Optimization of Transport Problems with Fuzzy Coefficients

In this paper, we concentrate on three kinds of fuzzy linear programming problems: linear programming problems with only fuzzy technological coefficients, linear programming problems with fuzzy right-hand sides and linear programming problems in which both the right-hand side and the technological coefficients are fuzzy numbers. We consider here only the case of fuzzy numbers with linear membership functions. The symmetric method of Bellman and Zadeh [2] is used for a defuzzification of these problems. The crisp problems obtained after the defuzzification are non-linear and even nonconvex in general. Finally, we give illustrative examples and their numerical solutions.


Introduction
In fuzzy decision making problems, the concept of maximizing decision was proposed by Bellman and Zadeh [2]. From the beginning of the theory, three periods in its development can be recognized: 1. The period between 1965 -1977. This period is also known as the academic phase, which is characterized by the development of fundamentals of Fuzzy Set Theory, and speculations about possible prospective applications of the theory. The outcome was a small number of publications of a predominantly theoretical nature, by a small number of contributors, primarily from the academic community.
2. The period between 1978 -1988. This period, also referred to as the transformation phase, is characterized by advances in Fuzzy Set Theory, but also by some practical applications of the theory. The number of contributors to the theory also increased which resulted in increase of publications, some of which discussed various emerging applications. It is also important to say that some journals devoted to follow the development of this theory as well as to contribute to it were established.
3. The period 1989 -nowadays. This period, which is the current period, also referred to as the fuzzy boom phase, is characterized by a rapid increase in successful industrial and business applications of the theory which resulted in increase of revenues. Major research center devoted to applications of the theory were established, and brand new area of research, and if we may say, science emerged -soft computing. In the soft computing area, main partner of the Fuzzy Set Theory are neural networks and genetic algorithms.
The founder of the Fuzzy Set Theory is Lotfi Asker Zadeh (Lotfi Aliasker Zadeh). He was born in Baku, Azerbaijan, on February 4, 1921 The fuzzy logic concept was adapted to problems of mathematical programming by Tanaka et al. [13]. Zimmermann [14] presented a fuzzy approach to multiobjective linear programming problems. He also studied the duality relations in fuzzy linear programming. Fuzzy linear programming problem with fuzzy coefficients was formulated by Negoita [9] and called robust programming. Dubois and Prade [3] investigated linear fuzzy constraints. Tanaka and Asai [12] also proposed a formulation of fuzzy linear programming with fuzzy constraints and gave a method for its solution which bases on inequality relations between fuzzy numbers. Shaocheng [11] considered the fuzzy linear programming problem with fuzzy constraints and defuzzificated it by first determining an upper bound for the objective function. Further he solved the so-obtained crisp problem by the fuzzy decisive set method introduced by Sakawa and Yana [10].
In this paper, we combine solution methods of Asai and Shaocheng, We first consider linear programming problems with fuzzy right-hand sides. Next we consider problems in which technological coefficients are fuzzy numbers and then finally linear programming problems in which both technological coefficients and right-hand-side numbers are fuzzy numbers. Each problem is first converted into an equivalent crisp problem. This is a problem of finding a point which satisfies the constraints and the goal with the maximum degree. The idea of this approach is due to Bellman and Zadeh [2]. The crisp problems, obtained by such a manner, can be non-linear (even non-convex), where the non-linearity arises in constraints. For solving these problems we use the computer algebra package MA-THEMATICA.
The paper is outlined as follows. Linear programming problem with fuzzy technological coefficients is considered in Section 2.
Linear programming problem with fuzzy right-hand sides is considered in Section 3. In section 4, we study the linear programming problem with fuzzy objective function and fuzzy right hand sides. In section 5, several solution methods of the defuzzified problems are discussed. Section 6 is devoted to applications to concrete examples.
where R x  and 0  The degree of satisfaction of the i th constraint by the n dimensional decision vector x is given by

Solution of defuzzified problems
In the previous sections fuzzy decisive sets are obtained and the maximization problem of these sets is transformed into defuzzified problems. Notice that, the constraints in problems (2.9), (3.9) and (4.11) are generally not convex. These problems may be solved either by the fuzzy decisive set method, which is presented by Sakawa and Yana [10], or by the linearization method of Kettani and Oral [6].
There are some disadvantages in using these methods. The fuzzy decisive set methodtakes a long time for solving the problem. On the other hand, the linearization method increases the number of the constraints.
Azimov and Yenilmez [5] presented a modified sub gradient method and used it for solving the defuzzificated problems in fuzzy linear programming problems with linear membership functions. This method is based on the duality theory developed by Azimov and Gasimov [1] for nonconvex constrained problems and can be applied for solving a large class of such problems.
In the following applications to two fuzzy transport problems, we prefer to use Linear Programming Package in MATHEMATI-CA.  The following table that shows the relation between production centers and market places will be helpful: