Maximal and supremal tolerances in multiobjective linear programming

► We consider stability of efficiency in multiobjective linear programming problems. ► We compute maximal tolerances for perturbations of objective function coefficients. ► We show that the problem is NP-hard, the method is exponential in the worst case. ► We extend the united tolerance to individua...

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Vydáno v:European journal of operational research Ročník 228; číslo 1; s. 93 - 101
Hlavní autoři: Hladík, Milan, Sitarz, Sebastian
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 01.07.2013
Elsevier
Elsevier Sequoia S.A
Témata:
Algorithms
Efficient solution
Linear programming
Mathematical functions
Multiobjective linear programming
Sensitivity analysis
Studies
Tolerance analysis
Transportation problem (Operations research)
Tolerance analysis
Sensitivity analysis
Multiobjective linear programming
Efficient solution
ISSN:0377-2217, 1872-6860
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Shrnutí:► We consider stability of efficiency in multiobjective linear programming problems. ► We compute maximal tolerances for perturbations of objective function coefficients. ► We show that the problem is NP-hard, the method is exponential in the worst case. ► We extend the united tolerance to individual tolerances. Let a multiobjective linear programming problem and any efficient solution be given. Tolerance analysis aims to compute interval tolerances for (possibly all) objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals. The known methods either yield tolerances that are not the maximal possible ones, or they consider perturbations of weights of the weighted sum scalarization only. We focus directly on perturbations of the objective function coefficients, which makes the approach independent on a scalarization technique used. In this paper, we propose a method for calculating the supremal tolerance (the maximal one need not exist). The main disadvantage of the method is the exponential running time in the worst case. Nevertheless, we show that the problem of determining the maximal/supremal tolerance is NP-hard, so an efficient (polynomial time) procedure is not likely to exist. We illustrate our approach on examples and present an application in transportation problems. Since the maximal tolerance may be small, we extend the notion to individual lower and upper tolerances for each objective function coefficient. An algorithm for computing maximal individual tolerances is proposed.
Bibliografie:SourceType-Scholarly Journals-1
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ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2013.01.045