Non-normal fuzzy number analysis in various levels using centroid method for fuzzy optimization

Abstract

In the present article, a level analysis has been improved for various types of fuzzy numbers. In spite of non-normal fuzzy number ranking with more parameters are difficult, this analysis gives a clear idea for the non-normal case. The rank value may vary for different levels of various fuzzy numbers. The authors of this study essentially deal with the ranking approach, which is suitable to analyze three different fuzzy numbers, namely TrapFN, HFN, and HDFN, in the entire possible levels. The varying rank value in the fuzzy numbers can be identified by using the centroid ranking approach. Finally, a comparative analysis is given to demonstrate the advantages of the proposed analysis for fuzzy numbers levels. It is shown that the variation in ranking values of TrapFN, HFN, and HDFN is computed in a more efficient way.

Appendix 1

The increasing part of the membership function of HDFN (in Definition 14) is

$$ p_{1} (x) = u\left( {\frac{{x - h_{1} }}{{h_{2} - h_{1} }}} \right){,}\,q_{1} (x) = u + (v - u)\left( {\frac{{x - h_{3} }}{{h_{4} - h_{3} }}} \right)\,and\,r_{1} (x) = v + (\omega - v)\left( {\frac{{x - h_{5} }}{{h_{6} - h_{5} }}} \right) $$

(3)

Which are bounded left continuous non-decreasing functions over $[0,\omega_{1} ]$_,$[u,\omega_{2} ]$ and $[v,\omega_{3} ]$, respectively, with $0 \le \omega_{1} \le u$,$u < \omega_{2} \le v$,$v < \omega_{3} \le \omega$.

The decreasing part of the membership function of HDFN (in Definition 14) is

$$ r_{2} (x) = v + (\omega - v)\left( {\frac{{h_{7} - x}}{{h_{7} - h_{6} }}} \right),\,q_{2} (x) = u + (v - u)\left( {\frac{{h_{9} - x}}{{h_{9} - h_{8} }}} \right)\,and,\,p_{2} (x) = u\left( {\frac{{h_{11} - x}}{{h_{11} - h_{10} }}} \right) \, $$

(4)

Which are bounded right continuous non-increasing functions over $[0,\omega_{1} ]$_,$[u,\omega_{2} ]$ and $[v,\omega_{3} ]$, respectively, with $0 \le \omega_{1} \le u$, $u < \omega_{2} \le v$, $v < \omega_{3} \le \omega$.

For $\alpha \in (0,1]$, the $\alpha - cut$ of HDFN, $\widetilde{HD} = (h_{1} ,h_{2} ,h_{3} ,h_{4} ,h_{5} ,h_{6} ,h_{7} ,h_{8} ,h_{9} ,h_{10} ,h_{11} ;u,v,\omega )$, using Eqs. 3 and 4, is defined as

$$ \left[ {\widetilde{HD}} \right]_{\alpha } = \left\{ \begin{gathered} [HD_{1}^{L\alpha} ,HD_{1}^{R\alpha}]\,{\text{for}}\,\alpha \in {\text{[0,u]}} \hfill \\ [HD_{2}^{L\alpha},HD_{2}^{R\alpha} ]\,{\text{for}}\,\alpha \in {\text{(u,}}v{]} \hfill \\ [HD_{3}^{L\alpha}, HD_{3}^{R\alpha}]\,{\text{for}}\,\alpha \in {\text{(v,}}\omega {]} \hfill \\ \end{gathered} \right. $$

where ${\text{HD}}_{1\alpha }^{L} = h_{1} + \frac{\alpha }{u}(h_{2} - h_{1} )$, $HD_{1\alpha }^{R} = h_{11} - \frac{\alpha }{u}(h_{11} - h_{10} )$,

$$ \begin{gathered} HD_{2\alpha }^{L} = h_{3} + \left( {\frac{\alpha - u}{{v - u}}} \right)(h_{4} - h_{3} ),\,HD_{2\alpha }^{R} = h_{9} - \left( {\frac{\alpha - u}{{v - u}}} \right)(h_{9} - h_{8} ) \hfill \\ HD_{3\alpha }^{L} = h_{5} + \left( {\frac{\alpha - v}{{\omega - v}}} \right)(h_{6} - h_{5} ),\,HD_{3\alpha }^{R} = h_{7} - \left( {\frac{\alpha - v}{{\omega - v}}} \right)(h_{7} - h_{6} ). \hfill \\ \end{gathered} $$

(5)

Using Eqs. (1) and (5) the centroid formulae $(\tilde{x}_{0} ,\tilde{y}_{0} )$ for the HDFN is defined as follows:

$$ \begin{aligned} & \tilde{x}_{0} (\widetilde{HD}) = \frac{{\int\limits_{ - \infty }^{\infty } {x\mu_{{\tilde{H}D}} (x)dx} }}{{\int\limits_{ - \infty }^{\infty } {\mu_{{\tilde{H}D}} (x)dx} }} = \frac{{\int\limits_{{h_{1} }}^{{h_{2} }} {xp_{1} (x)dx} + \int\limits_{{h_{2} }}^{{h_{3} }} {uxdx} + \int\limits_{{h_{3} }}^{{h_{4} }} {xq_{1} (x)dx + \int\limits_{{h_{4} }}^{{h_{5} }} {vxdx} + } \int\limits_{{h_{5} }}^{{h_{6} }} {xr_{1} (x)dx + } \int\limits_{{h_{6} }}^{{h_{7} }} {xr_{2} \left( x \right)dx} + \int\limits_{{h_{7} }}^{{h_{8} }} {vxdx} + \int\limits_{{h_{8} }}^{{h_{9} }} {xq_{2} (x)dx} + \int\limits_{{h_{9} }}^{{h_{10} }} {uxdx} + \int\limits_{{h_{10} }}^{{h_{11} }} {xp_{2} (x)dx} }}{{\int\limits_{{h_{1} }}^{{h_{2} }} {p_{1} (x)dx} + \int\limits_{{h_{2} }}^{{h_{3} }} {udx} + \int\limits_{{h_{3} }}^{{h_{4} }} {q_{1} (x)dx} + \int\limits_{{h_{4} }}^{{h_{5} }} {vdx} + \int\limits_{{h_{5} }}^{{h_{6} }} {r_{1} (x)dx} + \int\limits_{{h_{6} }}^{{h_{7} }} {r_{2} (x)dx + \int\limits_{{h_{7} }}^{{h_{8} }} {vdx + \int\limits_{{h_{8} }}^{{h_{9} }} {q_{2} (x)dx + \int\limits_{{h_{9} }}^{{h_{10} }} {udx + \int\limits_{{h_{10} }}^{{h_{11} }} {p_{2} (x)dx} } } } } }} \\ & and \\ & \tilde{y}_{0} (\widetilde{HD}) = \frac{{\int\limits_{0}^{\omega } {\alpha |HD_{\alpha } |d\alpha } }}{{\int\limits_{0}^{\omega } {|HD_{\alpha } |d\alpha } }} = \frac{{\int\limits_{0}^{u} {\alpha [HD_{1\alpha }^{R} - HD_{1\alpha }^{L} ]d\alpha } + \int\limits_{u}^{v} {\alpha [HD_{2\alpha }^{R} - HD_{2\alpha }^{L} ]d\alpha + \int\limits_{v}^{\omega } {\alpha [HD_{3\alpha }^{R} - HD_{3\alpha }^{L} ]d\alpha } } }}{{\int\limits_{0}^{u} {[HD_{1\alpha }^{R} - HD_{1\alpha }^{L} ]d\alpha } + \int\limits_{u}^{v} {[HD_{2\alpha }^{R} - HD_{2\alpha }^{L} ]d\alpha + \int\limits_{v}^{\omega } {[HD_{3\alpha }^{R} - HD_{3\alpha }^{L} ]d\alpha } } }} \\ \end{aligned} $$

(6)

Solving Eq. (6), we get the pair of centroid formulae $(\tilde{x}_{0} ,\tilde{y}_{0} )$ for the HDFN as:

$$ \begin{aligned} \tilde{x}_{0} (\widetilde{HD}) & = \frac{1}{3}\left[ {\frac{{u\left[ {\left( {h_{11}^{2} + h_{10}^{2} + h_{10} h_{11} } \right) - \left( {h_{1}^{2} + h_{2}^{2} + h_{1} h_{2} } \right)} \right] + \left( {v - u} \right)\left[ {\left( {h_{9}^{2} + h_{8}^{2} + h_{9} h_{8} } \right) - \left( {h_{3}^{2} + h_{4}^{2} + h_{3} h_{4} } \right)} \right] + \left( {\omega - v} \right)\left( {h_{5} + h_{6} + h_{7} } \right)\left( {h_{7} - h_{5} } \right)}}{{u\left( {h_{11} - h_{1} + h_{10} - h_{2} } \right) + \left( {v - u} \right)\left( {h_{9} - h_{3} + h_{8} - h_{4} } \right) + \left( {\omega - v} \right)\left( {h_{7} - h_{5} } \right)}}} \right] \\ \tilde{y}_{0} (\widetilde{HD}) & = \frac{1}{3}\left[ {\frac{{u^{2} \left( {h_{11} - h_{1} + h_{10} - h_{2} } \right) + \left( {v - u} \right)\left[ {\left( {v + 2u} \right)\left( {h_{9} - h_{3} } \right) + \left( {u + 2v} \right)\left( {h_{8} - h_{4} } \right)} \right] + \left( {\omega - v} \right)\left( {2v + \omega } \right)\left( {h_{7} - h_{5} } \right)}}{{u\left( {h_{11} - h_{1} + h_{10} - h_{2} } \right) + \left( {v - u} \right)\left( {h_{9} - h_{3} + h_{8} - h_{4} } \right) + \left( {\omega - v} \right)\left( {h_{7} - h_{5} } \right)}}} \right] \\ \end{aligned} $$

(7)