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In this paper, we establish an economic production quantity model for a manufacturer (or wholesaler) with defective items when its supplier offers an up-stream trade credit M while it in turn provides its buyers (or retailers) a down-stream trade credit N. The proposed model is in a general framework that includes numerous previous models as special cases. In contrast to the traditional differential calculus approach, we use a simple-to-understand and easy-to-apply arithmetic–geometric inequality method to find the optimal solution. Furthermore, we provide some theoretical results to characterize the optimal solution. Finally, several numerical examples are presented to illustrate the proposed model and the optimal solution.

In the classical inventory economic order quantity (EOQ) model, it is implicitly assumed that a buyer must pay for the purchased items immediately upon receiving the items. However, in practice, a seller frequently offers his/her buyers a delay of payment for settling the amount owed to him/her. Usually, there is no interest charge if the outstanding amount is paid within the permissible delay period. However, if the payment is not paid in full by the end of the permissible delay period, then interest is charged on the outstanding amount. The permissible delay in payment produces two benefits to the seller: (1) it attracts new buyers who may consider it to be a type of price reduction, and (2) it may be applied as an alternative to price discount because it does not provoke competitors to reduce their prices and thus introduce lasting price reductions. On the other hand, the policy of granting credit terms adds an additional dimension of default risk to the seller because the longer the permissible delay, the higher the default risk. During the past two decades, many researchers have studied various inventory models with trade credit financing. Goyal (1985) was the first proponent for developing an economic order quantity (EOQ) model under the conditions of permissible delay in payments. Aggarwal and Jaggi (1995) extended Goyal’s model to allow for deteriorating items. Then Jamal, Sarker, and Wang (1997) further generalized Aggarwal and Jaggi’s model to allow for shortages. Teng (2002) amended Goyal’s model by incorporating the fact that unit price is significantly higher than unit cost. Huang (2003) extended Goyal’s model to a supply chain in which the supplier offers the wholesaler the permissible delay period M (i.e., the upstream trade credit), and the wholesaler in turn provides the trade credit period N (with N < M) to its retailers (i.e., the downstream trade credit). Teng and Goyal (2007) amended Huang’s model by complementing his shortcomings. Liao (2008) extended Huang’s model to analyze the impact of the two-level trade credit financing on an economic production quantity (EPQ) model for deteriorating items. Soni and Shah (2008) presented an inventory model with a stock-dependent demand under progressive payment scheme. Teng (2009a) established optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. Teng and Chang (2009) developed optimal manufacturer’s replenishment policies under two levels of trade credit financing. Kreng and Tan (2010) studied optimal replenishment decisions under two-level trade credit policy depending on the order quantity. Teng, Krommyda, Skouri, and Lou (2011) extended the model by Soni and Shah (2008) to allow for: a nonzero ending-inventory, a profit-maximization objective, a limited warehouse’s capacity and deteriorating items. Many related articles can be found in Chang et al., 2008, Chang et al., 2010, Goyal et al., 2007, Huang, 2004, Huang, 2007, Huang and Hsu, 2008, Ouyang et al., 2012, Shinn and Hwang, 2003, Skouri et al., 2011, Yang et al., 2011 and Yang et al., 2012, and their references. Recently, Kreng and Tan (2011) proposed the optimal replenishment decisions to the manufacturer (or wholesaler) with finite replenishment rate and imperfect product quality in a supply chain, in which the manufacturer receives an up-stream trade credit M from its supplier while provides its retailers a down-stream trade credit N with N < M. They then developed four theoretical results. However, they ignored the fact that the manufacturer offers his/her retailers a permissible delay period N, and, hence, the manufacturer receives sales revenue from N to T + N, not from 0 to T as shown in their model. In this paper, we not only extend their EPQ model to complement the above mentioned shortcomings but also relax some dispensable assumptions of N < M and others. In our view the permissible delay period N offered by the manufacturer is independent of the permissible delay period M offered by the supplier. The manufacturer must choose an appropriate value of N based on the prevalent market conditions. In many situations manufacturers may be forced to offer a permissible delay period to their retailers while receiving no permissible delay period (M = 0) from their suppliers. As a result, the proposed model here is in a general framework that includes numerous previous models such as Goyal, 1985, Teng, 2002, Huang, 2003, Teng and Goyal, 2007, Liao, 2008, Chang et al., 2010 and Kreng and Tan, 2011 as special cases. The rest of this paper is organized as follows. In Section 2, we first define the assumptions and notation used throughout the entire paper, and then establish the manufacturer annual total profit in a supply chain with both up-stream and down-stream trade credits. To maximize the annual total profit for the manufacturer, we use a simple-to-understand and easy-to-apply arithmetic–geometric inequality method to obtain the optimal solution, instead of the traditional differential calculus approach in Section 3. Furthermore, some theoretical results are established to obtain the optimal solution. In Section 4, several numerical examples are provided to illustrate the theoretical results and managerial insights. Finally, the conclusions and suggestions for the future research are given in Section 5.

In this paper, we have proposed an EPQ model with up-stream and down-stream trade credits in a general framework that includes numerous previous models such as Goyal, 1985, Teng, 2002, Huang, 2003, Teng and Goyal, 2007, Liao, 2008, Chang et al., 2010 and Kreng and Tan, 2011 as special cases. In order to obtain the explicit closed-form solution without using traditional differential calculus, we have used a simple arithmetic–geometric inequality method to obtain the optimal solution for the manufacturer. In addition, we have established some theoretical results to characterize the optimal solution. Finally, we have provided several numerical examples to illustrate the proposed model and its optimal solution. The research presented in this paper can be extended in several ways. For instance, we may generalize the constant demand rate to any non-decreasing demand rate. Also, we could extend the model to allow for shortages. Finally, we can consider the effect of inflation rates on the economic order quantity.