Information Management Sciences Volume 16 Number 3 pp.97-107 005 A Note on EOQ Model under Cash Discount Payment Delay Yung-Fu Huang Chaoyang University of Technology R.O.C. Abstract In this note we correct the models of Ouyang et al. (00 Chang (00 discussed with economic order quantity (EOQ under conditions of cash discount payment delay. Mathematical models have been modified for obtaining the optimal cycle time optimal payment policy for item under cash discount payment delay so that the annual total cost is minimized. Then we provide two theorems to efficiently determine the optimal cycle time optimal payment policy. Finally numerical examples are solved to illustrate the results given in the paper. Keywords: EOQ Payment Delay Cash Discount Trade Credit Inventory. 1. Introduction In practice the supplier frequently offers the retailer a fixed delay period that is the trade credit period in settling the accounts. Before the end of trade credit period the retailer can sell the goods accumulate revenue earn interest. A higher interest is charged if the payment is not settled by the end of trade credit period. In real world the supplier often makes use of this policy to promote his commodities. Goyal (1985 has discussed the well-known case of when the inventory system includes conditions of permissible delay in payments. Many articles related to the inventory policy under payment delay can be found in Aggarwal Jaggi (1995 Chang Dye (001 Chang et al. (001 Chen Chuang (1999 Chu et al. (1998 Chung (1998a 1998b 000 Chung et al. (001 Chung Huang (003 Goyal (1985 Huang (003 Jamal et al. (1997 000 Khouja Mehrez (1996 Liao et al. (000 Sarker et al. (000a 000b Teng (00 their references. Received September 003; Revised March 004; Accepted September 004. Supported by own.
98 Information Management Sciences Vol. 16 No. 3 September 005 Therefore it makes economic sense for the retailer to delay the settlement of the replenishment account up to the last moment of the permissible period allowed by the supplier. From the viewpoint of the supplier the supplier hopes that the payment is paid from retailer as soon as possible. It can avoid the possibility of resulting in bad debt. So in most business transactions the supplier will offer the credit terms mixing cash discount payment delay to the retailer. The retailer can obtain the cash discount when the payment is paid within cash discount period offered by the supplier. Otherwise the retailer will pay full payment within the trade credit period. In general the cash discount period is shorter than the trade credit period. Many articles related to the inventory policy under cash discount payment delay can be found in Chang (00 Ouyang et al. (00 003. The objective of this note is to correct the models of Ouyang et al. (00 Chang (00 discussed with the EOQ under conditions of cash discount payment delay. Mathematical models have been modified for obtaining the optimal cycle time optimal payment policy for item under cash discount payment delay so that the annual total cost is minimized. Then we provide two theorems to efficiently determine the optimal cycle time optimal payment policy. Finally numerical examples are solved to illustrate the results given in the paper.. Ouyang et al. s Model.1. Model formulation To shorten the note we adopt the same notation assumptions as in Ouyang et al. (00. However we want to correct the item IP in T C 1 (T in Ouyang et al. (00. Since the supplier offers cash discount if payment is paid within t 1. Therefore in this case the retailer will pay the annual purchasing cost p(1 rd to the supplier. Then the annual cost of interest charges for the items kept in stock is based on the annual purchasing cost p(1 rd. So we correct the annual IP in T C 1 (T in Ouyang et al. (00 picd(t t 1. We use the following items to formulate the T to p(1 ricd(t t 1 T annual total cost functions. Annual total cost ordering cost + stock-holding cost + purchasing cost + interest payable interest earned. For convience we define T V C 1 (T the annual total cost when payment is paid at time t 1 T > 0
A Note on EOQ Model under Cash Discount Payment Delay 99 T V C 11 (T if T t 1 T V C 1 (T if T t 1 T V C (T the annual total cost when payment is paid at time t T > 0 T V C 1 (T if T t T V C (T if T t T V C(T the annual total cost when T > 0 T V C 1 (T if the payment is paid at time t 1 T V C (T if the payment is paid at time t T the optimal cycle time of T V C(T. Therefore we can obtain following four annual total cost functions. T V C 11 (T S T V C 1 (T S T V C 1 (T S T V C (T S So S T V C 1 (T S T V C (T S S + p(1 rd + p(1 ri cd(t t 1 ( T + p(1 rd + pdi d t 1 T + pd + pi cd(t t T pi ddt T pi ddt 1 T (1 ( ( + pd pdi d t T. (4 + p(1 rd + p(1 ricd(t t 1 + p(1 rd + pdi d (t 1 T + pd + picd(t t T pi ddt + pd + pdi d (t T T pi ddt 1 (3 T if t 1 T (5a if 0 < T t 1 (5b T if t T (6a if 0 < T t. (6b At T t 1 we find T V C 11 (t 1 T V C 1 (t 1. Hence T V C 1 (T is continuous welldefined. All T V C 11 (T T V C 1 (T T V C 1 (T are defined on T > 0. And at T t we find T V C 1 (t T V C (t. Hence T V C (T is continuous well-defined. All T V C 1 (T T V C (T T V C (T are defined on T > 0... Determination of the optimal cycle time T optimal payment policy From equations (1 ( (3 (4 yield T V C 11(T {S + Dpt 1 [(1 ri c I d ]} T + D[h + p(1 ri c] (7
100 Information Management Sciences Vol. 16 No. 3 September 005 T V C 11(T S + Dpt 1 [(1 ri c I d ] T 3 (8 T V C 1(T S T + D(h + pi d (9 T V C 1 S (T > 0 T 3 (10 T V C 1(T [S + Dpt (I c I d ] T + D(h + pi c (11 T V C 1(T S + Dpt (I c I d T 3 > 0 (1 T V C S (T T + D(h + pi d (13 T V C (T S > 0. (14 T 3 Equations (10 (1 (14 imply that all T V C 1 (T T V C 1 (T T V C (T are convex on T > 0. However equation (8 implies that T V C 11 (T is convex on T > 0 if S + Dpt 1 [(1 ri c I d ] > 0. Let T V C 11 (T 11 T V C 1 (T 1 T V C 1 (T 1 T V C (T 0 we can obtain T11 S + Dpt 1 [(1 ri c I d ] if S + Dpt 1 D[h + p(1 ri c ] [(1 ri c I d ] > 0 (15 T1 S D(h + pi d (16 T1 S + Dpt (I c I d (17 D(h + pi c T S D(h + pi d. (18 Then we find T 1 T. Equation (15 implies that the optimal for the case of T t 1 that is T 11 t 1. We substitute equation (15 into T 11 t 1 then we can obtain the optimal if only if S + Dt 1 (h + pi d 0. Similarly equation (16 implies that the optimal for the case of T t 1 that is T 1 t 1. We substitute equation (16 into T 1 t 1 then we can obtain the optimal if only if S + Dt 1 (h + pi d 0.
A Note on EOQ Model under Cash Discount Payment Delay 101 Likewise equation (17 implies that the optimal for the case of T t that is T1 t. We substitute equation (17 into T1 t then we can obtain the optimal if only if S + Dt (h + pi d 0. Finally equation (18 implies that the optimal for the case of T t that is T t. We substitute equation (18 into T t then we can obtain the optimal if only if S + Dt (h + pi d 0. Furthermore we let And 1 S + Dt 1(h + pi d (19 S + Dt (h + pi d. (0 Since t 1 < t equations (19 (0 we can get 1 <. From above arguments the optimal cycle time T optimal payment policy (t 1 or t can be obtained as follows. Theorem 1. (A If 1 > 0 then T V C(T min{t V C 1 (T 1 T V C (T }. Hence T T 1 T optimal payment time is t 1 or t associated with the least cost. (B If < 0 then T V C(T min{t V C 1 (T 11 T V C (T 1 }. Hence T is T 11 or T 1 optimal payment time is t 1 or t associated with the least cost. (C If 1 < 0 > 0 then T V C(T min{t V C 1 (T11 T V C (T }. Hence T is T11 or T optimal payment time is t 1 or t associated with the least cost. (D If 1 0 > 0 then T V C(T min{t V C 1 (t 1 T V C (T }. Hence T is t 1 or T optimal payment time is t 1 or t associated with the least cost. (E If 1 < 0 0 then T V C(T min{t V C 1 (T 11 T V C (t }. Hence T is T 11 or t optimal payment time is t 1 or t associated with the least cost. Theorem 1 immediately determines the optimal cycle time T optimal payment policy (t 1 or t after computing the numbers 1. simple. Theorem 1 is really very
10 Information Management Sciences Vol. 16 No. 3 September 005 3. Chang s Model 3.1 Model formulation In this section we want to correct Chang s model (00. In Chang s model (00 she extended Ouyang et al. s model (00 by considering the selling price per item higher than the purchasing cost per item. So we must define the extra notation as follow. Extra notation: s unit selling price per item s > p. Then we correct the interest payable per unit time in Case 1 in Chang (00 For convience we define pdi c (T t 1 T to p(1 rdi c(t t 1. T T RC 1 (T the annual total cost when payment is paid at time t 1 T > 0 T RC 11 (T if T t 1 T RC 1 (T if T t 1 T RC (T the annual total cost when payment is paid at time t T > 0 T RC 1 (T if T t T RC (T if T t T RC(T the annual total cost when T > 0 T RC 1 (T if the payment is paid at time t 1 T RC (T if the payment is paid at time t T the optimal cycle time of T RC(T. Therefore we can obtain following four annual total cost functions. T RC 11 (T S T RC 1 (T S T RC 1 (T S T RC (T S So + p(1 rd + p(1 ri cd(t t 1 ( T + p(1 rd sdi d t 1 T + pd + pi cd(t t T si ddt T si ddt 1 T (1 ( (3 ( + pd sdi d t T. (4
A Note on EOQ Model under Cash Discount Payment Delay 103 S T T RC 1 (T + hdt + p(1 rd + p(1 ricd(t t 1 T si ddt 1 T if t 1 T (5a S + p(1 rd sdi d (t 1 T if 0 < T t 1 (5b S T T RC (T + hdt + pd + picd(t t T si ddt T if t T (6a S + pd sdi d (t T if 0 < T t. (6b At T t 1 we find T RC 11 (t 1 T RC 1 (t 1. Hence T RC 1 (T is continuous welldefined. All T RC 11 (T T RC 1 (T T RC 1 (T are defined on T > 0. And at T t we find T RC 1 (t T RC (t. Hence T RC (T is continuous well-defined. All T RC 1 (T T RC (T T RC (T are defined on T > 0. 3.. Determination of the optimal cycle time T optimal payment policy From equations (1 ( (3 (4 yield T RC 11(T {S + Dt 1 [p(1 ri c si d ]} T + D[h + p(1 ri c] (7 T RC 11(T S + Dt 1 [p(1 ri c si d ] T 3 (8 T RC 1 S (T T + D(h + si d (9 T RC 1 S (T > 0 T 3 (30 T RC 1 (T [S + Dt (pi c si d ] T + D(h + pi c (31 T RC 1(T S + Dt (pi c si d T 3 (3 T RC S (T T + D(h + si d (33 T RC (T S > 0. (34 T 3 Equations (30 (34 imply that all T RC 1 (T T RC (T are convex on T > 0. However equation (8 implies that T RC 11 (T is convex on T > 0 if S+Dt 1 [p(1 ri c si d ] > 0. Equation (3 implies that T RC 1 (T is convex on T > 0 if S+Dt (pi c si d > 0. Let T RC 11 (T 11 T RC 1 (T 1 T RC 1 (T 1 T RC (T 0 we can obtain T S + Dt 1 11 [p(1 ri c si d ] if S + Dt D[h + p(1 ri c ] 1[p(1 ri c si d ] > 0 (35 T 1 S D(h + si d (36
104 Information Management Sciences Vol. 16 No. 3 September 005 T 1 S + Dt (pi c si d D(h + pi c if S + Dt (pi c si d > 0 (37 T S D(h + si d. (38 Then we find T 1 T. Equation (35 implies that the optimal for the case of T t 1 that is T 11 t 1. We substitute equation (35 into T 11 t 1 then we can obtain the optimal if only if S + Dt 1 (h + si d 0. Similarly equation (36 implies that the optimal for the case of T t 1 that is T 1 t 1. We substitute equation (36 into T 1 t 1 then we can obtain the optimal if only if S + Dt 1 (h + si d 0. Likewise equation (37 implies that the optimal for the case of T t that is T 1 t. We substitute equation (37 into T 1 t then we can obtain the optimal if only if S + Dt (h + si d 0. Finally equation (38 implies that the optimal for the case of T t that is T t. We substitute equation (38 into T t then we can obtain the optimal if only if S + Dt (h + si d 0. Furthermore we let 3 S + Dt 1(h + si d (39 4 S + Dt (h + si d. (40 Since t 1 < t equations (39 (40 we can get 3 < 4. From above arguments the optimal cycle time T optimal payment policy (t 1 or t can be obtained as follows. Theorem. (A If 3 > 0 then T RC(T min{t RC 1 (T 1 T RC (T }. Hence T T 1 T optimal payment time is t 1 or t associated with the least cost.
A Note on EOQ Model under Cash Discount Payment Delay 105 (B If 4 < 0 then T RC(T min{t RC 1 (T 11 T RC (T 1}. Hence T is T 11 or T 1 optimal payment time is t 1 or t associated with the least cost. (C If 3 < 0 4 > 0 then T RC(T min{t RC 1 (T 11 T RC (T }. Hence T is T 11 or T optimal payment time is t 1 or t associated with the least cost. (D If 3 0 4 > 0 then T RC(T min{t RC 1 (t 1 T RC (T }. Hence T is t 1 or T optimal payment time is t 1 or t associated with the least cost. (E If 3 < 0 4 0 then T RC(T min{t RC 1 (T 11 T RC (t }. Hence T is T 11 or t optimal payment time is t 1 or t associated with the least cost. Theorem immediately determines the optimal cycle time T optimal payment policy (t 1 or t after computing the numbers 3 4. simple. Theorem is really very 4. Numerical Examples To illustrate the results let us apply the proposed method to solve the following numerical examples. For convenience the numbers of the parameters are selected romly. Example 1. Let S $00/order D 3 000 units/year p $100/unit r 0.01 h $5/unit/year I c $0.15/$/year I d $0.1/$/year t 1 0.05 year t 0.1 year then 1 87.5 < 0 50 > 0 T V C 1 (T 11 $99 870. < T V C (T $301 4.6. Using Theorem 1-(C we get T T 11 0.085603 year the optimal order quantity will be DT 11 57 units. The optimal payment time is t 1 0.05 year. T V C(T T V C 1 (T11 $99 870.. Example. Let S $00/order D 000 units/year p $50/unit s $100/unit r 0.05 h $5/unit/year I c $0.15/$/year I d $0.05/$/year t 1 0.05 year t 0. year then 3 350 < 0 4 400 > 0 T RC 1 (T 11 $97 443.1 < T RC (T $100 88.4. Using Theorem -(C we get T T 11 0.13017 year the optimal order quantity will be DT 11 payment time is t 1 0.05 year. T RC(T T RC 1 (T 11 $97 443.1. 60 units. The optimal
106 Information Management Sciences Vol. 16 No. 3 September 005 5. Summary The supplier offers the trade credit to stimulate the dem of the retailer. However the supplier can also use the cash discount policy to attract retailer to pay the full payment of the amount of purchasing cost to shorten the collection period. The credit term that contains cash discount is very realistic in real-life business situations. This note corrects Ouyang et al. (00 Chang (00 provides two theorems to efficiently determine the optimal cycle time optimal payment policy. Theorem 1 provides the determination of T optimal payment policy depends on the numbers of 1. Theorem provides the determination of T optimal payment policy depends on the numbers of 3 4. Finally numerical examples are solved to illustrate the results given in this paper. Acknowledgements The author would like to thank anonymous referees for their valuable constructive comments suggestions that have led to a significant improvement on an earlier version of this paper. References [1] Aggarwal S. P. Jaggi C. K. Ordering policies of deteriorating items under permissible delay in payments Journal of the Operational Research Society Vol.46 pp.658-66 1995. [] Chang C. T. Extended economic order quantity model under cash discount payment delay International Journal of Information Management Sciences Vol.13 No.3 pp.57-69 00. [3] Chang H. J. Dye C. Y. An inventory model for deteriorating items with partial backlogging permissible delay in payments International Journal of Systems Science Vol.3 pp.345-35 001. [4] Chang H. J. Hung C. H. Dye C. Y. An inventory model for deteriorating items with linear trend dem under the condition of permissible delay in payments Production Planning Control Vol.1 pp.74-8 001. [5] Chen M. S. Chuang C. C. An analysis of light buyer s economic order model under trade credit Asia-Pacific Journal of Operational Research Vol.16 pp.3-34 1999. [6] Chu P. Chung K. J. Lan S. P. Economic order quantity of deteriorating items under permissible delay in payments Computers Operations Research Vol.5 pp.817-84 1998. [7] Chung K. J. A theorem on the determination of economic order quantity under conditions of permissible delay in payments Computers Operations Research Vol.5 pp.49-5 1998a. [8] Chung K. J. Economic order quantity model when delay in payments is permissible Journal of Information & Optimization Sciences Vol.19 pp.411-416 1998b. [9] Chung K. J. The inventory replenishment policy for deteriorating items under permissible delay in payments Opsearch Vol.37 pp.67-81 000. [10] Chung K. J. Chang S. L. Yang W. D. The optimal cycle time for exponentially deteriorating products under trade credit financing The Engineering Economist Vol.46 3-4 001.
A Note on EOQ Model under Cash Discount Payment Delay 107 [11] Chung K. J. Huang Y. F. The optimal cycle time for EPQ inventory model under permissible delay in payments International Journal of Production Economics Vol.84 pp.307-318 003. [1] Goyal S. K. Economic order quantity under conditions of permissible delay in payments Journal of the Operational Research Society Vol.36 pp.335-338 1985. [13] Huang Y. F. Optimal retailer s ordering policies in the EOQ model under trade credit financing Journal of the Operational Research Society Vol.54 pp.1011-1015 003. [14] Jamal A. M. M. Sarker B. R. Wang S. An ordering policy for deteriorating items with allowable shortages permissible delay in payment Journal of the Operational Research Society Vol.48 pp.86-833 1997. [15] Jamal A. M. M. Sarker B. R. Wang S. Optimal payment time for a retailer under permitted delay of payment by the wholesaler International Journal of Production Economics Vol.66 pp.59-66 000. [16] Khouja M. Mehrez A. Optimal inventory policy under different supplier credit policies Journal of Manufacturing Systems Vol.15 pp.334-339 1996. [17] Liao H. C. Tsai C. H. Su C. T. An inventory model with deteriorating items under inflation when a delay in payment is permissible International Journal of Production Economics Vol.63 pp.07-14 000. [18] Ouyang L. Y. Chen M. S. Chuang K. W. Economic order quantity model under cash discount payment delay International Journal of Information Management Sciences Vol.13 No.1 pp.1-10 00. [19] Ouyang L. Y. Wu C. C. Chuang K. W. Economic order quantity with partial backorders under supplier credit Journal of Information & Optimization Sciences Vol.4 pp.55-67 003. [0] Sarker B. R. Jamal A. M. M. Wang S. Supply chain model for perishable products under inflation permissible delay in payment Computers Operations Research Vol.7 pp.59-75 000a. [1] Sarker B. R. Jamal A. M. M. Wang S. Optimal payment time under permissible delay in payment for products with deterioration Production Planning Control Vol.11 pp.380-390 000b. [] Teng J. T. On the economic order quantity under conditions of permissible delay in payments Journal of the Operational Research Society Vol.53 pp.915-918 00. Author s Information Yung-Fu Huang is an associate professor of the Department of Business Administration at Chaoyang University of Technology Taiwan. He was awarded a Ph.D. in Industrial Management from National Taiwan University of Science Technology Taiwan. His research areas include inventory management engineering economics operations research service quality management. He has published/accepted in journals such as Applied Mathematical Modelling Asia-Pacific Journal of Operational Research International Journal of Production Economics International Journal of Systems Science Journal of Applied Sciences Journal of Information & Optimization Sciences Journal of the Operational Research Society Journal of the Operations Research Society of Japan Journal of Statistics & Management Systems omega Opsearch Production Planning Control Quality & Quantity Taiwan Academy of Management Journal. Department of Business Administration Chaoyang University of Technology Wufong Taichung Taiwan 413 R.O.C. E-mail: huf@mail.cyut.edu.tw Tel: +886-4-333000 ext. 4377