Table 4 Equations

\(Q\left(\left|u>\right.\right)=\varphi {e}^{j\theta }\) and \(\left|\varsigma >\right.=\left\{\left|{u}_{1}>\right.,\left|{u}_{2}>\right.,\dots ,\left|{u}_{n}>\right.\right\}\)

(1)

\(\sum_{\left|u>\subseteq \right|\varsigma >}\left|Q(\left|u>)\right.\right|=1\)

(2)

\({\widetilde{A}}_{S}=\left\{\langle u,({\mu }_{{\widetilde{A}}_{S}}\left(u\right),{v}_{{\widetilde{A}}_{S}}\left(u\right),{h}_{{\widetilde{A}}_{S}}\left.\left(u\right))\right|u\in U\right\}\)

(3)

\(0{\le \mu }_{{\widetilde{A}}_{S}}^{2}\left(u\right)+{v}_{{\widetilde{A}}_{S}}^{2}\left(u\right)+{h}_{{\widetilde{A}}_{S}}^{2}\left(u\right)\le 1\),\({\forall }_{u}\in U\)

(4)

\(\left|{\varsigma }_{{\widetilde{A}}_{S}}>\right.=\left\{\langle u,({\varsigma }_{{\mu }_{{\widetilde{A}}_{S}}}\left(u\right),{\varsigma }_{{v}_{{\widetilde{A}}_{S}}}\left(u\right),{\varsigma }_{{h}_{{\widetilde{A}}_{S}}}\left.\left(u\right))\right|u\in {2}^{\left|{\varsigma }_{{\widetilde{A}}_{S}}>\right.}\right\}\)

(5)

\(\varsigma =\left[{\varsigma }_{\mu }.{e}^{j2\pi .\alpha },{\varsigma }_{v}.{e}^{j2\pi .\gamma },{\varsigma }_{h}.{e}^{j2\pi .\beta }\right]\)

(6)

\({\varphi }^{2}=\left|{\varsigma }_{\mu }\left(\left|{u}_{i}>\right.\right)\right|\)

(7)

\({\varsigma }_{v}=\frac{{\varsigma }_{\mu }}{G}\)

(8)

\({\varsigma }_{h}=1-{\varsigma }_{\mu }-{\varsigma }_{v}\)

(9)

\(\alpha =\left|{\varsigma }_{\mu }\left(\left|{u}_{i}>\right.\right)\right|\)

(10)

\(\gamma =\frac{\alpha }{G}\)

(11)

\(\beta =1-\alpha -\gamma\)

(12)

\(\lambda *{\widetilde{A}}_{\varsigma }=\left\{\begin{array}{c}{\left(1-{\left(1-{{\varsigma }_{{\mu }_{\widetilde{A}}}}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}{e}^{j2\pi .{\left(1-{\left(1-{\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}}, \\ {{\varsigma }_{{v}_{\widetilde{A}}}}^{\lambda }{e}^{j2\pi .{\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{\lambda }}, \\ {\left({\left(1-{{\varsigma }_{{h}_{\widetilde{A}}}}^{2}\right)}^{\lambda }-{\left(1-{{\varsigma }_{{\mu }_{\widetilde{A}}}}^{2}-{{\varsigma }_{{h}_{\widetilde{A}}}}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}{e}^{j2\pi .{\left({\left(1-{\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}\right)}^{\lambda }-{\left(1-{\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)}^{2}-{\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}}\end{array}\right\}\), \(\lambda >0\)

(13)

\({{\widetilde{A}}_{\varsigma }}^{\lambda }=\left\{\begin{array}{c}{{\varsigma }_{{\mu }_{\widetilde{A}}}}^{\lambda }{e}^{j2\pi .{\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)}^{\lambda }},\\ {\left(1-{\left(1-{{\varsigma }_{{v}_{\widetilde{A}}}}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}{e}^{j2\pi .{\left(1-{\left(1-{\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}} ,\\ {\left({\left(1-{{\varsigma }_{{v}_{\widetilde{A}}}}^{2}\right)}^{\lambda }-{\left(1-{{\varsigma }_{{v}_{\widetilde{A}}}}^{2}-{{\varsigma }_{{h}_{\widetilde{A}}}}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}{e}^{j2\pi .{\left({\left(1-{\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{2}\right)}^{\lambda }-{\left(1-{\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{2}-{\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}\right)}^{\lambda }\right)}^\frac{1}{2}}\end{array}\right\}\),\(\lambda >0\)

(14)

\({\widetilde{A}}_{\varsigma }\oplus {\widetilde{B}}_{\varsigma }=\left\{\begin{array}{c}{\left({{\varsigma }_{{\mu }_{\widetilde{A}}}}^{2}+{{\varsigma }_{{\mu }_{\widetilde{B}}}}^{2}-{{\varsigma }_{{\mu }_{\widetilde{A}}}}^{2}{{\varsigma }_{{\mu }_{\widetilde{B}}}}^{2}\right)}^\frac{1}{2}{e}^{j2\pi .{\left({\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)}^{2}+{\left(\frac{{\alpha }_{\widetilde{B}}}{2\pi }\right)}^{2}-{\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)}^{2}{\left(\frac{{\alpha }_{\widetilde{B}}}{2\pi }\right)}^{2}\right)}^\frac{1}{2}} ,\\ {\varsigma }_{{v}_{\widetilde{A}}}{\varsigma }_{{v}_{\widetilde{B}}}{e}^{j2\pi .\left(\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)\left(\frac{{\gamma }_{\widetilde{B}}}{2\pi }\right)\right)},\\ {\left(\begin{array}{c}\left(1-{{\varsigma }_{{\mu }_{\widetilde{B}}}}^{2}\right){{\varsigma }_{{h}_{\widetilde{A}}}}^{2}+\\ \left(1-{{\varsigma }_{{\mu }_{\widetilde{A}}}}^{2}\right){{\varsigma }_{{h}_{\widetilde{B}}}}^{2}-{{\varsigma }_{{h}_{\widetilde{A}}}}^{2}{{\varsigma }_{{h}_{\widetilde{B}}}}^{2}\end{array}\right)}^\frac{1}{2}{e}^{j2\pi .{\left(\left(1-{\left(\frac{{\alpha }_{\widetilde{B}}}{2\pi }\right)}^{2}\right){\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}+\left(1-{\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)}^{2}\right){\left(\frac{{\beta }_{\widetilde{B}}}{2\pi }\right)}^{2}-{\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}{\left(\frac{{\beta }_{\widetilde{B}}}{2\pi }\right)}^{2}\right)}^\frac{1}{2}}\end{array}\right\}\)

(15)

\({\widetilde{A}}_{\varsigma }\otimes {\widetilde{B}}_{\varsigma }=\left\{\begin{array}{c}{\begin{array}{c}{\varsigma }_{{\mu }_{\widetilde{A}}}{\varsigma }_{{\mu }_{\widetilde{B}}}{e}^{j2\pi .\left(\frac{{\alpha }_{\widetilde{A}}}{2\pi }\right)\left(\frac{{\alpha }_{\widetilde{B}}}{2\pi }\right)},\\ \left({{\varsigma }_{{v}_{\widetilde{A}}}}^{2}+{{\varsigma }_{{v}_{\widetilde{B}}}}^{2}-{{\varsigma }_{{v}_{\widetilde{A}}}}^{2}{{\varsigma }_{{v}_{\widetilde{B}}}}^{2}\right)\end{array}}^\frac{1}{2}{e}^{j2\pi .{\left({\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{2}+{\left(\frac{{\gamma }_{\widetilde{B}}}{2\pi }\right)}^{2}-{\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{2}{\left(\frac{{\gamma }_{\widetilde{B}}}{2\pi }\right)}^{2}\right)}^\frac{1}{2}} ,\\ {\left(\begin{array}{c}\left(1-{{\varsigma }_{{v}_{\widetilde{B}}}}^{2}\right){{\varsigma }_{{h}_{\widetilde{A}}}}^{2}+\\ \left(1-{{\varsigma }_{{v}_{\widetilde{A}}}}^{2}\right){{\varsigma }_{{h}_{\widetilde{B}}}}^{2}-{{\varsigma }_{{h}_{\widetilde{A}}}}^{2}{{\varsigma }_{{h}_{\widetilde{B}}}}^{2}\end{array}\right)}^\frac{1}{2}{e}^{j2\pi .{\left(\left(1-{\left(\frac{{\gamma }_{\widetilde{B}}}{2\pi }\right)}^{2}\right){\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}+\left(1-{\left(\frac{{\gamma }_{\widetilde{A}}}{2\pi }\right)}^{2}\right){\left(\frac{{\beta }_{\widetilde{B}}}{2\pi }\right)}^{2}-{\left(\frac{{\beta }_{\widetilde{A}}}{2\pi }\right)}^{2}{\left(\frac{{\beta }_{\widetilde{B}}}{2\pi }\right)}^{2}\right)}^\frac{1}{2}}\end{array}\right\}\)

(16)

\({\varsigma }_{k}\)= \(\left[\begin{array}{cccccc}0& {\varsigma }_{12}& \cdots & & \cdots & {\varsigma }_{1n}\\ {\varsigma }_{21}& 0& \cdots & & \cdots & {\varsigma }_{2n}\\ \vdots & \vdots & \ddots & & \cdots & \cdots \\ \vdots & \vdots & \vdots & & \ddots & \vdots \\ {\varsigma }_{n1}& {\varsigma }_{n2}& \cdots & & \cdots & 0\end{array}\right]\)

(17)

\(\varsigma =\left\{\begin{array}{c}{\left[1-\prod_{i=1}^{k}{\left(1-{{\varsigma }_{{\mu }_{i}}}^{2}\right)}^\frac{1}{k}\right]}^\frac{1}{2}{e}^{2\pi .{\left[1-\prod_{i=1}^{k}{\left(1-{\left(\frac{{\alpha }_{i}}{2\pi }\right)}^{2}\right)}^\frac{1}{k}\right]}^\frac{1}{2}},\\ \prod_{i=1}^{k}{{\varsigma }_{{v}_{i}}}^\frac{1}{k}{e}^{2\pi .\prod_{i=1}^{k}{\left(\frac{{\gamma }_{i}}{2\pi }\right)}^\frac{1}{k}},\\ {\left[\begin{array}{c}\prod_{i=1}^{k}{\left(1-{{\varsigma }_{{\mu }_{i}}}^{2}\right)}^\frac{1}{k}-\\ \prod_{i=1}^{k}{\left(1-{{\varsigma }_{{\mu }_{i}}}^{2}-{{\varsigma }_{{h}_{i}}}^{2}\right)}^\frac{1}{k}\end{array}\right]}^\frac{1}{2}{e}^{2\pi .{\left[\prod_{i=1}^{k}{\left(1-{\left(\frac{{\alpha }_{i}}{2\pi }\right)}^{2}\right)}^\frac{1}{k}-\prod_{i=1}^{k}{\left(1-{\left(\frac{{\alpha }_{i}}{2\pi }\right)}^{2}-{\left(\frac{{\beta }_{i}}{2\pi }\right)}^{2}\right)}^\frac{1}{k}\right]}^\frac{1}{2}}\end{array}\right\}\)

(18)

\({Def\varsigma }_{i}={\varsigma }_{{\mu }_{i}}+{\varsigma }_{{h}_{i}}\left(\frac{{\varsigma }_{{\mu }_{i}}}{{\varsigma }_{{\mu }_{i}}+{\varsigma }_{{v}_{i}}}\right)+\left(\frac{{\alpha }_{i}}{2\pi }\right)+\left(\frac{{\gamma }_{i}}{2\pi }\right)\left(\frac{\left(\frac{{\alpha }_{i}}{2\pi }\right)}{\left(\frac{{\alpha }_{i}}{2\pi }\right)+\left(\frac{{\beta }_{i}}{2\pi }\right)}\right)\)

(19)

\({k}_{j}=\left\{\begin{array}{c}1 j=1\\ {s}_{j}+1 j>1\end{array}\right.\)

(20)

\({q}_{j}=\left\{\begin{array}{c}1 j=1\\ \frac{{q}_{j-1}}{{k}_{j}} j>1\end{array}\right.\)

(21)

\({w}_{j}=\frac{{q}_{j}}{\sum_{k=1}^{n}{q}_{k}}\)

(22)

\({X}_{k}\)= \(\left[\begin{array}{cccccc}0& {X}_{12}& \cdots & & \cdots & {X}_{1m}\\ {X}_{21}& 0& \cdots & & \cdots & {X}_{2m}\\ \vdots & \vdots & \ddots & & \cdots & \cdots \\ \vdots & \vdots & \vdots & & \ddots & \vdots \\ {X}_{n1}& {X}_{n2}& \cdots & & \cdots & 0\end{array}\right]\)

(23)

\({r}_{ij}= \frac{{X}_{ij}}{\sqrt{\sum_{i=1}^{m}{X}_{ij}^{2}}}\)

(24)

\({v}_{ij}={w}_{ij}\times {r}_{ij}\)

(25)

\({A}^{+}=\left\{{v}_{1j},{v}_{2j},\dots ,{v}_{mj}\right\}=\left\{\mathrm{max}{v}_{1j} for \forall jn\right\}\)

(26)

\({A}^{-}= \left\{{v}_{1j},{v}_{2j},\dots ,{v}_{mj}\right\}=\left\{\mathrm{min}{v}_{1j} for \forall jn\right\}\)

(27)

\({D}_{i}^{+}= \sqrt{\sum_{j=1}^{n}{\left({v}_{ij}-{A}_{j}^{+}\right)}^{2}}\)

(28)

\({D}_{i}^{-}= \sqrt{\sum_{j=1}^{n}{\left({v}_{ij}-{A}_{j}^{-}\right)}^{2}}\)

(29)

\({RC}_{i}= \frac{{D}_{i}^{-}}{{D}_{i}^{+}+{D}_{i}^{-}}\)

(30)