A DecisionMaking Approach Based on a Multi QHesitant Fuzzy Soft MultiGranulation Rough Model
Abstract
:1. Introduction
2. Preliminaries
3. Multi QHesitant Fuzzy Soft Sets
 ${h}_{{A}_{Q}^{c}}(uq)=\sim {h}_{{A}_{Q}}(uq)={\bigcup}_{\gamma \in {h}_{{A}_{Q}^{c}}(uq)}\{1\gamma \}$.
 ${A}_{Q}\cup {B}_{Q}=\{\langle (uq),{h}_{{A}_{Q}}(uq)\vee {h}_{{B}_{Q}}(uq)\rangle ,u\in U,q\in Q\}$.
 ${A}_{Q}\cap {B}_{Q}=\{\langle (uq),{h}_{{A}_{Q}}(uq)\wedge {h}_{{B}_{Q}}(uq)\rangle ,u\in U,q\in Q\}$.
 ${A}_{Q}\oplus {B}_{Q}={\bigcup}_{{\gamma}_{1}\in {h}_{{A}_{Q}}(uq),{\gamma}_{2}\in {h}_{{B}_{Q}}(uq)}\{{\gamma}_{1}+{\gamma}_{2}{\gamma}_{1}{\gamma}_{2}\}$.
 ${A}_{Q}\otimes {B}_{Q}={\bigcup}_{{\gamma}_{1}\in {h}_{{A}_{Q}}(uq),{\gamma}_{2}\in {h}_{{B}_{Q}}(uq)}\{{\gamma}_{1}{\gamma}_{2}\}$.
 $({H}_{Q},A)\subseteq (U,E),$
 $(\varphi ,A)\subseteq ({H}_{Q},B),$
 If $({H}_{Q},A)\subseteq ({F}_{Q},B)$ and $({F}_{Q},B)\subseteq ({G}_{Q},C)$, then $({H}_{Q},A)\subseteq ({G}_{Q},C).$
 $({H}_{Q},A)\cup ({H}_{Q},A)=({H}_{Q},A)$,
 $({H}_{Q},A)\cap ({H}_{Q},A)=({H}_{Q},A)$,
 $({H}_{Q},A)\cup {\varphi}_{A}^{k}=({H}_{Q},A)$,
 $({H}_{Q},A)\cap {\varphi}_{A}^{k}={\varphi}_{A}^{k}$,
 $({H}_{Q},A)\cup {U}_{A}^{k}={U}_{A}^{k}$,
 $({H}_{Q},A)\cap {U}_{A}^{k}=({H}_{Q},A)$,
 $({H}_{Q},A)\cup ({F}_{Q},B)=({F}_{Q},B)\cup ({H}_{Q},A)$,
 $({H}_{Q},A)\cap ({F}_{Q},B)=({F}_{Q},B)\cap ({H}_{Q},A)$.
4. Multi QHesitant Fuzzy Soft Rough Set
 $\underline{{R}_{Q}}({A}_{Q}^{c})={(\overline{{R}_{Q}}({A}_{Q}))}^{c},\overline{{R}_{Q}}({A}_{Q}^{c})={(\underline{{R}_{Q}}({A}_{Q}))}^{c},$
 ${A}_{Q}\subseteq {B}_{Q}\Rightarrow \underline{{R}_{Q}}({A}_{Q})\subseteq (\underline{{R}_{Q}}({B}_{Q})),{A}_{Q}\subseteq {B}_{Q}\Rightarrow \overline{{R}_{Q}}({A}_{Q})\subseteq (\overline{{R}_{Q}}({A}_{Q})),$
 $\underline{{R}_{Q}}({A}_{Q}\cap {B}_{Q})=\underline{{R}_{Q}}({A}_{Q})\cap (\underline{{R}_{Q}}({B}_{Q})),\overline{{R}_{Q}}({A}_{Q}\cup {B}_{Q})=\overline{{R}_{Q}}({A}_{Q})\cup (\overline{{R}_{Q}}({B}_{Q})),$
 $\underline{{R}_{Q}}({A}_{Q}\cup {B}_{Q})\supseteq \underline{{R}_{Q}}({A}_{Q})\cup (\underline{{R}_{Q}}({B}_{Q})),\overline{{R}_{Q}}({A}_{Q}\cap {B}_{Q})\subseteq \overline{{R}_{Q}}({A}_{Q})\cap (\overline{{R}_{Q}}({B}_{Q})).$
 By Definition 17, we have$\underline{{R}_{Q}}({A}_{Q}^{c})=\{\langle (uq),{h}_{\underline{{R}_{Q}}(\sim {A}_{Q})}^{i}(uq)\rangle :uq\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigwedge}_{e\in E}\{{h}_{\sim {R}_{Q}}^{i}(uq,eq)\vee {h}_{\sim {A}_{Q}}^{i}(eq)\rangle :uq\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),\sim ({\bigvee}_{e\in E}\{{h}_{{R}_{Q}}^{i}(uq,eq)\wedge {h}_{{A}_{Q}}^{i}(eq)\rangle :uq\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),\sim {h}_{\overline{{R}_{Q}}({A}_{Q})}^{i}(uq)\rangle :(uq)\in U\times Q,i=1,2,\dots ,k\}$$={(\overline{{R}_{Q}}({A}_{Q}))}^{c}.$Similarly, we can obtain that $\overline{{R}_{Q}}({A}_{Q}^{c})={(\underline{{R}_{Q}}({A}_{Q}))}^{c}.$
 If ${A}_{Q}\subseteq {B}_{Q}$, by Definition 8, ${h}_{{A}_{Q}}^{i}(uq)\le {h}_{{B}_{Q}}^{i}(uq)$ for all $u\in U,q\in Q$. Therefore, ${\bigwedge}_{e\in E}\{(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)\}\le {\bigwedge}_{e\in E}\{(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{B}_{Q}}^{i}(eq)\},$ thus ${h}_{\underline{{R}_{Q}}({A}_{Q})}^{i}(uq)\le {h}_{\underline{{R}_{Q}}({B}_{Q})}^{i}(uq).$ It follows that $\underline{{R}_{Q}}({A}_{Q})\subseteq \underline{{R}_{Q}}({B}_{Q}).$
 $\underline{{R}_{Q}}({A}_{Q}\cap {B}_{Q})=\{\langle (uq),{h}_{\underline{{R}_{Q}}({A}_{Q}\cap {B}_{Q})}^{i}(uq)\rangle :uq\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigwedge}_{e\in E}(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}\cap {B}_{Q}}^{i}(eq)\rangle :uq\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigwedge}_{e\in E}(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee ({h}_{{A}_{Q}}^{i}(eq)\wedge {h}_{{B}_{Q}}^{i}(eq))\rangle :uq\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),({\bigwedge}_{e\in E}((1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)))\wedge ({\bigwedge}_{e\in E}((1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{B}_{Q}}^{i}(eq)))\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{h}_{\underline{{R}_{Q}}({A}_{Q})}^{i}(uq)\wedge {h}_{\underline{{R}_{Q}}({B}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\underline{{R}_{Q}}({A}_{Q})\cap \underline{{R}_{Q}}({B}_{Q})$.Hence, $\underline{{R}_{Q}}({A}_{Q}\cap {B}_{Q})=\underline{{R}_{Q}}({A}_{Q})\cap \underline{{R}_{Q}}({B}_{Q})$.Similarly, we can prove that $\overline{{R}_{Q}}({A}_{Q}\cap {B}_{Q})=\overline{{R}_{Q}}({A}_{Q})\cap \overline{{R}_{Q}}({B}_{Q}).$
 $\underline{{R}_{Q}}({A}_{Q}\cup {B}_{Q})=\{\langle (uq),{h}_{\underline{{R}_{Q}}({A}_{Q}\cup {B}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigwedge}_{e\in E}(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}\cup {B}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigwedge}_{e\in E}(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee ({h}_{{A}_{Q}}^{i}(eq)\vee {h}_{{B}_{Q}}^{i}(eq))\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),({\bigwedge}_{e\in E}((1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)))\vee ({\bigwedge}_{e\in E}((1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{B}_{Q}}^{i}(eq)))\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{h}_{\underline{{R}_{Q}}({A}_{Q})}^{i}(uq)\vee {h}_{\underline{{R}_{Q}}({B}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\underline{{R}_{Q}}({A}_{Q})\cup (\underline{{R}_{Q}}({B}_{Q})).$Hence, $\underline{{R}_{Q}}({A}_{Q}\cup {B}_{Q})=\underline{{R}_{Q}}({A}_{Q})\cup \underline{{R}_{Q}}({B}_{Q}).$Similarly, we can prove that $\overline{{R}_{Q}}({A}_{Q}\cup {B}_{Q})=\overline{{R}_{Q}}({A}_{Q})\cup \overline{{R}_{Q}}({B}_{Q})$.
 $\underline{{R}_{Q}}({A}_{Q})\supseteq \underline{{S}_{Q}}({A}_{Q}),$
 $\overline{{R}_{Q}}({A}_{Q})\subseteq \overline{{S}_{Q}}({A}_{Q}).$
 If ${R}_{Q}\subseteq {S}_{Q}$, then, by Definition 8, we have ${h}_{{R}_{Q}}^{i}(uq,eq)\le {h}_{{S}_{Q}}^{i}(uq,eq)$ for all $uq\phantom{\rule{4pt}{0ex}}\in U\times Q$, $eq\in E\times Q$, then$\underline{{R}_{Q}}({A}_{Q})=\{\langle (uq),{h}_{\underline{{R}_{Q}}({A}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ..,k\}$$=\{\langle (uq),{\bigwedge}_{e\in E}\{(1{h}_{{R}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$\ge \{\langle (uq),{\bigwedge}_{e\in E}\{(1{h}_{{S}_{Q}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{h}_{\underline{{S}_{Q}}({A}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\underline{{S}_{Q}}({A}_{Q}).$
 Similarly, it can be proved.
5. Multi QHesitant Fuzzy Soft MultiGranulation Rough Set
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}^{c})={(\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}))}^{c},$$\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}^{c})={(\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}))}^{c}.$
 ${A}_{Q}\subseteq {B}_{Q}\Rightarrow \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\subseteq \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}),$${A}_{Q}\subseteq {B}_{Q}\Rightarrow \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\subseteq \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})=\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cap \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}),$$\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cup {B}_{Q})=\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cup \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cup {B}_{Q})\supseteq \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cup \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}),$$\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})\subseteq \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cap \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}).$
 By Definition 18, we have,$\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}^{c})=\{\langle (uq),{h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}(\sim {A}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}\{\sim {h}_{{R}_{{Q}_{j}}}^{i}(uq,eq)\vee {h}_{\sim {A}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\{\langle (uq),\sim ({\bigwedge}_{j=1}^{m}{\bigvee}_{i=1}^{k}\{{h}_{{R}_{{Q}_{j}}}^{i}(uq,eq)\wedge {h}_{{A}_{Q}}^{i}(eq))\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\{\langle (uq),\sim {h}_{\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$={(\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}))}^{c}$.Similarly, we can obtain that $\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}^{c})={(\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}))}^{c}.$
 If ${A}_{Q}\subseteq {B}_{Q}$, by Definition 8, ${h}_{{A}_{Q}}^{i}(u,q)\le {h}_{{B}_{Q}}^{i}(uq)$ for all $u\in U,q\in Q$, therefore, ${\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}\{(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}(e,q)\}\le {\bigvee}_{i=1}^{m}{\bigwedge}_{i=1}^{k}\{(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{B}_{Q}}^{i}(eq)\},$ thus ${h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})}^{i}(uq)\le {h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q})}^{i}(uq)$ it follows that $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\subseteq \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})=\{\langle (uq),{h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}\cap {B}_{Q}}^{i}(eq)\rangle :uq\in U\times Q\}$$=\{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee ({h}_{{A}_{Q}}^{i}(eq)\wedge {h}_{{B}_{Q}}^{i}(eq))\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\{\langle (uq),({\bigvee}_{i=1}^{m}{\bigwedge}_{i=1}^{k}((1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)))\wedge $$({\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}((1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{B}_{Q}}^{i}(eq)))\rangle :uq\in U\times Q\}$$=\{\langle (uq),{h}_{\underline{{{\sum}_{i=1}^{m}{R}_{{Q}_{i}}}^{o}}({A}_{Q})}^{i}(uq)\wedge {h}_{\underline{{{\sum}_{i=1}^{m}{R}_{{Q}_{i}}}^{o}}({B}_{Q})}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cap (\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q})).$Hence, $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})=\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cap \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q})$.Similarly, we can prove that $\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})=\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cap \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cup {B}_{Q})=\{\langle (uq),{h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cup {B}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}\cup {B}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee ({h}_{{A}_{Q}}^{i}(eq)\vee {h}_{{B}_{Q}}^{i}(eq))\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\{\langle (uq),({\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}((1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)))\vee $$({\bigvee}_{i=1}^{m}{\bigwedge}_{i=1}^{k}((1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{B}_{Q}}^{i}(eq)))\rangle :(uq)\in U\times Q\}$$=\{\langle (uq),{h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})}^{i}(uq)\vee {h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cup (\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}))$.Hence, $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cup {B}_{Q})=\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cup \underline{{R}_{Q}}({B}_{Q})$.Similarly, we can prove that $\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cup {B}_{Q})=\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\cup \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\supseteq \underline{{{\sum}_{j=1}^{m}{S}_{{Q}_{j}}}^{o}}({A}_{Q}),$
 $\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})\subseteq \overline{{{\sum}_{j=1}^{m}{S}_{{Q}_{j}}}^{o}}({A}_{Q}).$
 If ${R}_{{Q}_{j}}\subseteq {S}_{{Q}_{j}}$, then, by Definition 8, we have ${h}_{{R}_{{Q}_{j}}}^{i}(uq,eq)\le {h}_{{S}_{{Q}_{j}}}^{i}(uq,eq)$ for all $(u,q)\in U\times Q$, $eq\in E\times Q$, then$\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})=\{\langle (uq),{h}_{\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q,i=1,2,\dots ,k\}$$=\{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}\{(1{h}_{{R}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$\ge \{\langle (uq),{\bigvee}_{j=1}^{m}{\bigwedge}_{i=1}^{k}\{(1{h}_{{S}_{{Q}_{j}}}^{i})(uq,eq)\vee {h}_{{A}_{Q}}^{i}(eq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\{\langle (uq),{h}_{\underline{{{\sum}_{j=1}^{m}{S}_{{Q}_{j}}}^{o}}({A}_{Q})}^{i}(uq)\rangle :uq\phantom{\rule{4pt}{0ex}}\in U\times Q\}$$=\underline{{{\sum}_{j=1}^{m}{S}_{{Q}_{j}}}^{o}}({A}_{Q})$.
 It can be proved similarly to 1.
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q}^{c})={(\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q}))}^{c},$$\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q}^{c})={(\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q}))}^{c}.$
 ${A}_{Q}\subseteq {B}_{Q}\Rightarrow \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\subseteq \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({B}_{Q}),$${A}_{Q}\subseteq {B}_{Q}\Rightarrow \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\subseteq \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{o}}({A}_{Q}\cap {B}_{Q})=\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\cap \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({B}_{Q}),$$\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q}\cup {B}_{Q})=\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\cup \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({B}_{Q}).$
 $\underline{{{\sum}_{i=1}^{m}{R}_{{Q}_{i}}}^{p}}({A}_{Q}\cup {B}_{Q})\supseteq \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\cup \underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({B}_{Q}),$$\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q}\cap {B}_{Q})\subseteq \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\cap \overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({B}_{Q}).$
 $\underline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\supseteq \underline{{{\sum}_{j=1}^{m}{S}_{{Q}_{j}}}^{p}}({A}_{Q}),$
 $\overline{{{\sum}_{j=1}^{m}{R}_{{Q}_{j}}}^{p}}({A}_{Q})\subseteq \overline{{{\sum}_{j=1}^{m}{S}_{{Q}_{j}}}^{p}}({A}_{Q}).$
6. Photovoltaic Systems Fault Detection Approach
6.1. The Application Model
 If ${T}_{1}\cap {T}_{2}\cap {T}_{3}\ne \varphi $, then the decision maker will choose $(m,n)$ as the optimal object, where $(m,n)$$\in {T}_{1}\cap {T}_{2}\cap {T}_{3}$.
 If ${T}_{1}\cap {T}_{2}\cap {T}_{3}=\varphi $ and ${T}_{1}\cap {T}_{2}\ne \varphi $, then the decision maker will choose $(m,n)$ as the optimal object, where $(m,n)$$\in {T}_{1}\cap {T}_{2}$.
 If ${T}_{1}\cap {T}_{2}\cap {T}_{3}=\varphi $ and ${T}_{1}\cap {T}_{2}=\varphi $, then $(m,n)\in {T}_{3}$ is the determined fault type in level.
Algorithm 1. Photovoltaic systems fault detection 

6.2. Example
6.3. Comparative Analysis and Discussion
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
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${\mathit{R}}_{\mathit{Q}}$  ${\mathit{e}}_{1}{\mathit{q}}_{1}$  ${\mathit{e}}_{1}{\mathit{q}}_{2}$  ${\mathit{e}}_{2}{\mathit{q}}_{1}$  ${\mathit{e}}_{2}{\mathit{q}}_{2}$ 

$({u}_{1}{q}_{1})$  {(0.2)(0.6,0.4)}  {(0.3,0.7)(0.6)}  {(0.5,0.4,0.6)(0.6,0.5)}  {(0.4,0.2)(0.1,0.3)} 
$({u}_{1}{q}_{2})$  {(0.8,0.5)(0.2)}  {(0.6,0.9)(0.2,0.9)}  {(0.3)(0.2,0.7)}  {(0.5,0.2,0.1)(0.1,0.5)} 
$({u}_{2}{q}_{1})$  {(0.1,0.3)(0.9,0.7,0.2)}  {(0.5,0.1,)(0.6,0.2)}  {(0.4)(0.5)}  {(0.2,0.4)(0.2,0.8)} 
$({u}_{2}{q}_{2})$  {(0.5)(0.6)}  {(0.9,0.5)(0.6,0.7,0.4)}  {(0.6)(0.3,0.1)}  {(0.2)(0.6,0.1)} 
${\mathit{R}}_{{\mathit{Q}}_{1}}$  ${\mathit{e}}_{1}{\mathit{q}}_{1}$  ${\mathit{e}}_{1}{\mathit{q}}_{2}$  ${\mathit{e}}_{2}{\mathit{q}}_{1}$  ${\mathit{e}}_{2}{\mathit{q}}_{2}$  ${\mathit{e}}_{3}{\mathit{q}}_{1}$  ${\mathit{e}}_{3}{\mathit{q}}_{2}$ 

$({u}_{1}{q}_{1})$  {0.1,0.4,0.8}  {0.9,0.3}  {0.5,0.7,0.1}  {0.2,0.6,}  {0.9,0.3}  {0.1,0.2,0.4} 
$({u}_{1}{q}_{2})$  {0.5,0.2}  {0.2,0.6}  {0.3,0.1,0.4}  {0.1,0.8}  {0.1,0.3}  {0.4,0.9,0.3} 
$({u}_{2}{q}_{1})$  {0.8,0.1}  {0.1,0.8,0.7}  {0.8,0.3}  {0.2,0.6,0.3}  {0.2,0.4,0.9}  {0.6,0.3} 
$({u}_{2}{q}_{2})$  {0.3,0.4,0.5}  {0.8,0.3}  {0.5,0.4,0.3}  {0.1,0.6,0.7}  {0.2,0.9}  {0.3,0.1,0.6} 
$({u}_{3}{q}_{1})$  {0.2,0.1}  {0.4,0.7,0.8}  {0.6,0.9,0.4}  {0.7,0.1}  {0.8,0.7,0.2}  {(0.4,0.5)} 
$({u}_{3}{q}_{2})$  {0.1,0.2,0.4}  {0.7,0.2,0.5}  {0.5,0.6}  {0.1,0.2,0.8}  {0.4,0.2}  {0.7,0.3,0.1} 
${\mathit{R}}_{{\mathit{Q}}_{2}}$  ${\mathit{e}}_{1}{\mathit{q}}_{1}$  ${\mathit{e}}_{1}{\mathit{q}}_{2}$  ${\mathit{e}}_{2}{\mathit{q}}_{1}$  ${\mathit{e}}_{2}{\mathit{q}}_{2}$  ${\mathit{e}}_{3}{\mathit{q}}_{1}$  ${\mathit{e}}_{3}{\mathit{q}}_{2}$ 

$({u}_{1}{q}_{1})$  {0.6,0.2,0.7}  {0.3}  {0.4,0.8,0.2}  {0.1,0.4}  {0.2,0.7,0.3}  {0.5,0.9} 
$({u}_{1}{q}_{2})$  {0.2,0.6}  {0.3,0.4}  {0.2,0.3}  {0.6,0.2}  {0.3,0.9}  {0.1,0.6,0.3} 
$({u}_{2}{q}_{1})$  {0.4,0.2,0.6}  {0.1,0.2}  {0.7,0.5,0.7}  {0.8,0.3,0.9}  {0.9,0.8,0.4}  {0.4,0.3} 
$({u}_{2}{q}_{2})$  {0.9,0.6}  {0.4,0.8}  {0.3,0.1,0.9}  {0.6,0.5}  {0.7,0.3,0.6}  {0.1,0.7} 
$({u}_{3}{q}_{1})$  {0.2,0.1,0.2}  {0.7,0.4}  {0.1,0.5,0.6}  {0.7,0.1,0.3}  {0.2,0.1}  {0.5,0.9,0.6} 
$({u}_{3}{q}_{2})$  {0.7,0.8}  {0.3}  {0.4,0.8}  {0.1,0.2,0.4}  {0.2,0.7,0.3}  {0.4,0.5} 
${\mathit{R}}_{{\mathit{Q}}_{3}}$  ${\mathit{e}}_{1}{\mathit{q}}_{1}$  ${\mathit{e}}_{1}{\mathit{q}}_{2}$  ${\mathit{e}}_{2}{\mathit{q}}_{1}$  ${\mathit{e}}_{2}{\mathit{q}}_{2}$  ${\mathit{e}}_{3}{\mathit{q}}_{1}$  ${\mathit{e}}_{3}{\mathit{q}}_{2}$ 

$({u}_{1}{q}_{1})$  {0.6,0.2,0.1}  {0.2,0.3}  {0.1,0.2,0.9}  {0.2,0.8}  {0.8,0.5,0.6}  {0.7,0.3,0.6} 
$({u}_{1}{q}_{2})$  {0.5,0.3}  {0.3,0.1,0.4}  {0.2,0.3}  {0.9,0.1,0.6}  {0.5,0.4}  {0.2,0.7,0.1} 
$({u}_{2}{q}_{1})$  {0.4,0.6,0.5}  {0.5,0.1}  {0.2,0.8,0.7}  {0.8,0.7}  {0.5,0.2,0.1}  {0.4,0.3} 
$({u}_{2}{q}_{2})$  {0.3,0.4}  {0.8,0.2,0.5}  {0.4,0.9}  {0.1,0.2}  {0.8,0.5,0.3}  {0.5,0.3} 
$({u}_{3}{q}_{1})$  {0.4,0.3,0.6}  {0.5,0.4}  {0.4,0.7,0.5}  {0.4,0.6}  {0.7,0.6,0.2}  {0.8,0.9,0.2} 
$({u}_{3}{q}_{2})$  {0.8,0.2}  {0.3,0.1,0.3}  {0.9,0.1}  {0.4,0.6,0.7}  {0.3,0.8}  {0.6,0.4,0.7} 
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Alsager, K.M.; Alshehri, N.O.; Akram, M. A DecisionMaking Approach Based on a Multi QHesitant Fuzzy Soft MultiGranulation Rough Model. Symmetry 2018, 10, 711. https://doi.org/10.3390/sym10120711
Alsager KM, Alshehri NO, Akram M. A DecisionMaking Approach Based on a Multi QHesitant Fuzzy Soft MultiGranulation Rough Model. Symmetry. 2018; 10(12):711. https://doi.org/10.3390/sym10120711
Chicago/Turabian StyleAlsager, Kholood Mohammad, Noura Omair Alshehri, and Muhammad Akram. 2018. "A DecisionMaking Approach Based on a Multi QHesitant Fuzzy Soft MultiGranulation Rough Model" Symmetry 10, no. 12: 711. https://doi.org/10.3390/sym10120711