Fuzzy reasoning for solving fuzzy multiple objective linear programs

We interpret fuzzy multiple objective linear programming (FMOLP) problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes (MFR), where the antecedents of the scheme correspond to the constraints of the FMOLP problem and the facts of the scheme are the objectives of the FMOLP problem. Then the solution process consists of two steps: first, for every decision variable x ∈ IRn and objective s ∈ {1, . . . k}, we compute the effective attainment of the s-th objective function, Es(x), via sup-min convolution of the antecedents/constraints and the s-th fact/objective, then a solution to FMOLP problem is any point which is a good compromise solution to the crisp multiple objective problem max x∈IRn {H1(x), . . . Hk(x)}, where Hs is an application function (measuring the degree of fulfillment of the decision maker’s requirements about the s-th objective function) s ∈ {1, . . . k}. 1 Statement of FMOLP problems with fuzzy coefficients We consider MOLP problems, in which all of the coefficients are fuzzy quantities (i.e. fuzzy sets of the real line IR), of the form maximize (c̃1x, . . . , c̃kx) subject to ãix < ∼ b̃i, i = 1, . . . ,m, (1) where x ∈ IR is the vector of decision variables, ãi, b̃i and c̃s are vectors of fuzzy quantities, the operations addition and multiplication by a real number of fuzzy quantities are defined by Zadeh’s extension principle [13] and the inequality relation < ∼ for ∗The final version of this paper appeared in: R.Trappl ed., Cybernetics and Systems ’94, Proceedings of the Twelfth European Meeting on Cybernetics and Systems Research, World Scientific Publisher, London, 1994, vol.1, 295-301. †Partially supported by the Hungarian Research Foundation for Scientific Research (OTKA) under contract I/3-2152.