Advances in Decision Sciences

Volume 2014, Article ID 340135, 11 pages

http://dx.doi.org/10.1155/2014/340135

## An Economic Order Quantity Model with Completely Backordering and Nondecreasing Demand under Two-Level Trade Credit

^{1}Department of Mechanical and Manufacturing Engineering, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia^{2}Department of Computer Engineering, Islamic Azad University, Science and Research Branch, Simon Boulevard, Ashrafi Esfahani Highway, Tehran 1477893855, Iran^{3}Faculty of Manufacturing Engineering, Universiti Malaysia Pahang, 26600 Pekan, Pahang, Malaysia

Received 31 July 2014; Accepted 5 December 2014; Published 31 December 2014

Academic Editor: David Bulger

Copyright © 2014 Zohreh Molamohamadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the traditional inventory system, it was implicitly assumed that the buyer pays to the seller as soon as he receives the items. In today’s competitive industry, however, the seller usually offers the buyer a delay period to settle the account of the goods. Not only the seller but also the buyer may apply trade credit as a strategic tool to stimulate his customers’ demands. This paper investigates the effects of the latter policy, two-level trade credit, on a retailer’s optimal ordering decisions within the economic order quantity framework and allowable shortages. Unlike most of the previous studies, the demand function of the customers is considered to increase with time. The objective of the retailer’s inventory model is to maximize the profit. The replenishment decisions optimally are obtained using genetic algorithm. Two special cases of the proposed model are discussed and the impacts of parameters on the decision variables are finally investigated. Numerical examples demonstrate the profitability of the developed two-level supply chain with backorder.

#### 1. Introduction

Since the introduction of the classical economic order quantity (EOQ) model by Harris [1], many researchers have extended it in several ways. One of the discussed issues in this area is including delay in payment, as an incentive system, in the EOQ or economic production quantity (EPQ) models [2]. According to Piasecki [3] and Molamohamadi et al. [4], different types of delay in payment can be classified as pay as sold, pay as sold during a predefined period, pay after a predefined period, and pay at the next consignment order.

In the first type of delay in payment, so-called consignment inventory, the buyer defers paying for the items till they are sold to the customers. The second type refers to the case that the buyer pays off as soon as he sells the items to the customers during a predefined period. At the end of this period, he can either pay for the remaining items in his stock or return the unsold items to the vendor. According to the third type of delay in payment, which is known as trade credit in the literature, the buyer must pay to the vendor at the end of a predetermined period. During the credit period, the buyer sells the items to his customers and accumulates revenue and earns interest. After this period, however, he would be charged a higher interest if the payment is not settled. Based on the fourth type, the payment for each order would be settled at the time of the next replenishment order. Therefore, there is one replenishment cycle delay for each received order in this type. The advantage of delay in payment contract to the buyer is obvious; he does not need to invest his capital in inventory and can earn interest for the items he sells. Moreover, the vendor can apply this agreement as a sales promotional tool for attracting new buyers and selling new and unproven products.

As this paper focuses on the third type of delay in payment, we review the literature related to trade credit (please refer to Seifert et al. [5] and Molamohamadi et al. [6]). Goyal [7] presented an EOQ mathematical model for determining the economic order quantity where the supplier offers a fixed credit period to the retailer to settle the account. His paper is the infrastructure for its following studies. Aggarwal and Jaggi [8] extended Goyal [7] by considering deterioration rate and assuming that the customer accumulates the sales revenue and earns interest during the credit period and beyond it. Jamal et al. [9] included shortages in the proposed model by Aggarwal and Jaggi [8] to generalize it. Teng [10] modified Goyal’s [7] model by distinguishing between the unit purchase cost and the selling price. By applying an EPQ model, Chung and Huang [11] further developed Goyal [7] by assuming finite replenishment rate. Huang [12] considered a two-level trade credit and deduced Goyal [7] as a special case of his research. In a two-level trade credit, not only does the vendor offer trade credit to the buyer, but the retailer also provides a credit period to his customers.

Huang [13] investigated the retailer’s inventory policy under two-level trade credit with unequal selling and purchasing prices and extended Teng [10] and Huang [12] by considering the retailer’s limited storage space. Teng and Goyal [14] complemented the shortcoming of Huang [12]’s model in calculating the earned interest from the time the retailer is paid by his customers, not from time zero. They further extended his paper by relaxing the limitations on the selling and purchasing prices, as well as retailer and customer’s credit periods. Huang [15] established an economic order quantity model in which the supplier provides the retailer partially permissible delay in payment for the order quantities smaller than a predetermined quantity and offers him complete trade credit otherwise. Huang [16] differentiated between the purchase cost and the selling price and presented an EPQ model under two levels of trade credit to generalize Chung and Huang [11] and Huang [12]. Teng and Chang [17] reformulated Huang’s [16] model by calculating the retailer’s earned revenue from the time he is paid by the customers and further extended his model by assuming that the customer’s credit period is not inevitably smaller than the retailer’s delay period. Su [18] developed a supplier-buyer inventory model in which the supplier’s selling price is dependent on his productions cost. He further assumed that the latter is affected by the market demand and production rates, and the production rate is sensitive to the price dependent market demand. He finally obtained the optimal pricing, ordering, and inventory decisions of a profit maximizing system under trade credit contract.

Dye and Ouyang [19] proposed an EOQ mixed-integer nonlinear programming model under two levels of trade credit for deteriorating items with time-varying demand and applied a traditional particle swarm optimization (PSO) algorithm to determine the optimal selling price and replenishment policy. Dye [20] applied PSO algorithm to obtain the optimal replenishment decisions of an EOQ model with price and time dependent demand, partially backlogged items, and deterioration under two-level trade credit policy. Mahata [21] presented a generalization of Goyal [7], Chung and Huang [11], Huang [12], and Huang [16] where an economic production quantity model is formulated for exponentially deteriorating items under two levels of trade credit with the assumption that the customer’s partial credit period is not necessarily smaller than the retailer’s complete credit agreement. Lou and Wang [22] formulated an EPQ inventory model for defective items under two independent levels of trade credit to extend some of the previous studies including Goyal [7], Teng [10], Huang [12], and Teng and Goyal [14].

Having applied cuckoo search algorithm, Molamohamadi et al. [23] solved an EPQ model of an exponentially deteriorating item with price-sensitive demand under trade credit contract and allowable shortages. Chern et al. [24, 25] formulated a supply chain under trade credit financing with noncooperative Stackelberg and Nash equilibrium solutions, respectively. Chen and Teng [26] determined the retailer’s optimal cycle time by developing an EOQ model under trade credit policy for continuously deteriorating items with maximum lifetime. Chen et al. [27] reformulated Mahata’s [21] proposed model by calculating the earned and paid interest based on the facts that (i) the retailer earns interest from the time he is paid by the customers and (ii) the retailer’s interest payable must be calculated based on the total items in stock, not only on the unsold finished products. Some of the previous models such as Goyal [7] and Teng [10] are mentioned as special cases of their proposed model. Chen et al. [28] complemented some shortcomings of Huang’s [15] mathematical expressions and figures and proposed a simple method to solve the inventory problem.

Reviewing the literature clarifies that trade credit has received great attention of the researchers, while it has still outstanding space for further studies. For instance, it is mostly assumed that the demand rate is constant. However, recently, Teng et al. [29] developed an EOQ inventory model under trade credit contract in which demand has an increasing function of time. Although, their model can be considered as a generalization of its preceding studies, it has great potential for further extension. As it is stated in Jamal et al. [9], when delay in payment is assumed, shortages are more important as they affect the quantity ordered to benefit from the delay in payment. Moreover, in practices, not only does the supplier propose a delay period to the retailer, but the retailer also allows his customers to defer their payment. Thus, considering two levels of trade credit contributes to practical situations.

Considering the gaps in the literature, we extend the proposed model of Teng et al. [29] to the case of backorder and two-level trade credit. It is assumed here that the retailer’s credit period offered by the supplier is greater than the customer’s delay period offered by the retailer. The proper replenishment policy and the maximum profit of the retailer are then obtained by applying genetic algorithm (GA). It is finally deduced that the inventory system of Teng et al. [29] and the traditional inventory system are special cases of our proposed model and the results obtained in this paper are compared with these cases.

The rest of this paper is organized as follows. Section 2 lays out the notations and assumptions used in the modeling of the problem. The model is formulated in Section 3 and the two special cases of the presented model presented are discussed in Section 4. The genetic algorithm, used for solving the model, is described in Section 5. Regarding the numerical examples of Teng et al. [29], Section 6 provides some numerical examples and the conclusion is finally discussed in Section 7.

#### 2. Notations and Assumptions

The following notations and assumptions are used in this paper.

##### 2.1. Notations

:ordering cost per order,:unit purchasing cost,:unit selling price (with ),:unit stock holding cost per unit of time (excluding interest charges),:unit backorder cost of retailer per unit of time,:interest which can be earned per $ per unit of time by the retailer,:interest charges per $ in stocks per unit of time by the supplier,:the retailer’s trade credit period offered by supplier in years,:the customer’s trade credit period offered by retailer in years,:the net profit of the retailer per unit of time,:the inventory cycle time,:the inventory cycle time with positive stock,:the retailer’s order quantity,:the inventory consumed in .

##### 2.2. Assumptions

(1)The demand is assumed to have an increasing function of time and is defined by as follows: where and are nonnegative constants and is the growth stage of the product life cycle.(2)Shortages are allowed and completely backordered.(3)The lead time is zero.(4)The retailer is offered a delay period () by the supplier and provides the customers with a shorter credit period (). The retailer pays off to the supplier at the end of the credit period () and pays for the interest charges on the remaining items in his stock with rate during if . When the credit period is greater than the positive-stock replenishment cycle, the retailer would not be charged by any interest after settling the account.(5)The retailer accumulates revenue and earns interest with rate from to .

#### 3. Mathematical Formulation

According to the notations and assumptions discussed in previous section, the retailer’s inventory system is depicted in Figure 1 and can be explained as follows. The retailer orders and receives units of items at time zero and sends the previously backlogged orders to the customers immediately. The remaining inventory () depletes gradually due to the customers’ demand and becomes zero at time . From to there is no inventory on hand and the arriving orders would be backlogged to the next cycle.