On a Solution Set to a Mathematical Programming Problem With a Fuzzy Set of Constraint Indices

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Introduction

Mathematical programming (MP) is a classical area which is in high demand for many disciplines. Engineering, management, politics, operations research and many other fields are, in one way or another, concerned with optimization of solutions, structures or processes. Many specific problems that can be formulated in an optimization setting are full of fuzziness sources. According to Dubois and Prade (1994), a fuzzy optimization problem is specified by an objective function and a set of feasible alternatives, and at least one of these components has to be specified in a fuzzy way. Similar to classical optimization problems, two different types are distinguished. These are fuzzy extremum problems (without constraints) and fuzzy mathematical programming problems (with constraints). In this paper a fuzzy mathematical programming (FMP) problem has been considered.

Let be a crisp set of alternatives . The objective function is defined on , where is a fuzzy parameter. A Decision Maker (DM) needs to ‘maximize’ this function in a fuzzy sense, that is,

.(1)

Formula (1) is formally represented in the form

,where is the additional variable, and is a fuzzy relation of the (non-strict) total order on the real line . In particular, if is a crisp constant, then the objective function has a crisp set of values and be given in the form . In addition, when is the crisp relation (not smaller than) on , formula (1) takes the form of the maximization problem in a classical (crisp) sense.

Let be constraint functions on , where is the set of constraint indices and is a fuzzy parameter. The constraints define a set of feasible alternatives, where is a fuzzy relation of the (non-strict) total order on . If is the crisp relation (not larger than) on and is a crisp constant, then the constraints are usual (crisp) and they take the form , . Thus, the FMP problem has the form: