A New Mathematical Programming Model for a Vehicle Routing Problem in a Competitive Environment: A Real Case Study

Transportation, which has a specific division in production economics and service systems, has a great value in a gross national product (GNP) in every country. Thus, most researchers worked on improving routes, omitting unimportant transportations, or replacing shortcut routes. For these reasons, they have developed a traveling salesman problem (TSP) and a vehicle routing problem (VRP). The classical and deterministic VRP seeks routes with the lowest cost from a depot to a geographically dispersed customers’ set with known demands. Usually in the VRP, monopoly in an environment is the first assumption in solving the problem. Also attending to the competitors for getting better success is important. In the real-world situation, customer’s cash is limited, and a competitor that reach to customers earlier than others can receive the maximum cash. So, providing the specific service to customers earlier than other competitors is very important. In this paper, getting the maximum cash is considered in our presented model with a competitive environment. In the best of our knowledge, the VRP with maximizing cash in the objective function has not been considered in the literature. Thus, we consider a bi-objective model that minimizes the total distance travelled and maximizes the cash flow in a competitive environment. These objectives conflict with each other (i.e., the better one of them, the worse the other). So for facing to this difficulty, a weighting method is needed to integrate these objectives into a single minimization cost function. The presented mathematical model is solved by the Lingo 8 software for various test problems. Since exact solutions of VRPs that belong to the category of NP-hard problems are not practical in large scale, a hybrid simulated annealing (SA) algorithm embedded with genetic operators is proposed in order to obtain an efficient solution in a reasonable computational time. Furthermore, a real case study is carried out to validate our presented model and the related results are reported. Finally, the remarkable conclusion is presented.