Demonstrating the importance of using total time balance instead of route balance on a multi-objective vehicle routing problem with time windows

Abstract

‘Route balance’ is the difference between the longest and the shortest among all the route lengths. The route balance is frequently considered in a vehicle routing problem (VRP) for balancing the distance travelled among delivery vehicles. This is a common practice since in VRPs there are no time elements. However, in recent years, some research works have considered route balance in a vehicle routing problem with time windows (VRPTW). As there are more time elements in VRPTW, this paper refutes that route balance is sufficient for VRPTW by simultaneously optimising makespan and workload imbalance using ‘total time balance’ instead of route balance. On the other hand, ‘total time balance’ is the difference between the longest total time taken and the shortest total time taken among delivery vehicles. As such, makespan is the longest total time taken among vehicles. Total time taken for a vehicle is the sum of vehicle run time, waiting time and service time. In order to demonstrate the importance of using total time balance instead of route balance on VRPTW, three different multi-objective VRPTW models, namely, (1) only with general VRPTW objectives (i.e. ‘total distance travelled’ by all vehicles and ‘total number of vehicles’ used), (2) general VRPTW objectives with route balance and (3) general VRPTW objectives with total time balance are developed and solved by fitness aggregated genetic algorithm (FAGA) for 36 Solomon’s benchmark instances. By comparison of the makespan produced by the FAGA between the three cases, the importance of using total time balance instead of route balance on multi-objective VRPTW is demonstrated. Also, the makespan produced by the FAGA for the third case, i.e. general VRPTW objectives with total time balance is compared with the makespan produced by fitness aggregated differential evolution (FADE). By comparing makespan with statistical testing between FAGA and FADE, the outperformance of FAGA over FADE is verified. To check the practicality of the total time balance on multi-objective VRPTW, an instance with real time windows is also solved for the three cases by the FAGA and its makespan are compared and reported.