
The following multiobjective linear programming problem with fuzzy coefficients in the objective functions is considered: \[ \min[c^f_1 x,c^f_2 x,\dots, c^f_k x],\quad\text{s.t. }Ax\leq b,\;x\geq 0,\tag{1} \] where \(A\) is an \(m\times n\)-matrix, \(b\) an \(m\)-vector, \(x\) an \(n\)-vector and each \(c^f_j\), \(j= 1,\dots, k\), an \(n\)-vector of fuzzy numbers. By applying the constraint approach the problem (1) is replaced with the following one: \[ \min c^f_1 x,\quad \text{s.t. }c^f_jx\leq_f \lambda^f_j,\quad j= 2,\dots, k,\quad Ax\leq b,\;x\geq 0,\tag{2} \] where \(\lambda^f_j\), \(j= 2,\dots, k\), are fuzzy numbers (fuzzy goals) assigned to the corresponding objective functions and \(\leq_f\) is a certain relation between fuzzy numbers which may be defined in different ways. The authors propose to use for comparison of fuzzy numbers so-called ranking functions. If \(I: FN(\mathbb{R})\to [0,1]\) is any ranking function \((FN(\mathbb{R}))\) is the set of all fuzzy numbers) then \[ \forall A^f, B^f\in FN(\mathbb{R}) A^f\leq_f B^f\overset{\text{def}}\Leftrightarrow I(A^f)\leq I(B^f). \] Using the above way of comparison of fuzzy numbers the authors transform the problem (2) to the following crisp mathematical programming problem: \[ \min I_1(c^f_1 x),\quad\text{s.t. }I_j(c^f_j x)\leq I_j(\lambda^f_j),\quad j= 1,2,\dots, k,\quad Ax\leq b,\;x\geq 0.\tag{3} \] Since, the authors assume that the objective functions in (1) may come from different decision makers, the ranking functions \(I_j\) applied in (3) may be different for different \(j, j= 1,\dots, k\). Then the authors derive the form of the problem (3) for two special situations when \(I_j\) is for each \(j\) equal to the comparison index of Yager or that proposed by Adamo. In each case, assuming that the all fuzzy parameters in (2) and (3) are triangular fuzzy numbers, the problem (3) is reduced to a linear programming. The full usefulness of the given approach is illustrated by a biobjective linear programming problem for land use, involving fuzzy coefficients in the objective functions.
ranking function, fuzzy coefficients, Linear programming, crisp mathematical programming problem, Fuzzy and other nonstochastic uncertainty mathematical programming, multiobjective linear programming, Multi-objective and goal programming
ranking function, fuzzy coefficients, Linear programming, crisp mathematical programming problem, Fuzzy and other nonstochastic uncertainty mathematical programming, multiobjective linear programming, Multi-objective and goal programming
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