A comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity Liang-Yuh Ouyang a , Chih-Te Yang b,⇑ , Ya-Lan Chan c , Leopoldo Eduardo Cárdenas-Barrón d a Department of Management Sciences, Tamkang University, Tamsui, New Taipei City 251, Taiwan, ROC b Department of Industrial Management, Chien Hsin University of Science and Technology, Jung-Li 320, Taiwan, ROC c Department of International Business, Asia University, Taichung 41354, Taiwan, ROC d Department of Industrial and Systems Engineering, School of Engineering, Tecnológico de Monterrey, Ave. E. Garza Sada 2501 Sur, CP64849 Monterrey, Nuevo León, México article info Keywords: Inventory Finance Permissible delay in payments abstract Recently, Kreng and Tan [Expert Systems with Applications 37 (2010) 5514–5522] devel- oped an economic order quantity (EOQ) model under two levels of trade credit policy in which the supplier offers to the wholesaler a permissible delay period M, and the whole- saler also provides its retailers a permissible delay period N (with M > N). In this paper, we point out some inappropriate mathematical expressions in both interest charged and interest earned in Kreng and Tan. For generality, we then extend their model to allow the following facts: (1) the interest rate I c charged by the supplier is not necessarily higher than the interest rate I e earned by the wholesaler, and (2) the permissible delay period M offered by the supplier is independent of the permissible delay period N offered by the wholesaler. Furthermore, we study the necessary and sufficient conditions for finding the optimal solution, and thus establish several theoretical results to characterize the solu- tion that provides the minimum annual total relevant cost. Finally, numerical examples are given to illustrate the theoretical results and obtain some managerial insights. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction In practice, the supplier usually allows the wholesaler a fixed permissible delay period for settling the account (i.e., an up- stream trade credit) and the wholesaler in turn provides a similar credit period to its customers (i.e., a down-stream trade credit). It is well known that the permissible delay in payments has two benefits: (1) it invites new buyers who consider it to be a type of price reduction, and (2) it may be useful as an alternative to price discount because it does not aggravate com- petitors to decrease their prices and thus introduce permanent price reductions (e.g., Yang et al. [1]). Goyal [2] developed an EOQ model with a constant demand rate under an up-stream trade credit. Aggarwal and Jaggi [3] extended Goyal’s [2] model to consider the deteriorating items. Jamal et al. [4] further generalized Aggarwal and Jaggi’s [3] model to allow for shortages. Teng [5] amended Goyal’s [2] model by considering the difference between unit price and unit cost, and found that it makes economic sense for a well-established retailer to order less quantity and take the benefits of payment delay more frequently. Chang et al. [6] developed an EOQ model for deteriorating items under supplier’s up-stream trade credit linked to ordering quantity. Huang [7] first established an EOQ model under two levels of trade credit policy with 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.062 ⇑ Corresponding author. E-mail address: ctyang@uch.edu.tw (C.-T. Yang). Applied Mathematics and Computation 224 (2013) 268–277 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc