Published online: 18 Jun 2013.

This article was downloaded by: [Academia Sinica - Taiwan] On: 02 November 2014, At: 21:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Information and Optimization Sciences Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tios20 Economic order quantity and optimal payment time under supplier credit Chun-Tao Chang a & Shuo-Jye Wu a a Department of Statistics, Tamkang University, 251, Taiwan R.O.C. Published online: 18 Jun 2013. To cite this article: Chun-Tao Chang & Shuo-Jye Wu (2002) Economic order quantity and optimal payment time under supplier credit, Journal of Information and Optimization Sciences, 23:3, 551-561, DOI: 10.1080/02522667.2002.10699545 To link to this article: http://dx.doi.org/10.1080/02522667.2002.10699545 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

Economic order quantity and optimal payment time under supplier credit Chun-Tao Chang Shuo-Jye Wu Department of Statistics Tamkang University Tamsui, Taipei Taiwan, 251 R.O.C. ABSTRACT In today's business transactions, the buyer is allowed a permissible period to pay back the cost of prod ucts bought without paying any interest. The buyer can either pay the supplier at the end of the permissible period or incur interest charges on the unpaid balance after this permissible period. This paper sels up a mathematical model to determine an optimal payment period and inventory length of the buyer which minimize the total cost of the inventory system. Furthermore, we find two theorems to characterize the optimal solution. Finally, two numerical examples are discussed to illustrate these theorems. Keywords: Inventory; Delay Payments 1. INTRODUCTION In the classical inventory model, it is tacitly assumed that a supplier must be paid for the items as soon as the items are received. However, with today's competitive business transactions, the supplier allows credit for some fixed time period in settling the payment for the product and does not charge any interest from the customer on the amount owed during this period in order to stimulate the demand of the customer. But, if the payment is delayed beyond that period, interest will be charged. The customer can start to accumulate revenues on the sales or use of the product, and earn interest on that revenue. Thus, it is an advantage for the customer to defer the payment to the supplie!r until the end of the allowed period. Goyal (1985) developed Journal of Information & Optimization Sciences Vol. 23 (2002), No.3, pp. 551-561 Analytic Publishing Co. 0252-2667/02 $2.00+.25

552 C. T. CHANG AND S. J. WU an economic order quantity under the conditions of permissible delay in payments for an inventory system. Chand and Ward (1987) investigated the same model within the framework of classical economic order quantity model. In the model proposed by Goyal (1985), it is assumed that no deterioration is allowed to occur. Aggarwal and Jaggi (1995) extended Goyal's model to the case of deterioration. Recently, some studies of this topic can be referred to Jamal et al. (1997), Chung (1998), Sarker et al. (2000) and Teng (2001). The previous studies dealing with the inventory problems under permissible delay in payments discussed the case that customer pays the supplier at the end of the credit. In today's business transaction, the customer may not pay the supplier at the end of the credit period; instead he will invest the money until the interest payable to the supplier is larger than the interest earned. Since the payable interest rate generally higher than the earned interest, the customer is expected to settle the account at a time before the end of the inventory cycle time. This paper extends Goyal (1985)'s model to the case of determination of an optimal payment time to fit the real~life inventory system. Goyal (1985) assumed that the unit purchasing cost is the same as the selling price per unit. However, in practice, the unit selling price is not lower than the unit purchasing cost. Hence, we revise his model by considering the difference between unit price and unit cost. We then yield an easy analytical closed-form solution t.o the problem and determine the optimal payment time and inventory length so that the total variable cost per unit time is minimized. The rest of this paper is organized as follows. Section 2 describes the assumptions and notations adopted for this study. Section 3 develops a mathematical formulation whose objective is to minimize the total variable cost per unit time. In Section 4, theoretical results are presented and a theorem is established to determine the optimal payment time and inventory length. We also compare the optimal order quantity under the supplier credits with the classical economic order quantity. Numerical examples are provided in Section 5. Finally, some conclusions are made in Section 6. 2. ASSUMPTIONS AND NOTATIONS The following assumptions are similar to those in Goyal (1985). (1) The demand for the item is constant with time. (2) Shortages are not allowed. (3) Replenishment is instantaneous. (4) Time horizon is infinite.

ECONOMIC ORDER QUANTITY 553 In addition, the following notations are used throughout this paper. D h p = e = Ie the demand rate per unit time. the unit holding cost per unit time excluding interest charges. the selling price per unit. t.he unit purchasing cost, with e <po the interest charged per dollar in stocks per unit time by the supplier. the interest earned per dollar j)er unit time. the ordering cost per order. ld A = Q = the order quantity. let) == S = the period of payment. M T == TVe the level of inventory at time t, 0:::; t:::; T. the period of permissible delay in settling account S?:: Jvi. the length of the inventory cycle (time units). the total variable cost per unit time. 3. MATHEMATICAL FORMULATION The inventory level I(t) is depleted due to demand. Hence, the rate of change of inventory diet) _ D < < dt --, 0_ L T, (1) with the boundary conditions, J(O):::: Q and J(T) solution of (1) is given by O. Consequently, the J(t) == D(T - t), 0:::; t :::; T, (2) and the order quantity Q==DT. (3) The total variable cost per unit time consists of (a) cost of placing orders, (b) cost of stock holding (excluding interest charges), (c) cost of interest charges, and (d) interest earned. This first two elements of total variable cost per unit time can be written as: and Cost of placing orders =AIT, (4) T Cost of stock holding =h f J(t)dtIT hdti2. (5) o

554 C. T. CHANG AND S. J. WU Now, there are two distinct cases namely (1) T ~ M and (2) T> M. Figure 1 shows these two cases. Inventory Level T Q 1 Inventory Level T Q 1 T Case 1. T:?:M M=S Time Case 2. T<M Figure 1. Inventory system

ECONOMIC ORDER QUANTITY 555 Case 1. T?:.M In this case, the inventory cycle length is larger than the permissible period. So, the customer can either pay the supplier at time M or incur interest charges on the unpaid balance after this credit period. Instead of paying the supplier by time M, the customer will usually invest the money until the interest payable to the supplier is larger than the interest earned. Since the payable interest rate is generally higher than the earned interest rate, the customer is expected to settle the account at time 8 before the end of inventory cycle time T. Consequently, the cost of interest charges contain two parts. One is the interest yielded by unsold items after the payment time 8. The other is the interest charged by the supplier between M and S. Furthermore, the customer utilizes the sales revenue and earns interest during the payment period. Hence, the interest payable per unit time is c1c [ J~ DT dt + s: I(t)dt }T::= cd1c[(8 M) + (T - 8)2/211, (6) and the interest earned per unit time is where M~8~T. Therefore, we obtain the total variable cost per unit time hdt 2 p1dd 2 TVC1(T, S) ::=AlT+--+cDl c [(8 -M)+(T-S) ;2T] 2T 8 2... (8) The problem now is to determine the optimal values of T and 8 such that TVC1(T, S) is minimized. (7) Case 2. T<M In this case, the inventory cycle length is less than the permissible period. The customer pays at M (i.e. 8 = M). Hence, the customer earns interest on sales revenue up to the permissible period and pays no interest for the supplier and the items kept in stock. The interest earned per unit time is p1d [ s: Dt dt + DT(M - T) }T=:: p1dd(m - TI2). (9)

556 C. T. CHANG AND S. J. WU Therefore, the total variable cost per unit time is (10) The problem now is to determine the optimal value of T such that TVC 2 (T) is minimized. 4. THEORETICAL RESULTS Case 1. T;;:: M Taking the first derivative of TVC1(T, S) with respect to S, we get (11) Consequently, there are three different cases for finding the optimal T*. They are: Case 1-1. T;;::M and clc-p1d>o If elc - pld > 0, then otvc1/os> O. That is, when M::;; S::;; T, TVC l is an increasing function of S. Hence, we choose S =M and then (8) can be written as In order to obtain the minimum value of TVC1(T), we have to solve the first-order condition for TVC1(T). That is, the equation dtvc1(t)ldt= 0, whose solution for Tis -V2A + M2D(clc - p1d). D(h + elc) (13) Note that Tl actually minimizes TVC1(T) and can be easily checked by substituting (13) into the second-order condition (14) To ensure that Tl > M, we substitute (13) into it, and obtain that if and only if 2A > JJ(h + pldjm 2, then Tl > M. (15)

ECONOMIC ORDER QUANTITY 557 Case 1-2. T? M and cic - pid < 0 If cic - pid < 0, then otvc/os < O. That is, when M:;::; S:;::; T, TVC l is a decreasing function of S. Hence, we choose S", T and then (8) can be written as hdt p~d 2 TVC1(T) AIT+ 2 + cd~(t - M) - 2T T. (16) Likewise, for Case 1-2, we can easily obtain the optimal value of T which is and the second-order condition is Substituting (18) into the inequality Tz > M, we obtain that (17) (18) if and only if 2A > IXH + 2cIc - pid)m z, then Tz > M. (19) If ci e - pid =0, then OTVCl/OS =O. That is, TVC I is a constant function of S. Thus, S is an arbitrary number between M and T, and (8) can be written as hdt TVC 1 (1) = AIT+ -2- + cdic(t12 - M). (20) Similarly, the optimal value of T for this case is (21) and the second-order condition is (22) Substituting (21) into the inequality Ta > M, we obtain that

558 C. T. CHANG AND S. J. WU Case 2. T<M if and only if 2A > D(h + clc) M Z, then T3 > M. (23) In order to obtain the minimum value of TVCz(T), we have to solve the first-order condition for TVCz(T), that is, to solve the equation dtvcz(t)/dt = O. The solution for T is -VD(hz.:Pl d T4 ), (24) and the second-order condition is d 2 TVCz(T)/DT Z I T ;T = :3 A> O. (25) 4 4 Substituting (24) into the inequality T4 < M, we obtain that 'if and only if 2A < IXh +pi 4 )M 2, then T4 < M. (26) From (13), (17), (21), and (24), we know that a higher value of the ordering cost per order A leads to a longer inventory length T. and vice versa. Combining the above possible cases, we have the following theorem. THEOREM l. (1) If c/p > Ia/lc, we have the following results: (a) If 2A > D(h +pl d )M 2, then the optimal inventory length T' = Tl and the optimal payment period S' = M. (b) If 2A < D(h + pla)m z, then the optimal inventory length T' = T4 and the optimal payment period S' == M. (c) If 2A =D(h + pla)m z, then the optimal inventory length T' is equal to the optimal payment period S' == M(i.e. T' = S = M). (2) If c/p < Ia/lc, we get: (a) If 2A > D(h +pla)m2, then the optimal inventory length T' T2 and the optimal payment period S TO. (b) If 2A < D(h + 2cl c pla)m2, then the optimal inventory length T T4 and the optimal payment period S M. (c) If (h + 2eI c - pla)m2 < 2A < D(h +pla)m~, then we know: (i) if TVCl(l~)::; TVC z (T 4 ), then T* = T z and S == T'. (ii) Otherwise, T' = T4 and S = M.

ECONOMIC ORDER QUANTITY 559 (d) If 2A =(h + 2elc - pld)m z or 2A =D(h + pla)m z, then T* M=S*. (3) If elp = IdlIe' then the optimal inventory length T* T3 = T4 and M s S * s T*. PROOF. It immediately follows from (15), (19), (23), and (26). 0 In addition, using (3), (13), (17), (21) and (24), we can get the economic order quantity for each of these cases as follows: Q*(Tl) = DTI = V[2AD+ MD2(cI c - pld)]/(h + elc), (27) Q*CTz) = DTz = ;J2AD/Ch + 2elc - pia), (28) Q*CT3) DTa '>I2AD/(h + elc), (29) Q*(T,!) DT4 '>I2AD/(h + pi,!). (30) In the classical economic order quantity model, the supplier must be paid for the items immediately after the items are received. Hence, it is a special case with M 0, and its optimal order quantity is By comparing (27)-(31), we have the following theorem. THEOREM 2. (1) Ifelp>Ialle' then Q*(Tl»Q* and Q*(T4»Q*. (2) If clp < Ialle, then Q*CTz) > Q* and Q*(T4) < Q*. (3) If elp =Idlle' then Q*(T3) =Q*(T4) = Q*. PROOF. It is obvious from (27)-(31). 0 (31) From Theorem 2(2), we know that Q*(T4) < Q* when pia> el ' c This result means that by comparing with the classical optimal economic order quantity Q*, the customer will order a less quantity than Q* in order to take the benefits of the permissible delay more frequently. 5. NUMERICAL EXAMPLES Example 1. Given D = 1000 units/unit time, h = $4/unit/unit time, Ie = O.W/unit time, Ia 0.06/unit time, e =$20 per unit, p =$30 per unit, and M = 45 days = 45/365 years. Then 0.667 =elp > IalIe = 0.6 and D(h + pla)m2 =88.1591. Consequently, if A = 50, then the

560 C, T, CHANG AND S, J. \V1J 2A > D(h +pi d )M 2, and we know from Theorem 1(1) that T* = Tl and the optimal payment period S' == M == 45 days 45/365 years. If A == 25, then the 2A < D(h +pid)mz, T* = T4 and the optimal payment period S * == M == 45 days = 45/365 years. In addition, we also know that the economic order quantity Q*(T*) is more than the classical economic order quantity Q* from Theorem 2(1). The computational results are shown in Table 1. Table 1 Optimal solutions for different ordering costs (clp > IdlIc)._-,--, A T* S' Q* TVC(T*) 50 = 0.131047 0.123288 131.0470 129.0994 539.7069 25 T4 = 0.092848 0.123288 92.8477 91.2871 316.5987 --------.----.-~. Example 2. Given D 1000 units/unit time, h $4/unit/unit time, Ie O.lO/unit time, Id 0.08/unit time, C = $20 per unit, p = $30 per unit, and M 45 days 45/365 years. Then 0.667 = clp < IdlIe = 0.8, D(h +pi d )M 2 97.2790 and D(h + 2cI e - pi d )M 2 = 85.1192. Consequently, if A = 50, then the 2A > D(h +pirijmz, and we know from Theorem 1(2) that T* == Tz and the optimal payment period S* = T* == T 2. IfA =25, then 2A < D(h + 2cI c - pi d )M 2, T* T4 and the optimal payment period S' =M =45 days;::;; 45/365.rears. If A = 45 and 47, then D(h + pi d )M 2 > 2A > D(h + 2cI pid)m~, T* T2 or T4 and the optimal payment period S* = Tz or M = 45 days 45/365 years. Additionally, from Theorem 2(2), we also know that the economic order quantity Q*(T*) is more than (less than) the classical economic order quantity Q* when the optimal inventory cycle T* Tz (T 4 ). The computational results are shown in Table 2. Table 2 Optimal solutions for different ordering costs (clp < IdlIe) A T* S* Q*(T*) Q* TVC(T*) 50 Tz =0.133631 T z 0.133631 133.6306 129.0994 501.7561 47 T2 == 0.129560 Tz =0.129560 129.5597 125.1666 478.9589 45 T4 =0.118585 M 0.123288 118.5854 122.4745 463.0562 25 T4 = 0.088388 M =0.123288 88.3883 91.2871 269.7950 c 6. CONCLUSIONS We develop a modified economic order quantity model to determine an optimal ordering policy under permissible delay of payment. In

ECONOMIC ORDER QUA.1"JTITY 561 this policy, we not only determine the optimal inventory cycle, but also find the optimal payment time. We modify Goyal (1985)'s model by considering the difference between unit price and unit cost, and the payment period is not always equal to the credit period. These considerations are more realistic in today's business transactions. In this paper, we obtain the optimal inventory length, the optimal payment time and the corresponding optimal economic order quantity. We set up a theorem which provides us a simple way to determine the optimal inventory length and the optimal time of payment by examining some explicit conditions stated in Theorem L Furthermore, we compare the optimal economic order quantities with a permissible delay in payment with the classical economic order quantity. Theorem 2 shows that the quantity of the customer ordered is more than (equal to) the classical economic order quantity when the ratio of cost to price clp is more than (equal to) the ratio of interest earned to charged 1,1/Ie. Finally, the numerical examples are given to demonstrate these two theorems. REFERENCES 1. S. P. Aggarwal and C. K Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society, VoL 46 (1995), pp. 58-66. 2. S. Chand and J. Ward, A note on "Economic order quantity under conditions of permissible delay in payments", Journal of the Opemtional Research Society. Vol. 38 (1987), pp. 83-84. 3. K -J. Chung, A theorem on the determination of economic order quantity under conditions of permissible delay in payments, Computers and Operations Research, Vol. 25 (1998), pp. 49 52. 4. S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Joumal of the Operational Reseal'ch Society, Vol. 36 (1985), pp. 335-338. 5. A. M. Jamal, B. R. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and' permissible delay in payment, Joumal of.the Operational Research Society, Vol. 48 (1997), pp. 826-833. 6. B. R. Sarker, A. M. M. Jamal and S. Wand, Supply chain models for perishable products under inflation and permissible delay in payment, Computers & Operations Research, Vol. 27 (2000), pp. 59-75. 7. J. -T. Teng, On economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, Vol. 53 (2002), pp. 915-918. Received January, 2002