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Analysis of inverted PBI and comparison with other scalarizing functions in decomposition based MOEAs

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Abstract

MOEA/D is one of the promising evolutionary approaches for solving multi and many-objective optimization problems. MOEA/D decomposes a multi-objective optimization problem into a number of single objective optimization problems. Each single objective optimization problem is defined by a scalarizing function using a weight vector. In MOEA/D, there are several scalarizing approaches such as weighted Tchebycheff, reciprocal weighted Tchebycheff, weighted sum (WS) and penalty-based boundary intersection (PBI). Each scalarizing function has a characteristic effect on the search performance of MOEA/D and provides a scenario of multi-objective solution search. To improve the availability of MOEA/D framework for solving various kinds of problems, it is important to provide a new scalarizing function which has different characteristics from the conventional scalarizing functions. In particular, the conventional scalarizing approaches face a difficulty to approximate a widely spread Pareto front in some problems. To approximate the entire Pareto front by improving the spread of solutions in the objective space and enhance the search performance of MOEA/D in multi and many-objective optimization problems, in this work we propose the inverted PBI scalarizing approach which is an extension of the conventional PBI and WS. In this work, we analyze differences between inverted PBI and other scalarizing functions, and compare the search performances of NSGA-III and five MOEA/Ds using weighted Tchebycheff, reciprocal weighted Tchebycheff, WS, PBI and inverted PBI in many-objective knapsack problems and WFG4 problems with 2–8 objectives. As results, we show that the inverted PBI based MOEA/D achieves higher search performance than other algorithms in problems with many-objectives and the difficulty to approximate a widely spread Pareto front in the objective space. Also, we show the robustness of the inverted PBI on Pareto front geometry by using problems with four representative concave, linear, convex and discontinuous Pareto fronts.

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Notes

  1. In this experiment, we use \(\theta =\{0,0.1,0.2,0.3,0.4,0.6,0.8,1,1.5,2,3,4,5,6,8,10,50,100,150,200,500,1000\}\) for both PBI and IPBI.

  2. In this experiment, we use \(\theta =\{\)0, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1, 1.5, 2, 3, 4, 5, 6, 8, 10, 50, 100, 150, 200, 500, 1000\(\}\) for both PBI and IPBI.

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Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 26730129.

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  1. Faculty of Informatics and Engineering, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan

    Hiroyuki Sato

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Sato, H. Analysis of inverted PBI and comparison with other scalarizing functions in decomposition based MOEAs. J Heuristics 21, 819–849 (2015). https://doi.org/10.1007/s10732-015-9301-6

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