Vehicle routing problems (VRPs) have usually been studied with a single objective function defined by the distances associated with the routing of vehicles. The central problem is to design a set of routes to meet the demands of customers at minimum cost. However, in real life, it is necessary to take into account other objective functions, such as social functions, which consider, for example, the drivers' workload balance. This has led to growth in both the formulation of multiobjective models and exact and approximate solution techniques. In this article, to verify the quality of the results, first, a mathematical model is proposed that takes into account both economic and work balance objectives simultaneously and is solved using an exact method based on the decomposition approach. This method is used to compare the accuracy of the proposed approximate method in test cases of medium mathematical complexity. Second, an approximate method based on the Iterated Local Search (ILS) metaheuristic and Decomposition (ILS/D) is proposed to solve the biobjective Capacitated VRP (bi-CVRP) using test cases of medium and high mathematical complexity. Finally, the nondominated sorting genetic algorithm (NSGA-II) approximate method is implemented to compare both medium- and high-complexity test cases with a benchmark. The obtained results show that ILS/D is a promising technique for solving VRPs with a multiobjective approach.

The multi-objective problem of multi-depot vehicle routing (MOMDVRP) is proposed by considering the minimization of the traveled arc costs and the balance of routes. Seven mathematical models were reviewed to determine the route balance equation and the best-performing model is selected for this purpose. The solution methodology consists of three stages; in the first one, beginning solutions are built up by means of a constructive heuristic. In the second stage, fronts are constructed from each starting solution using the iterated local search multi-objective metaheuristics (ILSMO). In the third stage, we obtain a single front by using concepts of dominance, taking as a base the fronts of the previous stage. Thus, the first two fronts are taken and a single front is formed that corresponds to the current solution of the problem; next the third front is added to the current Pareto front of the problem, the procedure is repeated until exhaustion of the list of the fronts initially obtained. The resulting front is the solution to the problem. To validate the methodology we use instances from the specialized literature, which have been used for the multi-depot routing problem (MDVRP). The results obtained provide very good quality. Finally, decision criteria are used to select the most appropriate solution for the front, both from the point of view of the balance and the route cost.