A class of weighted Tchebycheff preference relations and multi-objective optimization

Hongjie Tan; Kequan Zhao; Yuanmei Xia; Hongjie Tan; Kequan Zhao; Yuanmei Xia

doi:10.3934/jimo.2026001

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 1: 1-23. doi: 10.3934/jimo.2026001

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Research article

A class of weighted Tchebycheff preference relations and multi-objective optimization

School of Mathematical Sciences, Chongqing Normal University, No. 37 University Town Middle Road, Shapingba District, Chongqing 401331, China

Received: 15 September 2025 Revised: 22 October 2025 Accepted: 31 October 2025 Published: 12 November 2025
90C26, 90C29, 90C30

In this paper, we have proposed a new class of binary preference relations by utilizing the weighted Tchebycheff scalarization method, and established several related properties in the objective space. Furthermore, we gave the definitions of the (weakly, strictly) efficient solution and (weakly, strictly) nondominated point for multi-objective optimization problems with respect to the proposed preference relations, and investigated the relationship between these solutions (nondominated points) and Pareto solutions (nondominated points). Moreover, we established the theoretical associations between the three types of solutions proposed in this paper and the optimal solutions of the weighted Tchebycheff scalarization model. Finally, two numerical examples were employed to further illustrate the significance of the proposed preference relations. The results indicated that, for certain specific multi-objective optimization problems, the number of weakly (strictly) nondominated points with respect to the relations presented by us is fewer than that with respect to the natural orders and the weighted aggregation preference relations. Meanwhile, the computational time, worst-case computational complexity, and sensitivity analysis for the weight vector were discussed in the second numerical example.
- Multi-objective optimization,
- binary preference relation,
- weighted Tchebycheff scalarization,
- optimality,
- (weakly, strictly) nondominated set
Citation: Hongjie Tan, Kequan Zhao, Yuanmei Xia. A class of weighted Tchebycheff preference relations and multi-objective optimization[J]. Journal of Industrial and Management Optimization, 2026, 22(1): 1-23. doi: 10.3934/jimo.2026001

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Abstract

In this paper, we have proposed a new class of binary preference relations by utilizing the weighted Tchebycheff scalarization method, and established several related properties in the objective space. Furthermore, we gave the definitions of the (weakly, strictly) efficient solution and (weakly, strictly) nondominated point for multi-objective optimization problems with respect to the proposed preference relations, and investigated the relationship between these solutions (nondominated points) and Pareto solutions (nondominated points). Moreover, we established the theoretical associations between the three types of solutions proposed in this paper and the optimal solutions of the weighted Tchebycheff scalarization model. Finally, two numerical examples were employed to further illustrate the significance of the proposed preference relations. The results indicated that, for certain specific multi-objective optimization problems, the number of weakly (strictly) nondominated points with respect to the relations presented by us is fewer than that with respect to the natural orders and the weighted aggregation preference relations. Meanwhile, the computational time, worst-case computational complexity, and sensitivity analysis for the weight vector were discussed in the second numerical example.

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