تعیین جواب‌های تقریباً کارای مسائل بهینه‌سازی چندهدفه با استفاده از روش اسکالرسازی مقید ترکیبی
محورهای موضوعی : آمارمهرداد غزنوی 1 , فرشته اکبری 2 , اسماعیل خرم 3
1 - گروه ریاضی کاربردی و علوم کامپیوتر، دانشکده علوم ریاضی، دانشگاه صنعتی شاهرود، شاهرود، ایران
2 - دانشکده ریاضی و علوم کامپیوتر، دانشگاه صنعتی امیرکبیر، تهران، ایران
3 - دانشکده علوم ریاضی و کامپیوتر، دانشگاه صنعتی امیر کبیر، تهران، ایران
کلید واژه: Scalarization method. Approximate efficient solutions, Approximate optimality. , Multiobjective optimization, Proper efficiency,
چکیده مقاله :
در این مقاله، جواب‌های تقریباً کارای ( -کارای) مسائل بهینه‌سازی چندهدفه مورد بررسی قرار می‌گیرند. یک دسته از مهم ترین روش‌ها برای حل مسائل چندهدفه، استفاده از تکنیک‌های اسکالرسازی است. در این روش‌ها یک مسأله تک‌هدفه متناظر با مسأله چندهدفه حل می‌شود و ارتباط بین جواب‌های بهینه مسأله‌ی تک‌هدفه و جواب‌های کارای (سره، ضعیف) مسأله‌ی چندهدفه بررسی می‌شود. در این مقاله، ترکیبی از روش‌های اسکالرسازی مقید اصلاح شده (modified constrained) و مقید انعطاف‌پذیر (elastic constrained) در نظر گرفته می‌شود و با کمک آن شرایطی لازم و کافی برای تولید جواب‌های تقریباً کارا (ضعیف، سره) ارائه خواهد شد. نتایج بدست آمده را با شرایط لازم و کافی حاصل از روش‌های مقید اصلاح شده و مقید انعطاف‌پذیر مقایسه می‌کنیم. قضایای ارائه شده بدون هیچ شرط تحدبی برای هر یک از توابع هدف در مسأله‌ی بهینه‌سازی چندهدفه برقرار هستند. برخلاف بسیاری از روش‌های قبلی، نتایج بدست آمده برای مسائل چندهدفه با فضای هدف بیکران نیز برقرار هستند.
In this paper, approximate efficient ( -efficient) solutions of multiobjective optimization problems are investigated. One of the most important methods for solving multiobjective optimization problems is to use scalarization techniques. In these methods, a single objective optimization problem corresponding to the multiobjective problem is solved, and the relationship between optimal solutions of the single objective problem and (weakly, properly) efficient solutions of the multiobjective problem is investigated. In this paper, a combination of the modified constrained and elastic constrained scalarization methods is considered, which will provide necessary and sufficient conditions for generating approximate (weakly, properly) efficient solutions. We compare the results with the necessary and sufficient conditions obtained from the modified constrained and the elastic constrained methods. The presented results can be applied for every multiobjective optimization problem without any convexity assumption for the objective functions. Unlike many of the previous methods, the obtained results are also consistent with multiobjective problems with unbounded criterion space.
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