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.
Highlights
We studied in [1] a class of combina- where li are given numbers; torial optimization problems of the following form: 2) a feasible solution that satisfies the condition: min(max) ∑si=1 ωiki(σ)
(1) under uncertainty only for the following case: for the problems of the form (1) there exists an exact algorithm which is qualitatively more efficient in terms of speed and/or accuracy than an arbitrary solving method for the problem (1) in which the structure of the feasible solutions domain differs from Ω of (1), e.g., in additional constraints on inequalities
For a problem of the form (1), solution of which uses expert coefficients, we formulated in [1] the combinatorial optimization problem statement under uncertainty as follows
Summary
We studied in [1] a class of combina- where li are given numbers; torial optimization problems of the following form: 2) a feasible solution that satisfies the condition: min(max) ∑si=1 ωiki(σ). For a problem of the form (1), solution of which uses expert coefficients, we formulated in [1] the combinatorial optimization problem statement under uncertainty as follows. Probabilities Pl > 0, l = 1̅̅,̅L, ∑l Pl = 1, may be specified for each possible set of weights (such probabilities do not exist if the uncertainty is not described by a probabilistic model).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Bulletin of National Technical University "KhPI". Series: System Analysis, Control and Information Technologies
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
