Continuous review inventory model with variable lead time in a fuzzy random environment

Inventory control is an important field in supply chain management, and a great deal of research efforts have been devoted to it over past few decades. In previous researches, there are some assumptions like that the lead time is an incontrollable variable, and all the items replenished are of perfect quality, and so on. However, those assumptions may not be fit for the real environments, and the inventory control problem needs to be considered in a more comprehensive sense. The aims of this paper is to establish the mathematical model and propose an solving approach for the reorder point inventory problems with partial backordered and partial lost sale situation in fuzzy random environment. Specially, the paper investigates the mixture inventory control system in which the lead time demands in different cycles are independent and identically distributed (iid) random variables, and the defective rates of the arrived order lot in different cycles are also iid random variables. Moreover, the backorder rate, ordering cost, shortage penalty cost and marginal profit per unit in different cycles are iid fuzzy variables, respectively. Then based on the fuzzy random renewal reward theory, a mathematical formulation about the expected annual total cost is presented, and some useful properties are analyzed for establishing an efficient solution procedure. In order to calculate the expected value of fuzzy expression and search the optimal values of order quantity, reorder point and lead time, a fuzzy simulation algorithm and an iterative algorithm are designed, respectively. Finally, a numerical example is given to illustrate the procedure of searching the optimal solution.