Economic order quantity under conditionally permissible delay in payments

Abstract

Within the economic order quantity (EOQ) framework, the main purpose of this paper is to investigate the retailer’s optimal replenishment policy under permissible delay in payments. All previously published articles dealing with optimal order quantity with permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this paper, we want to extend this extreme case by assuming that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity. Under this condition, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer. Some previously published results of other researchers can be deduced as special cases. Finally, numerical examples are given to illustrate all these theorems and to draw managerial insights.

Introduction

The traditional economic order quantity (EOQ) model assumes that the retailer must be paid for the items as soon as the items were received. In practice, the supplier hopes to stimulate his products and so he will offer the retailer a delay period, namely, the trade credit period: Before the end of the trade credit period, the retailer can sell the goods and accumulate revenue and earn interest. On the other hand, a higher interest is charged if the payment is not settled by the end of the trade credit period. 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.

Several papers discussing this topic have appeared in the literatures that investigate inventory problems under varying conditions. Some of the prominent papers are discussed below. Goyal [8] established a single-item inventory model for determining the economic ordering quantity in the case that the supplier offers the retailer the opportunity to delay his payment within a fixed time period. Chung [5] simplified the search of the optimal solution for the problem explored by Goyal [8]. Aggarwal and Jaggi [2] considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Jamal et al. [13] then further generalized the model to allow for shortages. Hwang and Shinn [12] developed the model for determining the retailer’s optimal price and lot-size simultaneously when the supplier permits delay in payments for an order of a product whose demand rate is a function of constant price elasticity. Jamal et al. [14] formulated a model where the retailer can pay the wholesaler either at the end of the credit period or later, incurring interest charges on the unpaid balances for the overdue period. They developed a retailer’s policy for the optimal cycle and payment times for a retailer in a deteriorating-item inventory scenario, in which a wholesaler allows a specified credit period for payment without penalty. Teng [17] assumed that the selling price is not equal to the purchasing price to modify Goyal’s model [8]. The important finding from Teng’s study [17] is that it makes economic sense for a well-established retailer to order small lot sizes and so take more frequently the benefits of the permissible delay in payments. Chung and Huang [6] extended Goyal [8] to consider the case that the units are replenished at a finite rate under permissible delay in payments and developed an efficient solution-finding procedure to determine the retailer’s optimal ordering policy. Huang [9] extended one-level trade credit into two-level trade credit to develop the retailer’s replenishment model from the viewpoint of the supply chain. He assumed that not only the supplier offers the retailer trade credit but also the retailer offers the trade credit to his/her customer. This viewpoint reflected more real-life situations in the supply chain model. Khouja [15] showed that for many supply chain configurations, complete synchronization would result in some members of the chain being ‘losers’ in terms of cost. He used the economic delivery and scheduling problem model and analyzed supply chains dealing with single and multiple components in developing his model. Huang and Chung [11] extended Goyal’s model [8] to discuss the replenishment and payment policies to minimize the annual total average cost under cash discount and payment delay from the retailer’s point of view. They assumed that the supplier could adopt a cash discount policy to attract retailer to pay the full payment of the amount of purchasing at an earlier time as a means to shorten the collection period. Arcelus et al. [3] modeled the retailer’s profit-maximizing retail promotion strategy, when confronted with a vendor’s trade promotion offer of credit and/or price discount on the purchase of regular or perishable merchandise. Abad and Jaggi [1] formulated models of seller-buyer relationship, they provided procedures for finding the seller’s and buyer’s best policies under non-cooperative and cooperative relationship respectively. Huang [10] extended Chung and Huang’s model [6], in allowing the retailer adopts different payment policy and finding differences between unit purchase and selling price, and developed an efficient solution-finding procedure to determine the retailer’s optimal cycle time and optimal order quantity.

All above published papers assumed that the supplier offer the retailer fully permissible delay in payments independent of the order quantity. Recently, Shinn and Hwang [16] determined the retailer’s optimal price and order size simultaneously under the condition of order-size-dependent delay in payments. They assumed that the length of the credit period is a function of the retailer’s order size, and also the demand rate is a function of the selling price. Chung and Liao [7] dealt with the problem of determining the economic order quantity for exponentially deteriorating items under permissible delay in payments depending on the ordering quantity and developed an efficient solution-finding procedure to determine the retailer’s optimal ordering policy. In this regard, Chang [4] extended Chung and Liao [7] by taking into account inflation and finite time horizon. However, all above published papers dealing with economic order quantity in the presence of permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer orders a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. We know that this policy of the supplier to stimulate the demands from the retailer is very practical. But this is just an extreme case. That is, the retailer would obtain 100% permissible delay in payments if the retailer ordered a large enough quantity. Otherwise, 0% permissible delay in payments would happen.

In reality, the supplier can relax this extreme case to offer the retailer partially permissible delay in payments rather than 0% permissible delay in payments when the order quantity is smaller than a predetermined quantity. That is, the retailer must make a partial payment to the supplier when the order is received to enjoy some portion of the trade credit. Then, the retailer must pay off the remaining balances at the end of the permissible delay period. For example, the supplier provides 100% delay payment permitted if the retailer ordered a sufficient quantity, otherwise only α% (0 ⩽ α ⩽ 100) delay payment permitted. From the viewpoint of supplier’s marketing policy, the supplier can use the fraction of the permissible delay in payments to agilely control the effects of stimulating the demands from the retailer. This viewpoint is a realistic and novel one in this research field, hence, forms the focus of the present study. Therefore, we ignore the effect of deteriorating item; inflation and finite time horizon similar to most previously published articles. Under these conditions, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer under the more general framework. Some previously published results of other researchers can be viewed as special cases. Finally, numerical examples are given to illustrate all these theorems and to draw managerial insights.