“…That is, when the order quantity of retailer Q is greater than or equal to a certain threshold, Q d , then the transportation cost is absorbed by the supplier; otherwise, it is paid by the retailer. On the other hand, to facilitate the transaction and receive payment as soon as possible, the supplier provides the retailer with a two-part trade credit, which allows the retailer to make payments at time M 2 and, then, provides the retailer with a cash discount and a discount rate (0 < < 1 [63], and Sarkar et al [71], the production-inventory system considers a single supplier, a single retailer, and single commodity; 2. Shortages are allowed for the retailer and all customers are willing to wait for the next delivery (see, for example [31,46]); 3.…”