Multi-Objective Factored Evolutionary Optimization and the Multi-Objective Knapsack Problem

We propose a factored evolutionary framework for multi-objective optimization that can incorporate any multi-objective population based algorithm. Our framework, which is based on Factored Evolutionary Algorithms, uses overlapping subpopulations to increase exploration of the objective space; however, it also allows for the creation of distinct subpopulations as in co-operative co-evolutionary algorithms (CCEA). We apply the framework with the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), resulting in Factored NSGA-II. We compare NSGA-II, CC-NSGA-II, and F-NSGA-II on two different versions of the multi-objective knapsack problem. The first is the classic binary multi-knapsack implementation introduced by Zitzler and Thiele, where the number of objectives equals the number of knapsacks. The second uses a single knapsack where, aside from maximizing profit and minimizing weight, an additional objective tries to minimize the difference in weight of the items in the knapsack, creating a balanced knapsack. We further extend this version to minimize volume and balance the volume. The proposed 3-to-5 objective balanced single knapsack problem poses a difficult problem for multi-objective algorithms. Our results indicate that the non-dominated solutions found by F-NSGA-II tend to cover more of the Pareto front and have a larger hypervolume.