases occurred in interest earned per year. When T P t1 From Assumption 5, we know the retailer can accumulate revenue and earn interest during the period t2 to t1 with rate Ie. There are two parts in the annual interest earned. The first part is the revenue from selling item during the PIeð1Y 1 ÞD1 ðt 21 t 22 Þ . The second part is the revsales period, i.e., 2T enue from selling item before the sales period (under ASD program), i.e., (Y1D1 + Y2D2)P(1 r)Ie(t1 t2) (Fig. 2(a)).
Annual interest earned ¼
P Ie ð1 Y 1 ÞD1 ðt 21 t22 Þ 2T þ ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt1 t2 Þ
Case 2: When t2 6 T 6 t1 The first part is the revenue from2 selling item during the PIeð1Y 1 ÞD1 ð2Tt1 T t 22 Þ . The second part is sales period, i.e., 2T the revenue from selling item before the sales period (under ASD program), i.e., (Y1D1 + Y2 D2)P(1 r)Ie(t1 t2) (Fig. 2(b)).
Annual interest earned ¼
P Ie ð1 Y 1 ÞD1 ð2Tt 1 T 2 t22 Þ 2T þ ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt 1 t2 Þ
Case 3: When T 6 t2 The first part is the revenue from selling item during the sales period, i.e., [(1 Y1)D1]P(1 r)Ie(t1 t2). The second part is the revenue from selling item before the sales period (under ASD program), i.e., (Y1D1 + Y2D2) P(1 r)Ie(t1 t2) (Fig. 2(c)).
Annual interest earned ¼ ½ðY 1 D1 þ Y 2 D2 ÞPð1 rÞ þ ð1 Y 1 ÞD1 P Ie ðt 1 t 2 Þ (4) There are three cases occurred in interest paid per year. Case 1: When T P t1
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Fig. 2. The total accumulation of interest earned.
Annual interest paid ¼
C Ip ð1 Y 1 ÞD1 ðT t1 Þ2 2T
Case 2: When t2 6 T 6 t1
Case 2: When t2 6 T 6 t1 Annual interest charge = 0. In this case, no interest charge is paid for the items. Case 3: When T 6 t2 Annual interest charge = 0. In this case, no interest charge is paid for the items. Therefore, total related cost TCi has three different expressions as follows: Case 1: When T P t1
C Ip ð1 Y 1 ÞD1 ðT t 1 Þ2 ; 2T
A ð1 Y 1 ÞD1 TH P Ie ð1 Y 1 ÞD1 ð2Tt 1 T 2 t 22 Þ þ T 2 2T ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt1 t 2 Þ;
ð2Þ
Case 3: When T 6 t2
TC3 ¼
A ð1 Y 1 ÞD1 TH þ ½ðY 1 D1 þ Y 2 D2 ÞPð1 rÞ þ ð1 T 2 Y 1 ÞD1 PIeðt1 t2 Þ:
ð3Þ
4. Determination of optimal ordering policy
A ð1 Y 1 ÞD1 TH P Ie ð1 Y 1 ÞD1 ðt21 t22 Þ TC1 ¼ þ T 2 2T ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt 1 t2 Þ þ
TC2 ¼
ð1Þ
In this section, the problem is to find an optimal cycle time T* while minimizing TCi, where TCi(T) = TC1(T) when T P t1; TCi(T) = TC2(T) when t2 6 T 6 t1 and TCi(T) = TC3(T) when T 6 t2. The first-order and second-order derivative of TC1(T), TC2(T) and TC3(T) with respective to T are as follows:
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dTC1 ðTÞ ð1 Y 1 ÞD1 ½HT2 þ CIpðT 2 t21 Þ þ IePðt21 t22 Þ 2A ; ¼ dT 2T 2
ð4Þ
dTC2 ðTÞ ð1 Y 1 ÞD1 ½HT2 þ IePðT 2 t 22 Þ 2A ; ¼ dT 2T 2
ð5Þ
dTC3 ðTÞ ð1 Y 1 ÞD1 HT2 2A ; ¼ dT 2T 2
ð6Þ
2
d TC1 ðTÞ dT 2 d TC2 ðTÞ
¼
2
2
dT 2 d TC3 ðTÞ dT 2
¼ ¼
2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ T3 2A þ ð1 Y 1 ÞD1 P Ie t 22 T 2A T3
3
ð7Þ
;
> 0;
ð8Þ
> 0:
ð9Þ
Eqs. (8) and (9) imply that TC2(T) and TC3(T) are convex on T > 0. Eq. is concave on T>0 if (7) implies that TC1(T) 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt21 t 22 Þ > 0. From TC1(t1) = TC2(t1) and TC2 (t2) = TC3(t2), TCi(T) is continuous and defined on T > 0. Therefore, we have the following lemma: Lemma 1 (a) If 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t22 Þ > 0, Pi(T) is concave on (0, 1). (b) If 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t 22 Þ 6 0, Pi(T) is concave on (0, t1] and convex on [t1, 1).
Proof. From Lemma 1 and the feasible cycle time interval (Eqs. (13)–(15)), it can be proved. h Then we discuss how to determine the optimal replenishment cycle time under the situation when 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt21 t 22 Þ 6 0 If 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ 6 0, we consider the following lemmas first. Lemma 3. If 2A þ qð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ 6 0, then T 2 < t 1 . rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Aþð1Y ÞD t 2 PIe
1 1 2 Proof. Assuming T 2 P t 1 , this means P t1 . Arrange ð1Y 1 ÞD1 ðHþPIeÞ the inequality, we obtain 2A þ ð1 Y 1 ÞD1 ðt22 t21 ÞPIe P
t21 ð1 Y 1 ÞD1 H. Then adding ð1 Y 1 ÞD1 CIpt21 into both sides, we obtain 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t 22 Þ P t21 ð1 Y 1 Þ D1 ðH þ CIpt21 Þ > 0. This is contradiction. Therefore, T 2 < t1 .
h
Lemma 4. T 2 6 t 2 if and only if T 3 6 t2 . Proof. If T 2 6 t2 , from Eq. (11), we get 2A 6 t22 ð1 Y 1 ÞD1 H. Then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we have ð1Y2A 6 t 2 , this means T 3 6 t 2 . Similarly, if T 3 6 t 2 , 1 ÞD1 H Eq. (12) implies 2A 6 t22 ð1 Y 1 ÞD1 H. Adding ð1 Y 1 ÞD1 t22 PIe into both sides, we obtain 2A þ ð1 Y 1 ÞD1 t 22 PIe 6 t 22 ð1 Y 1 ÞD1 Hþ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 Y 1 ÞD1 t22 PIe. Then we have
2Aþð1Y 1 ÞD1 t 22 PIe ð1Y 1 ÞD1 ðHþPIeÞ
6 t 2 , i.e., T 2 6 t 2 .
h
Then we develop Theorem 2 based on above analysis. First, we discuss how to determine the optimal replenishment cycle time under the situation when 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t 22 Þ > 0. If 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t 22 Þ > 0, we can obtain the optimal replenishment cycle time for each case by solving dTCdTi ðTÞ ¼ 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A þ ð1 Y 1 ÞD1 ½t 21 ðCIp PIeÞ þ t 22 PIe ; ¼ ð1 Y 1 ÞD1 ðH þ CIpÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A þ ð1 Y 1 ÞD1 t22 PIe ; T 2 ¼ ð1 Y 1 ÞD1 ðH þ PIeÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A T3 ¼ : ð1 Y 1 ÞD1 H
T 1
Proof. From Lemmas 3 and 4 and the feasible cycle time interval (t2 6 T 6 t1 in case 2 and T 6 t2 in case 3), it can be proved. h
ð11Þ 5. Determination of optimal ordering and discounting policies
ð12Þ
T P t 1 ) 2A þ ð1 Y 1 ÞD1 ½PIeðt22 t 21 Þ t 21 H P 0; t 2 6 T 6 t1 ) 2A þ ð1
t 21 Þ
6 2A 1 Y 1 D1 Ht 22 ; T 6 t2 ) 2A ð1
Y 1 ÞD1 Ht22
t 21 H
ð13Þ
60 ð14Þ
6 0:
Y 1 ÞD1 ½PIeðt 22
t 21 Þ
Let f1 ¼ 2A þ ð1 D1 Ht22 , we have the following lemma:
ð15Þ t 21 H
and f2 ¼ 2A ð1 Y 1 Þ
Lemma 2. f2 P f1. Proof. It can be proved from f2 f1 ¼ ð1 Y 1 ÞD1 ½PIeðt 21 t 22 Þ þ Hðt21 t 22 Þ P 0 since t1 > t2 and 0 6 Y 6 11. h We develop the following theorem to determine the optimal cycle time T* based on above analysis. Theorem 1. When 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt21 t 22 Þ > 0, (a) If f2 P f1 P 0, then T ¼ T 1 . (b) If f2 P 0 P f1, then T ¼ T 2 . (c) If 0 P f2 P f1, then T ¼ T 3 .
(a) If T 2 < t2 , then T ¼ T 3 . (b) If T 2 > t2 , then T ¼ T 2 . (c) If T 2 ¼ t2 , then T ¼ T 2 ¼ T 3 .
ð10Þ
Then from the relation T P t1 in case 1, t2 6 T 6 t1 in case 2 and T 6 t2 in case 3, and Eqs. (10)–(12), we have
Y 1 ÞD1 ½PIeðt 22
Theorem 2. When 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ 6 0,
In this section, we consider that the retail price r is endogenous, i.e., the retailer wants to determine the optimal replenishment cycle time T* and the optimal price discount r* to minimize his/her total related cost TCi(T, r). The problem is to minimize 8 < TC1 ðT; rÞ; when T P t 1 TCi ðT; rÞ ¼ TC2 ðT; rÞ; when t 2 6 T 6 t 1 which is a three: TC3 ðT; rÞ; when T 6 t2 branch function with two variables. The optimal TCi (T, r) occurs at the minimum point defined by the function of TC1(T, r), TC2(T, r) or TC3(T, r). In other words, the minimum value of TC1(T, r), TC2(T, r) and TC3(T, r) will be the solution. To solve this problem, we first find the closed form of the retail price ri(T), which minimizes TCi(T, r). Substituting ri(T) into the corresponding TCi(T, r), the model can be reduced to a three-branch function, with T as its single variable. We have already verified that TCi(T, r) is continuous at T = t1 and T = t2, from the fact that TC1(t1, r) = TC2(t1, r) and TC2(t2, r) = TC3(t2, r). When we search for T directly, the minimum value of TCi(T, r) is determined by the local minimum points or the boundary points of T. We need to check whether T is found within the valid range, i.e., T P t1, t2 6 T 6 t1 or T 6 t2. We select the optimal T* and r* such that TCi(T, r) = Min{TC1(T, r), TC2(T, r),TC3(T, r)}. We assume that Y1(r) = a r and Y2(r) = b r in this research. This indicates that when the price discount r increases, this will result
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in a higher of the fraction of retailer 1’s customers who use advance sales discount program and the fraction of other retailers’ customers who switch to retailer 1 under advance sales discount program. The second derivatives of TCi(T, r) with respect to r is
o2 TCi ðT; rÞ ¼ 2Ie Pðt 1 t 2 ÞðaD1 þ bD2 Þ: or 2
ð16Þ
It is obvious that Eq. (16) is larger than zero. Therefore, for a fixed T, TCi(T, r) is a convex function of r, and there exists a unique value, ri, which minimizes the cost. The following is calculated by setting oTCi ðT;rÞ to be zero: or
aD1 ½HT2 þ CIpðT t1 Þ2 þ IePð2T t1 t2 Þðt1 t2 Þ þ 2bD2 IePTðt1 t2 Þ ; 4IePTðt 1 t 2 ÞðaD1 þ bD2 Þ ð17Þ aD1 ½HT2 þ IePðT t2 Þ2 þ 2bD2 IePTðt1 t2 Þ r 2 ðTÞ ¼ ð18Þ ; 4IePTðt 1 t 2 ÞðaD1 þ bD2 Þ aD1 HT þ 2bD2 IePðt1 t2 Þ r 3 ðTÞ ¼ ð19Þ : 4IePðt 1 t2 ÞðaD1 þ bD2 Þ r 1 ðTÞ ¼
minimum points in T1, T2 and T3 falling into the valid interval T P t1, t2 6 T 6 t1 or T 6 t2, respectively, are kept for further analysis. We also need to consider the boundary points t1 or t2 for T. T 1 , T 2 and T 3 are the point for which TC1(T, r1(T)), TC2(T, r2(T)) or TC3 (T, r3(T)) are the smallest among these points. TC*(T, r) is determined by Min{TC1(T, r),TC2(T, r),TC3(T, r)}. We arrive at the following proposition: Proposition 2. We need to consider only boundary point T = t2 when searching for the T which minimizes TC3(T, r3(T)). Proof. Given two very small positive values e1 and e2, where e1 ? 0, e2 ? 0 and e1 < e2, because according to Eqs. (3) and (19), we know that
TC3 ðe1 ; r3 ðe1 ÞÞ
A þ ½ðY 1 ðr 3 ðe1 ÞÞ D1 þ Y 2 ðr 3 ðe1 ÞÞ D2 ÞPð1 r 3 ðe1 ÞÞ e1 þ ð1 Y 1 ðr3 ðe1 ÞÞÞ D1 PIeðt 1 t 2 Þ
The value ri(T) can be either negative or too big (ri(T) > 1) to be managerially acceptable. In order to avoid this type of situation, we introduce upper and lower bounds to ri(T), i.e., 0 < ri(T) < 1 should hold. We utilize Proposition 1 to show the condition sufficient for the optimal price discount within the acceptable interval.
and
Proposition 1
2bD2 IePðt 1 t 2 Þ where r 3 ðe1 Þ r3 ðe2 Þ 4IePðt . 1 t 2 ÞðaD1 þbD2 Þ Since e1 < e2, we know that TC3 (e1, r3(e1)) > TC3(e2,r3 (e2)). This means that when T is very close to 0, TC3(T, r3(T)) decreases as T increases. Therefore, to minimize TC3(T, r3(T)) when searching for T we need not consider boundary point values close to 0. h
(a) If aD1[HT2 + CIp(T t1)2 IeP(2T + t1 + t2) (t1 t2)] + 2b D2IePT(t1 t2) > 0, there exists a unique optimal solution r1 ðTÞ to oTC1orðT;rÞ ¼ 0, which satisfies 0 < r 1 ðTÞ < 1. (b) If aD1 fHT2 þ IeP½T 2 2Tðt 1 t 2 Þ t 22 g þ 2bD2 IePTðt 1 t2 Þ > 0, there exists a unique optimal solution r 2 ðTÞ to oTC2 ðT;rÞ ¼ 0, which satisfies 0 < r2 ðTÞ < 1. or (c) If 12 aD1 HT þ IePðt1 t2 Þð2aD1 þ bD2 Þ > 0, there exists a unique optimal solution r 3 ðTÞ to oTC3orðT;rÞ ¼ 0, which satisfies 0 < r 3 ðTÞ < 1.
1 1 ðT;rÞ ¼ 2T faD1 ½HT2 þ CIpðT t 1 Þ2 IeP Proof. (a) Let GðrÞ ¼ dTCdr ð2T þ 4rT þ t1 þ t 2 Þðt 1 t 2 Þ þ 2bD2 IePTð2r 1Þðt1 t2 Þg. Since 2 according to Eq. (16) o TCori 2ðT;rÞ ¼ 2Ie Pðt1 t2 ÞðaD1 þ bD2 Þ > 0, then TC1(t1,r) is convex in r. Hence, G(r) is a increasing function in r. When r = 0 or 1, G(r) can be obtained by
þCIpðTt 1 Þ2 þIePð2Tt 1 t 2 Þðt 1 t 2 Þ2bD2 IePTðt 1 t 2 Þ < 2T aD1 ½HT2 þCIpðTt 1 Þ2 IePð2Tþt1 þt 2 Þðt 1 t 2 Þþ2bD2 IePTðt 1 t 2 Þ :Gð0Þ 2T 2
Gð0Þ ¼ aD1 ½HT
0;
TC3 ðe2 ; r3 ðe2 ÞÞ
þ ð1 Y 1 ðr3 ðe2 ÞÞÞ D1 PIeðt 1 t 2 Þ;
Based on the above analysis and Proposition 2, the following algorithm can be used to determine the optimal values for T*. Algorithm 1 Step 1:
Step 2:
Step 3:
and
<0 is Gð1Þ ¼ always satisfied because T > t1 > t2 in Case 1. If G(1) > 0, this implies that there exists a r 1 ðTÞ 2 ð0:1Þ such that Gðr1 ðTÞÞ ¼ 0. This completes the proof. The proofs of (b) and (c) are similar to that in (a). h After obtaining the optimal price discount ri(T) we now want to find the optimal value of cycle time T for each branch. Since we do not know which branch, TC1(T, r), TC2(T, r)or TC3 (T, r), will determine the optimal solution for TCi(T, r), we need to substitute the optimal price r1(T) to TC1(T, r), r2 (T) to TC2(T, r) and r3(T) to TC3(T, r). Each of the three branches becomes a function with a single variable. We then, respectively, determine the optimal TC1 ðT; rðTÞÞ, TC2 ðT; rðTÞÞ and TC3 ðT; rðTÞÞ for each branch. After this we select the TCi(T, r(T)) = Min{TC1(T, r(T)),TC2(T, r(T)),TC3(T, r(T))}. To minimize TC1(T, r1(T)), TC2(T, r2(T)) or TC3(T,r3 (T)) we first find T 1 , T 2 and T 3 , and then select the best one. A graphic illustration of TCi for Examples 1–3 discussed in Section 6.1 are shown in Fig. 2. Please note that TC(T, r) is continuous at T=t2 and T = t1, and that the ri(T) in TCi(T, ri(T)) can be determined by looking at T in Eqs. (17), (18) or (19). To find T i , i = 1, 2, 3, we first find all stationary i ðTÞÞ ¼ 0. The local minimum points of TCi(T, ri(T)) by solving dTCi ðT;r dT 2 ðT;r i ðTÞÞ > 0. The local points are determined by verifying that d TCidT 2
A þ ½ðY 1 ðr 3 ðe2 ÞÞ D1 þ Y 2 ðr 3 ðe2 ÞÞ D2 ÞPð1 r 3 ðe2 ÞÞ e2
Step 4:
Step 5:
Step 6:
Step 7:
Determine local minimum points by solving for 2 dTC1 ðT;r 1 ðTÞÞ ðT;r 1 ðTÞÞ ¼ 0 and by verifying that d TC1dT > 0, such 2 dT that the T satisfies T P t1 with respect to each local minimum point. Let TC1 ðT 1 ; rÞ associate with the local minimum point or the boundary point t1 which gives the smallest value of TC1(T1, r). Determine local minimum points by solving for 2 dTC2 ðT;r 2 ðTÞÞ ðT;r 2 ðTÞÞ ¼ 0 and by verifying that d TC2dT > 0, such 2 dT that the T satisfies t2 6 T 6 t1 with respect to each local minimum point. Let TC2 ðT 2 ; rÞ associate with the local minimum point or any of two boundary points t1 and t2 which gives the smallest value of TC2(T2, r). Determine local minimum points by solving for 2 dTC3 ðT;r 3 ðTÞÞ ðT;r 3 ðTÞÞ ¼ 0 and by verifying that d TC3dT > 0, such 2 dT that the T satisfies T 6 t2 with respect to each local minimum point. Let TC3 ðT 3 ; rÞ associate with the local minimum point or the boundary point t2 which gives the smallest value of TC3(T3,r). Let TC ðT ; r Þ ¼ MinfTCs1 ðT 1 ; rÞ; TC2 ðT 2 ; rÞ; TC3 ðT 3 ; rÞg.
6. Computational analyses The purposes of the numerical analysis are as follows: 1. To illustrate the solution procedure. 2. To use sensitivity analysis to highlight the influence of the length of the credit period, the ordering cost and the holding cost.
Y.-C. Tsao / Computers & Industrial Engineering 56 (2009) 208–215
6.1. Numerical examples To illustrate the solution procedures, we consider the following numerical examples: Example 1. Consider A = 100 dollars/order, P = 6 dollars/U, C = 3 dollars/U, H = 0.08 dollars/U/year, Ip = 0.15 per dollars, Ie = 0.1 per dollars, t1 = 0.3 years, t2 = 0.15 years, D1 = 3000 U/year, D2 = 3000 U/year, Y1 = 1.02 r, Y2 = 1.01 r. Since 2A þ ð1 Y 1 Þ D1 ½C Ip t21 Ie Pðt21 t22 Þ ¼ 200 > 0, then we can utilize Theorem 1 to get the optimal cycle time T* = T1 = 0.374 year, the total related cost TC1 = 121.363 dollars. If r is a decision variable, we utilize Algorithm to get the optimal cycle time T* = T1 = 0.505 year, the
700
TC3 (T , r 3 (T ))
600 500 400
TC2 (T , r 2 (T ))
300
optimal price discount r* = 0.497 and the total related cost TC1 = 59.262 dollars. A graphic representation of TCi is shown in Fig. 3(a). Example 2. Consider A = 50 dollars/order, P = 6 dollars/U, C = 3 dollars/U, H = 0.4 dollars/U/year, Ip = 0.15 per dollars, Ie = 0.1 per dollars, t1 = 0.3 years, t2 = 0.15 years, D1 = 3000 U/year, Y1 = 1.02 r, Y2 = 1.01 r. Since D2 = 3000 U/year, 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t22 Þ ¼ 100 > 0, then we can utilize Theorem 1 to get the optimal cycle time T* = T2 = 0.225 year, the total related cost TC1 = 71.868 dollars. If r is a decision variable, we utilize Algorithm to get the optimal cycle time T* = T2 = 0.274 year, the optimal price discount r* = 0.448 and the total related cost TC2 = 17.020 dollars. A graphic representation of TCi is shown in Fig. 3(b). Example 3. Consider A = 20 dollars/order, P = 6 dollars/U, C = 3 dollars/U, H = 1.3 dollars/U/year, Ip = 0.15 per dollars, Ie = 0.1 per dollars, t1 = 0.3 years, t2 = 0.15 years, D1 = 8000 U/year, Y1 = 1.02 r, Y2 = 1.01 r. Since D2 = 8000 U/year, 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t22 Þ ¼ 60 > 0, then we can utilize Theorem 1 to get the optimal cycle time T* = T3 = 0.107 year, the total related cost TC3 = 82.494 dollars. If r is a decision variable, we utilize Algorithm to get the optimal cycle time T* = T3 = 0.147 year, the optimal price discount r* = 0.516 and the total related cost TC2 = 7.056 dollars. A graphic representation of TCi is shown in Fig. 3(c).
(a) Example 1 TCi
213
200
TC1 (T , r 1 (T ))
100 0.2
t2
0.4
6.2. Effects of credit period, ordering cost and holding cost
0.6
T
0.8
t1
(b) Example 2 250
TCi 200
TC3 (T , r 3 (T ))
150 100
TC2 (T , r 2 (T ))
TC1 (T , r 1 (T ))
50
t2
0.2
0.3
0.4
t1
0.5
T
(c) Example 3
It is also important to discuss the influence of the credit period, the ordering cost, and the holding cost. The computational results shown in Fig. 4 indicate the following managerial phenomena: (1) When the supplier provides a longer credit period t1, the retailer will replenish the goods more often. In other words the retailer will shorten the replenishment cycle time to take advantage of the longer credit period. (2) When the retailer provides a longer credit period t2, the retailer’s replenishment cycle time will be increased. This implies the retailer will replenish the goods not often to decrease the loss. An examination of Fig. 5 gives the following the results: (1) When the holding cost H increases, the retailer will increase the price discount and shorten the replenishment cycle time. It is reasonable that when the holding cost increases the retailer will then shorten the cycle time in an effort to lower his inventory cost. (2) When the ordering cost A increases, the retailer will increase his price discount and the replenishment cycle time to reflect this. If the ordering cost increases, it is reasonable that the retailer lengthens the cycle time to reduce the frequency of replenishment.
TCi 200 7. Conclusion
TC3 (T , r 3 (T ))
150
TC1 (T , r 1 (T )) 100
TC2 (T , r2 (T ))
50
0.05
0.1
0.15
t2
0.2
0.25
Fig. 3. Graphic illustration of TCi versus T.
0.3
t1
0.35
T
This paper considers retailer’s promotion and replenishment policies with advance sales discount under supplier’s and retailer’s trade credits. We calculate the interest earned based on retail price instead of purchase cost in most previous researches. The first model deals with the replenishment policy when promotional effort is predetermined. In the second model, we present an algorithm to determine the optimal promotional effort and replenishment cycle time simultaneously. Several examples are given to illustrate the solution procedures and discuss the impact of various system parameters. We conclude with a computational analysis that leads to variety of managerial insights.
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(a) Effect of supplier's credit period t 1 250 200 Total cost
Replenishment cycle time
0.386 0.384 0.382 0.38 0.378 0.376 0.374 0.372 0.37 0.368
150 100 50
0.25
0.275
0
0.3
0.25
Supplier's credit period
0.275
0.3
Supplier's credit period
0.41 0.4 0.39 0.38 0.37 0.36 0.35 0.34 0.33
200 150 Total cost
Replenishment cycle time
(b) Effect of retailer’s credit period t 2
100 50
0.1
0.15
0
0.2
0.1
Retailer's credit period
0.15
0.2
Retailer's credit period
Fig. 4. Effects of credit period.
0.385
135
0.38
130 125
0.375
Total cost
Replenishment cycle time
(a) Effect of holding cost
0.37 0.365
115 110
0.36 0.355
120
105 0.06
0.08
100
0.1
0.06
0.08
0.1
Holding cost
Holding cost
0.4
160
0.39
150
0.38
140 Total cost
Replenishment cycle time
(b) Effect of ordering cost
0.37 0.36
130 120 110
0.35
100
0.34
90
0.33
90
100
110
80
90
Orderinging cost
100
110
Orderinging cost
Fig. 5. Effects of holding cost and ordering cost.
For further research, this paper can be extended to consider other realistic situations, such as for deteriorating item, allowable shortages, permissible order cancellation or stochastic demand.
References Aggarwal, S. P., & Jaggi, C. K. (1995). Ordering policies of deteriorating items under permissible delay in payments. Journal of the Operational Research Society, 46, 658–662.
Y.-C. Tsao / Computers & Industrial Engineering 56 (2009) 208–215 Chang, H. J., & Dye, C. Y. (2001). An inventory model for deteriorating items with partial backlogging and permissible delay in payments. International Journal of Systems Science, 32, 345–352. Chang, H. J., Hung, C. H., & Dye, C. Y. (2001). An inventory model for deteriorating items with linear demand under condition of permissible delay in payments. Production Planning and Control, 12, 274–282. Chang, H. J., Hung, C. H., & Dye, C. Y. (2002). A finite time horizon inventory model with deterioration and time-value of money under the conditions of permissible delay in payments. International Journal of Systems Science, 33, 141–151. Chang, C. T., Ouyang, L. Y., & Teng, J. T. (2003). An EOQ model for deteriorating items under supplier credits linked to ordering quantity. Applied Mathematical Modeling, 27, 983–996. Chapman, C. B., Ward, S. C., Cooper, D. F., & Page, M. J. (1984). Credit policy and inventory control. Journal of the Operational Research Society, 35, 1055–1065. Chu, P., Chung, K. J., & Lan, S. P. (1998). Economic order quantity of deteriorating items under permissible delay in payments. Computers and Operations Research, 25, 817–824. Chung, K. J. (1998). A theorem on determination of economic order quantity under conditions of permissible in payments. Computers and Operations Research, 25, 49–52. 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. Chung, K. J., & Liao, J. J. (2004). Lot-sizing decisions under trade credit depending on the ordering quantity. Computers and Operations Research, 31, 909–928.
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Chung, K. J., Goyal, S. K., & Huang, Y. F. (2005). The optimal inventory policies under permissible delay in payments depending on ordering quantity. International Journal of Production Economics, 95, 203–213. Daellenbach, H. G. (1986). Inventory control and trade credit. Journal of the Operational Research Society, 37, 525–528. Goyal, S. K. (1985). Economics order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36, 335–338. Haley, C. W., & Higgins, R. C. (1973). Inventory policy and trade credit financing. Management Science, 20, 464–471. 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. 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. Jamal, A. M. M., Sarker, B. R., & Wang, S. (2000). Optimal payment time for a retailer under permitted delay of payment by the wholesaler. International Journal of Production Economics, 66, 59–66. Rachamadugu, R. (1989). Effect of delayed payments (trade credit) on order quantities. Journal of the Operational Research Society, 40, 805–813. Shinn, S. W., & Hwang, H. (2003). Optimal pricing and ordering policies for retailers under order size-dependent delay in payments. Computers and Operations Research, 30, 35–50. 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.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Retailer’s optimal ordering and discounting policies under advance sales discount and trade credits Yu-Chung Tsao * Department of Business Management, Tatung University, No. 300, Jhongda Road, Taipei 104, Taiwan
a r t i c l e
i n f o
Article history: Received 8 September 2007 Received in revised form 13 April 2008 Accepted 21 May 2008 Available online 28 May 2008 Keywords: EOQ Ordering Discounting Trade credits Advance sales discount Two-echelon supply chain
a b s t r a c t Providing advance sales discount program to attract consumers or eliminate demand uncertainty is commonly in today’s business, such as music disk, apparel, books, etc. Also, trade credit is a popular payment behavior in B2B and B2C transactions. In this research, we use EOQ to model the decisions under advance sales discount and two-echelon trade credits, which involve how much to order and how much to discount to minimize the total related cost. Several examples are given to illustrate the solution procedures and discuss the impact of various system parameters. We conclude with a computational analysis that leads to variety of managerial insights. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction To date, companies have provided several activities to boost consumer demand. For examples, retailers Wall-Mart and Costco often try to stimulate demand for specific types of electric equipment by offering price discounts; clothiers A&F and American Eagle make shelf space for specific clothes items available for longer periods; fast-food restaurants McDonald’s and Burger King often use coupons to attract consumers. In this paper, we consider such a strategy in which the retailer develops a program called an ‘‘advance sales discount” (ASD) program that entices customers to commit to their orders at a discount price prior to the selling season. This means the retailer offers his/her customers a price discount if they can commit their orders prior to the sales period. Due to the progress of information technique, many companies provide on-line transaction to their customers on internet. This accelerates the use of ASD program. Now customers can commit their orders on-line before the sales period to get the price discount. For companies, the ASD strategy is common and useful in reality to decrease the estimation error in demand and to increase the market share. Also, customers can save their money if they make sure that they need the product earlier. In addition, traditional EOQ model tacitly assumes that the payment must be made to the supplier for the items immediately after the retailer receive the products. However, such an assumption is not quite practical in the real world. In practice, the supplier allows * Tel.: +886 2 25925252 2435 19. E-mail addresses: [email protected], [email protected] 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.05.006
the retailer a credit period for settling the amount owed and does not charge any interest on the amount owed during this period. Over years, a number of researches have been published which dealt with the inventory model under permissible delay in payments. Haley and Higgins (1973), Chapman, Ward, Cooper, and Page (1984), Goyal (1985), Daellenbach (1986) and Rachamadugu (1989) examined the effect of the credit period on the optimal inventory policy. Recently, Chung (1998) simplified the search of the optimal solution for the problem explored by Goyal (1985). Jamal, Sarker, and Wang (2000) developed a model for optimal cycle and payment time with deteriorating products. Teng (2002) then amended Goyal (1985) model by considering the difference between unit price and unit cost. For deteriorating items, Aggarwal and Jaggi (1995) and Chu et al. (1998) researched the ordering policy for deteriorating items under permissible delay in payment. Jamal, Sarker, and Wang (1997) generalized Aggarwal and Jaggi (1995) model to allow for shortage. Chang and Dye (2001) extended the model of Jamal et al. (1997) to allow for a varying deteriorating rate of time and partial backlogging rate. Chang, Hung, and Dye (2001) considered the linear demand for deteriorating items under credit period. Then Chang, Hung, and Dye (2002) developed a finite time horizon inventory model with deterioration and time-value of money when payment periods are offered. Chung and Huang (2003) considered the case that the units are replenished at a finite rate under permissible delay in payments. Recently, Chang, Ouyang, and Teng (2003), Shinn and Hwang (2003), Chung and Liao (2004) and Chung et al. (2005) considered the inventory model under the condition of order-size-dependent credit period. All these above researches assumed the supplier pro-
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vided a credit period to the retailer, but the customer must pay for the goods as soon as the goods are received. In most business transactions, the retailer may adopt the credit period policy to stimulate his customer demand. Huang (2003) considered not only the supplier offers a credit period to the retailer, but also the retailer offers a credit period to his customers. In this note, this paper both considers the condition of advance sales discount and two-echelon trade credits in a one supplier-one retailer supply chain. In this model, the retailer offers his/her customers a price discount if they can commit their orders prior to the sales period. The supplier offers a credit period to the retailer and the retailer offers a credit period to his customers. To the best of our knowledge, this research is the first to incorporate the conditions of permissible delay in payments and advance sales discount into EOQ model. This research not only deals with the inventory replenishment decisions but also determines the discounting policy in the ASD program. The objective is to determine the optimal replenishment cycle time and the optimal price discount while minimizing the total related cost. We conclude with a computational analysis that leads to variety of managerial insights. The remainder of this paper is given as follows. First, we describe the assumptions and notations. We formulate the model in Section 3. In Section 4, we develop theorems based on lemmas to determine the optimal replenishment cycle time when price discount is exogenous. Then we develop an algorithm based on properties to determine ordering and discounting decisions simultaneously in Section 5. In Section 6, some computational analyses with respect to various system parameters are also conducted. Finally, some conclusions are made. 2. Notations and assumptions The following notation is used in this study: unit retail price unit purchase cost ordering cost per order unit inventory holding cost per unit time retailer’s credit period provided by supplier customer’s credit period provided by retailer, where t2 < t1 price discount, 0 6 r < 1 the interest paid per dollar per unit time, Ip > 0 the interest earned per dollar per unit time, Ie > 0 replenishment cycle time annual demand for the retailer, we call the retailer in this paper retailer 1 annual demand for other retailers, i.e., not retailer 1 D2 fraction of retailer 1’s customers who use advance sales Y1 discount program, where Y1 is a function of r, i.e., Y1 (r) fraction of other retailers’ customers who switch to retailer Y2 1 under advance sales discount program, where Y2 is a function of r, i.e., Y2 (r). Y1D1 + Y2D2 annual demand of customers who use advance sales discount program, where Y2D2 is the increasing demand when provide ASD program (Fig. 1) (1 Y1)D1 annual demand of customers who are not use advance sales discount program (Fig. 1) D total annual demand, D = Y1D1 + Y2D2 + (1 Y1)D1 TCi total related cost, i = 1 when T P t1, i = 2 when t2 6 T 6 t1, i = 3 when T 6 t2.
P C A H t1 t2 r Ip Ie T D1
The mathematical model in this paper is developed on the following assumptions: 1. We consider the problem of single product. 2. The retailer offers a price discount r to his customers if they can commit their orders prior to the sales period. 3. Y1 percentage of the retailer’s (retailer 1’s) customers use advance sales discount program and Y2 percentage of other retailers’ customers use advance sales discount program.
D1 1 − Y1
D2 Y1
Y2
(1 − Y1 ) D1
Y1 D1 + Y2 D2
Not use ASD program
Use ASD program
Fig. 1. The impact of ASD program on end customer demand.
4. All customers (include who use ASD program) should settle his/ her payment at t2. 5. The unit retail price of the products sold during the period t2 to t1 is deposited in an interest bearing account with rate Ie. At the end of t1, the credit is settled and the retailer starts paying for the interest charges for the items in stocks with rate Ip. 6. Since the retailer cannot earn any interest in this situation t2 > t1, it is reasonable to assume that the retailer’s credit period provided by supplier t1 is longer than the customer’s credit period provided by retailer t2. (Huang (2003) made the same assumption.) 3. Model formulation The total related cost consists of the following elements: (1) (2) (3) Case 1:
Annual ordering cost = A/T. Annual inventory holding cost = (1 Y1)D1TH/2. There are three cases occurred in interest earned per year. When T P t1 From Assumption 5, we know the retailer can accumulate revenue and earn interest during the period t2 to t1 with rate Ie. There are two parts in the annual interest earned. The first part is the revenue from selling item during the PIeð1Y 1 ÞD1 ðt 21 t 22 Þ . The second part is the revsales period, i.e., 2T enue from selling item before the sales period (under ASD program), i.e., (Y1D1 + Y2D2)P(1 r)Ie(t1 t2) (Fig. 2(a)).
Annual interest earned ¼
P Ie ð1 Y 1 ÞD1 ðt 21 t22 Þ 2T þ ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt1 t2 Þ
Case 2: When t2 6 T 6 t1 The first part is the revenue from2 selling item during the PIeð1Y 1 ÞD1 ð2Tt1 T t 22 Þ . The second part is sales period, i.e., 2T the revenue from selling item before the sales period (under ASD program), i.e., (Y1D1 + Y2 D2)P(1 r)Ie(t1 t2) (Fig. 2(b)).
Annual interest earned ¼
P Ie ð1 Y 1 ÞD1 ð2Tt 1 T 2 t22 Þ 2T þ ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt 1 t2 Þ
Case 3: When T 6 t2 The first part is the revenue from selling item during the sales period, i.e., [(1 Y1)D1]P(1 r)Ie(t1 t2). The second part is the revenue from selling item before the sales period (under ASD program), i.e., (Y1D1 + Y2D2) P(1 r)Ie(t1 t2) (Fig. 2(c)).
Annual interest earned ¼ ½ðY 1 D1 þ Y 2 D2 ÞPð1 rÞ þ ð1 Y 1 ÞD1 P Ie ðt 1 t 2 Þ (4) There are three cases occurred in interest paid per year. Case 1: When T P t1
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Fig. 2. The total accumulation of interest earned.
Annual interest paid ¼
C Ip ð1 Y 1 ÞD1 ðT t1 Þ2 2T
Case 2: When t2 6 T 6 t1
Case 2: When t2 6 T 6 t1 Annual interest charge = 0. In this case, no interest charge is paid for the items. Case 3: When T 6 t2 Annual interest charge = 0. In this case, no interest charge is paid for the items. Therefore, total related cost TCi has three different expressions as follows: Case 1: When T P t1
C Ip ð1 Y 1 ÞD1 ðT t 1 Þ2 ; 2T
A ð1 Y 1 ÞD1 TH P Ie ð1 Y 1 ÞD1 ð2Tt 1 T 2 t 22 Þ þ T 2 2T ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt1 t 2 Þ;
ð2Þ
Case 3: When T 6 t2
TC3 ¼
A ð1 Y 1 ÞD1 TH þ ½ðY 1 D1 þ Y 2 D2 ÞPð1 rÞ þ ð1 T 2 Y 1 ÞD1 PIeðt1 t2 Þ:
ð3Þ
4. Determination of optimal ordering policy
A ð1 Y 1 ÞD1 TH P Ie ð1 Y 1 ÞD1 ðt21 t22 Þ TC1 ¼ þ T 2 2T ðY 1 D1 þ Y 2 D2 ÞPð1 rÞIeðt 1 t2 Þ þ
TC2 ¼
ð1Þ
In this section, the problem is to find an optimal cycle time T* while minimizing TCi, where TCi(T) = TC1(T) when T P t1; TCi(T) = TC2(T) when t2 6 T 6 t1 and TCi(T) = TC3(T) when T 6 t2. The first-order and second-order derivative of TC1(T), TC2(T) and TC3(T) with respective to T are as follows:
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dTC1 ðTÞ ð1 Y 1 ÞD1 ½HT2 þ CIpðT 2 t21 Þ þ IePðt21 t22 Þ 2A ; ¼ dT 2T 2
ð4Þ
dTC2 ðTÞ ð1 Y 1 ÞD1 ½HT2 þ IePðT 2 t 22 Þ 2A ; ¼ dT 2T 2
ð5Þ
dTC3 ðTÞ ð1 Y 1 ÞD1 HT2 2A ; ¼ dT 2T 2
ð6Þ
2
d TC1 ðTÞ dT 2 d TC2 ðTÞ
¼
2
2
dT 2 d TC3 ðTÞ dT 2
¼ ¼
2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ T3 2A þ ð1 Y 1 ÞD1 P Ie t 22 T 2A T3
3
ð7Þ
;
> 0;
ð8Þ
> 0:
ð9Þ
Eqs. (8) and (9) imply that TC2(T) and TC3(T) are convex on T > 0. Eq. is concave on T>0 if (7) implies that TC1(T) 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt21 t 22 Þ > 0. From TC1(t1) = TC2(t1) and TC2 (t2) = TC3(t2), TCi(T) is continuous and defined on T > 0. Therefore, we have the following lemma: Lemma 1 (a) If 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t22 Þ > 0, Pi(T) is concave on (0, 1). (b) If 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t 22 Þ 6 0, Pi(T) is concave on (0, t1] and convex on [t1, 1).
Proof. From Lemma 1 and the feasible cycle time interval (Eqs. (13)–(15)), it can be proved. h Then we discuss how to determine the optimal replenishment cycle time under the situation when 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt21 t 22 Þ 6 0 If 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ 6 0, we consider the following lemmas first. Lemma 3. If 2A þ qð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ 6 0, then T 2 < t 1 . rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Aþð1Y ÞD t 2 PIe
1 1 2 Proof. Assuming T 2 P t 1 , this means P t1 . Arrange ð1Y 1 ÞD1 ðHþPIeÞ the inequality, we obtain 2A þ ð1 Y 1 ÞD1 ðt22 t21 ÞPIe P
t21 ð1 Y 1 ÞD1 H. Then adding ð1 Y 1 ÞD1 CIpt21 into both sides, we obtain 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t 22 Þ P t21 ð1 Y 1 Þ D1 ðH þ CIpt21 Þ > 0. This is contradiction. Therefore, T 2 < t1 .
h
Lemma 4. T 2 6 t 2 if and only if T 3 6 t2 . Proof. If T 2 6 t2 , from Eq. (11), we get 2A 6 t22 ð1 Y 1 ÞD1 H. Then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we have ð1Y2A 6 t 2 , this means T 3 6 t 2 . Similarly, if T 3 6 t 2 , 1 ÞD1 H Eq. (12) implies 2A 6 t22 ð1 Y 1 ÞD1 H. Adding ð1 Y 1 ÞD1 t22 PIe into both sides, we obtain 2A þ ð1 Y 1 ÞD1 t 22 PIe 6 t 22 ð1 Y 1 ÞD1 Hþ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 Y 1 ÞD1 t22 PIe. Then we have
2Aþð1Y 1 ÞD1 t 22 PIe ð1Y 1 ÞD1 ðHþPIeÞ
6 t 2 , i.e., T 2 6 t 2 .
h
Then we develop Theorem 2 based on above analysis. First, we discuss how to determine the optimal replenishment cycle time under the situation when 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t 22 Þ > 0. If 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t 22 Þ > 0, we can obtain the optimal replenishment cycle time for each case by solving dTCdTi ðTÞ ¼ 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A þ ð1 Y 1 ÞD1 ½t 21 ðCIp PIeÞ þ t 22 PIe ; ¼ ð1 Y 1 ÞD1 ðH þ CIpÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A þ ð1 Y 1 ÞD1 t22 PIe ; T 2 ¼ ð1 Y 1 ÞD1 ðH þ PIeÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A T3 ¼ : ð1 Y 1 ÞD1 H
T 1
Proof. From Lemmas 3 and 4 and the feasible cycle time interval (t2 6 T 6 t1 in case 2 and T 6 t2 in case 3), it can be proved. h
ð11Þ 5. Determination of optimal ordering and discounting policies
ð12Þ
T P t 1 ) 2A þ ð1 Y 1 ÞD1 ½PIeðt22 t 21 Þ t 21 H P 0; t 2 6 T 6 t1 ) 2A þ ð1
t 21 Þ
6 2A 1 Y 1 D1 Ht 22 ; T 6 t2 ) 2A ð1
Y 1 ÞD1 Ht22
t 21 H
ð13Þ
60 ð14Þ
6 0:
Y 1 ÞD1 ½PIeðt 22
t 21 Þ
Let f1 ¼ 2A þ ð1 D1 Ht22 , we have the following lemma:
ð15Þ t 21 H
and f2 ¼ 2A ð1 Y 1 Þ
Lemma 2. f2 P f1. Proof. It can be proved from f2 f1 ¼ ð1 Y 1 ÞD1 ½PIeðt 21 t 22 Þ þ Hðt21 t 22 Þ P 0 since t1 > t2 and 0 6 Y 6 11. h We develop the following theorem to determine the optimal cycle time T* based on above analysis. Theorem 1. When 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt21 t 22 Þ > 0, (a) If f2 P f1 P 0, then T ¼ T 1 . (b) If f2 P 0 P f1, then T ¼ T 2 . (c) If 0 P f2 P f1, then T ¼ T 3 .
(a) If T 2 < t2 , then T ¼ T 3 . (b) If T 2 > t2 , then T ¼ T 2 . (c) If T 2 ¼ t2 , then T ¼ T 2 ¼ T 3 .
ð10Þ
Then from the relation T P t1 in case 1, t2 6 T 6 t1 in case 2 and T 6 t2 in case 3, and Eqs. (10)–(12), we have
Y 1 ÞD1 ½PIeðt 22
Theorem 2. When 2A þ ð1 Y 1 ÞD1 ½C Ip t 21 Ie Pðt 21 t22 Þ 6 0,
In this section, we consider that the retail price r is endogenous, i.e., the retailer wants to determine the optimal replenishment cycle time T* and the optimal price discount r* to minimize his/her total related cost TCi(T, r). The problem is to minimize 8 < TC1 ðT; rÞ; when T P t 1 TCi ðT; rÞ ¼ TC2 ðT; rÞ; when t 2 6 T 6 t 1 which is a three: TC3 ðT; rÞ; when T 6 t2 branch function with two variables. The optimal TCi (T, r) occurs at the minimum point defined by the function of TC1(T, r), TC2(T, r) or TC3(T, r). In other words, the minimum value of TC1(T, r), TC2(T, r) and TC3(T, r) will be the solution. To solve this problem, we first find the closed form of the retail price ri(T), which minimizes TCi(T, r). Substituting ri(T) into the corresponding TCi(T, r), the model can be reduced to a three-branch function, with T as its single variable. We have already verified that TCi(T, r) is continuous at T = t1 and T = t2, from the fact that TC1(t1, r) = TC2(t1, r) and TC2(t2, r) = TC3(t2, r). When we search for T directly, the minimum value of TCi(T, r) is determined by the local minimum points or the boundary points of T. We need to check whether T is found within the valid range, i.e., T P t1, t2 6 T 6 t1 or T 6 t2. We select the optimal T* and r* such that TCi(T, r) = Min{TC1(T, r), TC2(T, r),TC3(T, r)}. We assume that Y1(r) = a r and Y2(r) = b r in this research. This indicates that when the price discount r increases, this will result
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in a higher of the fraction of retailer 1’s customers who use advance sales discount program and the fraction of other retailers’ customers who switch to retailer 1 under advance sales discount program. The second derivatives of TCi(T, r) with respect to r is
o2 TCi ðT; rÞ ¼ 2Ie Pðt 1 t 2 ÞðaD1 þ bD2 Þ: or 2
ð16Þ
It is obvious that Eq. (16) is larger than zero. Therefore, for a fixed T, TCi(T, r) is a convex function of r, and there exists a unique value, ri, which minimizes the cost. The following is calculated by setting oTCi ðT;rÞ to be zero: or
aD1 ½HT2 þ CIpðT t1 Þ2 þ IePð2T t1 t2 Þðt1 t2 Þ þ 2bD2 IePTðt1 t2 Þ ; 4IePTðt 1 t 2 ÞðaD1 þ bD2 Þ ð17Þ aD1 ½HT2 þ IePðT t2 Þ2 þ 2bD2 IePTðt1 t2 Þ r 2 ðTÞ ¼ ð18Þ ; 4IePTðt 1 t 2 ÞðaD1 þ bD2 Þ aD1 HT þ 2bD2 IePðt1 t2 Þ r 3 ðTÞ ¼ ð19Þ : 4IePðt 1 t2 ÞðaD1 þ bD2 Þ r 1 ðTÞ ¼
minimum points in T1, T2 and T3 falling into the valid interval T P t1, t2 6 T 6 t1 or T 6 t2, respectively, are kept for further analysis. We also need to consider the boundary points t1 or t2 for T. T 1 , T 2 and T 3 are the point for which TC1(T, r1(T)), TC2(T, r2(T)) or TC3 (T, r3(T)) are the smallest among these points. TC*(T, r) is determined by Min{TC1(T, r),TC2(T, r),TC3(T, r)}. We arrive at the following proposition: Proposition 2. We need to consider only boundary point T = t2 when searching for the T which minimizes TC3(T, r3(T)). Proof. Given two very small positive values e1 and e2, where e1 ? 0, e2 ? 0 and e1 < e2, because according to Eqs. (3) and (19), we know that
TC3 ðe1 ; r3 ðe1 ÞÞ
A þ ½ðY 1 ðr 3 ðe1 ÞÞ D1 þ Y 2 ðr 3 ðe1 ÞÞ D2 ÞPð1 r 3 ðe1 ÞÞ e1 þ ð1 Y 1 ðr3 ðe1 ÞÞÞ D1 PIeðt 1 t 2 Þ
The value ri(T) can be either negative or too big (ri(T) > 1) to be managerially acceptable. In order to avoid this type of situation, we introduce upper and lower bounds to ri(T), i.e., 0 < ri(T) < 1 should hold. We utilize Proposition 1 to show the condition sufficient for the optimal price discount within the acceptable interval.
and
Proposition 1
2bD2 IePðt 1 t 2 Þ where r 3 ðe1 Þ r3 ðe2 Þ 4IePðt . 1 t 2 ÞðaD1 þbD2 Þ Since e1 < e2, we know that TC3 (e1, r3(e1)) > TC3(e2,r3 (e2)). This means that when T is very close to 0, TC3(T, r3(T)) decreases as T increases. Therefore, to minimize TC3(T, r3(T)) when searching for T we need not consider boundary point values close to 0. h
(a) If aD1[HT2 + CIp(T t1)2 IeP(2T + t1 + t2) (t1 t2)] + 2b D2IePT(t1 t2) > 0, there exists a unique optimal solution r1 ðTÞ to oTC1orðT;rÞ ¼ 0, which satisfies 0 < r 1 ðTÞ < 1. (b) If aD1 fHT2 þ IeP½T 2 2Tðt 1 t 2 Þ t 22 g þ 2bD2 IePTðt 1 t2 Þ > 0, there exists a unique optimal solution r 2 ðTÞ to oTC2 ðT;rÞ ¼ 0, which satisfies 0 < r2 ðTÞ < 1. or (c) If 12 aD1 HT þ IePðt1 t2 Þð2aD1 þ bD2 Þ > 0, there exists a unique optimal solution r 3 ðTÞ to oTC3orðT;rÞ ¼ 0, which satisfies 0 < r 3 ðTÞ < 1.
1 1 ðT;rÞ ¼ 2T faD1 ½HT2 þ CIpðT t 1 Þ2 IeP Proof. (a) Let GðrÞ ¼ dTCdr ð2T þ 4rT þ t1 þ t 2 Þðt 1 t 2 Þ þ 2bD2 IePTð2r 1Þðt1 t2 Þg. Since 2 according to Eq. (16) o TCori 2ðT;rÞ ¼ 2Ie Pðt1 t2 ÞðaD1 þ bD2 Þ > 0, then TC1(t1,r) is convex in r. Hence, G(r) is a increasing function in r. When r = 0 or 1, G(r) can be obtained by
þCIpðTt 1 Þ2 þIePð2Tt 1 t 2 Þðt 1 t 2 Þ2bD2 IePTðt 1 t 2 Þ < 2T aD1 ½HT2 þCIpðTt 1 Þ2 IePð2Tþt1 þt 2 Þðt 1 t 2 Þþ2bD2 IePTðt 1 t 2 Þ :Gð0Þ 2T 2
Gð0Þ ¼ aD1 ½HT
0;
TC3 ðe2 ; r3 ðe2 ÞÞ
þ ð1 Y 1 ðr3 ðe2 ÞÞÞ D1 PIeðt 1 t 2 Þ;
Based on the above analysis and Proposition 2, the following algorithm can be used to determine the optimal values for T*. Algorithm 1 Step 1:
Step 2:
Step 3:
and
<0 is Gð1Þ ¼ always satisfied because T > t1 > t2 in Case 1. If G(1) > 0, this implies that there exists a r 1 ðTÞ 2 ð0:1Þ such that Gðr1 ðTÞÞ ¼ 0. This completes the proof. The proofs of (b) and (c) are similar to that in (a). h After obtaining the optimal price discount ri(T) we now want to find the optimal value of cycle time T for each branch. Since we do not know which branch, TC1(T, r), TC2(T, r)or TC3 (T, r), will determine the optimal solution for TCi(T, r), we need to substitute the optimal price r1(T) to TC1(T, r), r2 (T) to TC2(T, r) and r3(T) to TC3(T, r). Each of the three branches becomes a function with a single variable. We then, respectively, determine the optimal TC1 ðT; rðTÞÞ, TC2 ðT; rðTÞÞ and TC3 ðT; rðTÞÞ for each branch. After this we select the TCi(T, r(T)) = Min{TC1(T, r(T)),TC2(T, r(T)),TC3(T, r(T))}. To minimize TC1(T, r1(T)), TC2(T, r2(T)) or TC3(T,r3 (T)) we first find T 1 , T 2 and T 3 , and then select the best one. A graphic illustration of TCi for Examples 1–3 discussed in Section 6.1 are shown in Fig. 2. Please note that TC(T, r) is continuous at T=t2 and T = t1, and that the ri(T) in TCi(T, ri(T)) can be determined by looking at T in Eqs. (17), (18) or (19). To find T i , i = 1, 2, 3, we first find all stationary i ðTÞÞ ¼ 0. The local minimum points of TCi(T, ri(T)) by solving dTCi ðT;r dT 2 ðT;r i ðTÞÞ > 0. The local points are determined by verifying that d TCidT 2
A þ ½ðY 1 ðr 3 ðe2 ÞÞ D1 þ Y 2 ðr 3 ðe2 ÞÞ D2 ÞPð1 r 3 ðe2 ÞÞ e2
Step 4:
Step 5:
Step 6:
Step 7:
Determine local minimum points by solving for 2 dTC1 ðT;r 1 ðTÞÞ ðT;r 1 ðTÞÞ ¼ 0 and by verifying that d TC1dT > 0, such 2 dT that the T satisfies T P t1 with respect to each local minimum point. Let TC1 ðT 1 ; rÞ associate with the local minimum point or the boundary point t1 which gives the smallest value of TC1(T1, r). Determine local minimum points by solving for 2 dTC2 ðT;r 2 ðTÞÞ ðT;r 2 ðTÞÞ ¼ 0 and by verifying that d TC2dT > 0, such 2 dT that the T satisfies t2 6 T 6 t1 with respect to each local minimum point. Let TC2 ðT 2 ; rÞ associate with the local minimum point or any of two boundary points t1 and t2 which gives the smallest value of TC2(T2, r). Determine local minimum points by solving for 2 dTC3 ðT;r 3 ðTÞÞ ðT;r 3 ðTÞÞ ¼ 0 and by verifying that d TC3dT > 0, such 2 dT that the T satisfies T 6 t2 with respect to each local minimum point. Let TC3 ðT 3 ; rÞ associate with the local minimum point or the boundary point t2 which gives the smallest value of TC3(T3,r). Let TC ðT ; r Þ ¼ MinfTCs1 ðT 1 ; rÞ; TC2 ðT 2 ; rÞ; TC3 ðT 3 ; rÞg.
6. Computational analyses The purposes of the numerical analysis are as follows: 1. To illustrate the solution procedure. 2. To use sensitivity analysis to highlight the influence of the length of the credit period, the ordering cost and the holding cost.
Y.-C. Tsao / Computers & Industrial Engineering 56 (2009) 208–215
6.1. Numerical examples To illustrate the solution procedures, we consider the following numerical examples: Example 1. Consider A = 100 dollars/order, P = 6 dollars/U, C = 3 dollars/U, H = 0.08 dollars/U/year, Ip = 0.15 per dollars, Ie = 0.1 per dollars, t1 = 0.3 years, t2 = 0.15 years, D1 = 3000 U/year, D2 = 3000 U/year, Y1 = 1.02 r, Y2 = 1.01 r. Since 2A þ ð1 Y 1 Þ D1 ½C Ip t21 Ie Pðt21 t22 Þ ¼ 200 > 0, then we can utilize Theorem 1 to get the optimal cycle time T* = T1 = 0.374 year, the total related cost TC1 = 121.363 dollars. If r is a decision variable, we utilize Algorithm to get the optimal cycle time T* = T1 = 0.505 year, the
700
TC3 (T , r 3 (T ))
600 500 400
TC2 (T , r 2 (T ))
300
optimal price discount r* = 0.497 and the total related cost TC1 = 59.262 dollars. A graphic representation of TCi is shown in Fig. 3(a). Example 2. Consider A = 50 dollars/order, P = 6 dollars/U, C = 3 dollars/U, H = 0.4 dollars/U/year, Ip = 0.15 per dollars, Ie = 0.1 per dollars, t1 = 0.3 years, t2 = 0.15 years, D1 = 3000 U/year, Y1 = 1.02 r, Y2 = 1.01 r. Since D2 = 3000 U/year, 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t22 Þ ¼ 100 > 0, then we can utilize Theorem 1 to get the optimal cycle time T* = T2 = 0.225 year, the total related cost TC1 = 71.868 dollars. If r is a decision variable, we utilize Algorithm to get the optimal cycle time T* = T2 = 0.274 year, the optimal price discount r* = 0.448 and the total related cost TC2 = 17.020 dollars. A graphic representation of TCi is shown in Fig. 3(b). Example 3. Consider A = 20 dollars/order, P = 6 dollars/U, C = 3 dollars/U, H = 1.3 dollars/U/year, Ip = 0.15 per dollars, Ie = 0.1 per dollars, t1 = 0.3 years, t2 = 0.15 years, D1 = 8000 U/year, Y1 = 1.02 r, Y2 = 1.01 r. Since D2 = 8000 U/year, 2A þ ð1 Y 1 ÞD1 ½C Ip t21 Ie Pðt 21 t22 Þ ¼ 60 > 0, then we can utilize Theorem 1 to get the optimal cycle time T* = T3 = 0.107 year, the total related cost TC3 = 82.494 dollars. If r is a decision variable, we utilize Algorithm to get the optimal cycle time T* = T3 = 0.147 year, the optimal price discount r* = 0.516 and the total related cost TC2 = 7.056 dollars. A graphic representation of TCi is shown in Fig. 3(c).
(a) Example 1 TCi
213
200
TC1 (T , r 1 (T ))
100 0.2
t2
0.4
6.2. Effects of credit period, ordering cost and holding cost
0.6
T
0.8
t1
(b) Example 2 250
TCi 200
TC3 (T , r 3 (T ))
150 100
TC2 (T , r 2 (T ))
TC1 (T , r 1 (T ))
50
t2
0.2
0.3
0.4
t1
0.5
T
(c) Example 3
It is also important to discuss the influence of the credit period, the ordering cost, and the holding cost. The computational results shown in Fig. 4 indicate the following managerial phenomena: (1) When the supplier provides a longer credit period t1, the retailer will replenish the goods more often. In other words the retailer will shorten the replenishment cycle time to take advantage of the longer credit period. (2) When the retailer provides a longer credit period t2, the retailer’s replenishment cycle time will be increased. This implies the retailer will replenish the goods not often to decrease the loss. An examination of Fig. 5 gives the following the results: (1) When the holding cost H increases, the retailer will increase the price discount and shorten the replenishment cycle time. It is reasonable that when the holding cost increases the retailer will then shorten the cycle time in an effort to lower his inventory cost. (2) When the ordering cost A increases, the retailer will increase his price discount and the replenishment cycle time to reflect this. If the ordering cost increases, it is reasonable that the retailer lengthens the cycle time to reduce the frequency of replenishment.
TCi 200 7. Conclusion
TC3 (T , r 3 (T ))
150
TC1 (T , r 1 (T )) 100
TC2 (T , r2 (T ))
50
0.05
0.1
0.15
t2
0.2
0.25
Fig. 3. Graphic illustration of TCi versus T.
0.3
t1
0.35
T
This paper considers retailer’s promotion and replenishment policies with advance sales discount under supplier’s and retailer’s trade credits. We calculate the interest earned based on retail price instead of purchase cost in most previous researches. The first model deals with the replenishment policy when promotional effort is predetermined. In the second model, we present an algorithm to determine the optimal promotional effort and replenishment cycle time simultaneously. Several examples are given to illustrate the solution procedures and discuss the impact of various system parameters. We conclude with a computational analysis that leads to variety of managerial insights.
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(a) Effect of supplier's credit period t 1 250 200 Total cost
Replenishment cycle time
0.386 0.384 0.382 0.38 0.378 0.376 0.374 0.372 0.37 0.368
150 100 50
0.25
0.275
0
0.3
0.25
Supplier's credit period
0.275
0.3
Supplier's credit period
0.41 0.4 0.39 0.38 0.37 0.36 0.35 0.34 0.33
200 150 Total cost
Replenishment cycle time
(b) Effect of retailer’s credit period t 2
100 50
0.1
0.15
0
0.2
0.1
Retailer's credit period
0.15
0.2
Retailer's credit period
Fig. 4. Effects of credit period.
0.385
135
0.38
130 125
0.375
Total cost
Replenishment cycle time
(a) Effect of holding cost
0.37 0.365
115 110
0.36 0.355
120
105 0.06
0.08
100
0.1
0.06
0.08
0.1
Holding cost
Holding cost
0.4
160
0.39
150
0.38
140 Total cost
Replenishment cycle time
(b) Effect of ordering cost
0.37 0.36
130 120 110
0.35
100
0.34
90
0.33
90
100
110
80
90
Orderinging cost
100
110
Orderinging cost
Fig. 5. Effects of holding cost and ordering cost.
For further research, this paper can be extended to consider other realistic situations, such as for deteriorating item, allowable shortages, permissible order cancellation or stochastic demand.
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