Bi-objective mixed-integer nonlinear programming for multi-commodity tri-echelon supply chain networks

The competitive market and declined economy have increased the relevant importance of making supply chain network efficient. Up to now, this has resulted in great motivations to reduce the cost of services, and simultaneously, to improve their quality. A mere network model, as a tri-echelon, consists of Suppliers, Warehouses or Distribution Centers (DCs), and Retailer nodes. To bring it closer to reality, the majority of parameters in this network involve retailer demands, lead-time, warehouses holding and shipment costs, and also suppliers procuring and stocking costs which are all assumed to be stochastic. The aim is to determine the optimum service level so that total cost is minimized. Obtaining such conditions requires determining which supplier nodes, and which DC nodes in network should be active to satisfy the retailers’ needs, an issue which is a network optimization problem per se. The proposed supply chain network for this paper is formulated as a mixed-integer nonlinear programming, and to solve this complicated problem, since the literature for the related benchmark is poor, three numbers of genetic algorithm called Non-dominated Sorting Genetic Algorithm (NSGA-II), Non-dominated Ranking Genetic Algorithm (NRGA), and Pareto Envelope-based Selection Algorithm (PESA-II) are applied and compared to validate the obtained results. The Taguchi method is also utilized for calibrating and controlling the parameters of the applied triple algorithms.