Multiple Attribute Decision Making Algorithm via Picture Fuzzy Nano Topological Spaces
Abstract
:1. Introduction
Motivation and Objective
2. Preliminaries
 1.
 The union of ${\mathfrak{S}}_{\mathfrak{1}}$ and ${\mathfrak{S}}_{\mathfrak{2}}$ is$${\mathfrak{S}}_{\mathfrak{1}}\cup {\mathfrak{S}}_{\mathfrak{2}}=\{(\zeta ,\langle {\mathfrak{m}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\vee {\mathfrak{m}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right),{\mathfrak{a}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\wedge {\mathfrak{a}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right),{\mathfrak{n}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\wedge {\mathfrak{n}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right)\rangle ):\zeta \in \mathfrak{U}\},$$
 2.
 The intersection of ${\mathfrak{S}}_{\mathfrak{1}}$ and ${\mathfrak{S}}_{\mathfrak{2}}$ is$${\mathfrak{S}}_{\mathfrak{1}}\cap {\mathfrak{S}}_{\mathfrak{2}}=\{(\zeta ,\langle {\mathfrak{m}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\wedge {\mathfrak{m}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right),{\mathfrak{a}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\vee {\mathfrak{a}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right),{\mathfrak{n}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\vee {\mathfrak{n}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right)\rangle ):\zeta \in \mathfrak{U}\},$$
 3.
 The symmetric difference of ${\mathfrak{S}}_{\mathfrak{1}}$ and ${\mathfrak{S}}_{\mathfrak{2}}$ is$${\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}=\{(\zeta ,\langle {\mathfrak{m}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right),{\mathfrak{a}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right),{\mathfrak{n}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right)\rangle ):\zeta \in \mathfrak{U}\},\phantom{\rule{0.166667em}{0ex}}where$$$${\mathfrak{m}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right)=0\vee {\mathfrak{m}}_{\mathfrak{S}}\mathfrak{1}{\mathfrak{m}}_{\mathfrak{S}}\mathfrak{2},{\mathfrak{n}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right)=0\vee {\mathfrak{n}}_{\mathfrak{S}}\mathfrak{1}{\mathfrak{n}}_{\mathfrak{S}}\mathfrak{2},$$$${\mathfrak{a}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right)=\left\{\begin{array}{c}1{\mathfrak{m}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right){\mathfrak{n}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right),\text{}if\phantom{\rule{3.33333pt}{0ex}}{\mathfrak{a}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right){\mathfrak{a}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right)\hfill \\ \{1+{\mathfrak{a}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right){\mathfrak{a}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right)\}\wedge \{1{\mathfrak{m}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right){\mathfrak{n}}_{{\mathfrak{S}}_{\mathfrak{1}}{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right)\},\text{}if\phantom{\rule{3.33333pt}{0ex}}{\mathfrak{a}}_{\mathfrak{S}}\mathfrak{1}\left(\zeta \right)\le {\mathfrak{a}}_{\mathfrak{S}}\mathfrak{2}\left(\zeta \right)\hfill \end{array}\right.$$
 4.
 ${\mathfrak{S}}_{\mathfrak{1}}\subseteq {\mathfrak{S}}_{\mathfrak{2}}$ if and only if$${\mathfrak{m}}_{{\mathfrak{S}}_{\mathfrak{1}}}\left(\zeta \right)\le {\mathfrak{m}}_{{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right),{\mathfrak{a}}_{{\mathfrak{S}}_{\mathfrak{1}}}\left(\zeta \right)\ge {\mathfrak{a}}_{{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right)\text{}and\text{}{\mathfrak{n}}_{{\mathfrak{S}}_{\mathfrak{1}}}\left(\zeta \right)\ge {\mathfrak{n}}_{{\mathfrak{S}}_{\mathfrak{2}}}\left(\zeta \right),\forall \zeta \in \mathfrak{U}.$$
 ${0}_{S}$ and ${1}_{S}$ are member of $\mathfrak{S}$.
 Arbitrary union of picture fuzzy set S in $\mathfrak{S}$ if each S in $\mathfrak{S}$
 Finite intersection of picture fuzzy set S in $\mathfrak{S}$ if each S in $\mathfrak{S}$
 (i)
 The upper approximation of S with respect to R is denoted by ${\mathfrak{PFU}}_{\mathfrak{R}}\left(\mathfrak{S}\right)$, i.e.,${\mathfrak{PFU}}_{\mathfrak{R}}\left(\mathfrak{S}\right)=\left\{\u2329\zeta ,{\mathfrak{m}}_{\overline{\mathfrak{R}}\mathfrak{S}}\left(\zeta \right),{\mathfrak{a}}_{\overline{\mathfrak{R}}\mathfrak{S}}\left(\zeta \right),{\mathfrak{n}}_{\overline{\mathfrak{R}}\mathfrak{S}}\left(\zeta \right)\u232a{\xi \in \left[\zeta \right]}_{R},\zeta \in \mathfrak{U}\right\}$
 (ii)
 The lower approximation of S with respect to R is the set is denoted by ${\mathfrak{PFL}}_{\mathfrak{R}}\left(\mathfrak{S}\right)$, i.e., ${\mathfrak{PFL}}_{\mathfrak{R}}\left(\mathfrak{S}\right)=\left\{\u2329\zeta ,{\mathfrak{m}}_{\underline{\mathfrak{R}}\mathfrak{S}}\left(\zeta \right),{\mathfrak{a}}_{\underline{\mathfrak{R}}\mathfrak{S}}\left(\zeta \right),{\mathfrak{n}}_{\underline{\mathfrak{R}}\mathfrak{S}}\left(\zeta \right)\u232a{\xi \in \left[\zeta \right]}_{R},\zeta \in \mathfrak{U}\right\}$
 (iii)
 The boundary region of S with respect to R is the set of all objects which can be classified neither as S nor as not S with respect to R and is denoted by ${\mathfrak{PFB}}_{\mathfrak{R}}\left(\mathfrak{S}\right)$. ${\mathfrak{PFB}}_{\mathfrak{R}}\left(\mathfrak{S}\right)={\mathfrak{PFU}}_{\mathfrak{R}}\left(\mathfrak{S}\right){\mathfrak{PFL}}_{\mathfrak{R}}\left(\mathfrak{S}\right)$, where
3. Picture Fuzzy Nano Topological Spaces
 1.
 ${0}_{\mathfrak{p}},{1}_{\mathfrak{p}}\in {\tau}_{\mathfrak{R}}$
 2.
 If ${\mathfrak{A}}_{\mathfrak{i}}\in {\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)$, for $\mathfrak{a}=1,2,3,..$, then$$\bigcup _{\mathfrak{a}=1}^{\infty}{\mathfrak{A}}_{\mathfrak{i}}\in {\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)$$
 3.
 If ${\mathfrak{A}}_{\mathfrak{i}}\in {\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)$, for $\mathfrak{a}=1,2,3,..n$, then$$\bigcap _{\mathfrak{a}=1}^{n}{\mathfrak{A}}_{\mathfrak{i}}\in {\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)$$The complement ${\mathfrak{A}}^{\mathfrak{c}}$ of a PFNOS $\mathfrak{A}$ in a PFNTS. $(\mathfrak{U},{\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right))$ is called a PFNCS in $\mathfrak{S}$.
 1.
 The collection ${\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)=\{{0}_{\mathfrak{p}},{1}_{\mathfrak{p}}\}$, is the indiscrete picture fuzzy nano topology on $\mathfrak{U}$.
 2.
 If ${\mathfrak{PFL}}_{\mathfrak{R}}={\mathfrak{PFU}}_{\mathfrak{R}}={\mathfrak{PF}}_{\mathfrak{R}}$, then the picture fuzzy nano topology is${\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)=\{{0}_{\mathfrak{p}},{1}_{\mathfrak{p}},{\mathfrak{PFL}}_{\mathfrak{R}}\left(\mathfrak{A}\right),{\mathfrak{PFB}}_{\mathfrak{R}}\left(\mathfrak{A}\right)\}$.
 3.
 If ${\mathfrak{PFL}}_{\mathfrak{R}}={\mathfrak{PFB}}_{\mathfrak{R}}$, then ${\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)=\{{0}_{\mathfrak{p}},{1}_{\mathfrak{p}},{\mathfrak{PFL}}_{\mathfrak{R}}\left(\mathfrak{A}\right),{\mathfrak{PFU}}_{\mathfrak{R}}\left(\mathfrak{A}\right)\}$ is a picture fuzzy nano topology.
 4.
 If ${\mathfrak{PFU}}_{\mathfrak{R}}={\mathfrak{PFB}}_{\mathfrak{R}}$, then the picture fuzzy nano topology is${\tau}_{\mathfrak{R}}\left(\mathfrak{A}\right)=\{{0}_{\mathfrak{p}},{1}_{\mathfrak{p}},{\mathfrak{PFL}}_{\mathfrak{R}}\left(\mathfrak{A}\right),{\mathfrak{PFB}}_{\mathfrak{R}}\left(\mathfrak{A}\right)\}$
 1.
 ${\mathfrak{A}}^{\mathfrak{o}}$ = $\cup \{\mathfrak{G}:\mathfrak{G}$ is a PFNOS in $\mathfrak{S}$ and $\mathfrak{G}\subseteq \mathfrak{A}\}$,
 2.
 ${\mathfrak{A}}^{}$ = $\cap \{\mathfrak{G}:\mathfrak{G}$ is a PFNCS in $\mathfrak{S}$ and $\mathfrak{G}\supseteq \mathfrak{A}\}$.
 1.
 ${\left[{\mathfrak{A}}^{c}\right]}^{}$ = ${\left[{\mathfrak{A}}^{\mathfrak{o}}\right]}^{c}$.
 2.
 ${\left[{\mathfrak{A}}^{c}\right]}^{\mathfrak{o}}$ = ${\left[{\mathfrak{A}}^{}\right]}^{c}$.
 3.
 $\mathfrak{A}$ is a PFNCS if and only if ${\mathfrak{A}}^{}=\mathfrak{A}$.
 4.
 $\mathfrak{A}$ is a PFNOS if and only if ${\mathfrak{A}}^{\mathfrak{o}}=\mathfrak{A}$.
 5.
 ${\mathfrak{A}}^{}$ is a PFNCS in $\mathfrak{U}$.
 6.
 ${\mathfrak{A}}^{\mathfrak{o}}$ is a PFNOS in $\mathfrak{U}$.
 1.
 $\mathfrak{A}\subseteq {\mathfrak{A}}^{}$.
 2.
 $\mathfrak{A}$ is picture fuzzy nano closed if and only if ${\mathfrak{A}}^{}=\mathfrak{A}$.
 3.
 ${0}_{\mathfrak{p}}^{}$ = ${0}_{\mathfrak{p}}$ and ${1}_{\mathfrak{p}}^{}$ = ${1}_{\mathfrak{p}}$.
 4.
 ${\mathfrak{A}}_{\mathfrak{1}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}\Rightarrow $${\mathfrak{A}}_{\mathfrak{1}}^{}\subseteq {\mathfrak{A}}_{\mathfrak{2}}^{}$.
 5.
 ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{}$ = ${\mathfrak{A}}_{\mathfrak{1}}^{}\cup {\mathfrak{A}}_{\mathfrak{2}}^{}$.
 6.
 ${({\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}})}^{}$ = ${\mathfrak{A}}_{\mathfrak{1}}^{}\cap {\mathfrak{A}}_{\mathfrak{2}}^{}$.
 7.
 ${\left({\mathfrak{A}}^{}\right)}^{}$ =${\mathfrak{A}}^{}$.
 1.
 By definition of picture fuzzy nano closure, $\mathfrak{A}\subseteq $${\mathfrak{A}}^{}$
 2.
 If $\mathfrak{A}$ is a picture fuzzy nano closed set, then $\mathfrak{A}$ is the smallest picture fuzzy nano closed set containing itself and hence ${\mathfrak{A}}^{}=\mathfrak{A}$. Conversely, if ${\mathfrak{A}}^{}$ = $\mathfrak{A}$, then $\mathfrak{A}$ is the smallest picture fuzzy nano closed set containing itself and hence $\mathfrak{A}$ is a picture fuzzy nano closed set.
 3.
 Since ${0}_{\mathfrak{p}}$ and ${1}_{\mathfrak{p}}$ are picture fuzzy nano closed sets in $(\mathfrak{U};{\tau}_{\mathfrak{R}})\left(\mathfrak{S}\right)$, ${0}_{\mathfrak{p}}^{}={0}_{\mathfrak{p}}$ and ${1}_{\mathfrak{p}}^{}={1}_{\mathfrak{p}}$.
 4.
 If PFN set ${\mathfrak{A}}_{\mathfrak{1}}$ is a subset of PFN set ${\mathfrak{A}}_{\mathfrak{2}}$, since PFN set ${\mathfrak{A}}_{\mathfrak{2}}$ is a subset of ${\mathfrak{A}}_{\mathfrak{2}}^{}$, then PFN set ${\mathfrak{A}}_{\mathfrak{1}}$ is a subset of ${\mathfrak{A}}_{\mathfrak{2}}^{}$, i.e., ${\mathfrak{A}}_{\mathfrak{2}}^{}$ is a PFNCS containing ${\mathfrak{A}}_{\mathfrak{1}}$. However, ${\mathfrak{A}}_{\mathfrak{1}}^{}$ is the smallest PFNCS containing ${\mathfrak{A}}_{\mathfrak{1}}$. Therefore, ${\mathfrak{A}}_{\mathfrak{1}}^{}\subseteq {\mathfrak{A}}_{\mathfrak{2}}^{}$
 5.
 Since PFN set ${\mathfrak{A}}_{\mathfrak{1}}$ is a subset of union of two PFN sets ${\mathfrak{A}}_{\mathfrak{1}}$ and ${\mathfrak{A}}_{\mathfrak{2}}$ and PFN set ${\mathfrak{A}}_{\mathfrak{2}}$ is a subset of union of two PFN sets ${\mathfrak{A}}_{\mathfrak{1}}$ and ${\mathfrak{A}}_{\mathfrak{2}}$, ${\mathfrak{A}}_{\mathfrak{1}}^{}\subseteq {({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{}$. Then closure of PFN set ${\mathfrak{A}}_{\mathfrak{1}}$ is a subset of closure of union of two PFN sets ${\mathfrak{A}}_{\mathfrak{1}}$ and ${\mathfrak{A}}_{\mathfrak{2}}$ and closure of PFN set ${\mathfrak{A}}_{\mathfrak{2}}$ is a subset of closure of union of two PFN sets ${\mathfrak{A}}_{\mathfrak{1}}$ and ${\mathfrak{A}}_{\mathfrak{2}}$. Therefore, union of closure of PFN sets ${\mathfrak{A}}_{\mathfrak{1}}^{}$, ${\mathfrak{A}}_{\mathfrak{2}}^{}$ is a subset of closure of union of $({\mathfrak{A}}_{\mathfrak{1}}^{}$, ${\mathfrak{A}}_{\mathfrak{2}}{)}^{}$. By the fact that ${\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}^{}\cup {\mathfrak{A}}_{\mathfrak{2}}^{}$, and since ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{}$ is the smallest picture fuzzy nano closed set containing ${\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}}$, so ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{}\subseteq {\mathfrak{A}}_{\mathfrak{1}}^{}\cup {\mathfrak{A}}_{\mathfrak{2}}^{}$. Thus, ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{}={\mathfrak{A}}_{\mathfrak{1}}^{}\cup {\mathfrak{A}}_{\mathfrak{2}}^{}$.
 6.
 Since ${\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}$ and ${\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}$, ${({\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}})}^{}\subseteq {\mathfrak{A}}_{\mathfrak{1}}^{}\cap {\mathfrak{A}}_{\mathfrak{2}}^{}$.
 7.
 Since ${\mathfrak{A}}^{}$ is a picture fuzzy nano closed set, then ${\left({\mathfrak{A}}^{}\right)}^{}={\mathfrak{A}}^{}$.
 1.
 ${1}_{\mathfrak{p}}{\mathfrak{A}}^{\mathfrak{o}}$ = ${({1}_{\mathfrak{p}}\mathfrak{A})}^{}$.
 2.
 ${1}_{\mathfrak{p}}{\mathfrak{A}}^{}$ = ${({1}_{\mathfrak{p}}\mathfrak{A})}^{\mathfrak{o}}$.
 1.
 $\mathfrak{A}$ is picture fuzzy nano open if and only if ${\mathfrak{A}}^{\mathfrak{o}}=\mathfrak{A}$.
 2.
 ${0}_{\mathfrak{p}}^{\mathfrak{o}}={0}_{\mathfrak{p}}$ and ${1}_{\mathfrak{p}}^{\mathfrak{o}}={1}_{\mathfrak{p}}$.
 3.
 ${\mathfrak{A}}_{\mathfrak{1}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}\Rightarrow {\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$.
 4.
 ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}={\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cup {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$.
 5.
 ${({\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}={\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cap {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$.
 6.
 ${\left({\mathfrak{A}}^{\mathfrak{o}}\right)}^{\mathfrak{o}}={\mathfrak{A}}^{\mathfrak{o}}$.
 1.
 $\mathfrak{A}$ is a picture fuzzy nano open set if and only if ${1}_{\mathfrak{p}}\mathfrak{A}$ is a picture fuzzy nano closed set, if and only if ${({1}_{\mathfrak{p}}\mathfrak{A})}^{}={1}_{\mathfrak{p}}\mathfrak{A}$, if and only if ${1}_{\mathfrak{p}}{({1}_{\mathfrak{p}}\mathfrak{A})}^{}=\mathfrak{A}$ if and only if ${\mathfrak{A}}^{\mathfrak{o}}=\mathfrak{A}$.
 2.
 Since ${0}_{\mathfrak{p}}$ and ${1}_{\mathfrak{p}}$ are picture fuzzy nano open sets in $(\mathfrak{U};{\tau}_{\mathfrak{R}})\left(\mathfrak{S}\right)$, ${0}_{\mathfrak{p}}^{\mathfrak{o}}={0}_{\mathfrak{p}}$ and ${1}_{\mathfrak{p}}^{\mathfrak{o}}={1}_{\mathfrak{p}}$.
 3.
 If ${\mathfrak{A}}_{\mathfrak{1}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}$, since ${\mathfrak{A}}_{\mathfrak{2}}\supseteq {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$, then ${\mathfrak{A}}_{\mathfrak{1}}\supseteq {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$, i.e., ${\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$ is a picture fuzzy nano open set containing ${\mathfrak{A}}_{\mathfrak{1}}$. However, ${\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}$ is the largest picture fuzzy nano open set contained in ${\mathfrak{A}}_{\mathfrak{1}}$. Therefore, ${\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$
 4.
 Since ${\mathfrak{A}}_{\mathfrak{1}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}}$ and ${\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}}$, ${\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\subseteq {({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}$ and ${\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}\subseteq {({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}$. Therefore, ${\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cup {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}\subseteq {({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}$. By the fact that ${\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cup {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$, and since ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}$ is the largest picture fuzzy nano open set containing ${\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}}$, so ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cup {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$. Thus, ${({\mathfrak{A}}_{\mathfrak{1}}\cup {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}={\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cup {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$.
 5.
 Since ${\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}$ and ${\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}}\subseteq {\mathfrak{A}}_{\mathfrak{2}}$, ${({\mathfrak{A}}_{\mathfrak{1}}\cap {\mathfrak{A}}_{\mathfrak{2}})}^{\mathfrak{o}}\subseteq {\mathfrak{A}}_{\mathfrak{1}}^{\mathfrak{o}}\cap {\mathfrak{A}}_{\mathfrak{2}}^{\mathfrak{o}}$.
 6.
 Since ${\mathfrak{A}}^{\mathfrak{o}}$ is a picture fuzzy nano open set, then ${\left({\mathfrak{A}}^{\mathfrak{o}}\right)}^{\mathfrak{o}}$ = ${\mathfrak{A}}^{\mathfrak{o}}$.
 1.
 if ${\mathfrak{Sr}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{1}}\right)>{\mathfrak{Sr}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{2}}\right)$, then ${\mathfrak{A}}_{\mathfrak{1}}\succ {\mathfrak{A}}_{\mathfrak{2}}$
 2.
 if ${\mathfrak{Sr}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{1}}\right)<{\mathfrak{Sr}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{2}}\right)$, then if ${\mathfrak{A}}_{\mathfrak{1}}\prec {\mathfrak{A}}_{\mathfrak{2}}$
 3.
 ${\mathfrak{Sr}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{1}}\right)={\mathfrak{Sr}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{2}}\right)$, then
 (i)
 if ${\mathfrak{H}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{1}}\right)>{\mathfrak{H}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{2}}\right)$, then ${\mathfrak{A}}_{\mathfrak{1}}\succ {\mathfrak{A}}_{\mathfrak{2}}$
 (ii)
 if ${\mathfrak{H}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{1}}\right)={\mathfrak{H}}_{\mathfrak{p}}\left({\mathfrak{A}}_{\mathfrak{2}}\right)$, then ${\mathfrak{A}}_{\mathfrak{1}}\sim {\mathfrak{A}}_{\mathfrak{2}}$
4. Picture Fuzzy Nano Topology in Multiple Attribute DecisionMaking
Proposed Algorithm and Flowchart
Algorithm 1: Ideal decision making with PFTSs 

Algorithm 2: Ideal decision making with PFNTSs 

5. Numerical Example
 1.
 ${\tau}_{1}\left({\mathfrak{b}}_{\mathfrak{1}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.7,0.1,0.1\rangle ,\langle 0.5,0.2,0.1\rangle ,\langle 0.3,0.4,0.2\rangle ,\langle 0.9,0.0,0.1\rangle ,\langle 0.3,0.1,0.5\rangle ,$$\langle 0.3,0.2,0.5\rangle ,\langle 0.3,0.4,0.5\rangle ,\langle 0.5,0.1,0.1\rangle ,\langle 0.3,0.1,0.2\rangle \}$
 2.
 ${\tau}_{1}\left({\mathfrak{b}}_{\mathfrak{2}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.4,0.2,0.2\rangle ,\langle 0.3,0.2,0.3\rangle ,\langle 0.6,0.1,0.3\rangle ,\langle 0.6,0.1,0.2\rangle ,\langle 0.6,0.2,0.1\rangle ,\langle 0.4,0.2,0.3\rangle ,\langle 0.6,0.2,0.3\rangle ,\langle 0.6,0.2,0.2\rangle ,\langle 0.6,0.1,0.1\rangle \}$
 3.
 ${\tau}_{1}\left({\mathfrak{b}}_{\mathfrak{3}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.2,0.4,0.3\rangle ,\langle 0.6,0.2,0.1\rangle ,\langle 0.3,0.2,0.5\rangle ,\langle 0.5,0.2,0.1\rangle ,\langle 0.2,0.4,0.3\rangle ,\langle 0.3,0.2,0.3\rangle \}$
 4.
 ${\tau}_{1}\left({\mathfrak{b}}_{\mathfrak{4}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.3,0.3,0.2\rangle ,\langle 0.5,0.1,0.2\rangle ,\langle 0.2,0.6,0.1\rangle ,\langle 0.4,0.2,0.3\rangle ,\langle 0.7,0.0,0.2\rangle ,\langle 0.2,0.6,0.2\rangle ,\langle 0.3,0.3,0.3\rangle ,\langle 0.2,0.6,0.3\rangle ,\langle 0.3,0.3,0.1\rangle ,\langle 0.4,0.2,0.2\rangle ,\langle 0.5,0.1,0.1\rangle ,\langle 0.4,0.2,0.1\rangle ,\langle 0.7,0.0,0.1\rangle \}$
 1.
 ${\tau}_{2}\left({\mathfrak{f}}_{\mathfrak{1}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.3,0.1,0.3\rangle ,\langle 0.4,0.3,0.2\rangle ,\langle 0.6,0.1,0.3\rangle ,\langle 0.3,0.4,0.2\rangle ,\langle 0.5,0.2,0.2\rangle \langle 0.3,0.3,0.3\rangle ,\langle 0.3,0.4,0.3\rangle ,\langle 0.3,0.2,0.3\rangle ,\langle 0.4,0.3,0.3\rangle ,\langle 0.5,0.2,0.3\rangle ,\langle 0.4,0.1,0.2\rangle ,\langle 0.3,0.1,0.2\rangle ,\langle 0.5,0.1,0.2\rangle ,\langle 0.6,0.1,0.2\rangle \}$
 2.
 ${\tau}_{2}\left({\mathfrak{f}}_{\mathfrak{2}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.7,0.1,0.1\rangle ,\langle 0.4,0.2,0.2\rangle ,\langle 0.2,0.4,0.3\rangle ,\langle 0.3,0.3,0.2\rangle ,\langle 0.6,0.1,0.2\rangle \}$
 3.
 ${\tau}_{2}\left({\mathfrak{f}}_{\mathfrak{3}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.9,0.1,0.0\rangle ,\langle 0.6,0.2,0.2\rangle ,\langle 0.3,0.2,0.3\rangle ,\langle 0.6,0.0,0.0\rangle ,\langle 0.4,0.2,0.3\rangle ,\langle 0.6,0.1,0.0\rangle ,\langle 0.9,0.0,0.0\rangle \}$
 4.
 ${\tau}_{2}\left({\mathfrak{f}}_{\mathfrak{4}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.5,0.1,0.1\rangle ,\langle 0.7,0.2,0.1\rangle ,\langle 0.7,0.0,0.2\rangle ,\langle 0.5,0.3,0.1\rangle ,\langle 0.5,0.2,0.2\rangle ,\langle 0.5,0.2,0.1\rangle ,\langle 0.5,0.1,0.2\rangle ,\langle 0.7,0.2,0.2\rangle ,\langle 0.5,0.3,0.2\rangle ,\langle 0.7,0.1,0.1\rangle ,\langle 0.7,0.0,0.1\rangle ,\langle 0.7,0.2,0.1\rangle \}$
 1.
 ${\mathfrak{C}}_{{\tau}_{1}}^{*}\left({\mathfrak{b}}_{\mathfrak{1}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.9,0.0,0.1\rangle ,\langle 0.5,0.2,0.1\rangle ,\langle 0.4,0.5,0.1\rangle ,\langle 0.3,0.1,0.2\rangle ,\langle 0.3,0.4,0.5\rangle ,\langle 0.0,0.7,0.0\rangle \}$
 2.
 ${\mathfrak{C}}_{{\tau}_{1}}^{*}\left({\mathfrak{b}}_{\mathfrak{2}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.6,0.1,0.2\rangle ,\langle 0.3,0.2,0.3\rangle ,\langle 0.3,0.7,0.0\rangle ,\langle 0.6,0.1,0.1\rangle ,\langle 0.6,0.2,0.3\rangle ,\langle 0.0,0.9,0.0\rangle \}$
 3.
 ${\mathfrak{C}}_{{\tau}_{1}}^{*}\left({\mathfrak{b}}_{\mathfrak{3}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.6,0.2,0.1\rangle ,\langle 0.2,0.4,0.5\rangle ,\langle 0.4,0.6,0.0\rangle ,\langle 0.6,0.2,0.1\rangle ,\langle 0.5,0.2,0.1\rangle ,\langle 0.1,0.9,0.0\rangle \}$
 4.
 ${\mathfrak{C}}_{{\tau}_{1}}^{*}\left({\mathfrak{b}}_{\mathfrak{4}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.5,0.1,0.2\rangle ,\langle 0.3,0.3,0.3\rangle ,\langle 0.2,0.8,0.0\rangle ,\langle 0.7,0.0,0.1\rangle ,\langle 0.2,0.6,0.2\rangle ,\langle 0.5,0.4,0.0\rangle \}$
 1.
 ${\mathfrak{C}}_{{\tau}_{2}}^{*}\left({\mathfrak{f}}_{\mathfrak{1}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.4,0.1,0.2\rangle ,\langle 0.3,0.4,0.3\rangle ,\langle 0.1,0.7,0.0\rangle ,\langle 0.6,0.1,0.2\rangle ,\langle 0.5,0.2,0.3\rangle ,\langle 0.1,0.9,0.0\rangle \}$
 2.
 ${\mathfrak{C}}_{{\tau}_{2}}^{*}\left({\mathfrak{f}}_{\mathfrak{2}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.7,0.1,0.1\rangle ,\langle 0.3,0.3,0.2\rangle ,\langle 0.4,0.6,0.0\rangle ,\langle 0.6,0.1,0.2\rangle ,\langle 0.2,0.4,0.3\rangle ,\langle 0.4,0.6,0.0\rangle \}$
 3.
 ${\mathfrak{C}}_{{\tau}_{2}}^{*}\left({\mathfrak{f}}_{\mathfrak{3}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.9,0.0,0.0\rangle ,\langle 0.6,0.2,0.2\rangle ,\langle 0.3,0.7,0.0\rangle ,\langle 0.4,0.2,0.3\rangle ,\langle 0.3,0.2,0.3\rangle ,\langle 0.1,0.9,0.0\rangle \}$
 4.
 ${\mathfrak{C}}_{{\tau}_{2}}^{*}\left({\mathfrak{f}}_{\mathfrak{4}}\right)=\{{1}_{\mathfrak{p}},{0}_{\mathfrak{p}},\langle 0.7,0.1,0.1\rangle ,\langle 0.5,0.3,0.1\rangle ,\langle 0.2,0.8,0.0\rangle ,\langle 0.7,0.0,0.2\rangle ,\langle 0.5,0.2,0.2\rangle ,\langle 0.2,0.8,0.0\rangle \}$
6. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
MADM  multiple attribute decision making 
RFS  Rough Fuzzy Set 
NS  neutrosophic set 
IFS  intuitionistic fuzzy sets 
NT  nano topology 
PFS  picture fuzzy set 
PFNT  picture fuzzy nano Topological spaces 
PFNCS  picture fuzzy nano closed set 
PFN  picture fuzzy nano 
PFNOS  picture fuzzy nano open set 
PFNTS  picture fuzzy nano topological space 
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${\mathfrak{b}}_{\mathbf{1}}$  ${\mathfrak{b}}_{\mathbf{2}}$  ${\mathfrak{b}}_{\mathbf{3}}$  ${\mathfrak{b}}_{\mathbf{4}}$  

${\zeta}_{1}$  $\langle 0.7,0.1,0.1\rangle $  $\langle 0.4,0.2,0.2\rangle $  $\langle 0.2,0.4,0.3\rangle $  $\langle 0.3,0.3,0.2\rangle $ 
${\zeta}_{2}$  $\langle 0.5,0.2,0.1\rangle $  $\langle 0.3,0.2,0.3\rangle $  $\langle 0.6,0.2,0.1\rangle $  $\langle 0.5,0.1,0.2\rangle $ 
${\zeta}_{3}$  $\langle 0.3,0.4,0.2\rangle $  $\langle 0.6,0.1,0.3\rangle $  $\langle 0.6,0.2,0.1\rangle $  $\langle 0.2,0.6,0.1\rangle $ 
${\zeta}_{4}$  $\langle 0.9,0.0,0.1\rangle $  $\langle 0.6,0.1,0.2\rangle $  $\langle 0.3,0.2,0.5\rangle $  $\langle 0.4,0.2,0.3\rangle $ 
${\zeta}_{5}$  $\langle 0.3,0.1,0.5\rangle $  $\langle 0.6,0.2,0.1\rangle $  $\langle 0.5,0.2,0.1\rangle $  $\langle 0.7,0.0,0.2\rangle $ 
${\mathit{\zeta}}_{\mathbf{1}}$  ${\mathit{\zeta}}_{\mathbf{2}}$  ${\mathit{\zeta}}_{\mathbf{3}}$  ${\mathit{\zeta}}_{\mathbf{4}}$  ${\mathit{\zeta}}_{\mathbf{5}}$  

${\mathfrak{f}}_{1}$  $\langle 0.3,0.1,0.3\rangle $  $\langle 0.4,0.3,0.2\rangle $  $\langle 0.6,0.1,0.3\rangle $  $\langle 0.3,0.4,0.2\rangle $  $\langle 0.5,0.2,0.2\rangle $ 
${\mathfrak{f}}_{2}$  $\langle 0.7,0.1,0.1\rangle $  $\langle 0.4,0.2,0.2\rangle $  $\langle 0.2,0.4,0.3\rangle $  $\langle 0.3,0.3,0.2\rangle $  $\langle 0.6,0.1,0.2\rangle $ 
${\mathfrak{f}}_{3}$  $\langle 0.9,0.1,0.0\rangle $  $\langle 0.6,0.2,0.2\rangle $  $\langle 0.3,0.2,0.3\rangle $  $\langle 0.6,0.0,0.0\rangle $  $\langle 0.4,0.2,0.3\rangle $ 
${\mathfrak{f}}_{4}$  $\langle 0.5,0.1,0.1\rangle $  $\langle 0.7,0.2,0.1\rangle $  $\langle 0.7,0.0,0.2\rangle $  $\langle 0.5,0.3,0.1\rangle $  $\langle 0.5,0.2,0.2\rangle $ 
Sets  Uncertainty  Truth Value of an Element  False Value of an Element  Abstinence of an Element  Roughness & Boundary of a Set 

Zafer refInt. J. Fuzzy Syst. RFS  ✓  ✓      ✓ 
Atanassov refFuzzy sets and systems IFT  ✓  ✓  ✓     
Wei refIranian Journal of Fuzzy Systems PFS  ✓  ✓  ✓  ✓   
Proposed Algorithm PFT  ✓  ✓  ✓  ✓   
Proposed Algorithm PFNT  ✓  ✓  ✓  ✓  ✓ 
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Alshammari, I.; Mani, P.; Ozel, C.; Garg, H. Multiple Attribute Decision Making Algorithm via Picture Fuzzy Nano Topological Spaces. Symmetry 2021, 13, 69. https://doi.org/10.3390/sym13010069
Alshammari I, Mani P, Ozel C, Garg H. Multiple Attribute Decision Making Algorithm via Picture Fuzzy Nano Topological Spaces. Symmetry. 2021; 13(1):69. https://doi.org/10.3390/sym13010069
Chicago/Turabian StyleAlshammari, Ibtesam, Parimala Mani, Cenap Ozel, and Harish Garg. 2021. "Multiple Attribute Decision Making Algorithm via Picture Fuzzy Nano Topological Spaces" Symmetry 13, no. 1: 69. https://doi.org/10.3390/sym13010069