حل مسئله برنامه‌ریزی کسری خطی در محیط عدم قطعیت با پارامترهای خاکستری

نویسنده
فرید پورافقی
ستادیار گروه ریاضی ، دانشگاه پیام نور، گروه ریاضی، تهران، ایران
چکیده
مسئله برنامه ریزی کسری، یک ابزار برنامه‌ریزی غیرخطی مهم است که در زمینه‌های مختلف از جمله تخصیص منابع، حمل و نقل، برنامه‌ریزی تولید و غیره استفاده می‌شود. با توجه به عدم قطعیت در مسائل دنیای واقعی، تعیین ضرایب قطعی برای مدل ریاضی مسائل بسیار دشوار است. بنابراین، ضرایب مدل ریاضی مسائل به صورت غیر قطعی در نظر گرفته می‌شوند. یکی از رویکردهای مقابله با عدم قطعیت، در کنار نظریه احتمال، تصادفی، فازی ، نظریه سیستم‌های خاکستری است. در این مقاله، مسئله برنامه‌ریزی کسری خطی با ضرایب خاکستری در تابع هدف در نظر گرفته شده است. برای حل این مسائل اغلب از روش سفیدسازی پارامترهای خاکستری استفاده می‌شود که باعث می‌شود جواب‌ به دست آمده در این روش، عدم قطعیت پارامترهای خاکستری را در جواب بهینه منعکس نکند. برای حل این مشکل، در این مقاله، الگوریتمی ارائه شده است که بر اساس آن با توجه به نوع تابع هدف مسئله برنامه‌ریزی کسری خطی خاکستری، آن را به دو زیر مسئله برنامه‌ریزی خطی خاکستری تبدیل کرده و سپس جواب مسئله برنامه‌ریزی کسری خطی خاکستری، از طریق حل آن دو زیر مسئله تعیین شده است. با اجرای روش پیشنهادی جواب مسئله برنامه‌ریزی کسری خطی خاکستری به صورت اعداد خاکستری بازه‌ای تعیین می‌شود و در نتیجه عدم قطعیت در تابع هدف در نتیجه نهایی منعکس می‌شود. در پایان برای نشان دادن اثربخشی روش پیشنهادی، یک مثال با روش پیشنهادی حل شده است. برای نمایش دادن کارایی روش پیشنهادی، جواب به دست آمده از روش پیشنهادی با جواب‌های بدست آمده از روش‌های دیگر برای مثال ارائه شده با دو روش رتبه‌بندی اعداد خاکستری هو و وانگ و مرکز و درجه خاکستری ارزیابی شد. و نشان داده شد که جواب بدست آمده از روش پیشنهادی بهتر از روش‌های حل دیگر می‌باشد.




کلیدواژه‌ها
عدم قطعیت
برنامه ریزی کسری
سیستم خاکستری
اعداد بازه خاکستری

عنوان مقاله English

Solving linear fractional programming problem in uncertainty environment with grey parameters

نویسنده English

farid pourofoghi
Department of Mathematics, Payame Noor University, Tehran, Iran
چکیده English

The fractional programming problem is an important nonlinear programming tool used in various fields such as resource allocation, transportation, production planning, etc. Due to the uncertainty in real-world problems, it is very difficult to determine the definite coefficients for the mathematical model of the problems. Therefore, the coefficients of the mathematical model of the problems are considered as uncertain. One of the approaches to dealing with uncertainty, in addition to probability, stochastic, fuzzy theory, is the theory of grey systems. In this paper, a linear fractional programming problem with grey coefficients in the objective function is considered. To solve these problems, the grey parameter whitening method is often used, which causes the solution obtained in this method not to reflect the uncertainty of the grey parameters in the optimal solution. To solve this problem, in this paper, an algorithm is presented, based on which, according to the type of objective function of the grey linear fractional programming problem, it is converted into two grey linear programming subproblems, and then the solution of the grey linear fractional programming problem is determined by solving those two subproblems. By implementing the proposed method, the solution of the grey linear fractional programming problem is determined as interval grey numbers, and as a result, the uncertainty in the objective function is reflected in the final result. Finally, to demonstrate the effectiveness of the proposed method, an example has been solved with the proposed method. To demonstrate the effectiveness of the proposed method, the solution obtained from the proposed method was evaluated with the solutions obtained from other methods, for example, the two grey number ranking methods of Hu and Wang and the center and grey degree. It was shown that the solution obtained from the proposed method is better than other solution methods.

کلیدواژه‌ها English

Uncertainty
Fractional Programming
Grey System
Grey Interval Numbers
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