An Adapted Fuzzy Multi-Objective Programming Algorithm for Vehicle Routing

Abstract

The vehicle routing problem (VRP) is a well-known problem in the logistics sector. In this study, two objectives, minimizing the total distance and maximizing the saving value, were considered in VRP with a fuzzy environment. The game theory approach is proposed for determining the weights of objectives when decision-makers have insufficient knowledge of assigning the weights. Thus, a fuzzy pay-off matrix is proposed for determining the weights of objectives by combining the fuzzy two-person zero-sum game with mixed strategies (FTZG with MS) and membership functions. Therefore, the fuzzy multi-objective programming (FMOP) model is adapted to the VRP model, which is named Adapted FMOP algorithm for VRP. Proposed algorithm clusters customers according to two objectives and by using four fuzzy operators, and routes customers with the traveling salesman problem (TSP) model in order to avoid the non-deterministic polynomial-time hardness (NP-hard) structure of VRP. In the end, the results are improved using local search methods. The main contribution of the Adapted FMOP algorithm for VRP is that it provides a solution that considers more than one objective without the need for decision makers’ view on the weights of objectives in all decision models in the fuzzy environment. Also, the proposed algorithm can find the solution with the help of a mathematical model without requiring any heuristics or metaheuristics, since it primarily performs clustering. Firstly, the efficiency of this algorithm was tested on problems in the literature. The Adapted FMOP algorithm for VRP achieved the best-known solutions by some small margins and exceeded the best-known solution for one problem in the literature. After seeing that the performance of the algorithm was sufficient, a data set of a firm in the construction sector was implemented to see how the algorithm works in real life and the obtained results were discussed. The solutions demonstrate that the Adapted FMOP algorithm for VRP also works well for real-world problems.