Fuzzy risk analysis using similarity measure of interval-valued fuzzy numbers and its application in poultry farming

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.

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

$$\begin{array}{@{}rcl@{}} (I_{x})_{R_{1}}&=&\int\limits_{R_{1}}y^{2}dR_{1}\\ &=&\int\limits_{0}^{\omega_{1}}y^{2}\left\{a_{2}-a_{1}-\frac{y(a_{2}-a_{1})}{\omega_{1}} \right\}dy\\ &=&\frac{(a_{2}-a_{1}){\omega_{1}^{3}}}{12}. \end{array} $$

$$\begin{array}{@{}rcl@{}} (I_{y})_{R_{1}}&=&\int\limits_{R_{1}}x^{2}dR_{1}\\ &=&\int\limits_{a_{1}}^{{a_{2}}}x^{2}\frac{\omega_{1}(x-a_{1})}{a_{2}-a_{1}}dx\\ &=&\frac{\omega_{1}}{12}\left( 3{a_{2}^{3}}-{a_{1}^{3}}-{a_{1}^{2}}a_{2}-a_{1}{a_{2}^{2}}\right). \end{array} $$

$$(I_{x})_{R_{2}}= \left\{\begin{array}{lll} \frac{(a_{3}-a_{2}){\omega_{1}^{3}}}{3}, &\text{if } \omega_{1} \leq \omega_{2};\\ \frac{(a_{3}-a_{2}){\omega_{2}^{3}}}{3}, &\text{if } \omega_{1} \geq \omega_{2}. \end{array}\right. $$

$$(I_{x})_{R_{3}}= \left\{\begin{array}{lll} \frac{(a_{3}-a_{2})}{12}\left( {\omega_{2}^{3}}-3{\omega_{1}^{3}}+{\omega_{1}^{2}}\omega_{2}+\omega_{1}{\omega_{2}^{2}} \right), &\text{ if } \omega_{1} \leq \omega_{2};\\ \frac{(a_{3}-a_{2})}{12}\left( {\omega_{1}^{3}}-3{\omega_{2}^{3}}+{\omega_{1}^{2}}\omega_{2}+\omega_{1}{\omega_{2}^{2}} \right), &\text{if } \omega_{1} \geq \omega_{2}. \end{array}\right. $$

$$(I_{x})_{R_{4}}=\frac{(a_{4}-a_{3}){\omega_{2}^{3}}}{12}. $$

$$(I_{y})_{(R_{2}+R_{3})}\,=\,\frac{\omega_{1}}{3}\left( {a_{3}^{3}}\!-{a_{2}^{3}}\right)+\frac{(\omega_{2}\,-\,\omega_{1})}{12}\left( 3{a_{3}^{3}}\,-\,{a_{2}^{3}}\!-{\!a_{3}^{2}}a_{2}\,-\,a_{3}{a_{2}^{2}}\right)\!. $$

$$(I_{y})_{R_{4}}=\frac{\omega_{2}}{12}\left( {a_{4}^{3}}-3{a_{3}^{3}}+{a_{3}^{2}}a_{4}+a_{3}{a_{4}^{2}}\right). $$

Hence, the ROG point of the GFN A can be obtained using the above equations as

$$\begin{array}{@{}rcl@{}} {r_{x}^{A}}&=&\sqrt{\frac{(I_{x})_{R_{1}}+(I_{x})_{R_{2}}+(I_{x})_{R_{3}}+(I_{x})_{R_{4}}}{\text{ar}(A)}}, \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} {r_{y}^{A}}&=&\sqrt{\frac{(I_{y})_{R_{1}}+(I_{y})_{(R_{2}+R_{3})}+(I_{y})_{R_{4}}}{\text{ar}(A)}}. \end{array} $$

(24)

where ar(A) is the area of the GFN A.

Fig. 11

Graphical representation of GFN A with different left height and right height

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].