Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain

In a competitive market, the retailers, in order to encourage the customers to increase their orders, give them the opportunity to pay a fraction of the purchasing cost after delivery of the ordered items (i.e., down-stream partial delayed payment). On the other hand, the suppliers, in order to reduce the risk of cancellations of orders from buyers, may ask the retailers to pay a portion of the purchasing cost before delivery of products (i.e., up-stream partial prepayment). In this paper, an EOQ model with down-stream partial delayed payment and up-stream partial prepayment under three different scenarios (without shortage, with full backordering and with partial backordering) is presented. In order to find the optimal solutions of the models developed for different scenarios, the convexity of the objective functions (i.e., total cost functions) are proved and then closed-form optimal solutions are derived. Also, a solution algorithm is proposed for the model of the third scenario. To demonstrate the applicability of the framework, some numerical examples are presented. Finally, sensitivity analyses are made on several key parameters, in order to gain some managerial insight.