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Computing Min-Max Regret Solutions in Possibilistic Combinatorial Optimization Problems

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Book cover Fuzzy Optimization

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 254))

Abstract

In this chapter we discuss a wide class of combinatorial optimization problems with a linear sum and a bottleneck cost function. We first investigate the case when the weights in the problem are modeled as closed intervals. We show how the notion of optimality can be extended by using a concept of a deviation interval. In order to choose a solution we adopt a robust approach. We seek a solution that minimizes the maximal regret, that is the maximal deviation from optimum over all weight realizations, called scenarios, which may occur. We then explore the case in which the weights are specified as fuzzy intervals. We show that under fuzzy weights the problem has an interpretation consistent with possibility theory. Namely, fuzzy weights induce a possibility distribution over the scenario set and the possibility and necessity measures can be used to extend the optimality evaluation and the min-max regret approach.

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Author information

Authors and Affiliations

  1. Institute of Industrial Engineering and Management, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Adam Kasperski

  2. Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Paweł Zieliński

Authors
  1. Adam Kasperski
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  2. Paweł Zieliński
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Editors and Affiliations

  1. Department of Mathemaitical and Statistical Sciences, University of Colorado Denver, PO Box 173364, 80217-3364, Colorado, Denver, USA

    Weldon A. Lodwick

  2. Systems Research Institute Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

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Kasperski, A., Zieliński, P. (2010). Computing Min-Max Regret Solutions in Possibilistic Combinatorial Optimization Problems. In: Lodwick, W.A., Kacprzyk, J. (eds) Fuzzy Optimization. Studies in Fuzziness and Soft Computing, vol 254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13935-2_13

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  • DOI: https://doi.org/10.1007/978-3-642-13935-2_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13934-5

  • Online ISBN: 978-3-642-13935-2

  • eBook Packages: EngineeringEngineering (R0)

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