A Study on the Specification of a Scalarizing Function in MOEA/D for Many-Objective Knapsack Problems

In recent studies on evolutionary multiobjective optimization, MOEA/D has been frequently used due to its simplicity, high computational efficiency, and high search ability. A multiobjective problem in MOEA/D is decomposed into a number of single-objective problems, which are defined by a single scalarizing function with evenly specified weight vectors. The number of the single-objective problems is the same as the number of weight vectors. The population size is also the same as the number of weight vectors. Multiobjective search for a variety of Pareto optimal solutions is realized by single-objective optimization of a scalarizing function in various directions. In this paper, we examine the dependency of the performance of MOEA/D on the specification of a scalarizing function. MOEA/D is applied to knapsack problems with 2-10 objectives. As a scalarizing function, we examine the weighted sum, the weighted Tchebycheff, and the PBI (penalty-based boundary intersection) function with a wide range of penalty parameter values. Experimental results show that the weighted Tchebycheff and the PBI function with an appropriate penalty parameter value outperformed the weighted sum and the PBI function with no penalty parameter in computational experiments on two-objective problems. However, better results were obtained from the weighted sum and the PBI function with no penalty parameter for many-objective problems with 6-10 objectives. We discuss the reason for these observations using the contour line of each scalarizing function. We also suggest potential usefulness of the PBI function with a negative penalty parameter value for many-objective problems.