COMBINATORIAL OPTIMIZATION UNDER UNCERTAINTY AND FORMAL MODELS OF EXPERT ESTIMATION

Abstract

Previously, the author formalized the concepts of uncertainty, compromise solution, compromise criteria and conditions for a quite general class of combinatorial optimization problems. The functional of the class’ problems contains linear convolution of weights and arbitrary numerical characteris­tics of a feasible solution. It was shown that the efficiency of the presented algorithms for the uncertainty resolution is largely determined by the effi­ciency of solving the combinatorial optimization problem in a deterministic formulation. A part of the formulated compromise criteria and conditions uses expert weights. Previously, the author and his disciples also formulated combinatorial optimization models, optimality criteria, criteria for deci­sions’ consistency. The models allow to evaluate and justify the degree of stability and reliability of the estimated values of empirical coefficients using a formally ill-conditioned empirical pairwise comparison matrix of arbitrary dimension. The matrix may contain zero elements. The theoretical research and statistical experiments allowed to choose the most efficient of these optimization models. In this article, on the base of earlier results by the author and his disciples, we formalize and substantiate the efficiency of the proposed sequential procedure for expert estimation of weights that determine compromise criteria and conditions. The procedure is an integral part of the algorithm introduced by the author to solve combinatorial optimization problems under uncertainty of the mentioned class. We give unified algorithm for efficient uncertainty resolution that includes original and efficient formal procedure for expert coefficients’ estimation using empirical matrices of pairwise comparisons.