Improved population-based incremental learning algorithm for vehicle routing problems with soft time windows

An improved population-based incremental learning algorithm, in short IPBIL, is proposed to solve thevehicle routing problem with soft time windows (VRPSTW) with an objective to minimize the count of vehicles as well as the total travel distance.VRPSTW is subject to the soft time window constraint that allows to be violated but with penalty.In this paper, the constraint is embedded into a probability selection function and the original probability model of population-based incremental learning (PBIL) algorithm becomes 3-dimensional. This improvement guarantees that the population search of individuals is more effective by escaping from a bad solution space. Simulation of Solomon benchmark shows that the results average vehicle counts with IPBIL is reduced very significantly contrasted to those with Genetic Algorithm (GA) and PBIL, respectively. Both the average travel length and total time window violations by IPBIL are the least among these tested methods.IPBIL is more effective and adaptive than PBIL and GA.