عنوان مقاله [English]
In this paper, a stochastic bi-objective inventory control model is developed, in which its objectives are to minimize the total inventory and transportation costs and minimize the expected number of items stocked out annually. The demand within the lead time is a random variable with a normal distribution. In this paper, multi-mode transportation is used. Since the fixed transportation costs are high, coordination of orders and full truckload shipments can benefit from economies of scale. Bin packing problems have been used for allocation product to trucks, which belong to a class of well-studied and highly popular combinatorial optimization problems. In general, bin packing problems are motivated by a large number of real-world applications. The problem is to find a best assignment of objects to bins such that weight of the objects in each bin does not exceed its capacity and the number of bins used is minimized. Variable-sized bin packing problem and bin packing problem with over-sized items are generalizations of the bin packing problem. The first problem is to pack a given set of items into a minimum-cost set of bins of variable sizes and costs. In the second problem, some item sizes are larger than the largest size of bins. Because the presented model is a bi-objective nonlinear programming type and NP-hard one to solve it in reasonable time, a well-known multi-objective evolutionary algorithm, namely a non-dominated sorting genetic algorithm (NSGA-II), is proposed. To verify the obtained solution and evaluate the performance of the proposed algorithm in small-size problems, we use the varepsilon-constraint as an exact method that has been developed for general multi-objective problems. It solved varepsilon-constraint problems obtained by transforming one of the objectives into a constraint. In large-sized problems, 10 problems are solved with the proposed NSGA-II. Then, the Pareto-optimal solutions are evaluated. Finally, the consultation is provided.