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
References
 Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef][Green Version]
 Atanassov, K.T. Intuitionstic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
 Molodtsov, D. Soft set theory—First results. Comput. Math. Appl. 1999, 37, 19–31. [Google Scholar] [CrossRef][Green Version]
 Maji, P.K. Neutrosophic soft sets. Ann. Fuzzy Math. Inf. 2013, 5, 157–168. [Google Scholar]
 Maji, P. K.; Roy, A.R.; Biswass, R. An application of soft sets in a decision making problem. Comput. Math. Appl. 2002, 44, 1077–1083. [Google Scholar] [CrossRef]
 Parimala, M.; Smarandache, F.; Jafari, S.; Udhayakumar, R. On Neutrosophic αψ Closed Sets. Information 2018, 9, 103. [Google Scholar] [CrossRef]
 Parimala, M.; Karthika, M.; Dhavaseelan, R.; Jafari, S. On neutrosophic supra precontinuous functions in neutrosophic topological spaces. In New Trends in Neutrosophic Theory and Applications; European Union: Brussels, Belgium, 2018; Volume 2, pp. 371–383. [Google Scholar]
 Smarandache, F. Neutrosophy. Neutrosophic Probability, Set, and Logic; ProQuest Information & Learning: Ann Arbor, MI, USA, 1998; 105p. [Google Scholar]
 Broumi, S.; Smarandache, F. Intuitionistic neutrosophic soft set. J. Inf. Comput. Sc. 2013, 8, 130–140. [Google Scholar]
 Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Neutrosophic Book Series, No. 5; Hexis: Staffordshire, UK, 2005. [Google Scholar]
 Thao, N.X.; Smarandache, F.; Dinh, N.V. SupportNeutrosophic Set: A New Concept in Soft Computing. Neutrosophic Sets Syst. 2017, 16, 93–98. [Google Scholar]
 Deli, I. Intervalvalued neutrosophic soft sets and its decision making. Int. J. Mach. Learn. Cyber. 2017, 8, 665–676. [Google Scholar] [CrossRef]
 Cagman, N.; Citak, F.; Enginoglu, S. FPsoft set theory and its applications. Ann. Fuzzy Math. Inform. 2011, 2, 219–226. [Google Scholar]
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
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