Fuzzy resource allocation
The allocation problem of J resources to I tasks is studied by means of a fuzzy set (FS) theory approach. The total consumption \(y_ j\) of resource j is described by an FS \(A_ j\) and another FS \(B_ j\) permits to model the effectiveness of activity k. Take \(\mu_ j\) and \(r_ k\) as the corresponding membership functions. It is assumed that \(\mu_ j\) is a continuous decreasing function and \(r_ k\) a continuous increasing one. A fuzzy decision in this case, is related to an allocation matrix x with membership function \(z(x)\) and the objective is to obtain a decision \(\bar x\) such that \(z(\bar x)\) is maximum. This is an optimization problem D. A solution algorithm is proposed for obtaining \(z(\bar x)\) in an other optimization problem DA. A lemma proves that the optimal solution x' of DA yields the optimum of D and if \(\bar x\) has the objective value \(\bar z\) the matrices and the membership functions of D and DA coincide. A theorem states that an optimal solution of D can be determined by an algorithm which is developed in this paper. A numerical example illustrates the behaviour of it. If \(\mu_ j\) and \(r_ k\) are concave, for any j and k, the solution of D is obtained by using linear programming techniques. Under certain situations the hypothesis of ''strictly decreasing'' function may be removed for the \(\mu_ j\)'s. Some applications to specific types of resource allocation problems are given.
scientific article; zbMATH DE number 3974718