DecisionMaking via Neutrosophic Support Soft Topological Spaces
Abstract
:1. Introduction
2. Preliminaries
X  ${x}_{1}$  ${x}_{2}$  ${x}_{3}$  ${x}_{4}$  ${x}_{5}$  ${x}_{6}$ 
${y}_{1}$  1  1  0  0  0  0 
${y}_{2}$  1  0  0  1  1  1 
${y}_{3}$  0  0  0  0  0  0 
${y}_{4}$  1  1  1  1  1  1 
${y}_{5}$  1  1  1  1  1  0 
 (1)
 The complement of ${F}_{X}$ is defined by ${F}_{{X}^{c}}\left(x\right)=X\setminus {f}_{X}\left(x\right)$ for all $x\in A;$
 (2)
 The union of two soft sets is defined by ${f}_{X\cup Y}\left(x\right)={f}_{X}\left(x\right)\cup {f}_{Y}\left(x\right)$ for all $x\in A;$
 (3)
 The intersection of two soft sets is defined by ${f}_{X\cap Y}\left(x\right)={f}_{X}\left(x\right)\cap {f}_{Y}\left(x\right)$ for all $x\in A.$
3. Interval Valued Neutrosophic Support Soft Set
 (i)
 An empty set A, denoted by $A=\varnothing $, is defined by $\varnothing =\{\langle (0,0),(1,1),(0,0),(1,1)\rangle /x:x\in X\}.$
 (ii)
 The universal set is defined by $U=\{\langle (1,1),(0,0),(1,1),(0,0)\rangle /x:x\in X\}.$
 (iii)
 The complement of A is defined by ${A}^{c}=\{\langle (inf{\gamma}_{A}\left(x\right),sup{\gamma}_{A}\left(x\right)),(1sup{\sigma}_{A}\left(x\right),1inf{\sigma}_{A}\left(x\right)),(1sup{\omega}_{A}\left(x\right),1inf{\omega}_{A}\left(x\right)),(inf{\mu}_{A}\left(x\right),sup{\mu}_{A}\left(x\right))\rangle /x:x\in X\}.$
 (iv)
 A and B are two intervalvalued neutrosophic support sets of X. A is a subset of B if ${\mu}_{A}\left(x\right)\le {\mu}_{B}\left(x\right),{\sigma}_{A}\left(x\right)\ge {\sigma}_{B}\left(x\right),{\omega}_{A}\left(x\right)\le {\omega}_{B}\left(x\right),{\gamma}_{A}\left(x\right)\ge {\gamma}_{B}\left(x\right).$
 (v)
 Two intervalvalued neutrosophic support sets A and B in X are said to be equal if $A\subseteq B$ and $B\subseteq A$.
 (i)
 The intersection of A and B is defined by $A\cap B=\{\langle (min[inf{\mu}_{A}\left(x\right),inf{\mu}_{B}\left(x\right)],min[sup{\mu}_{A},sup{\mu}_{B}\left(x\right)]),(max[inf{\sigma}_{A}\left(x\right),inf{\sigma}_{B}\left(x\right)],max[sup{\sigma}_{A}\left(x\right),sup{\sigma}_{B}\left(x\right)]),(min[inf{\omega}_{A}\left(x\right),inf{\omega}_{B}\left(x\right)],min[sup{\omega}_{A}\left(x\right),sup{\omega}_{B}\left(x\right)]),(max[inf{\gamma}_{A}\left(x\right),inf{\gamma}_{B}\left(x\right)],max[sup{\gamma}_{A}\left(x\right),sup{\gamma}_{B}\left(x\right)])\rangle /x:x\in X\}$.
 (ii)
 The union of A and B is defined by $A\cup B=\{\langle (max[inf{\mu}_{A}\left(x\right),inf{\mu}_{B}\left(x\right)],max[sup{\mu}_{A}\left(x\right),sup{\mu}_{B}\left(x\right)]),(min[inf{\sigma}_{A}\left(x\right),inf{\sigma}_{B}\left(x\right)],min[sup{\sigma}_{A}\left(x\right),{\sigma}_{B}\left(x\right)]),(max[inf{\omega}_{A}\left(x\right),inf{\omega}_{B}\left(x\right)],max[sup{\omega}_{A}\left(x\right),sup{\omega}_{B}\left(x\right)]),(min[inf{\gamma}_{A}\left(x\right),inf{\gamma}_{B}\left(x\right)],min[sup{\gamma}_{A}\left(x\right),sup{\gamma}_{B}\left(x\right)])\rangle /x:x\in X\}$.
 (iii)
 A difference, B, is defined by $A\setminus B$=$\{\langle (min[inf{\mu}_{A}\left(x\right),inf{\gamma}_{B}\left(x\right)],min[sup{\mu}_{A}\left(x\right),sup{\gamma}_{B}\left(x\right)]),(max[inf{\sigma}_{A}\left(x\right),1sup{\sigma}_{B}\left(x\right)],max[sup{\sigma}_{A}\left(x\right),1inf{\sigma}_{B}\left(x\right)]),(min[inf{\omega}_{A}\left(x\right),1sup{\omega}_{B}\left(x\right)],min[sup{\omega}_{A}\left(x\right),1inf{\omega}_{B}\left(x\right)]),(max[inf{\gamma}_{A}\left(x\right),inf{\mu}_{B}\left(x\right)],max[sup{\gamma}_{B}\left(x\right),sup{\mu}_{B}\left(x\right)])\rangle /x:x\in X\}$.
 (iv)
 Scalar multiplication of A is defined by $A.a=\{\langle (min[inf{\mu}_{A}\left(x\right).a,1],min[sup{\mu}_{A}\left(x\right).a,1]),(min[inf{\sigma}_{A}\left(x\right).a,1],min[sup{\sigma}_{A}\left(x\right).a,1]),(min[inf{\omega}_{A}\left(x\right).a,1],min[sup{\omega}_{A}\left(x\right).a,1]),(min[inf{\gamma}_{A}\left(x\right).a,1],min[sup{\gamma}_{A}\left(x\right).a,1])\rangle /x:x\in X\}$.
 (v)
 Scalar division of A is defined by $A/a=\{\langle (min[inf{\mu}_{A}\left(x\right)/a,1],min[sup{\mu}_{A}\left(x\right)/a,1]),(min[inf{\sigma}_{A}\left(x\right)/a,1],min[sup{\sigma}_{A}\left(x\right)/a,1]),(min[inf{\omega}_{A}\left(x\right)/a,1],min[sup{\omega}_{A}\left(x\right)/a,1]),(min[inf{\gamma}_{A}\left(x\right)/a,1],min[sup{\gamma}_{A}\left(x\right)/a,1])\rangle /x:x\in X\}$.
${G}_{i}$  a  b 
${y}_{1}$  [0.6,0.8],[0.8,0.9][0.5,0.6][0.1,0.5]  [0.6,0.8][0.1,0.8][0.3,0.7][0.1,0.7] 
${y}_{2}$  [0.2,0.4][0.5,0.8][0.4,0.3][0.3,0.8]  [0.2,0.8][0.6,0.9][0.5,0.8][0.2,0.3] 
${y}_{3}$  [0.1,0.9][0.2,0.5][0.5,0.7][0.6,0.8]  [0.4,0.9][0.2,0.6][0.5,0.6][0.5,0.7] 
${y}_{4}$  [0.6,0.8][0.8,0.9][0.1,0.9][0.8,0.9]  [0.5,0.7][0.6,0.8][0.7,0.9][0.1,0.8] 
${y}_{5}$  [0.0,0.9][1.0,0.1][1.0,0.9][1.0,1.0]  [0.0,0.9][0.8,1.0][0.3,0.5][0.2,0.5] 
${G}_{i}$  a  b 
${y}_{1}$  [0.6,0.8],[0.8,0.9][0.5,0.6][0.1,0.5]  [0.6,0.8][0.1,0.8][0.3,0.7][0.1,0.7] 
${y}_{2}$  [0.2,0.4][0.5,0.8][0.4,0.3][0.3,0.8]  [0.2,0.8][0.6,0.9][0.5,0.8][0.2,0.3] 
${y}_{3}$  [0.1,0.9][0.2,0.5][0.5,0.7][0.6,0.8]  [0.4,0.9][0.2,0.6][0.5,0.6][0.5,0.7] 
${y}_{4}$  [0.6,0.8][0.8,0.9][0.1,0.9][0.8,0.9]  [0.5,0.7][0.6,0.8][0.7,0.9][0.1,0.8] 
${y}_{5}$  [0.0,0.9][1.0,0.1][1.0,0.9][1.0,1.0]  [0.0,0.9][0.8,1.0][0.3,0.5][0.2,0.5] 
${G}_{j}$  a  b 
${y}_{1}$  [0.7,0.8],[0.7,0.9][0.6,0.6][0.1,0.5]  [0.7,0.9][0.0,0.8][0.4,0.8][0.1,0.6] 
${y}_{2}$  [0.3,0.6][0.5,0.5][0.5,0.3][0.2,0.6]  [0.4,0.8][0.6,0.9][0.5,0.8][0.1,0.2] 
${y}_{3}$  [0.2,1.0][0.2,0.5][0.5,0.7][0.5,0.7]  [0.5,0.9][0.2,0.6][0.6,0.6][0.5,0.5] 
${y}_{4}$  [0.6,0.8][0.8,0.9][0.1,0.7][0.8,0.9]  [0.6,0.8][0.6,0.8][0.9,0.9][0.1,0.4] 
${y}_{5}$  [0.1,1.0][0.9,0.1][1.0,1.0][0.9,0.8]  [0.2,0.9][0.7,0.9][0.3,0.5][0.2,0.5] 
 (1)
 Each ${G}_{n}$ is a subset of ${G}_{X}$, where n= i,j,k;
 (2)
 Each ${G}_{n}$ is a superset of ${G}_{\varnothing}$, where n= i,j,k;
 (3)
 If ${G}_{i}$ is a subset of ${G}_{j}$ and ${G}_{j}$ is a subset of ${G}_{k}$, then, ${G}_{i}$ is a subset of ${G}_{k}$.
 (i)
 The complement of the empty interval valued neutrosophic support soft set of X is the universal interval valued neutrosophic support soft setof X.
 (ii)
 The complement of the universal interval valued neutrosophic support soft set of X is the empty interval valued neutrosophic support soft set of X.
${G}_{{i}^{c}}$  a  b 
${y}_{1}$  [0.1,0.5],[0.1,0.2][0.4,0.5][0.6,0.8]  [0.1,0.7][0.2,0.9][0.3,0.7][0.6,0.8] 
${y}_{2}$  [0.3,0.8][0.2,0.5][0.6,0.7][0.2,0.4]  [0.2,0.3][0.1,0.4][0.2,0.5][0.2,0.8] 
${y}_{3}$  [0.6,0.8][0.5,0.8][0.3,0.5][0.1,0.9]  [0.5,0.7][0.4,0.8][0.4,0.5][0.4,0.9] 
${y}_{4}$  [0.8,0.9][0.1,0.2][0.3,0.9][0.6,0.8]  [0.1,0.8][0.2,0.4][0.1,0.3][0.5,0.7] 
${y}_{5}$  [1.0,1.0]0.0,0.9][0.0,0.1][0.0,0.9]  [0.2,0.5][0.0,0.2][0.5,0.7][0.0,0.8] 
 (i)
 ${G}_{i}\cup {G}_{\varnothing}={G}_{i}$.
 (ii)
 ${G}_{i}\cup {G}_{X}={G}_{X}$.
 (iii)
 ${G}_{i}\cup {G}_{j}={G}_{j}\cup {G}_{i}$.
 (iv)
 $({G}_{i}\cup {G}_{j})\cup {G}_{k}={G}_{i}\cup ({G}_{j}\cup {G}_{k})$.
${G}_{i}\cup {G}_{j}$  a  b 
${y}_{1}$  [0.7,0.8],[0.7,0.9][0.6,0.6][0.1,0.5]  [0.7,0.9][0.0,0.8][0.4,0.8][0.1,0.6] 
${y}_{2}$  [0.3,0.6][0.5,0.5][0.5,0.3][0.2,0.6]  [0.4,0.8][0.6,0.9][0.5,0.8][0.1,0.2] 
${y}_{3}$  [0.2,1.0][0.2,0.5][0.5,0.7][0.5,0.7]  [0.5,0.9][0.2,0.6][0.6,0.6][0.5,0.5] 
${y}_{4}$  [0.6,0.8][0.8,0.9][0.1,0.7][0.8,0.9]  [0.6,0.8][0.6,0.8][0.9,0.9][0.1,0.4] 
${y}_{5}$  [0.1,1.0]0.1,0.9][1.0,1.0][0.8,0.9]  [0.2,0.9][0.7,0.9][0.3,0.5][0.2,0.5] 
 (i)
 ${G}_{i}\cap {G}_{\varnothing}={G}_{\varnothing}$.
 (ii)
 ${G}_{i}\cap {G}_{X}={G}_{i}$.
 (iii)
 ${G}_{i}\cap {G}_{j}={G}_{j}\cap {G}_{i}$.
 (iv)
 $({G}_{i}\cap {G}_{j})\cap {G}_{k}={G}_{i}\cap ({G}_{j}\cap {G}_{k})$.
${G}_{i}\cap {G}_{j}$  a  b 
${y}_{1}$  [0.6,0.8],[0.8,0.9][0.5,0.6][0.1,0.5]  [0.6,0.8][0.1,0.8][0.3,0.7][0.1,0.7] 
${y}_{2}$  [0.2,0.4][0.5,0.8][0.3,0.4][0.3,0.8]  [0.2,0.8][0.6,0.9][0.5,0.8][0.2,0.3] 
${y}_{3}$  [0.1,0.9][0.2,0.5][0.5,0.7][0.6,0.8]  [0.4,0.9][0.2,0.6][0.5,0.6][0.5,0.7] 
${y}_{4}$  [0.6,0.8][0.8,0.9][0.1,0.7][0.8,0.9]  [0.5,0.7][0.6,0.8][0.7,0.9][0.1,0.8] 
${y}_{5}$  [0.0,0.9]0.1,0.9][0.9,1.0][1.0,1.0]  [0.0,0.8][0.8,1.0][0.3,0.5][0.2,0.5] 
 (i)
 ${({G}_{i}\cup {G}_{j})}^{c}={G}_{i}^{c}\cap {G}_{j}^{c}.$
 (ii)
 ${({G}_{i}\cap {G}_{j})}^{c}={G}_{i}^{c}\cup {G}_{j}^{c}.$
 (i)
 ${G}_{i}\cup ({G}_{j}\cap {G}_{k})=({G}_{i}\cup {G}_{j})\cap ({G}_{i}\cap {G}_{k})$.
 (ii)
 ${G}_{i}\cap ({G}_{i}\cup {G}_{j})=({G}_{i}\cap {G}_{j})\cup ({G}_{i}\cap {G}_{k})$
4. DecisionMaking
Algorithm 1: 

${G}_{i}$  a  b 
${y}_{1}$  [0.4,0.7][0.8,0.8][0.4,0.8][0.3,0.5]  [0.3,0.6][0.3,0.8][0.3,0.7][0.3,0.8] 
${y}_{2}$  [0.1,0.3][0.6,0.7][0.2,0.3][0.3,0.8]  [0.2,0.7][0.7,0.9][0.3,0.6][0.3,0.4] 
${y}_{3}$  [0.2,0.6][0.4,0.5][0.1,0.5][0.7,0.8]  [0.4,0.9][0.1,0.6][0.3,0.8][0.5,0.7] 
${y}_{4}$  [0.6,0.9][0.6,0.9][0.6,0.9][0.6,0.9]  [0.5,0.9][0.6,0.8][0.2,0.8][0.1,0.7] 
${y}_{5}$  [0.0,0.9]1.0,1.0][1.0,1.0][1.0,1.0]  [0.0,0.9][0.8,1.0][0.1,0.4][0.2,0.5] 
${G}_{i}$  c  d 
${y}_{1}$  [0.5,0.7][0.8,0.9][0.4,0.8][0.2,0.5]  [0.3,0.6][0.3,0.9][0.2,0.8][0.2,0.8] 
${y}_{2}$  [0.0,0.3][0.6,0.8][0.1,0.4][0.3,0.9]  [0.1,0.8][0.8,0.9][0.2,0.9][0.3,0.5] 
${y}_{3}$  [0.1,0.7][0.4,0.5][0.2,0.8][0.8,0.9]  [0.2,0.5][0.5,07][0.3,0.6][0.6,0.8] 
${y}_{4}$  [0.2,0.4][0.7,0.9][0.6,0.8][0.6,0.9]  [0.3,0.9][0.6,0.9][0.2,0.8][0.3,0.9] 
${y}_{5}$  [0.0,0.2][1.0,1.0][1.0,1.0][1.0,1.0]  [0.0,0.1][0.9,1.0][0.2,0.2][0.2,0.9] 
 1.
 The average interval valued neutrosophic support soft set is determined as follows:$$\langle \mu ,\sigma ,\omega ,\gamma \rangle Av{g}_{{G}_{i}}=\{\langle (0.375,0.65),(0.55,0.85),(0.325,0.775),(0.25,0.6),\rangle /{y}_{1},\langle (0.125,0.575),$$$$(0.675,0.825),(0.2,0.5),(0.3,0.65)\rangle /{y}_{2},\langle (0.225,0.675),(0.35,0.575),(0.225,0.675),(0.65,0.8)\rangle $$$$/{y}_{3},\langle (0.4,0.775),(0.625,0.875),(0.4,0.825),(0.4,0.85)\rangle /{y}_{4},\langle (0.0,0.525),(0.825,1.0),$$$$(0.575,0.625),(0.6,0.85)\rangle /{y}_{5}\};$$
 2.
 $\{{G}_{i};\langle \mu ,\sigma ,\omega ,\gamma \rangle Av{g}_{{G}_{i}}\}=\{({y}_{2},b),({y}_{3},b),({y}_{4},a),({y}_{5},b)\};$
 3.
 The averagelevel support soft set, $\{{G}_{i};\langle \mu ,\sigma ,\omega ,\gamma \rangle Av{g}_{{G}_{i}}\}$ is represented in tabular form.
X a b c d ${y}_{1}$ 0 0 0 0 ${y}_{2}$ 0 1 0 0 ${y}_{3}$ 0 1 0 0 ${y}_{4}$ 1 0 0 0 ${y}_{5}$ 0 1 0 0  4.
 Compute the choice value, ${C}_{{v}_{i}}$, of ${a}_{i}$ for all ${a}_{i}\in X$ as$${C}_{{v}_{3}}={C}_{{v}_{4}}=\sum _{j=1}^{4}{a}_{3j}=\sum _{j=1}^{4}{a}_{4j}=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{C}_{{v}_{1}}=\sum _{j=1}^{4}{a}_{1j}=1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{C}_{{v}_{2}}=\sum _{j=1}^{4}{a}_{2j}=3;$$
 5.
 ${C}_{{v}_{2}}$ gives the maximum value. Therefore b is the optimum choice.Now, we conclude that there are a few ways to get rid of cancer, but surgery chemotherapy is preferred by most of the physicians with respect to the cost of treatment and extending the life of the patient with the least side effects. Moreover, side effects will be reduced or vanish completely after finished chemotherapy, and the cancer and its growth will be controlled.
5. Conclusions and Future Work
Author Contributions
Conflicts of Interest
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Mani, P.; Muthusamy, K.; Jafari, S.; Smarandache, F.; Ramalingam, U. DecisionMaking via Neutrosophic Support Soft Topological Spaces. Symmetry 2018, 10, 217. https://doi.org/10.3390/sym10060217
Mani P, Muthusamy K, Jafari S, Smarandache F, Ramalingam U. DecisionMaking via Neutrosophic Support Soft Topological Spaces. Symmetry. 2018; 10(6):217. https://doi.org/10.3390/sym10060217
Chicago/Turabian StyleMani, Parimala, Karthika Muthusamy, Saeid Jafari, Florentin Smarandache, and Udhayakumar Ramalingam. 2018. "DecisionMaking via Neutrosophic Support Soft Topological Spaces" Symmetry 10, no. 6: 217. https://doi.org/10.3390/sym10060217