non-dominated genetic algorithm (nsga- ii)

In this paper, we develop a multi-period mathematical model involving economic and environmental considerations. A vehicle-routing problem is considered an essential matter due to decreasing the routing cost, especially in the concerned bioenergy supply chain. A few of the optimization model recognized the vehicle routing to design the bioenergy supply chain. In this study, a bi-objective mixed-integer linear programming (MILP) model is presented.  The economic objective function minimizes the transportation, capacity expanding, fixed and variable costs, and the locating routing cost in this problem. The proposed bi-objective model is solved through a Non-Dominated Genetic Algorithm (NSGA- II).  Furthermore, the small-sized problem is solved by the CPLEX solver and augmented ɛ-constraint method.