The Biased Multi-Objective Optimization using the Reference Point: Toward the industrial logistics network Eriko Azuma, Tomohiro Shimada, Keiki Takadama, Hiroyuki Sato, and Kiyohiko Hattori The University of Electro-Communications, Chofu, Japan. Tel: +81-42-443-5808, Email: {azuma@cas., shimada@cas., keiki@, sato@, hattori@}hc.uec.ac.jp Abstract— This paper explores the multi-objective evolutionary algorithm that can effectively solve a multi-objective problem where an importance of the objective differs each other unlike the conventional problem which concerns each objective evenly. Since such a type of a problem is often found in industrial problems (e.g., logistics network), we propose the biased multi- objective optimization using the reference point (i.e., the factor of strongly concerned). Intensive experiment on the multi-objective knapsack problem had revealed that our proposed method was more superior and had higher diversity than the conventional multi-objective optimization method. Keywords-component; multi-objective optimization, genetic algorithm, logistics I. INTRODUCTION Most of problems in the real world have two or more objectives, and are required to solve the trade-off among those objectives at the same time. These problems are called the multi-objective optimization problem. When focusing on the industrial logistics network in airplanes, ships, and trucks, for example, we have to address the trade-off problem between the profit and CO 2 emissions of a company. Since CO 2 emissions are regarded as the biggest problem in the earth in terms of the global warming, the companies must try to reduce the CO 2 emissions, but the most important objective of them is to earn more profit than the current situation. Although, these companies in the transportation industry have to earn more profit while reducing the amount of CO 2 emissions, the importance of these objectives is not equally, i.e., the profit is more important than CO 2 emissions. From these features, the conventional multi-objective optimization algorithms (e.g., Elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II) [3]) cannot work efficiently due to the fact that all objectives at the same importance. To tackle the problem where an importance of the objectives differs each other unlike the conventional problem, this research proposes the biased multi-objective optimization using the reference point (i.e., the factor of strongly concerned). In detail, it searches near the reference point which has the best value of the most important objective, in order to find the solution which has the higher value of the most important objective. To verify the effectiveness of our proposed method, we apply it into the 2-objective knapsack problem. This paper is organized as follows. Section 2 describes multi-objective optimization. Section 3 introduces our proposed method. Experiments are conducted in Section 4 and their results are discussed in Section 5. Finally, our conclusion is given in Section 6. II. MULTI-OBJECTIVE OPTIMIZATION Multi-objective optimization is the process of optimizing two or more conflicting objectives simultaneously subject to certain constraints. The optimal solutions are located at the Pareto optimal front as shown in Fig. 1, where the horizontal and vertical axes are objective functions f 1 (x) and f 2 (x). The Pareto front solutions are a set of solutions which are not dominated by other solutions in terms of f 1 (x) and f 2 (x). Figure 1. The Pareto optimal front One of the major multi-objective optimization algorithms is NSGA-II (Elitist Non-Dominated Sorting Genetic Algorithm), which follows the following algorithms. (i) Each solution is ranked by the non-dominated sort (i.e., a smaller rank is better). Fig. 2 shows the process of the non-dominated sort. The solutions which are not dominated by other solutions for all objectives are regarded as the rank 1 in the non- dominated sort ranking. The same process is conducted without the solutions of the rank 1 to calculate the solution of the rank 2, 3, and more. (ii) For i-th solutions of the same ranked, the density of the solutions in its neighbor is calculated by the 2011 10th International Conference on Machine Learning and Applications 978-0-7695-4607-0/11 $26.00 © 2011 IEEE DOI 10.1109/ICMLA.2011.138 27