There are many cases that a nonlinear fractional PROGRAMMING, generated as a result of studying fractional stochastic PROGRAMMING, must be solved. Sometimes an approximate solution may be sufficient enough to start a new process of calculations. To this end, this author introduces a new linear approximation technique for solving a fractional chance constrained PROGRAMMING (CCP) problem. After introducing the problem, the equivalent deterministic form of the fractional nonlinear PROGRAMMING problem is developed. To solve the problem, a fuzzy GOAL PROGRAMMING model of the equivalent deterministic form of the fractional chance constrained PROGRAMMING is provided and then the process of defuzzification and linearization of the problem is started. A sample test problem is solved for presentation purposes.

We are concerned with solving Fuzzy Flexible Linear PROGRAMMING (FFLP) problems. Even though, this model is very practical and is useful for many applications, but there are only a few methods for its situation. In most approaches proposed in the literature, the solution process needs at least, two phases where each phase needs to solve a linear PROGRAMMING problem. Here, we propose a method to solve the given problem in just one phase using only one problem. Furthermore, using our approach, sensitivity analysis of Fuzzy Flexible Linear PROGRAMMING (FFLP) problem is simpler. For an illustration of our method, some numerical examples given. In particular, a practical problem is formulated and is solved by our method and several other methods and the obtained results are compared.

This paper presents a fuzzy GOAL PROGRAMMING (FGP) methodology for solving bi-level quadratic PROGRAMMING (BLQP) problems. In the FGP model formulation, firstly the objectives are transformed into fuzzy GOALs (membership functions) by means of assigning an aspiration level to each of them, and suitable membership function is defined for each objectives, and also the membership functions for vector of fuzzy GOALs of the decision variables controlled by decision maker at the first level are developed in the model formulation of the problem. To achieve the highest membership value of each of the fuzzy GOALs, we formulate the problem by minimizing the negative deviational variables and thereby obtaining the most satisfactory solution for all decision makers. A numerical example is given to demonstrate the proposed approach.