Simulation optimization for the vehicle routing problem with time windows using a Bayesian network as a probability model

The main purpose of the vehicle routing problem (VRP) is to deliver a set of customers with known demands on minimum-travel routes and terminating at the same depot. The vehicle routing problem with time windows (VRPTW) requires the delivery made in a specific time window for every customer and returning to the depot before a due time. Contrary to current research, an estimation of distribution-algorithm-based approach coupled with a simulation model is proposed and developed to solve the problem and implement the solution. The approach mentioned makes use of a Bayesian network as a probability model to describe the distribution of the solution space. Furthermore, the approach taken in this study combines the key advantages of both estimation of distribution algorithms (EDA) and simulation. The simulation is used to model the VRPTW environment, while the EDA is used to guide the overall search process to identify the best performing ones. Solomon’s (Oper Res 35:254–265, 1987 ) instances served as input and test parameters in order to show that there exists a relationship and interaction between vertices and positions on the sequence of the VRPTW solution. A better position for each vertex on the sequence can be estimated through a Bayesian network. Experimental results show that the EDA performance was better in 70 % of the cases, as average, for the number of vehicles used in all the trails with respect the other algorithms proposed as a benchmark for comparison with the EDA scheme.