Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions
Bi-objective optimization seeks to obtain Pareto optimal solutions that balance two trade-off objectives, providing guidance for decision making in various fields, particularly in the field of transportation. The novelty of this study lies in two aspects. On the one hand, considering that Pareto opt...
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2025-02-01
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| author | Hongyu Zhang Qingfang Ruan Yong Jin Shuaian Wang |
| author_facet | Hongyu Zhang Qingfang Ruan Yong Jin Shuaian Wang |
| author_sort | Hongyu Zhang |
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| description | Bi-objective optimization seeks to obtain Pareto optimal solutions that balance two trade-off objectives, providing guidance for decision making in various fields, particularly in the field of transportation. The novelty of this study lies in two aspects. On the one hand, considering that Pareto optimal solutions are often numerous, finding the full set of Pareto optimal solutions is often computationally challenging and unnecessary for practical purposes. Therefore, we shift the focus of bi-objective optimization to finding a subset of Pareto optimal solutions whose resulting set of nondominated objective vectors is the same as, or at least a good approximation of, the full set of nondominated objective vectors for the problem. In particular, we elaborate three methods for generating a near-optimal subset of Pareto optimal solutions, including the revised <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-constraint method, the improved revised <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-constraint method, and the augmented <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-constraint method. More importantly, the near-optimality of the Pareto optimal solution subset obtained by these methods is rigorously analyzed and proved from a mathematical point of view. This study helps to offer theoretical support for future studies to find the subset of Pareto optimal solutions, which reduces the unnecessary workload and improves the efficiency of solving bi-objective optimization problems while guaranteeing a pre-specified tolerance level. |
| format | Article |
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| language | English |
| publishDate | 2025-02-01 |
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| series | Applied Sciences |
| spelling | doaj-art-b7ed76bfefd84d8d985bfb22c527239a2025-08-20T02:05:23ZengMDPI AGApplied Sciences2076-34172025-02-01155251910.3390/app15052519Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal SolutionsHongyu Zhang0Qingfang Ruan1Yong Jin2Shuaian Wang3Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 999077, ChinaShenzhen Guangming District Institute of Administration, Shenzhen 518106, ChinaFaculty of Business, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 999077, ChinaDepartment of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 999077, ChinaBi-objective optimization seeks to obtain Pareto optimal solutions that balance two trade-off objectives, providing guidance for decision making in various fields, particularly in the field of transportation. The novelty of this study lies in two aspects. On the one hand, considering that Pareto optimal solutions are often numerous, finding the full set of Pareto optimal solutions is often computationally challenging and unnecessary for practical purposes. Therefore, we shift the focus of bi-objective optimization to finding a subset of Pareto optimal solutions whose resulting set of nondominated objective vectors is the same as, or at least a good approximation of, the full set of nondominated objective vectors for the problem. In particular, we elaborate three methods for generating a near-optimal subset of Pareto optimal solutions, including the revised <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-constraint method, the improved revised <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-constraint method, and the augmented <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-constraint method. More importantly, the near-optimality of the Pareto optimal solution subset obtained by these methods is rigorously analyzed and proved from a mathematical point of view. This study helps to offer theoretical support for future studies to find the subset of Pareto optimal solutions, which reduces the unnecessary workload and improves the efficiency of solving bi-objective optimization problems while guaranteeing a pre-specified tolerance level.https://www.mdpi.com/2076-3417/15/5/2519bi-objective optimizationtransportation optimizationPareto optimal solutionsnear-optimal subset |
| spellingShingle | Hongyu Zhang Qingfang Ruan Yong Jin Shuaian Wang Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions Applied Sciences bi-objective optimization transportation optimization Pareto optimal solutions near-optimal subset |
| title | Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions |
| title_full | Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions |
| title_fullStr | Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions |
| title_full_unstemmed | Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions |
| title_short | Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions |
| title_sort | bi objective optimization for transportation generating near optimal subsets of pareto optimal solutions |
| topic | bi-objective optimization transportation optimization Pareto optimal solutions near-optimal subset |
| url | https://www.mdpi.com/2076-3417/15/5/2519 |
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