OPTIMAL RETAILER’S INVENTORY POLICY UNDER SUPPLIER CREDITS LINKED TO RETAILER PAYMENT TIME

- This paper wants to investigate the retailer’s optimal cycle time and optimal payment time under supplier credits including conditionally permissible delay in payments and cash discount depending on retailer payment time. That is, the retailer can obtain fully permissible delay in payments and cash discount if the payment is paid before the period of full delay payments permitted by the supplier. Otherwise, the retailer will just obtain partially permissible delay in payments within the period of partial delay payments permitted by the supplier. The supplier uses this policy to attract retailer to pay the payment as soon as possible to shorten the collection period. One theorem is developed to efficiently determine the optimal replenishment and payment policy for the retailer.


INTRODUCTION
The traditional EOQ (Economic Order Quantity) model assumes the payment for the quantity ordered is made when the quantity is received. However, in practice it is found that supplier allows a certain fixed credit period to the retailer to promote his/her commodities. The effect of supplier credit policy on inventory problem has received the attention of many researchers. Goyal [7] established a single-item inventory model under permissible delay in payments. Chung [4] developed an alternative approach to determine the economic order quantity under condition of permissible delay in payments. Aggarwal and Jaggi [1] considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Teng [12] assumed that the selling price not equal to the purchasing price to modify Goyal's model [7]. Huang [8] extended this issue under two levels of trade credit and developed an efficient solution-finding procedure to determine the optimal lot-sizing policy of the retailer. Chung and Huang [5] investigated this issue within EPQ framework and developed an efficient solution-finding procedure to determine the optimal cycle time for the retailer. Chang et al. [3] and Chung and Liao [6] deal with the problem of determining the economic order quantity for exponentially deteriorating items under permissible delay in payments depending on the ordering quantity. Huang [9] investigated that the unit selling price and the unit purchasing price are not necessarily equal within the EPQ framework under supplier's trade credit policy.
The timing of cash flows of an investment proposal is important because the sooner the money becomes available, the sooner it can be used for other worthwhile purposes. Therefore, it makes economic sense for the retailer to delay the settlement of the replenishment account up to the last moment of the permissible delay 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. Recently, Chang [2], Ouyang et al. [11] and Huang and Chung [10] investigated the inventory policy under cash discount and trade credit.
What the above statement describes is just one of ways of attracting the retailer to pay the payment as soon as possible. This paper tries to develop another more effective supplier's credit policy to shorten the collection period. We assume that the retailer can obtain fully permissible delay in payments and cash discount if the payment is paid before the period of full delay payments permitted by the supplier. Otherwise, the retailer will just obtain partially permissible delay in payments within the period of partial delay payments permitted by the supplier. In the policy of partial delay payments, the retailer must make partial payment at the time the retailer places a replenishment order and the rest of the total amount is payable before or at the end of the permissible credit period. The supplier uses the credits policy depending on payment time to attract the retailer to pay the payment as soon as possible to shorten the collection period. Under this condition, we model the retailer's inventory system as a cost minimization problem and prove one theorem to efficiently determine the retailer's optimal replenishment and optimal payment policy.

MODEL FORMULATION
2.1 Notation: D = demand rate per year A = cost of placing one order c = unit purchasing price h = unit stock holding cost per year excluding interest charges r = cash discount rate, 0 ≤ r < 1 α = the fraction of the total amount owed payable at the time of placing an order, 0≤α≤1 I e = interest earned per $ per year I k = interest charges per $ investment in inventory per year M 1 = the period of full delay payments permitted in years M 2 = the period of partial delay payments permitted in years, M 1 < M 2 T = the cycle time in years TRC(T)= the annual total relevant cost when T > 0 T 1 * = the optimal cycle time of TRC 1 (T) T 2 * = the optimal cycle time of TRC 2 (T) T* = the optimal cycle time of TRC(T) Q* = the optimal order quantity=DT*.

Assumptions:
(1) Demand rate is known and constant.
(4) Replenishments are instantaneous. (5) I k ≥ I e . (6) Supplier offers a cash discount and fully delayed payment to the retailer if payment is paid within M 1 , otherwise just partially delayed payment if payment is paid within M 2 . (7) If payment is paid within M 1 , when the account is settled the retailer starts paying for the interest charges on the items in stock. If payment is paid behind M 1 but within M 2 , as the order is received, the retailer must make a partial payment αcDT to the supplier. Then the retailer must pay off the remaining balance (1−α)cDT at the end of the partially permissible delay period M 2 . (8) During the time the account is not settled, generated sales revenue is deposited in an interest-bearing account.

The model:
The annual total relevant cost consists of the following elements. In this case, annual interests payable = 0. Case 2: Payment is paid at time M 2 , according to assumption (7); there are three sub-cases in terms of interests charged per year.
, as shown in Figure 1.

NUMERICAL EXAMPLES
To illustrate the theoretical results, let us apply the proposed method to solve the following numerical examples. The optimal solutions for different parameters of α, r and c are shown in Table 1.The following inferences can be made based on Table 1.
(1). For fixed r and c, the larger value of α is, the smaller value of the optimal cycle time and the higher value of the annual total relevant cost will be as the optimal payment time is M 2 ; however, if the optimal payment time is M 1 , the optimal cycle time is independent of the value of α. (2). For fixed α and c, the larger the value of r is, the larger value of the optimal cycle time and the lower value of the annual total relevant cost will be as the (1). When the fraction of the total amount owed payable is increasing, the retailer will order less quantity. Therefore, the supplier can use the higher fraction of permitted delay policy to stimulate the retailer's demands when the optimal payment time is M 2 .
(2). When the cash discount rate is increasing, the retailer will adopt the optimal payment time in M 1 to shorten the delay period.
(3). When the unit purchasing price is increasing, the retailer will order less quantity to take the benefits of the permissible delay in payments more frequently whatever fully or partially permissible delay in payments.
In future research, we would like to extend to allow for shortages, deteriorating items or finite replenishment rate.