Optimization of LR -Type Fully Bipolar Fuzzy Linear Programming Problems

In this study, we present a technique to solve LR -type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define LR -type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of LR -type bipolar fuzzy numbers. Furthermore, we develop a method to solve LR -type FBFLPPs with equality constraints involving LR -type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for LR -type bipolar fuzzy numbers which transform the LR -type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.

Solution of Fully Bipolar Fuzzy Linear Programming Models

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.

Generalised Single-Valued Neutrosophic Number and Its Application to Neutrosophic Linear Programming

In this chapter, the concept of single valued neutrosophic number (SVN-Number) is presented in a generalized way. Using this notion, a crisp linear programming problem (LP-problem) is extended to a neutrosophic linear programming problem (NLP-problem). The coefficients of the objective function of a crisp LP-problem are considered as generalized single valued neutrosophic number (GSVN-Number). This modified form of LP-problem is here called an NLP-problem. An algorithm is developed to solve NLP-problem by simplex method. Finally, this simplex algorithm is applied to a real-life problem. The problem is illustrated and solved numerically.

A Study of Fully Fuzzy Linear Fractional Programming Problems by Signed Distance Ranking Technique

The aim of this chapter is to study fully fuzzy linear fractional programming (FFLFP) problems where all coefficients of the decision variables and parameters are characterized by triangular fuzzy numbers. To deal with this, the authors have first to transform FFLFP problems to fuzzy linear programming (FLP) problems by using Charnes and Cooper method and then use signed distance ranking to convert fuzzy linear programming (FLP) problems to crisp linear programming (LP) problems. The proposed method is solved by using the simplex method to find the optimal solution of the problem. The authors have studied sensitivity analysis to determine changes in the optimal solution of the fully fuzzy linear fractional programming (FFLFP) problems resulting from changes in the parameters. To demonstrate the proposed method, one numerical example is solved.

A New Approach for Solving Fully Fuzzy Linear Programming by Using the Lexicography Method

The fully fuzzy linear programming (FFLP) problem has many different applications in sciences and engineering, and various methods have been proposed for solving this problem. Recently, some scholars presented two new methods to solve FFLP. In this paper, by considering theL-Rfuzzy numbers and the lexicography method in conjunction with crisp linear programming, we design a new model for solving FFLP. The proposed scheme presented promising results from the aspects of performance and computing efficiency. Moreover, comparison between the new model and two mentioned methods for the studied problem shows a remarkable agreement and reveals that the new model is more reliable in the point of view of optimality.

بناء أنموذج برمجة خطية ضبابية مع تطبيق عملي على المنتجات النفطية في مصفى الدورة

يتضمن هذا البحث بناء وحل نموذج البرمجة الخطية الضبابية , لتحديد الدخل الاجمالي وتحديد كميات الانتاج المثلى لمنتوجات الوقود في مصفى الدورة و المتكونة من سبع منتوجات , تؤثر بشكل مباشر في الاستهلاك اليومي , ونظراً لان كميات الانتاج اليومي بالمتر المكعب حسب ما ورد في بيانات المصفى تتراوح بين الطاقة التصميمية والطاقة المتاحة والطاقة المخططة والطاقة الفعلية , لذلك ارتأينا ان يكون انموذج برمجة خطية ضبابية , يعاني من صفة الضبابية ومن نوع شبه المنحرف في قيود الانتاج وقيود الطلب , ويتكون النموذج الخطي الضبابي من دالة هدف وقيود , حيث ان دالة الهدف متمثلة في اسعار البيع لكل منتوج وبالمتر المكعب , كذلك كميات الانتاج و الطلب على هذه المنتوجات , كانت ضبابية رباعية , وايضا مُقاسة بالمتر المكعب , لكل من ( ماء التبريد , بخار الماء , الهواء المضغوط , ) في حين كانت الطاقة الكهربائية مُقاسة بالواط لكل متر مكعب . بعد بناء الانموذج التطبيقي اعتمدت طريقة الرتب الحصينة لتحويله الى انموذج برمجة خطية اعتيادية (CLP)(Crisp Linear Programming) وتم حل الانموذج بأستخدام البرنامج الجاهز (WIN QSB) ووضعت النتائج في جداول خاصة تتضمن الحل الامثل .

BOUNDED LINEAR PROGRAMS WITH TRAPEZOIDAL FUZZY NUMBERS

Recently Ganesan and Veeramani introduced a new approach for solving a kind of linear programming problems involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems. But their approach is not efficient for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. In this paper, by a natural extension of their approach we obtain some new results leading to a new method to overcome this shortcoming.