A New MultiAttribute DecisionMaking Method Based on mPolar Fuzzy Soft Rough Sets
Abstract
:1. Introduction
2. Soft Rough $\mathit{m}$Polar Fuzzy Sets
 1.
 $\underline{\xi}\left(Q\right)=\sim \overline{\xi}(\sim Q),$
 2.
 $Q\subseteq R\Rightarrow \underline{\xi}\left(Q\right)\subseteq \underline{\xi}\left(R\right),$
 3.
 $\underline{\xi}(Q\cap R)=\underline{\xi}\left(Q\right)\cap \underline{\xi}\left(R\right),$
 4.
 $\underline{\xi}(Q\cup R)\supseteq \underline{\xi}\left(Q\right)\cup \underline{\xi}\left(R\right),$
 5.
 $\overline{\xi}\left(Q\right)=\sim \underline{\xi}(\sim Q),$
 6.
 $Q\subseteq R\Rightarrow \overline{\xi}\left(Q\right)\subseteq \overline{\xi}\left(R\right),$
 7.
 $\overline{\xi}(Q\cup R)=\overline{\xi}\left(Q\right)\cup \overline{\xi}\left(R\right),$
 8.
 $\overline{\xi}(Q\cap R)\subseteq \overline{\xi}\left(Q\right)\cap \overline{\xi}\left(R\right),$
 From Definition 5, we have$$\begin{array}{cc}\sim \overline{\xi}(\sim Q)\hfill & =\left\{\u2329v,\left(\mathbf{1}{(\sim Q)}_{\overline{\xi}}\left(v\right)\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\{\langle v,(\mathbf{1}{\displaystyle \underset{w\in {\xi}_{s}\left(v\right)}{\bigvee}}{p}_{i}\circ (\sim Q)\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,(\mathbf{1}\wedge \underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}{p}_{i}\circ Q\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}{p}_{i}\circ Q\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\left\{\u2329v,{Q}_{\underline{\xi}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\underline{\xi}\left(Q\right).\hfill \end{array}$$It follows that $\underline{\xi}\left(Q\right)=\sim \overline{\xi}(\sim Q).$
 It can be easily proved by Definition 5.
 By Definition 5,$$\begin{array}{cc}\underline{\xi}(Q\cap R)\hfill & =\left\{\u2329v,{(Q\cap R)}_{\underline{\xi}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}{p}_{i}\circ (Q\cap R)\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}({p}_{i}\circ Q\left(w\right)\wedge {p}_{i}\circ R\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}({p}_{i}\circ Q\left(w\right))\wedge \underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}({p}_{i}\circ R\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\left\{\u2329v,{Q}_{\underline{\xi}}\left(v\right)\wedge {R}_{\underline{\xi}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\underline{\xi}\left(Q\right)\cap \underline{\xi}\left(R\right).\hfill \end{array}$$Hence, $\underline{\xi}(Q\cap R)=\underline{\xi}\left(Q\right)\cap \underline{\xi}\left(R\right).$
 From Definition 5,$$\begin{array}{cc}\underline{\xi}(Q\cup R)\hfill & =\left\{\u2329v,{(Q\cup R)}_{\underline{\xi}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}{p}_{i}\circ (Q\cup R)\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}({p}_{i}\circ Q\left(w\right)\vee {p}_{i}\circ R\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & \supseteq \{\langle v,\underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}\left({p}_{i}\circ Q\left(w\right)\right)\vee \underset{w\in {\xi}_{s}\left(v\right)}{\bigwedge}({p}_{i}\circ R\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\left\{\u2329v,{Q}_{\underline{\xi}}\left(v\right)\vee {R}_{\underline{\xi}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\underline{\xi}\left(Q\right)\cup \underline{\xi}\left(R\right).\hfill \end{array}$$Hence, $\underline{\xi}(Q\cup R)\supseteq \underline{\xi}\left(Q\right)\cup \underline{\xi}\left(R\right).$
 1.
 $\sim \left(\underline{\xi}\left(Q\right)\cup \underline{\xi}\left(R\right)\right)=\overline{\xi}(\sim Q)\cap \overline{\xi}(\sim R),$
 2.
 $\sim \left(\underline{\xi}\left(Q\right)\cup \overline{\xi}\left(R\right)\right)=\overline{\xi}(\sim Q)\cap \underline{\xi}(\sim R),$
 3.
 $\sim \left(\overline{\xi}\left(Q\right)\cup \underline{\xi}\left(R\right)\right)=\underline{\xi}(\sim Q)\cap \overline{\xi}(\sim R),$
 4.
 $\sim \left(\overline{\xi}\left(Q\right)\cup \overline{\xi}\left(R\right)\right)=\underline{\xi}(\sim Q)\cap \underline{\xi}(\sim R),$
 5.
 $\sim \left(\underline{\xi}\left(Q\right)\cap \underline{\xi}\left(R\right)\right)=\overline{\xi}(\sim Q)\cup \overline{\xi}(\sim R),$
 6.
 $\sim \left(\underline{\xi}\left(Q\right)\cap \overline{\xi}\left(R\right)\right)=\overline{\xi}(\sim Q)\cup \underline{\xi}(\sim R),$
 7.
 $\sim \left(\overline{\xi}\left(Q\right)\cap \underline{\xi}\left(R\right)\right)=\underline{\xi}(\sim Q)\cup \overline{\xi}(\sim R),$
 8.
 $\sim \left(\overline{\xi}\left(Q\right)\cap \overline{\xi}\left(R\right)\right)=\underline{\xi}(\sim Q)\cup \underline{\xi}(\sim R).$
3. $\mathit{m}$F Soft Rough Sets
 1.
 $\underline{\zeta}\left(Q\right)=\sim \overline{\zeta}(\sim Q),$
 2.
 $Q\subseteq R\Rightarrow \underline{\zeta}\left(Q\right)\subseteq \underline{\zeta}\left(R\right),$
 3.
 $\underline{\zeta}(Q\cap R)=\underline{\zeta}\left(Q\right)\cap \underline{\zeta}\left(R\right),$
 4.
 $\underline{\zeta}(Q\cup R)\supseteq \underline{\zeta}\left(Q\right)\cup \underline{\zeta}\left(R\right),$
 5.
 $\overline{\zeta}\left(Q\right)=\sim \underline{\zeta}(\sim Q),$
 6.
 $Q\subseteq R\Rightarrow \overline{\zeta}\left(Q\right)\subseteq \overline{\zeta}\left(R\right),$
 7.
 $\overline{\zeta}(Q\cup R)=\overline{\zeta}\left(Q\right)\cup \overline{\zeta}\left(R\right),$
 8.
 $\overline{\zeta}(Q\cap R)\subseteq \overline{\zeta}\left(Q\right)\cap \overline{\zeta}\left(R\right),$
 From Definition 8,$$\begin{array}{cc}\sim \overline{\zeta}(\sim Q)\hfill & =\left\{\u2329v,\left(\mathbf{1}{(\sim Q)}_{\overline{\zeta}}\left(v\right)\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =(\langle v,\mathbf{1}\underset{w\in T}{\bigvee}({p}_{i}\circ {(\sim Q)}_{\zeta}(v,w)\wedge {p}_{i}\circ (\sim Q)\left(w\right))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\mathbf{1}\wedge \underset{w\in T}{\bigwedge}(\mathbf{1}{p}_{i}\circ {Q}_{\zeta}(v,w))\vee {p}_{i}\circ Q\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in T}{\bigwedge}(\mathbf{1}{p}_{i}\circ {Q}_{\zeta}(v,w))\vee {p}_{i}\circ Q\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\left\{\u2329v,{Q}_{\underline{\zeta}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\underline{\zeta}\left(Q\right).\hfill \end{array}$$
 It can be proved directly by Definition 8.
 By Definition 8,$$\begin{array}{cc}\underline{\zeta}(Q\cap R)\hfill & =\{\langle v,{(Q\cap R)}_{\underline{\zeta}}\left(v\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in T}{\bigwedge}(\mathbf{1}{p}_{i}\circ (Q\cap R)(v,w))\vee {p}_{i}\circ (Q\cap R)\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,\underset{w\in T}{\bigwedge}(\mathbf{1}{p}_{i}\circ (Q(v,w)\wedge R(v,w)))\vee {p}_{i}\circ (Q(w)\wedge R(w))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\{\langle v,{Q}_{\underline{\zeta}}\left(v\right)\wedge {R}_{\underline{\zeta}}\left(v\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\underline{\zeta}\left(Q\right)\cap \underline{\zeta}\left(R\right).\hfill \end{array}$$
 Using Definition 8,$$\begin{array}{cc}\underline{\zeta}(Q\cup R)\hfill & =\left\{\u2329v,{(Q\cup R)}_{\underline{\zeta}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\{\langle v,\underset{w\in T}{\bigwedge}(\mathbf{1}{p}_{i}\circ (Q\cup R)(v,w))\vee {p}_{i}\circ (Q\cup R)\left(w\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & \supseteq \{\langle v,\underset{w\in T}{\bigwedge}(\mathbf{1}{p}_{i}\circ (Q(v,w)\vee R(v,w)))\vee {p}_{i}\circ (Q(w)\vee R(w))\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\},\hfill \\ & =\left\{\u2329v,{Q}_{\underline{\zeta}}\left(v\right)\vee {R}_{\underline{\zeta}}\left(v\right)\u232a\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in Y\right\},\hfill \\ & =\underline{\zeta}\left(Q\right)\cup \underline{\zeta}\left(R\right).\hfill \end{array}$$
 ${v}_{1}$ denotes the Fuel efficiency,
 ${v}_{2}$ denotes the Price,
 ${v}_{3}$ denotes the Technology.
 1.
 $\sim \left(\underline{\zeta}\left(Q\right)\cup \underline{\zeta}\left(R\right)\right)=\overline{\zeta}(\sim Q)\cap \overline{\zeta}(\sim R),$
 2.
 $\sim \left(\underline{\zeta}\left(Q\right)\cup \overline{\zeta}\left(R\right)\right)=\overline{\zeta}(\sim Q)\cap \underline{\zeta}(\sim R),$
 3.
 $\sim \left(\overline{\zeta}\left(Q\right)\cup \underline{\zeta}\left(R\right)\right)=\underline{\zeta}(\sim Q)\cap \overline{\zeta}(\sim R),$
 4.
 $\sim \left(\overline{\zeta}\left(Q\right)\cup \overline{\zeta}\left(R\right)\right)=\underline{\zeta}(\sim Q)\cap \underline{\zeta}(\sim R),$
 5.
 $\sim \left(\underline{\zeta}\left(Q\right)\cap \underline{\zeta}\left(R\right)\right)=\overline{\zeta}(\sim Q)\cup \overline{\zeta}(\sim R),$
 6.
 $\sim \left(\underline{\zeta}\left(Q\right)\cap \overline{\zeta}\left(R\right)\right)=\overline{\zeta}(\sim Q)\cup \underline{\zeta}(\sim R),$
 7.
 $\sim \left(\overline{\zeta}\left(Q\right)\cap \underline{\zeta}\left(R\right)\right)=\underline{\zeta}(\sim Q)\cup \overline{\zeta}(\sim R),$
 8.
 $\sim \left(\overline{\zeta}\left(Q\right)\cap \overline{\zeta}\left(R\right)\right)=\underline{\zeta}(\sim Q)\cup \underline{\zeta}(\sim R).$
 1.
 $$\begin{array}{cc}{Q}_{\overline{\zeta}}\left(v\right)\hfill & =\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\left(v\right)\right)=\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma +}\right)\left(v\right)\right),\hfill \\ & =\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma}\right)\left(v\right)\right)=\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)\left(v\right)\right).\hfill \end{array}$$
 2.
 ${\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma +}\subseteq {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)\subseteq {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma}\right)\subseteq {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\subseteq {\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma}.$
 For all $v\in Y$,$$\begin{array}{cc}\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\left(v\right)\right)\hfill & =\mathrm{sup}\{\sigma \in {[0,1]}^{m}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}v\in {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\},\hfill \\ & =\mathrm{sup}\{\sigma \in {[0,1]}^{m}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\zeta}_{\sigma}\left(v\right)\cap {Q}_{\sigma}\},\hfill \\ & =\mathrm{sup}\{\sigma \in {[0,1]}^{m}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\exists \phantom{\rule{3.33333pt}{0ex}}w\in T[w\in {\zeta}_{\sigma}\left(v\right),w\in {Q}_{\sigma}]\},\hfill \\ & =\mathrm{sup}\{\sigma \in {[0,1]}^{m}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\exists \phantom{\rule{3.33333pt}{0ex}}w\in T[{p}_{i}\circ {Q}_{\zeta}(v,w)\ge \sigma ,{p}_{i}\circ Q\left(w\right)\ge \sigma ]\},\hfill \\ & =\underset{w\in T}{\bigvee}\left({p}_{i}\circ {Q}_{\zeta}(v,w)\wedge {p}_{i}\circ Q\left(w\right)\right),\hfill \\ & ={Q}_{\overline{\zeta}}\left(v\right).\hfill \end{array}$$By similar arguments, we can compute$${Q}_{\overline{\zeta}}\left(v\right)=\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma +}\right)\left(v\right)\right)=\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma}\right)\left(v\right)\right)=\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left(\sigma \wedge {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)\left(v\right)\right).$$
 By Definitions 9 and 10, we directly verified that ${\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)\subseteq {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma}\right)\subseteq {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right).$ Now, it is sufficient to show that ${\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma +}\subseteq {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)$ and ${\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\subseteq {\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma}.$For all $v\in {\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma +}$, we have ${Q}_{\overline{\zeta}}\left(v\right)>\sigma $. By Definition 8, $\underset{w\in T}{\bigvee}\left({p}_{i}\circ {Q}_{\zeta}(v,w)\wedge {p}_{i}\circ Q\left(w\right)\right)>\sigma $. Then, there exists ${w}_{0}\in T$, such that ${p}_{i}\circ {Q}_{\zeta}(v,{w}_{0})\wedge {p}_{i}\circ Q\left({w}_{0}\right)>\sigma $, that is, ${p}_{i}\circ {Q}_{\zeta}(v,{w}_{0})>\sigma $ and ${p}_{i}\circ Q\left({w}_{0}\right)>\sigma $. Thus, ${w}_{0}\in {\zeta}_{\sigma +}\left(v\right)$ and ${w}_{0}\in {Q}_{\sigma}$. It follows that ${\zeta}_{\sigma +}\left(v\right)\cap {Q}_{\sigma}\ne \varnothing $. By Definition 4, we have $v\in {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)$. Hence, ${\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma +}\subseteq {\overline{\zeta}}_{\sigma +}\left({Q}_{\sigma +}\right)$.To prove ${\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\subseteq {\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma}$, let an arbitrary $v\in {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)$, we have ${\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\left(v\right)=\mathbf{1}$. Since ${Q}_{\overline{\zeta}}\left(v\right)=\underset{\sigma \in {[0,1]}^{m}}{\bigvee}\left[{\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\left(v\right)\right]\ge \sigma \wedge {\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\left(v\right)=\sigma $, we obtain $v\in {\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma}$. Hence, ${\overline{\zeta}}_{\sigma}\left({Q}_{\sigma}\right)\subseteq {\left[\overline{\zeta}\left(Q\right)\right]}_{\sigma}.$
 1.
 $\underline{\zeta}(\varnothing )=\varnothing .\phantom{\rule{3.33333pt}{0ex}}\overline{\zeta}\left(T\right)=Y,$
 2.
 $\underline{\zeta}\left(Q\right)\subseteq \overline{\zeta}\left(Q\right),$ for all $Q\in m\left(T\right).$
4. Applications to DecisionMaking
4.1. Selection of a Hotel
 ‘${z}_{1}$’ represents the Location,
 ‘${z}_{2}$’ represents the Meal Options,
 ‘${z}_{3}$’ represents the Services.
 The “Location” of the hotel include close to main road, in the green surroundings, in the city center.
 The “Meal options” of the hotel include fast food, fast casual, casual dining.
 The “Services” of the hotel include WiFi connectivity, fitness center, room service.
Algorithm 1: Selection of a suitable hotel 

4.2. Selection of a Place
 ‘${a}_{1}$’ represents the Environment,
 ‘${a}_{2}$’ represents the Tour Cost.
 The “Environment” of the place includes built environment, natural environment, and social environment.
 The “Tour Cost” of the place may be low, medium, or high.
Algorithm 2: Selection of a suitable place 

4.3. Selection of a House
 ‘${t}_{1}$’ represents the Size,
 ‘${t}_{2}$’ represents the Location,
 ‘${t}_{3}$’ represents the Price.
 The “Size” of the house include small , large, and very large.
 The “Location” of the house include close to the main road, in the green surroundings, and in the city center.
 The “Price” of the house includes low, medium, and high.
Algorithm 3: Selection of a suitable house 

5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
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Akram, M.; Ali, G.; Alshehri, N.O. A New MultiAttribute DecisionMaking Method Based on mPolar Fuzzy Soft Rough Sets. Symmetry 2017, 9, 271. https://doi.org/10.3390/sym9110271
Akram M, Ali G, Alshehri NO. A New MultiAttribute DecisionMaking Method Based on mPolar Fuzzy Soft Rough Sets. Symmetry. 2017; 9(11):271. https://doi.org/10.3390/sym9110271
Chicago/Turabian StyleAkram, Muhammad, Ghous Ali, and Noura Omair Alshehri. 2017. "A New MultiAttribute DecisionMaking Method Based on mPolar Fuzzy Soft Rough Sets" Symmetry 9, no. 11: 271. https://doi.org/10.3390/sym9110271