The article discusses the study of a mathematical model of execution of the production task in the presence of fuzzy information about the matrixes of direct costs and final demand. This is a variant of Komiya's inverse of Berge's theorem we strengthened the assertion of convexity by assuming the costs to be nonparameterized, though we only consider the case of linear equation constraints. The second theorem states that any optimal solution function with a certain continuity can be realized by a continuous convex objective function if the feasible solutions form an affine subspace. ![]() This is a variant of the Berge maximum theorem, wherein we replace the condition of compact-valuedness of the feasible solution function with other assumptions. The first theorem states that any continuous quasi-convex objective function results in an upper semicontinuous optimal solution function. pair of relationships between classes of objective functions and optimal solution functions in the form of two theorems. This mapping, either single-valued or set-valued, is called an optimal solution function in the literature of parametric optimization and a generalized inverse mapping in the literature of inverse problems. When an optimization problem has a set of parameters besides decision variables, the mapping from the parameters to the corresponding optimal solutions is determined accordingly. Some numerical examples clarify the ability of our heuristics. Finally we extend a new method employing Linear Programming (LP) for solving square and non-square (over-determined) fuzzy systems. We propose some new methods to solve this system that are comparable to the well known methods such as the Cramer’s rule, Gaussian elimination, LU decomposition method (Doolittle algorithm) and its simplification. So we employ some heuristics based methods on Dubois and Prade’s approach, finding some positive fuzzy vector x˜ which satisfies A˜x˜=b˜, where A˜ and b˜ are a fuzzy matrix and a fuzzy vector respectively. The same result can similarly be derived for fuzzy systems. It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard. In continuation to our previous work, we focus on fuzzy systems in this paper. extended some matrix computations on fuzzy matrices, where a fuzzy matrix appears as a rectangular array of fuzzy numbers. One of the most practicable subjects in recent studies is based on LR fuzzy numbers, which are defined and used by Dubois and Prade with some useful and easy approximation arithmetic operators on them. In addition, this is an important sub-process in determining inverse, eigenvalue and some other useful matrix computations, too. ![]() Fuzzy systems have an essential role in this fuzzy modelling, which can formulate uncertainty in actual environment. Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process.
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