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On identifying fuzzy knees in fuzzy multi-criteria optimization problems

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Abstract

This paper introduces and analyzes the idea of fuzzy knee in fuzzy multi-criteria optimization problems. The fuzzy decision feasible region of the problem is constructed under a fuzzy inequality relation that is defined with the help of same points in fuzzy geometry. Then, fuzzy criteria feasible region is obtained through the image of the fuzzy decision feasible region by the criteria-vector-valued mapping. For the constructed fuzzy criteria feasible region, we define fuzzy knee and then propose a method to capture the fuzzy knee regions, along with the complete fuzzy Pareto set. All the studied ideas and methodologies are supported with suitable examples and pictorial illustrations. An engineering application of the presented method is also given.

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Notes

  1. A real-valued function \(\mu \) on a metric space M is called upper semi-continuous if for each real \(\alpha \), the set \(\{x \in M: \mu (x) \ge \alpha \}\) is closed in M (see [5, p. 67]).

  2. For two fuzzy sets \(\widetilde{A}\) and \(\widetilde{B}\) in X, the relation \(\widetilde{A} \subseteq \widetilde{B}\) holds when \(\mu (x|\widetilde{A}) \le \mu (x|\widetilde{B})\)\(\forall x \in X\).

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Acknowledgements

The author is truly thankful to the anonymous reviewers and editors for their valuable comments and suggestions to improve the paper. The author gratefully acknowledges the financial support through Early Career Research Award (ECR/2015/000467), Science & Engineering Research Board, Government of India.

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  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, 221005, India

    Debdas Ghosh

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  1. Debdas Ghosh
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Correspondence to Debdas Ghosh.

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Ghosh, D. On identifying fuzzy knees in fuzzy multi-criteria optimization problems. SeMA 76, 343–364 (2019). https://doi.org/10.1007/s40324-018-0179-8

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