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Retailer’s optimal replenishment decisions with credit-linked demand under permissible delay in payments
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European Journal of Operational Research 190 (2008) 130–135 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Retailer’s optimal replenishment decisions with credit-linked demand under permissible delay in payments Chandra K. Jaggi
a,*
, S.K. Goyal b, S.K. Goel
a
a
b
Department of Operational Research, University of Delhi, Delhi 110 007, India Decision Sciences and MIS, John Molson School of Business, Concordia University Montreal, Que., Canada H3G 1M8 Received 13 June 2006; accepted 26 May 2007 Available online 2 June 2007
Abstract Usually it is assumed that the supplier would offer a fixed credit period to the retailer but the retailer in turn would not offer any credit period to its customers, which is unrealistic, because in real practice retailer might offer a credit period to its customers in order to stimulate his own demand. Moreover, it is observed that credit period offered by the retailer to its customers has a positive impact on demand of an item but the impact of credit period on demand has received a very little attention by the researchers. To incorporate this phenomenon, we assume that demand is linked to credit period offered by the retailer to the customers. This paper incorporates the concept of credit-linked demand and develops a new inventory model under two levels of trade credit policy to reflect the real-life situations. An easy-to-use algorithm is developed to determine the optimal credit as well as replenishment policy jointly for the retailer. Finally, numerical example is presented to illustrate the theoretical results followed by the sensitivity of various parameters on the optimal solution. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Inventory; Credit-linked demand; Two level credit policy; Delay in payments
1. Introduction In classical inventory analysis, it was tacitly assumed that the supplier is paid for the items as soon as the retailer receives the items. But in practice, the supplier allows a certain fixed credit period to settle the account for stimulating retailer’s demand. During this credit period the retailer can start to accumulate revenues on the sales and earn
*
Corresponding author. Tel./fax: +91 11 27666672. E-mail address: [email protected] (C.K. Jaggi).
interest on that revenue, but beyond this period the supplier charges interest. Hence, paying later indirectly reduces the cost of holding stock. On the other hand, trade credit offered by the supplier encourages the retailer to buy more and it is also a powerful promotional tool that attracts new customers, who consider it as an alternative incentive policy to quantity discounts. Hence, trade credit can play a major role in inventory control for both the supplier as well as retailer. Owing to this fact, during the past few years, many articles dealing with various inventory models under trade credit have appeared in various research
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.05.042
C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
journals. Haley and Higgins [7] introduced the first model to consider the economic order quantity under conditions of permissible delay in payments with deterministic demand, no shortages, and zero-lead time. Goyal [5] considered a model similar to that of Haley and Higgins with the exclusion of the penalty cost due to a late payment. Chung [3] presented the discounted cash flows (DCF) approach for the analysis of the optimal inventory policy in the presence of the trade credit. Shah [12] and Aggarwal and Jaggi [1] extended the Goyal’s model to the case of deterioration. Jamal et al. [11] further generalized the model to allow shortages. Jaggi and Aggarwal [10] extended Chung [3] to develop an inventory model for obtaining the optimal order quantity of deteriorating items in the presence of trade credit using the DCF approach. Hwang and Shinn [9] considered the problem of determining the retailer’s optimal price and lot size simultaneously when the supplier permits delay in payments. Dye [4] in their paper considered the stockdependent demand for deteriorating items for partial backlogging and condition of permissible delay in payment. They assumed initial stock-dependent demand function. Teng [13] provided an alternative conclusion from Goyal [5], and mathematically proved that it makes economic sense for a buyer to order less quantity and take benefits of the permissible delay more frequently. Chang et al. [2] considered an inventory model for deteriorating items with instantaneous stock-dependent demand and timevalue of money when credit period is provided. Teng et al. [14] developed the optimal pricing and lot sizing under permissible delay in payments by considering the difference between the selling price and the purchase cost and demand is a function of price. Recently, Goyal et al. [6] established an EOQ model for a retailer when the supplier offers a progressive interest charge, and provided an easy-to-use closedform solution to the problem. All the aforementioned inventory models implicitly assumed that the customer would pay for the items as soon as the items are received from the retailer. But, in most business transactions, this assumption is unrealistic and usually the supplier offers a credit period to the retailer and the retailer, in turn, passes on this credit period to his/her customers. Recently, Huang [8] presented an inventory model assuming that the retailer also offers a credit period to his/her customer which is shorter than the credit period offered by the supplier, in order to stimulate the demand. Moreover, in all the above articles, although the presence of credit period has been incor-
131
porated in the mathematical models but the impact of credit period on demand is unfortunately ignored. In reality, it is observed that demand of an item does depend upon the length of the credit period offered by the retailer. In order to incorporate the above phenomena’s a new form of credit-linked demand function has been coined using which a new inventory model has been formulated to determine the retailer’s optimal credit as well as replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. 2. Assumptions and notations The following assumptions are made to develop the mathematical model: 1. The supplier provides a fixed credit period M to settle the accounts to the retailer and the retailer, in turn, also offers a credit period N to each of its customers to settle the accounts. 2. The demand rate is a function of the customer’s credit-period offered by the retailer (N). The demand function for any N can be represented as a differential difference equation: DðN þ 1Þ DðN Þ ¼ r½S DðN Þ; where D = D(N) demand for any N per unit time S maximum demand r rate of saturation of demand (which can be estimated using the past data) under the assumption that the marginal effect of credit period on sales is proportional to the unrealized potential of the market demand without any delay. The solution of the above difference equation, under the condition that at N = 0, D(0) = s (initial demand), keeping other attributes like price, quantity, etc. at constant level, is given by D ¼ sð1 rÞN þ Sð1 ð1 rÞN Þ; N
i:e: D ¼ S ðS sÞð1 rÞ : 3. Replenishment rate is instantaneous. 4. Shortages are not allowed. 5. Lead-time is negligible. In addition, following notation are used: Q T
order quantity inventory cycle length
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C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
q(t) A C P I
the inventory level at time t ordering cost per order unit purchase cost of the item unit selling price of the item out-of-pocket inventory carrying charge per $ per unit time Ie interest that can be earned per $ per unit time Ip interest payable per $ per unit time M retailer’s credit period offered by the supplier for settling the accounts N customer’s credit period offered by the retailer for settling the accounts Z(T, N) retailer’s profit per unit time which is a function of two variable; T and N where Z(T, N) = (a) revenue from sales (b) cost of purchasing units (c) cost of placing orders (d) cost of carrying inventory (excluding interest charges) + (e) interest earned from the sales during the permissible period (f) cost of interest charges for the unsold items after the permissible delay
InventoryLevel
Interest paid
Interest earned
Time 0
N
M
T
T+N
Fig. 1. When N 6 M 6 T + N.
to M and earns interest on average sales revenue for the time-period (M N). At M accounts are settled, if all the credit sales are not realized, the finances are to be arranged to make the payment to the supplier: ðeÞ Consequently the interest earned per unit time is ¼
I e PDðM N Þ 2T
2
ð8Þ
and 3. Mathematical formulation ðfÞ Interest payable per unit time is As the demand function is
2
N
D ¼ DðN Þ ¼ S ðS sÞð1 rÞ : Therefore, the order quantity is Z T D dt ¼ DT Q¼
ð1Þ
¼
I p CDðT þ N MÞ : 2T
ð9Þ
Using Eqs. (4)–(9), the retailer’s profit per unit time Z1(T, N) can be expressed as ð2Þ
A 1 þ fI e P ðDðM N Þ2 Þ T 2T 2 ICDT 2 I p CDðT þ N MÞ g: ð10Þ
Z 1 ðT ; N Þ ¼ ðP CÞD
0
and the inventory level at any time t during the cycle is Z t D dt ¼ DðT tÞ; 0 6 t 6 T : ð3Þ qðtÞ ¼ Q 0
The retailer’s profit per unit time consists of the following elements: ðaÞ Sales revenue ¼ PD;
ð4Þ
ðbÞ Cost of placing orders ¼ A=T ;
ð5Þ
ðcÞ Cost of purchasing units ¼ CD; ICDT ðdÞ Cost of carrying inventory ¼ : 2
ð6Þ
N 6 T + N 6 M (Fig. 2). In this case, the retailer earns interest on average sales revenues received during the period (N, T + N) and on full sales revenue (PQ) for a period of (M T N) but no interest payable is payable by the retailer.
Inventory Level
Int earned
ð7Þ
The computation for interest earned and payable (i.e. (e) and (f)) will depend on the following three possible cases based on the lengths of T, N and M: N 6 M 6 T + N (Fig. 1). In this case, the retailer starts getting actual sales revenues from time N
Time 0
N
T
T+N
Fig. 2. When N 6 T + N 6 M.
M
C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
ðeÞ Consequently the interest earned per unit time is
133
4. Solution procedure
2
I e P fDT =2 þ DT ðM T N Þg T ¼ I e PDðM N T =2Þ:
¼
ð11Þ As a result, using Eqs. (4)–(7) and (11), the retailer’s profit per unit time Z2(T, N) in this case is A ICDT Z 2 ðT ; N Þ ¼ ðP CÞD T 2 T þ I e PD M N : 2
ð12Þ
M 6 N 6 T + N (Fig. 3). In this case, the retailer earns no interest but pays interest on full order quantity (Q) for a period of (N M) and on average stock held during the cycle length (T): ðfÞ Consequently the interest payable per unit time is I p CfDT ðN MÞ þ DT 2 =2g T ¼ I p CDðN M þ T =2Þ: ¼
ð13Þ As a result, using Eqs. (4)–(7) and (13), the retailer’s profit per unit time Z3(T, N) in this case is A ICDT Z 3 ðT ; N Þ ¼ ðP CÞD T 2 T I p CD N M þ : 2 Therefore, the retailer’s is 8 > < Z 1 ðT ; N Þ ZðT ; N Þ ¼ Z 2 ðT ; N Þ > : Z 3 ðT ; N Þ
ð14Þ
profit per unit time Z(T, N)
if N 6 M 6 T þ N ; if N 6 T þ N 6 M;
Our problem is to determine the optimum value of T and N which maximizes Z(T, N). For a fixed value of N, taking the first and second order derivatives of Z1(T, N), Z2(T, N) and Z3(T, N) with respect to T, we get Z 01 ðT ; N Þ ¼
A ICD PI e DðM N Þ2 2 T2 2T 2 CI P D 2 fT ðN MÞ2 g; 2T 2 2
2A DðM N Þ ðCI p PI e Þ ; T3 A ICD PI e D ; Z 02 ðT ; N Þ ¼ 2 2 2 T 2A Z 002 ðT ; N Þ ¼ 3 ; T A ICD CI P D ; Z 03 ðT ; N Þ ¼ 2 2 2 T 2A Z 003 ðT ; N Þ ¼ 3 : T Z 001 ðT ; N Þ ¼
if M 6 N 6 T þ N ;
which is a function of two variable T and N where T is continuous and N is discrete. Inventory Level
Time M
N
Fig. 3. When M 6 N 6 T + N.
ð19Þ ð20Þ ð21Þ
T 1 would satisfy the condition 0 6 (M N) 6 T provided ð23Þ
Substituting Eq. (22) into (2) and (10), we can get the optimal values of Q and Z1(T) (say Q1 and Z 1 Þ. Similarly, there exists a unique value of T (say T 2 ) which maximizes Z2(T) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A T 2 ¼ : ð24Þ ðIC þ I e P ÞD
2A DðM N Þ2 ½IC þ I e P 6 0:
T
ð18Þ
T 2 would satisfy the condition 0 6 T 6 (M N) provided
Int paid
0
ð17Þ
For fixed N, Eqs. (19) and (21) imply that Z2(T, N) and Z3(T, N) are concave on T > 0. However, Z1(T, N) is concave on T > 0 if CIp > PIe. Thus, there exists a unique value of T (say T 1 ) which maximizes Z1(T) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2A DðM N Þ ðPI e CI p Þ T1 ¼ : ð22Þ ðI þ I p ÞC
2A DðM N Þ2 ½IC þ I e P P 0: ð15Þ
ð16Þ
T+N
ð25Þ
Substituting Eq. (24) into (2) and (12), we can get the optimal values of Q and Z2(T) (say Q2 and Z 2 ). Likewise, we can obtain the optimal value of T (say T 3 ) which maximizes Z3(T) as
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C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A T 3 ¼ : ðI þ I p ÞCD
ð26Þ
T 3 would satisfy the condition (M N) 6 0 6 T provided 2
2A CDðM N Þ ½I þ I p P 0:
ð27Þ
Substituting Eq. (26) into (2) and (14), we can get the optimal values of Q and Z3(T) (say Q3 and Z 3 ). Combining the three possible cases, we obtain the following theorem. Theorem 1. For a fixed value of N (a) If 2A D(M N)2[IC + IeP] P 0 then T ¼ T 1 . (b) If 2A D(M N)2[IC + IeP] 6 0 then T ¼ T 2 . (c) If 2A CD(M N)2[I + Ip] P 0 and (M N) < 0 then T ¼ T 3 .
demand (r) = 0.12, A = $1000/order, M = 30 days, C = $30/unit, P = $40/unit, Ip = 15% per year, Ie = 10% per year and I = 15% per year. Using the proposed algorithm, we obtained the optimal results as follows: Optimal cycle length (T*) = 28.63 days, optimal credit period offered by retailer (N*) = 33 days and profit per day (Z*) = $916. Further, the sensitivity analysis on M, A, r and Ip is shown in Tables 1–4, respectively. It is observed from Table 1 that as M increases, there is marginal increase in cycle length as well as credit period offered by the retailer but there is significant increase in retailer’s profit, which implies that credit period offered to customers has positive impact on the unrealized demand. Table 2 shows that a hike in ordering cost (A) causes higher cycle length and lower profit. Table 3 suggests that
Table 1 Effects of changing M on the optimal solution
Proof. It immediately follows from (23), (25) and (27). h In order to jointly optimize Q and N, we propose the following algorithm: 1. Put N = 1. 2. Determine the optimal values of T (i.e. T 1 or T 2 or T 3 ) using Theorem 1. 3. If T P M NP 0 then calculate Z1(T, N) else go to step 5. 4. If Z1(T, N) > Z1(T, N 1), increment the value of N by 1 and go to step 2 else current value of N is optimal and the corresponding value of T and Z(T, N) can be calculated. 5. If 0 6 T 6 M N then calculate Z2(T, N) else go to step 7. 6. If Z2(T, N) > Z2(T, N 1), increment the value of N by 1 and go to step 2 else current value of N is optimal and the corresponding value of T and Z(T, N) can be calculated. 7. If M N 6 0 6 T then calculate Z3(T, N). 8. If Z3(T, N) > Z3(T, N 1), increment the value of N by 1 and go to step 2 else current value of N is optimal and the corresponding value of T and Z(T, N) can be calculated. 5. Numerical example Let max demand (S) = 100 units/day, min demand (s) = 30 units/day, rate of saturation of
M (days)
T * (days)
N * (days)
Z(T *, N *) ($)
30 45 60 75
28.63 28.73 29.26 29.42
33 34 34 35
916 934 951 968
Table 2 Effects of changing A on the optimal solution A ($)
T * (days)
N * (days)
Z(T *, N *) ($)
500 1000 1500
20.24 28.63 35.06
33 33 33
937 916 900
Table 3 Effects of changing r on the optimal solution r
T * (days)
N * (days)
Z(T *, N *) ($)
0.09 0.12 0.15
28.67 28.63 28.59
42 33 28
902 916 925
Table 4 Effects of changing Ip on the optimal solution Ip
T * (days)
N * (days)
Z(T *, N *) ($)
0.12 0.15 0.18
30.17 28.63 27.30
33 33 33
920 916 913
C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
retailer should offer lower credit period to customers (N) as the rate of saturation of demand (r) increases. Finally, from Table 4 it is observed that increase in the interest payable (Ip) causes lower cycle length and lower profit. 6. Conclusion This paper investigates the impact of creditlinked demand on the retailer’s optimal replenishment policy. An easy-to-use algorithm is proposed which gives the optimal values of both cycle length as well as credit period offered by the retailer. Finally, numerical example is presented to illustrate the theoretical results. Results suggest that credit period offered to customers has positive impact on the unrealized demand. In further research, we would like to extend the proposed model to deteriorating item, time-value of money, etc. Acknowledgements The authors thank the anonymous referees for their valuable suggestions and comments, which helped in improving the paper. References [1] S.P. Aggarwal, C.K. Jaggi, Ordering policies of deteriorating item under permissible delay in payments, Journal of the Operational Research Society 46 (1995) 658–662. [2] H.J. Chang, C.H. Hung, C.Y. Dye, An inventory model with stock-dependent demand and time-value of money when credit period is provided, Journal of Information and Optimization Sciences 25 (2) (2004) 237–254.
135
[3] K.H. Chung, Inventory control and trade credit revisited, Journal of the Operational Research Society 40 (1989) 495– 498. [4] Chung-Yuan Dye, A deteriorating inventory model with stock-dependent demand rate and partial backlogging under conditions of permissible delay in payments, Operations Research 39 (2002) 189–198. [5] S.K. Goyal, Economic ordering quantity under conditions of permissible delay in payments, Journal of the Operational Research Society 36 (1985) 335–343. [6] S.K. Goyal, J.T. Teng, C.T. Chang, Optimal ordering policies when the supplier provides a progressive interest scheme, European Journal of Operational Research 179 (2) (2007) 404–413. [7] C.W. Haley, R.C. Higgins, Inventory policy and trade credit financing, Management Science 20 (1973) 464–471. [8] Y.-F. Huang, Optimal retailer’s ordering policies in the EOQ model under trade credit financing, Journal of the Operational Research Society 54 (2003) 1011–1015. [9] H. Hwang, S.W. Shinn, Retailer’s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments, Computers and Operations Research 24 (1997) 539–547. [10] C.K. Jaggi, S.P. Aggarwal, Credit financing in economic ordering policies of deteriorating items, International Journal of Production Economics 34 (1994) 151–155. [11] A.M.M. Jamal, B.R. Sarkar, S. Wang, Optimal payment time for a retailer under permitted delay of payment by the wholesaler, International Journal of Production Economics 66 (2000) 59–66. [12] N.H. Shah, A lot size model for exponentially decaying inventory when delay in payments is permissible, Cashiers CERO, Belgium 35 (1993) 115–123. [13] J.T. Teng, On economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society 53 (2002) 915–918. [14] J.T. Teng, C.T. Chang, S.K. Goyal, Optimal pricing and ordering policy under permissible delay in payments, International Journal of Production Economics 97 (2005) 121– 129.
European Journal of Operational Research 190 (2008) 130–135 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Retailer’s optimal replenishment decisions with credit-linked demand under permissible delay in payments Chandra K. Jaggi
a,*
, S.K. Goyal b, S.K. Goel
a
a
b
Department of Operational Research, University of Delhi, Delhi 110 007, India Decision Sciences and MIS, John Molson School of Business, Concordia University Montreal, Que., Canada H3G 1M8 Received 13 June 2006; accepted 26 May 2007 Available online 2 June 2007
Abstract Usually it is assumed that the supplier would offer a fixed credit period to the retailer but the retailer in turn would not offer any credit period to its customers, which is unrealistic, because in real practice retailer might offer a credit period to its customers in order to stimulate his own demand. Moreover, it is observed that credit period offered by the retailer to its customers has a positive impact on demand of an item but the impact of credit period on demand has received a very little attention by the researchers. To incorporate this phenomenon, we assume that demand is linked to credit period offered by the retailer to the customers. This paper incorporates the concept of credit-linked demand and develops a new inventory model under two levels of trade credit policy to reflect the real-life situations. An easy-to-use algorithm is developed to determine the optimal credit as well as replenishment policy jointly for the retailer. Finally, numerical example is presented to illustrate the theoretical results followed by the sensitivity of various parameters on the optimal solution. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Inventory; Credit-linked demand; Two level credit policy; Delay in payments
1. Introduction In classical inventory analysis, it was tacitly assumed that the supplier is paid for the items as soon as the retailer receives the items. But in practice, the supplier allows a certain fixed credit period to settle the account for stimulating retailer’s demand. During this credit period the retailer can start to accumulate revenues on the sales and earn
*
Corresponding author. Tel./fax: +91 11 27666672. E-mail address: [email protected] (C.K. Jaggi).
interest on that revenue, but beyond this period the supplier charges interest. Hence, paying later indirectly reduces the cost of holding stock. On the other hand, trade credit offered by the supplier encourages the retailer to buy more and it is also a powerful promotional tool that attracts new customers, who consider it as an alternative incentive policy to quantity discounts. Hence, trade credit can play a major role in inventory control for both the supplier as well as retailer. Owing to this fact, during the past few years, many articles dealing with various inventory models under trade credit have appeared in various research
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.05.042
C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
journals. Haley and Higgins [7] introduced the first model to consider the economic order quantity under conditions of permissible delay in payments with deterministic demand, no shortages, and zero-lead time. Goyal [5] considered a model similar to that of Haley and Higgins with the exclusion of the penalty cost due to a late payment. Chung [3] presented the discounted cash flows (DCF) approach for the analysis of the optimal inventory policy in the presence of the trade credit. Shah [12] and Aggarwal and Jaggi [1] extended the Goyal’s model to the case of deterioration. Jamal et al. [11] further generalized the model to allow shortages. Jaggi and Aggarwal [10] extended Chung [3] to develop an inventory model for obtaining the optimal order quantity of deteriorating items in the presence of trade credit using the DCF approach. Hwang and Shinn [9] considered the problem of determining the retailer’s optimal price and lot size simultaneously when the supplier permits delay in payments. Dye [4] in their paper considered the stockdependent demand for deteriorating items for partial backlogging and condition of permissible delay in payment. They assumed initial stock-dependent demand function. Teng [13] provided an alternative conclusion from Goyal [5], and mathematically proved that it makes economic sense for a buyer to order less quantity and take benefits of the permissible delay more frequently. Chang et al. [2] considered an inventory model for deteriorating items with instantaneous stock-dependent demand and timevalue of money when credit period is provided. Teng et al. [14] developed the optimal pricing and lot sizing under permissible delay in payments by considering the difference between the selling price and the purchase cost and demand is a function of price. Recently, Goyal et al. [6] established an EOQ model for a retailer when the supplier offers a progressive interest charge, and provided an easy-to-use closedform solution to the problem. All the aforementioned inventory models implicitly assumed that the customer would pay for the items as soon as the items are received from the retailer. But, in most business transactions, this assumption is unrealistic and usually the supplier offers a credit period to the retailer and the retailer, in turn, passes on this credit period to his/her customers. Recently, Huang [8] presented an inventory model assuming that the retailer also offers a credit period to his/her customer which is shorter than the credit period offered by the supplier, in order to stimulate the demand. Moreover, in all the above articles, although the presence of credit period has been incor-
131
porated in the mathematical models but the impact of credit period on demand is unfortunately ignored. In reality, it is observed that demand of an item does depend upon the length of the credit period offered by the retailer. In order to incorporate the above phenomena’s a new form of credit-linked demand function has been coined using which a new inventory model has been formulated to determine the retailer’s optimal credit as well as replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. 2. Assumptions and notations The following assumptions are made to develop the mathematical model: 1. The supplier provides a fixed credit period M to settle the accounts to the retailer and the retailer, in turn, also offers a credit period N to each of its customers to settle the accounts. 2. The demand rate is a function of the customer’s credit-period offered by the retailer (N). The demand function for any N can be represented as a differential difference equation: DðN þ 1Þ DðN Þ ¼ r½S DðN Þ; where D = D(N) demand for any N per unit time S maximum demand r rate of saturation of demand (which can be estimated using the past data) under the assumption that the marginal effect of credit period on sales is proportional to the unrealized potential of the market demand without any delay. The solution of the above difference equation, under the condition that at N = 0, D(0) = s (initial demand), keeping other attributes like price, quantity, etc. at constant level, is given by D ¼ sð1 rÞN þ Sð1 ð1 rÞN Þ; N
i:e: D ¼ S ðS sÞð1 rÞ : 3. Replenishment rate is instantaneous. 4. Shortages are not allowed. 5. Lead-time is negligible. In addition, following notation are used: Q T
order quantity inventory cycle length
132
C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
q(t) A C P I
the inventory level at time t ordering cost per order unit purchase cost of the item unit selling price of the item out-of-pocket inventory carrying charge per $ per unit time Ie interest that can be earned per $ per unit time Ip interest payable per $ per unit time M retailer’s credit period offered by the supplier for settling the accounts N customer’s credit period offered by the retailer for settling the accounts Z(T, N) retailer’s profit per unit time which is a function of two variable; T and N where Z(T, N) = (a) revenue from sales (b) cost of purchasing units (c) cost of placing orders (d) cost of carrying inventory (excluding interest charges) + (e) interest earned from the sales during the permissible period (f) cost of interest charges for the unsold items after the permissible delay
InventoryLevel
Interest paid
Interest earned
Time 0
N
M
T
T+N
Fig. 1. When N 6 M 6 T + N.
to M and earns interest on average sales revenue for the time-period (M N). At M accounts are settled, if all the credit sales are not realized, the finances are to be arranged to make the payment to the supplier: ðeÞ Consequently the interest earned per unit time is ¼
I e PDðM N Þ 2T
2
ð8Þ
and 3. Mathematical formulation ðfÞ Interest payable per unit time is As the demand function is
2
N
D ¼ DðN Þ ¼ S ðS sÞð1 rÞ : Therefore, the order quantity is Z T D dt ¼ DT Q¼
ð1Þ
¼
I p CDðT þ N MÞ : 2T
ð9Þ
Using Eqs. (4)–(9), the retailer’s profit per unit time Z1(T, N) can be expressed as ð2Þ
A 1 þ fI e P ðDðM N Þ2 Þ T 2T 2 ICDT 2 I p CDðT þ N MÞ g: ð10Þ
Z 1 ðT ; N Þ ¼ ðP CÞD
0
and the inventory level at any time t during the cycle is Z t D dt ¼ DðT tÞ; 0 6 t 6 T : ð3Þ qðtÞ ¼ Q 0
The retailer’s profit per unit time consists of the following elements: ðaÞ Sales revenue ¼ PD;
ð4Þ
ðbÞ Cost of placing orders ¼ A=T ;
ð5Þ
ðcÞ Cost of purchasing units ¼ CD; ICDT ðdÞ Cost of carrying inventory ¼ : 2
ð6Þ
N 6 T + N 6 M (Fig. 2). In this case, the retailer earns interest on average sales revenues received during the period (N, T + N) and on full sales revenue (PQ) for a period of (M T N) but no interest payable is payable by the retailer.
Inventory Level
Int earned
ð7Þ
The computation for interest earned and payable (i.e. (e) and (f)) will depend on the following three possible cases based on the lengths of T, N and M: N 6 M 6 T + N (Fig. 1). In this case, the retailer starts getting actual sales revenues from time N
Time 0
N
T
T+N
Fig. 2. When N 6 T + N 6 M.
M
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ðeÞ Consequently the interest earned per unit time is
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4. Solution procedure
2
I e P fDT =2 þ DT ðM T N Þg T ¼ I e PDðM N T =2Þ:
¼
ð11Þ As a result, using Eqs. (4)–(7) and (11), the retailer’s profit per unit time Z2(T, N) in this case is A ICDT Z 2 ðT ; N Þ ¼ ðP CÞD T 2 T þ I e PD M N : 2
ð12Þ
M 6 N 6 T + N (Fig. 3). In this case, the retailer earns no interest but pays interest on full order quantity (Q) for a period of (N M) and on average stock held during the cycle length (T): ðfÞ Consequently the interest payable per unit time is I p CfDT ðN MÞ þ DT 2 =2g T ¼ I p CDðN M þ T =2Þ: ¼
ð13Þ As a result, using Eqs. (4)–(7) and (13), the retailer’s profit per unit time Z3(T, N) in this case is A ICDT Z 3 ðT ; N Þ ¼ ðP CÞD T 2 T I p CD N M þ : 2 Therefore, the retailer’s is 8 > < Z 1 ðT ; N Þ ZðT ; N Þ ¼ Z 2 ðT ; N Þ > : Z 3 ðT ; N Þ
ð14Þ
profit per unit time Z(T, N)
if N 6 M 6 T þ N ; if N 6 T þ N 6 M;
Our problem is to determine the optimum value of T and N which maximizes Z(T, N). For a fixed value of N, taking the first and second order derivatives of Z1(T, N), Z2(T, N) and Z3(T, N) with respect to T, we get Z 01 ðT ; N Þ ¼
A ICD PI e DðM N Þ2 2 T2 2T 2 CI P D 2 fT ðN MÞ2 g; 2T 2 2
2A DðM N Þ ðCI p PI e Þ ; T3 A ICD PI e D ; Z 02 ðT ; N Þ ¼ 2 2 2 T 2A Z 002 ðT ; N Þ ¼ 3 ; T A ICD CI P D ; Z 03 ðT ; N Þ ¼ 2 2 2 T 2A Z 003 ðT ; N Þ ¼ 3 : T Z 001 ðT ; N Þ ¼
if M 6 N 6 T þ N ;
which is a function of two variable T and N where T is continuous and N is discrete. Inventory Level
Time M
N
Fig. 3. When M 6 N 6 T + N.
ð19Þ ð20Þ ð21Þ
T 1 would satisfy the condition 0 6 (M N) 6 T provided ð23Þ
Substituting Eq. (22) into (2) and (10), we can get the optimal values of Q and Z1(T) (say Q1 and Z 1 Þ. Similarly, there exists a unique value of T (say T 2 ) which maximizes Z2(T) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A T 2 ¼ : ð24Þ ðIC þ I e P ÞD
2A DðM N Þ2 ½IC þ I e P 6 0:
T
ð18Þ
T 2 would satisfy the condition 0 6 T 6 (M N) provided
Int paid
0
ð17Þ
For fixed N, Eqs. (19) and (21) imply that Z2(T, N) and Z3(T, N) are concave on T > 0. However, Z1(T, N) is concave on T > 0 if CIp > PIe. Thus, there exists a unique value of T (say T 1 ) which maximizes Z1(T) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2A DðM N Þ ðPI e CI p Þ T1 ¼ : ð22Þ ðI þ I p ÞC
2A DðM N Þ2 ½IC þ I e P P 0: ð15Þ
ð16Þ
T+N
ð25Þ
Substituting Eq. (24) into (2) and (12), we can get the optimal values of Q and Z2(T) (say Q2 and Z 2 ). Likewise, we can obtain the optimal value of T (say T 3 ) which maximizes Z3(T) as
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A T 3 ¼ : ðI þ I p ÞCD
ð26Þ
T 3 would satisfy the condition (M N) 6 0 6 T provided 2
2A CDðM N Þ ½I þ I p P 0:
ð27Þ
Substituting Eq. (26) into (2) and (14), we can get the optimal values of Q and Z3(T) (say Q3 and Z 3 ). Combining the three possible cases, we obtain the following theorem. Theorem 1. For a fixed value of N (a) If 2A D(M N)2[IC + IeP] P 0 then T ¼ T 1 . (b) If 2A D(M N)2[IC + IeP] 6 0 then T ¼ T 2 . (c) If 2A CD(M N)2[I + Ip] P 0 and (M N) < 0 then T ¼ T 3 .
demand (r) = 0.12, A = $1000/order, M = 30 days, C = $30/unit, P = $40/unit, Ip = 15% per year, Ie = 10% per year and I = 15% per year. Using the proposed algorithm, we obtained the optimal results as follows: Optimal cycle length (T*) = 28.63 days, optimal credit period offered by retailer (N*) = 33 days and profit per day (Z*) = $916. Further, the sensitivity analysis on M, A, r and Ip is shown in Tables 1–4, respectively. It is observed from Table 1 that as M increases, there is marginal increase in cycle length as well as credit period offered by the retailer but there is significant increase in retailer’s profit, which implies that credit period offered to customers has positive impact on the unrealized demand. Table 2 shows that a hike in ordering cost (A) causes higher cycle length and lower profit. Table 3 suggests that
Table 1 Effects of changing M on the optimal solution
Proof. It immediately follows from (23), (25) and (27). h In order to jointly optimize Q and N, we propose the following algorithm: 1. Put N = 1. 2. Determine the optimal values of T (i.e. T 1 or T 2 or T 3 ) using Theorem 1. 3. If T P M NP 0 then calculate Z1(T, N) else go to step 5. 4. If Z1(T, N) > Z1(T, N 1), increment the value of N by 1 and go to step 2 else current value of N is optimal and the corresponding value of T and Z(T, N) can be calculated. 5. If 0 6 T 6 M N then calculate Z2(T, N) else go to step 7. 6. If Z2(T, N) > Z2(T, N 1), increment the value of N by 1 and go to step 2 else current value of N is optimal and the corresponding value of T and Z(T, N) can be calculated. 7. If M N 6 0 6 T then calculate Z3(T, N). 8. If Z3(T, N) > Z3(T, N 1), increment the value of N by 1 and go to step 2 else current value of N is optimal and the corresponding value of T and Z(T, N) can be calculated. 5. Numerical example Let max demand (S) = 100 units/day, min demand (s) = 30 units/day, rate of saturation of
M (days)
T * (days)
N * (days)
Z(T *, N *) ($)
30 45 60 75
28.63 28.73 29.26 29.42
33 34 34 35
916 934 951 968
Table 2 Effects of changing A on the optimal solution A ($)
T * (days)
N * (days)
Z(T *, N *) ($)
500 1000 1500
20.24 28.63 35.06
33 33 33
937 916 900
Table 3 Effects of changing r on the optimal solution r
T * (days)
N * (days)
Z(T *, N *) ($)
0.09 0.12 0.15
28.67 28.63 28.59
42 33 28
902 916 925
Table 4 Effects of changing Ip on the optimal solution Ip
T * (days)
N * (days)
Z(T *, N *) ($)
0.12 0.15 0.18
30.17 28.63 27.30
33 33 33
920 916 913
C.K. Jaggi et al. / European Journal of Operational Research 190 (2008) 130–135
retailer should offer lower credit period to customers (N) as the rate of saturation of demand (r) increases. Finally, from Table 4 it is observed that increase in the interest payable (Ip) causes lower cycle length and lower profit. 6. Conclusion This paper investigates the impact of creditlinked demand on the retailer’s optimal replenishment policy. An easy-to-use algorithm is proposed which gives the optimal values of both cycle length as well as credit period offered by the retailer. Finally, numerical example is presented to illustrate the theoretical results. Results suggest that credit period offered to customers has positive impact on the unrealized demand. In further research, we would like to extend the proposed model to deteriorating item, time-value of money, etc. Acknowledgements The authors thank the anonymous referees for their valuable suggestions and comments, which helped in improving the paper. References [1] S.P. Aggarwal, C.K. Jaggi, Ordering policies of deteriorating item under permissible delay in payments, Journal of the Operational Research Society 46 (1995) 658–662. [2] H.J. Chang, C.H. Hung, C.Y. Dye, An inventory model with stock-dependent demand and time-value of money when credit period is provided, Journal of Information and Optimization Sciences 25 (2) (2004) 237–254.
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[3] K.H. Chung, Inventory control and trade credit revisited, Journal of the Operational Research Society 40 (1989) 495– 498. [4] Chung-Yuan Dye, A deteriorating inventory model with stock-dependent demand rate and partial backlogging under conditions of permissible delay in payments, Operations Research 39 (2002) 189–198. [5] S.K. Goyal, Economic ordering quantity under conditions of permissible delay in payments, Journal of the Operational Research Society 36 (1985) 335–343. [6] S.K. Goyal, J.T. Teng, C.T. Chang, Optimal ordering policies when the supplier provides a progressive interest scheme, European Journal of Operational Research 179 (2) (2007) 404–413. [7] C.W. Haley, R.C. Higgins, Inventory policy and trade credit financing, Management Science 20 (1973) 464–471. [8] Y.-F. Huang, Optimal retailer’s ordering policies in the EOQ model under trade credit financing, Journal of the Operational Research Society 54 (2003) 1011–1015. [9] H. Hwang, S.W. Shinn, Retailer’s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments, Computers and Operations Research 24 (1997) 539–547. [10] C.K. Jaggi, S.P. Aggarwal, Credit financing in economic ordering policies of deteriorating items, International Journal of Production Economics 34 (1994) 151–155. [11] A.M.M. Jamal, B.R. Sarkar, S. Wang, Optimal payment time for a retailer under permitted delay of payment by the wholesaler, International Journal of Production Economics 66 (2000) 59–66. [12] N.H. Shah, A lot size model for exponentially decaying inventory when delay in payments is permissible, Cashiers CERO, Belgium 35 (1993) 115–123. [13] J.T. Teng, On economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society 53 (2002) 915–918. [14] J.T. Teng, C.T. Chang, S.K. Goyal, Optimal pricing and ordering policy under permissible delay in payments, International Journal of Production Economics 97 (2005) 121– 129.