A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items

In practice, installment of cost of lots is very common between manufacturers and distributors, and many distributors pay the purchasing cost in equal installments (i.e., an upstream partial prepayment). In a similar fashion, distributors allow retailers to pay the cost of purchased goods after such goods are received (i.e., a downstream partial delay payment). In this research we apply a model of economic order quantity (EOQ) in supply chains with partial downstream delayed payment and partial upstream advance payment for a deteriorating item under three conditions: 1) shortage is not allowed, 2) full back ordering is allowed, and 3) partial back ordering is allowed. We prove the convexity of the cost functions and derive closed form global optimal solutions for the decision variables of all models. Eventually, to demonstrate the theoretical results and managerial options, we perform some computational examples.