Optimizing an inventory model with fuzzy demand, backordering, and discount using a hybrid imperialist competitive algorithm

• An inventory mathematical model with fuzzy demand is developed regarding backordering and discounts.
• A hybrid algorithm including ICA, GA, PSO, and boundary operator is used to solve.
• A simulated annealing (SA) is presented to solve the problem as well.
• TOPSIS method determines the better algorithm in solving the problem.
• Sensitivity analyses are performed.

Vendor-managed inventory (VMI) is one of the popular policies in retailer-supplier partnerships to reduce total inventory costs in supply chain management (SCM). As there is a lack of studies on simultaneous consideration of backordering and discount in the VMI literature, this paper develops a constrained VMI model with fuzzy demand for a single-vendor multi-retailer supply chain, in which the centroid defuzzification method defuzzifies trapezoidal fuzzy numbers of the demand. While there are two constraints, namely warehouse space and number of orders, the aim of this paper is to find a near-optimal solution including the order size, the replenishment rate of the retailers, the maximum backorder quantities of retailers' item, and the suitable price of the item to minimize the total inventory cost. As the developed problem is shown to be NP-complete, a hybrid imperialist competitive algorithm (HICA) is used to find a near-optimum solution. As there is no benchmark available in the literature and that HICA is a population-based meta-heuristic, a single-solution-based meta-heuristic is employed to validate the results obtained. The technique for order preference by similarity to ideal solution (TOPSIS) determines the better algorithm in solving the problem in terms of six criteria. Moreover, the effects of some cost parameters on the solution obtained are investigated to provide managerial insights. Finally, conclusion and future research are presented.