Abstract
Similarity measure plays an important role in the decision-making process under an uncertain environment where parameters involved are linguistics terms. Mostly, similarity measure is discussed on generalized fuzzy numbers. However, a few efforts have been made to study this measure on interval-valued fuzzy numbers. Sometimes, methods involving interval-valued fuzzy numbers depict limitations and drawbacks. Moreover, some of the methods are just confined to interval-valued fuzzy numbers. Hence, these methods fail when similarity has to be determined between crisp-valued fuzzy numbers and interval-valued fuzzy numbers. Hence, a new method of similarity measure has been developed based on the concepts of geometric distance, heights and the radius of gyration of the interval-valued fuzzy numbers. Although the method is being discussed on interval-valued fuzzy numbers, yet it is not just confined to such numbers. This method can be applied efficiently to generalized fuzzy numbers too. The method seems to out-perform in many situations and overcome the drawbacks and limitations of existing methods. A few sets of fuzzy numbers are considered for a comparative study and draw out the out-performance of the proposed method. A real-life problem of risk analysis in poultry farming has been discussed using the proposed similarity measure.
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Appendix
Appendix
Consider a GFN A=(a1,a2,a3,a4; ω1,ω2) with membership function as given in Eq. 4. The graphical representations of the GFN A is shown in Fig. 11 depending on the heights ω1 and ω2. The ROG point of the GFN A is denoted as \(({r_{x}^{A}},{r_{y}^{A}})\) whose value can be obtained by using the Eqs. 13 and 14 in Definition 2.11. To evaluate the moment of inertia, the GFN A is divided into regions R1,R2,R3 and R4. Hence, the moment of inertia of the areas R1,R2,R3 and R4 about the x and y axis can be calculated, according to Eqs. 11 and 12 in Definition 2.10, as
Hence, the ROG point of the GFN A can be obtained using the above equations as
where ar(A) is the area of the GFN A.
If the GFN A is such that a1 = a2 = a3 = a4 and ω1 = ω2 = ω, then the ROG point is given by \({r_{x}^{A}}=\frac {\omega }{\sqrt {3}}\) and \(r_{y}^{A_{\omega }}=a\). For the detail derivation one may refer to Yong et al. [25].
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Chutia, R. Fuzzy risk analysis using similarity measure of interval-valued fuzzy numbers and its application in poultry farming. Appl Intell 48, 3928–3949 (2018). https://doi.org/10.1007/s10489-018-1178-2
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DOI: https://doi.org/10.1007/s10489-018-1178-2