A new geometric approach to multiobjective linear programming problems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| author | Kaci, Mustapha Radjef, Sonia |
| author_facet | Kaci, Mustapha Radjef, Sonia |
| contents | In this paper, we present a novel method for solving multiobjective linear programming problems (MOLPP) that overcomes the need to calculate the optimal value of each objective function. This method is a follow-up to our previous work on sensitivity analysis, where we developed a new geometric approach. The first step of our approach is to divide the space of linear forms into a finite number of sets based on a fixed convex polygonal subset of $\mathbb{R}^{2}$. This is done using an equivalence relationship, which ensures that all the elements from a given equivalence class have the same optimal solution. We then characterize the equivalence classes of the quotient set using a geometric approach to sensitivity analysis. This step is crucial in identifying the ideal solution to the MOLPP. By using this approach, we can determine whether a given MOLPP has an ideal solution without the need to calculate the optimal value of each objective function. This is a significant improvement over existing methods, as it significantly reduces the computational complexity and time required to solve MOLPP.
To illustrate our method, we provide a numerical example that demonstrates its effectiveness. Our method is simple, yet powerful, and can be easily applied to a wide range of MOLPP. This paper contributes to the field of optimization by presenting a new approach to solving MOLPP that is efficient, effective, and easy to implement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_10829 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A new geometric approach to multiobjective linear programming problems Kaci, Mustapha Radjef, Sonia Optimization and Control 49K40, 90C05 In this paper, we present a novel method for solving multiobjective linear programming problems (MOLPP) that overcomes the need to calculate the optimal value of each objective function. This method is a follow-up to our previous work on sensitivity analysis, where we developed a new geometric approach. The first step of our approach is to divide the space of linear forms into a finite number of sets based on a fixed convex polygonal subset of $\mathbb{R}^{2}$. This is done using an equivalence relationship, which ensures that all the elements from a given equivalence class have the same optimal solution. We then characterize the equivalence classes of the quotient set using a geometric approach to sensitivity analysis. This step is crucial in identifying the ideal solution to the MOLPP. By using this approach, we can determine whether a given MOLPP has an ideal solution without the need to calculate the optimal value of each objective function. This is a significant improvement over existing methods, as it significantly reduces the computational complexity and time required to solve MOLPP. To illustrate our method, we provide a numerical example that demonstrates its effectiveness. Our method is simple, yet powerful, and can be easily applied to a wide range of MOLPP. This paper contributes to the field of optimization by presenting a new approach to solving MOLPP that is efficient, effective, and easy to implement. |
| title | A new geometric approach to multiobjective linear programming problems |
| topic | Optimization and Control 49K40, 90C05 |
| url | https://arxiv.org/abs/2210.10829 |