EFFICIENCY OF THE IDEAL POINT METHOD FOR SOLVING TWO-CRITERION OPTIMIZATION PROBLEMS ; ЕФЕКТИВНІСТЬ МЕТОДУ ІДЕАЛЬНОЇ ТОЧКИ РОЗВ’ЯЗУВАННЯ ДВОКРИТЕРІАЛЬНИХ ЗАДАЧ ОПТИМІЗАЦІЇ
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| Název: | EFFICIENCY OF THE IDEAL POINT METHOD FOR SOLVING TWO-CRITERION OPTIMIZATION PROBLEMS ; ЕФЕКТИВНІСТЬ МЕТОДУ ІДЕАЛЬНОЇ ТОЧКИ РОЗВ’ЯЗУВАННЯ ДВОКРИТЕРІАЛЬНИХ ЗАДАЧ ОПТИМІЗАЦІЇ |
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| Autoři: | Цегелик, Г. Г., Цегелик, М. Г., Грипинська, Н. В., Ярецька, Н. О. |
| Zdroj: | Computer Science and Applied Mathematics; No. 1 (2025): Computer Science and Applied Mathematics; 63-67 ; Computer Science and Applied Mathematics; № 1 (2025): Computer Science and Applied Mathematics; 63-67 ; 2518-1785 ; 2413-6549 ; 10.26661/2786-6254-2025-1 |
| Informace o vydavateli: | Zaporizhzhia National University |
| Rok vydání: | 2025 |
| Témata: | decision-making methods, multi-criteria optimization, ideal point method, Sylvester function convexity test, Lagrange multiplier method, методи прийняття рішень, багатокритеріальна оптимізація, метод ідеальної точки, ознака опуклості функцій Сільвестера, метод множників Лагранжа |
| Popis: | The paper considers the ideal point method for solving two-criteria optimization problems. It is proven for the first time that when the number of variables on which the optimization criteria depend exceeds two, the objective function of a scalar problem loses convexity. This makes it impossible to use the Lagrange multiplier method, since the necessary optimality conditions are not met.A feature of the ideal point method is that it does not require additional information from the decision-maker regarding the preference of one criterion over another in the set of possible alternatives. However, the method is based on the assumption that there is a certain optimal solution to the two- criteria optimization problem, which can be found by transforming it into the corresponding scalar single-criteria problem. The basic principle of choosing a compromise solution is to determine the alternative whose value is closest to the ideal point in a certain metric. Thus, the method provides the possibility of obtaining an approximate optimal solution without the need to introduce additional weighting factors or subjective estimates.It is shown that there have been attempts to solve a two-criteria optimization problem in the field of production planning in the case of four variables. It has been established that such a problem cannot be solved using the ideal point method. This is explained by the fact that the saddle point conditions used to check optimality cannot be applied due to the lack of convexity of the objective function.So, the ideal point method is effective for analyzing two-criteria optimization problems, but its application is limited to cases with the number of variables not exceeding two. In more complex problems with a larger number of variables, difficulties arise due to the loss of convexity of the objective function, which makes it impossible to use classical solution methods.Therefore, further research will be aimed at developing approaches that will allow the search for alternative optimization methods ... |
| Druh dokumentu: | article in journal/newspaper |
| Popis souboru: | application/pdf |
| Jazyk: | Ukrainian |
| Relation: | http://journalsofznu.zp.ua/index.php/comp-science/article/view/4559/4345 |
| DOI: | 10.26661/2786-6254-2025-1-08 |
| Dostupnost: | http://journalsofznu.zp.ua/index.php/comp-science/article/view/4559 https://doi.org/10.26661/2786-6254-2025-1-08 |
| Přístupové číslo: | edsbas.4FD80B69 |
| Databáze: | BASE |
Nájsť tento článok vo Web of Science | Abstrakt: | The paper considers the ideal point method for solving two-criteria optimization problems. It is proven for the first time that when the number of variables on which the optimization criteria depend exceeds two, the objective function of a scalar problem loses convexity. This makes it impossible to use the Lagrange multiplier method, since the necessary optimality conditions are not met.A feature of the ideal point method is that it does not require additional information from the decision-maker regarding the preference of one criterion over another in the set of possible alternatives. However, the method is based on the assumption that there is a certain optimal solution to the two- criteria optimization problem, which can be found by transforming it into the corresponding scalar single-criteria problem. The basic principle of choosing a compromise solution is to determine the alternative whose value is closest to the ideal point in a certain metric. Thus, the method provides the possibility of obtaining an approximate optimal solution without the need to introduce additional weighting factors or subjective estimates.It is shown that there have been attempts to solve a two-criteria optimization problem in the field of production planning in the case of four variables. It has been established that such a problem cannot be solved using the ideal point method. This is explained by the fact that the saddle point conditions used to check optimality cannot be applied due to the lack of convexity of the objective function.So, the ideal point method is effective for analyzing two-criteria optimization problems, but its application is limited to cases with the number of variables not exceeding two. In more complex problems with a larger number of variables, difficulties arise due to the loss of convexity of the objective function, which makes it impossible to use classical solution methods.Therefore, further research will be aimed at developing approaches that will allow the search for alternative optimization methods ... |
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| DOI: | 10.26661/2786-6254-2025-1-08 |