On a Multi-objective Optimization Problem Arising from Production Theory

The paper presents a natural application of multi-objective programming to household production and consumption theory. A contribution to multi-objective programming theory is also included. 1. A multi-objective optimization problem. The following multiobjective optimization problem will be considered and applied in the present paper: Given a k×nmatrix A, anm×nmatrix C and anm-vector b, we consider the set of all k-vectors of the form Ax for some x ∈ R+, satisfying the constraint Cx = b; such x’s are called solutions of the problem P = (A,C, b). The set of all solutions of P will be denoted by X(P ); it is a convex set. We are specially interested in characterizing the existence of efficient and weakly efficient solutions: a solution x∗ is called efficient whenever there exists no x ∈ X(P ) for which Ax > Ax∗ (by y > z we always mean that the respective coordinates of the vector z do not exceed those of y and the vectors are different; y z will mean that all respective coordinates of y are larger than those of z); a solution x∗ is called weakly efficient whenever there exists no x ∈ X(P ) for which Ax Ax∗. Obviously, every efficient solution is also weakly efficient. Isermann has proven in [2] the following result, actually reducing the problem of finding all efficient solutions to solving a class of linear maximization problems: Theorem 1. A vector x∗ is an efficient solution of a problem P = (A,C, b) if and only if there exists a vector w ∈ R+, w 0, such that 1991 Mathematics Subject Classification: 90A11, 90A12, 90A15. Journal of Economic Literature Classification: C62.