Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers Jiang Wu a , Ya-Lan Chan b,n a School of Statistics, Southwestern University of Finance & Economics, Chengdu 611130, China b Department of International Business, Asia University, Taichung, Taiwan 41354, ROC article info Article history: Received 29 December 2013 Accepted 31 March 2014 Keywords: EOQ Trade credit Deteriorating item Expiration date Credit-risk customer abstract In practice, a credit-worthy retailer frequently receives a permissible delay on the entire purchase amount without collateral deposits from his/her supplier (i.e., an up-stream full trade credit). By contrast, a retailer usually requests his/her credit-risk customers to pay a fraction of the purchase amount at the time of placing an order, and then grants a permissible delay on the remaining balance (i.e., a down-stream partial trade credit). In addition, many products such as blood banks, pharmaceuticals, fruits, vegetables, volatile liquids, and others deteriorate constantly and have their expiration dates. However, not many researchers have taken the expiration date of a deteriorating item into consideration. The purpose of this paper is to establish optimal lot-sizing policies for a retailer who sells a deteriorating item to credit-risk customers by offering partial trade credit to reduce his/her risk. The proposed model is a generalized case of many previous models. By applying theorems in pseudo-convex fractional functions, we can easily prove that the optimal solution not only exists but is also unique. Moreover, we propose three discrimination terms, which can easily identify the optimal solution among all possible alternatives. Finally, some numerical examples are presented to highlight the theoretical results and managerial insights. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Harris (1913) established the classical economic order quantity (EOQ) model based on the assumptions that the purchase items are non-perishable and can be sold indefinitely, and the retailer must pay for the entire purchase cost as soon as the purchase items are received. In reality, many products (e.g., fruits, vegetables, medicines, volatile liquids, blood banks and others) not only deteriorate continuously (e. g., evaporation, obsolescence, and spoilage) but also have their expiration dates. Ghare and Schrader (1963) derived a revised form of the economic order quantity (hereafter EOQ) model by assuming exponential decay. Then Covert and Philip (1973) extended Ghare and Schrader's constant deterioration rate to a two-parameter Weibull distribution. Dave and Patel (1981) considered an EOQ model for dete- riorating items with time-proportional demand when shortages were prohibited. Sachan (1984) further extended the model to allow for shortages. Goswami and Chaudhuri (1991) generalized an EOQ model for deteriorating items from a constant demand pattern to a linear trend in demand. Hariga (1996) established optimal EOQ models for deteriorating items with time-varying demand. Goyal and Giri (2001) provided a survey on the recent trends in modeling of deteriorating inventory. Skouri et al. (2009) considered inventory models with ramp-type demand rate and Weibull deterioration rate. Skouri et al. (2011) further generalized the model to add a permissible delay in payments under consideration. Dye (2013) provided some results on finding the optimal replenishment and preservation technology stra- tegies for a non-instantaneous deteriorating inventory model. Recen- tly, Chen et al. (2013b) proposed economic production quantity (EPQ) models for deteriorating items. All the above mentioned papers did not consider the fact that deteriorating items have their expiration dates. In fact, the study of deteriorating items with expiration dates has received a relatively little attention in the literature. Currently, Bakker et al. (2012) provided an excellent review of inventory systems with deterioration since 2001. In practice, a seller frequently offers his/her buyers a permissible delay in payment (i.e., trade credit) for settling the purchase amount. 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. Goyal (1985) proposed an EOQ model under conditions of permissible delay in payments. Aggarwal and Jaggi (1995) extended Goyal's model for deteriorating items. Jamal et al. (1997) further generalized Aggarwal and Jaggi's model to allow for shortages. Chang et al. (2003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Huang (2003) proposed an EOQ model in which the supplier offers Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics http://dx.doi.org/10.1016/j.ijpe.2014.03.023 0925-5273/& 2014 Elsevier B.V. All rights reserved. n Corresponding author. Tel.: þ886 4 2332 3456x48038. E-mail address: yalan@asia.edu.tw (Y.-L. Chan). Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎