Membership functions, some mathematical programming models and production scheduling

Abstract The authors have studied in [5] alternative real variable models based on the function d(x) = x (α + x) , α >0 , for certain integer or mixed-interger programming problems. Mainly, we have shown that there exists a vector α > 0 such that the solution to the problem min σ 1 (x, α) = Σ i=1 n x i (ga i +x i ) , Ax = b, x ⩾ 0 , is a solution to the problem min σxσ + , Ax = b , x ⩾ 0, where σxσ + denotes the cardinal of x , i.e. the number of strictly positive components of x , thus obtaining a new model for solving in real numbers a Generalized Lattice Point Problem (Cabot, [3]). The function d ( x ) has been introduced by use as a general tool for solving integer or mixed-integer problems due to its property of having almost everywhere almost discrete values. In the meantime we noticed that this function may represent a membership function of a fuzzy set. In this paper, we study in detail the features of this membership function and show that Cabot's results [3] may be derived in this more general setting using the complementary function s(x) = 1 − x (α + x) = α (α+x) . At the same time, in the paper there are some production scheduling models within the framework of fuzzy-sets theory. To this end, a nonconvex production model is presented and it is shown that the value of the objective function μ 2 = 1 − σ 1 n for a production programming model whose deman and/or resource vector components are parametrized, may be considered as a grade of membership of the solution of the parametrized model to the feasible set of the nonparametrized production programming model. Consequently, we get a nonconvex production programming model whose convex envelope is linear with coefficients which are in an inverse proportior to the magnitude of the nonparametrized demand or resource vector components. This result agrees with the intuitive idea that a high level of demand or resource allows a greater interval of variation in the production process than a lower level of demand or resource.