Backorders case in inventory model with fuzzy demand and fuzzy lead time

In this paper, we present an inventory model with fuzzy parameters and back ordering is allowed. If the demands occurring when the system is out of stock, then there is shortage cost associated with incurring backorders to meet the demand. The annual demand and lead time are triangular fuzzy numbers. The cost parameters are crisps. Since the lead time is fuzzy, so holding cost and shortage cost are fuzzy. To determine holding cost and shortage cost, we need fuzzy integral. Hence, previously we define fuzzy integral through alpha-level. The total cost is summation of purchasing cost, ordering cost, holding cost and shortage cost. Furthermore, we get average annual cost with non linear membership function that depend on order quantity. We will determine the optimal order quantity and reorder point to minimize a fuzzy total cost by graded mean integration. We apply non linear optimization method by MATLAB to find order quantity. We give a numerical example. © 2019 International Association of Engineers.