Fixed points of fuzzy neutrosophic soft mapping with decision-making

Transcription

1 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 R E S E R C H Open ccess Fixed points of fuzzy neutrosophic soft mapping with decision-making Muhammad Riaz * and Masooma Raza Hashmi * Correspondence: mriaz.math@pu.edu.pk Faculty of Science, Department of Mathematics, University of the Punjab, Lahore, Pakistan bstract In this paper, we introduce some operations on a fuzzy neutrosophic soft set (fns-set) by utilizing the theories of fuzzy sets, soft sets and neutrosophic sets. We introduce fns-mappings by using a cartesian product with relations on fns-sets and establish some results on fixed points of an fns-mapping. We present an algorithm to deal with uncertainties in the multi-criteria decision making to slenderize energy crises by using an fns-average operator and a comparison table for fns-sets. MSC: 03B99; 03E99; 47H0 Keywords: fns-set; fns-cartesian product; fns-mapping;fixedpointsof fns-mapping; decision making Introduction Most of the problems in engineering, medical science, economics, environments etc. have various uncertainties. To deal with uncertainties there are different theories including the fuzzy set introduced by Zadeh [], the soft set introduced by Molodtsov [2], the fuzzy soft set (fs-sets) [3] and the fuzzy parameterized fuzzy soft set (fpfs-sets) [4, 5]. The intuitionistic fuzzy set (if-set) introduced by tanassov [6], which is an abstraction of fuzzy set. Smarandache [7] introduced the neutrosophic logic and neutrosophic set with its operations and some significant outcomes. fuzzy set[] is a significant mathematical model to characterize an assembling of objects whose boundary is obscure. soft set [2] is a mathematical tool to handle the uncertainties associated with real world data-based problems. Fuzzy parameterized fuzzy soft sets introduced in [4, 5] provide a very interesting extension of fuzzy sets and soft sets. fpfs-sets provide a suitable degree of membership to both parameters and elements of the initial universe. This is motivated by the reality that humans tend to convey their views using a simple language, which is always indeterminate, imprecise, incomplete, and inconsistent. This paper introduces the concept of fuzzy neutrosophic soft (fns), in which truth-membership, falsity-membership, and indeterminacy-membership are represented. That is why fnssets have a sound logic to represent the rationale of human choice and are a very useful technique for finding fixed points of fns-mapping and modeling uncertainties in multicriteria decision making. bbas et al. [8, 9] introduced the notion of soft contraction mapping based on the theory of soft elements of soft metric spaces. They studied fixed points of soft contraction The uthor(s) 208. This article is distributed under the terms of the Creative Commons ttribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 2 of 0 mappings and obtained among others results, a soft Banach contraction principle. kram et al. [0, ] presented certain types of soft graphs based on the soft set and some novel applications of fuzzy soft graphs and m-polar fuzzy hypergraphs. rockiarani et al. [2] presented fns-topologicalspacesandshowedsomeimportantresolutionsonit. Fenget al. studied soft sets, rough sets, fuzzy soft sets and presented an attribute analysis of information systems based on elementary soft implications. They also established an adjustable approach to fuzzy soft set-based decision making (see [3 6]). Smarandache et al. [7] suggested the idea of single valued neutrosophic set. Karaaslan [8] investigated the neutrosophic soft set (ns-set) by Maji [9] and then redefine the notion of ns-set and certain operations with some changes and corrections. He showed some applications based on ns-sets to decision-making problems. Riaz et al. [20 25] establishedsomeconcepts of soft sets together with soft algebra, soft σ -algebra, soft σ -ring, measurable soft set and measurable soft mappings. They established certain properties of soft metric spaces. They also studied a fuzzy parameterized fuzzy soft set (fpfs-set), anfpfs-topology and an fpfsmetric space and presented certain applications based on fpfs-sets and fpfs-mappings to thedecision-making problems.recently, Samet [26] defined generalized Meir-Keeler type functions and proved some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Jleli and Samet [27] introduced the concept of G-metric spaces and the fixed point existing results of contractive mappings defined on such spaces. Chen and Lin [28] obtained a soft metric version of the celebrated Meir-Keeler fixed point theorem. Wardowski [29] familiarized a concept of mapping on soft sets and determined its fixed point. vast amount of mathematical activity has been carried out to obtain fixed points of various mappings, as studied by many authors [8, 9, 26, 28 3]. In the present study, we obtain some resultsonfixedpointsof fns-mappings. We introduce fns-mappings by using the cartesian product with relations on fns-sets. We establish an outranking approach of fpfs-set to reduce energy crises. The purpose of this research is to explore some occurrences of fixed points of fns-mapping and a very solid application of fpfs-sets, which will be helpful to diminish energy crises. Definition. ([7, 7]) Let X be the universal set and a neutrosophic set N is defined by N = { δ, T N (δ), I N (δ), F N (δ), δ X},whereT, I, F : X ] 0, + [and 0 T N (δ)+i N (δ)+f N (δ) 3 +, where T N (δ) is the degree of membership, I N (δ) is the degree of indeterminacy and F N (δ) is the degree of falsity of elements of the given set. The neutrosophic set yields the value from real standard or non-standard subsets of ] 0, + [. It is difficult to utilize these values in daily life science and technology problems. We consider the fuzzy neutrosophic set to be given by [2], which takes the values of the degree of membership, the degree of indeterminacy and the degree of falsity from the subset of [0, ]. Definition.2 ([]) fuzzy set F in X is measured up by a mathematical mapping with the domain as X and membership degrees in [0, ]. The aggregation of all fuzzy sets in universal set X is symbolized by F(X).

3 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 3 of 0 Definition.3 ([2]) fuzzy neutrosophic set (fn-set) on the universal set X is defined as = { ζ, T (ζ ), I (ζ ), F (ζ ), ζ X},whereT, I, F : X [0, ] and 0 T (ζ )+I (ζ )+F (ζ ) 3. Definition.4 ([2]) Let X be the universal set and R be the set of parameters or attributes with R,thepair(F, ) is said to be a soft set over X,whereF is a mathematical function given by F : P(X). It can be written as (F, )= {( δ, F(δ) ) : δ }. Definition.5 ([2]) Let X be the universal set and R be the set of attributes. We consider the non-empty set R.Let P(X) denotes the assembling of all fuzzy neutrosophic sets of X.Theaggregation is called the fuzzy neutrosophic soft set (fns-set) over X,where is a mathematical function given by : P(X). We can write it as = {( δ, { ϕ, T (δ) (ϕ), I (δ) (ϕ), F (δ) (ϕ) : ϕ X }) : δ }. Note that if (δ)={ ϕ,0,, : ϕ X},thefns-element (δ, (δ)) does not seem to be in the fns-set. The assembling of all fns-sets over X is symbolized by fns(x R )orfns(x, R). We define some operations for fns-sets which are different from operations of the fns-sets in [2]. Definition.6 Let fns(x R ). If T (ζ ) (ρ) =0,I (ζ ) (ρ) =,F (ζ ) (ρ) = ζ R, ρ X, then is named as null fns-set and symbolized by φ. Definition.7 Let fns(x R ). If T (ζ ) (ρ) =,I (ζ ) (ρ) =0,F (ζ ) (ρ) =0 ζ R, ρ X, then is named as universal fns-set and symbolized by R. Definition.8 Let, B fns(x R ). is said to be fns-subset of B,ifT (ζ ) (ρ) T B(ζ ) (ρ), I (ζ ) (ρ) I B(ζ ) (ρ), F (ζ ) (ρ) F B(ζ ) (ρ), ζ R, ρ X. Wedenoteitby B. B is said to be an fns-superset of. Definition.9 Let fns(x R ). Then the complement of the fns-set is symbolized by c and delineated as follows: c = {( ζ, { ρ, F (ζ ) (ρ), I (ζ ) (ρ), T (ζ ) (ρ) : ρ X }) : ζ R }. Definition.0 Let, B fns(x R ). Then the fns-union of the fns-sets and B is symbolized by B and delineated as follows: B = {( ζ, { ρ, T (ζ ) (ρ) T B(ζ ) (ρ), I (ζ ) (ρ) I B(ζ ) (ρ), F (ζ ) (ρ) F B(ζ ) (ρ) : ρ X }) : ζ R }.

4 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 4 of 0 Definition. Let, B fns(x R ). Then the fns-intersection of fns-sets and B is symbolized by B and delineated as follows: B = {( ζ, { ρ, T (ζ ) (ρ) T B(ζ ) (ρ), I (ζ ) (ρ) I B(ζ ) (ρ), F (ζ ) (ρ) F B(ζ ) (ρ) : ρ X }) : ζ R }. 2 Some results on fns-mapping In this section, we introduce the idea of fns-mappings and present some important definitions and properties of fns-mappings. Definition 2. n fns-topological space ( N, τ) is called fns-hausdorff space if for distinct fns-elements ζ, ζ B of N,thereexistdisjointfns-open sets and B such that ζ and ζ B B. Proposition 2.2 Let ( N, τ) be an fns-topological space. n fns-set N is an fnsopen if and only if for every ζ B there exists an fns-set C τ such that ζ B C. Proof Let τ.thenobviouslyforevery ζ we have ζ.let N be such that for every ζ there subsists an fns-open set ζ such that ζ ζ, which means that ζ (ζ ) ζ (ζ ) (ζ )foreachζ R, (ζ )={ ζ : : ζ R} τ. ζ } ζ (ζ ) (ζ ). Therefore, = { ζ Definition 2.3 The cartesian product of two fns-sets and B is defined as an fnsset C = B where C = B and C : C fns(x, R) isdelineatedby C (ζ, ζ )= (ζ ) B (ζ ) for all (ζ, ζ ) C, where (ζ ) B (ζ )={ ρ, min{t (ζ ) (ρ), T B(ζ )(ρ)}, max{i (ζ ) (ρ), I B(ζ )(ρ)}, max{f (ζ ) (ρ), F B(ζ )(ρ)} : ρ X}. Example 2.4 Let X = {ρ, ρ 2 } and R = {ζ, ζ 2 } = = B. Definefns-sets and B as follows: = {(ζ, { ρ,0.8,0.,0.3, ρ 2,0.6,0.7,0.4 }), (ζ 2, { ρ,0.3,0.7,0.6, ρ 2,0.,0.9, 0.3 })}, B = {(ζ, { ρ,0.9,0.7,0.2, ρ 2,0.3,0.4,0.2 }), (ζ 2, { ρ,0.,0.3,0.6, ρ 2,0.7,0.3, 0.9 })}.Weusefns-sets in tabular form to make the calculations easy. ζ ζ 2 and ρ (0.8, 0., 0.3) (0.3, 0.7, 0.6) ρ 2 (0.6, 0.7, 0.4) (0., 0.9, 0.3) B ζ ζ 2 ρ (0.9, 0.7, 0.2) (0., 0.3, 0.6) ρ 2 (0.3, 0.4, 0.2) (0.7, 0.3, 0.9) Then B = C where C = B and (ζ ) B (ζ )calculatedas C (ζ, ζ ) (ζ, ζ 2 ) (ζ 2, ζ ) (ζ 2, ζ 2 ) ρ (0.8, 0.7, 0.3) (0., 0.3, 0.6) (0.3, 0.7, 0.6) (0., 0.7, 0.6) ρ 2 (0.3, 0.7, 0.4) (0.6, 0.7, 0.9) (0., 0.9, 0.3) (0., 0.9, 0.9) Definition 2.5 Let and B be fns-sets in fns(x, R). n fns-set R is said to be an fnsrelation from to B if R = D where D C = B and D : D fns(x, R)onD.

5 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 5 of 0 Example 2.6 Let and B be fns-sets in Example 2.4. ThenR = { (ζ ) B (ζ ), (ζ ) B (ζ 2 ), (ζ 2 ) B (ζ )} is a fns-relation from to B which itself is an fns-set with {(ζ, ζ ), (ζ, ζ 2 ), (ζ 2, ζ )} as a set of parameters. By R B we mean that (ζ ) B (ζ 2 ) R. In the next definition, we introduce an fns-mapping. Definition 2.7 Let and B be fns-sets in fns(x, R). n fns-relation ϒ from to B is said to be an fns-mapping from to B symbolized by ϒ : B if these properties are gratified. C :Foreveryfns-element ζ, there exists only one fnselement ζ B B such that ϒ ( ζ ) = ζ B. C 2 :Foreachemptyfns-element ζ φ, ϒ( ζ φ )isanemptyfns-element for B. Definition 2.8 Let and B be fns-sets in fns(x, R) andϒ : B be an fnsmapping. The image of C under fns-mapping ϒ is the fns-set ϒ( C )defined by ϒ( C )={ ζ ϒ( ζ C ):ζ R}. Itisobviousthatϒ( φ )= φ for every fnsmapping ϒ. Definition 2.9 Let and B be fns-sets in fns(x, R) andϒ : B be an fnsmapping. The inverse image of D B under fns-mapping ϒ is the fns-set symbolized by ϒ ( D ) and delineated as ϒ ( D )={{ ζ ϒ( ζ ):ζ R} : ϒ( ζ ) D for each ζ R}. Example 2.0 Let and B be given in Example 2.4. Defineϒ as ϒ( ζ )= ζ B for each ζ R, where ζ B is the greatest fns-element for every attribute ζ R, that is, if ζ B is an arbitrary fns-element in B then ζ B ζ B.So,ϒ( ζ )= ζ B = { ρ,0.9,0.7,0.2, ρ 2,0.3,0.4,0.2 } for all ζ and ϒ( ζ 2 )= ζ 2 ρ 2,0.7,0.3,0.9 } for all ζ 2. Moreover, ϒ( ) ={ ζ C ϒ( ζ {{ ζ ϒ( ζ C )}, { ζ 2 ϒ( ζ 2 C )}} = { ζ B, ζ 2 B } = B. B = { ρ,0.,0.3,0.6, ):ζ R} = Definition 2. Let (, τ)beanfns-topological space and B.nfns-open cover for B is an assembling of fns-open sets { α } α I τ whose fns-union carries B. Definition 2.2 n fns-topological space (, τ) isfns-compact if for every fns-open cover { α } α I of B there subsists α, α 2, α 3,...,α k I, k N such that B k n= α n. Definition 2.3 Let (, τ), ( B, τ )befns-topological spaces and ϒ : B an fnsmapping. Then ϒ be an fns-continuous function if for every B τ, ϒ ( B ) τ,that is, the inverse image of an fns-open set is an fns-open set. n fns-set C is an fnscompact in (, τ)ifthefns-topological space ( C, τ C )isfns-compact. Proposition 2.4 Let (, τ) be an fns-compact topological space and ϒ : an fns-continuous function. Then ϒ( ) is an fns-compact set in (, τ).

6 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 6 of 0 Proof Consider that ϒ( ) i i,where{ i } is an assembling of fns-open sets in. Thentakingthepre-image,wehave ϒ ( i i). s ϒ ( i )isfns-open in i so there must exists fns-open i ϒ( )suchthatϒ ( i )= i.so i ( i )impliesthat i i.since is an fns-compact set, there exist i, i 2,...,i α such that α n= i n.thus = i ( i )= α n= ϒ ( in )thisimpliesthatϒ( ) α n= ( i n ). Hence ϒ( )isfns-compact. 3 Fixed points of fns-mappings Definition 3. Let fns(x, R) beanfns-set and ϒ : an fns-mapping. n fns-element ζ is said to be a fixed point of ϒ if ϒ( ζ )= ζ. Example 3.2 If ϒ : is an identity mapping, then every fns-element of is a fixed point. Proposition 3.3 Let (, τ) be an fns-compact topological space and { α : α N} a countable assembling of fns-subsets of obeying: (a) α φ for every α N, (b) α is fns-closed for each α N, (c) α+ α for each α N. Then α N α φ. Proof Suppose on contrary that α N α = φ. We know that ( α N α ) c = α N ( α ) c.from(b),( α ) c is fns-open set for each α N.Hence α R =( φ ) c = ( α N α ) c = α N ( α ) c.s is fns-compact, there exists i, i 2,...,i k N, i < i 2 < < i k, k N such that c i c i2 c ik.hencefrom(c)wehave ik ( i i2 ik ) c =( ik ) c = R/ ik,whichisnotpossibleby(a). Example 3.4 Let (, τ)beanfns-topological space and given by = {(ζ, { ρ,0.7, 0.2, 0., ρ 2,0.5,0.3,0.4 }), (ζ 2, { ρ,0.9,0.,0.3, ρ 2,0.6,0.4,0.2 })}. Thetabularformis given by ζ ζ 2 ρ (0.7, 0.2, 0.) (0.9, 0., 0.3) ρ 2 (0.5, 0.3, 0.4) (0.6, 0.4, 0.2) Let two fns-subsets of namely and 2 be given as = {(ζ, { ρ,0.6,0.3,0.4, ρ 2,0.2,0.9,0.7 }), (ζ 2, { ρ,0.7,0.5,0.6, ρ 2,0.3,0.6,0.3 })} 2 = {(ζ, { ρ,0.7,0.4,0.5, ρ 2,0.3,0.9,0.8 }), (ζ 2, { ρ,0.8,0.6,0.7, ρ 2,0.5,0.7,0.4 })} The tabular forms of these sets are given below: ζ ζ 2 and ρ (0.6, 0.3, 0.4) (0.7, 0.5, 0.6) ρ 2 (0.2, 0.9, 0.7) (0.3, 0.6, 0.3) 2 ζ ζ 2 ρ (0.7, 0.4, 0.5) (0.8, 0.6, 0.7) ρ 2 (0.3, 0.9, 0.8) (0.5, 0.7, 0.4) These sets gratify the properties of Proposition 3.3.Moreover, 2 and φ. 2 α= α =

7 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 7 of 0 Proposition 3.5 Let (, τ) be an fns-topological space and ϒ : be an fnsmapping such that for every non-empty fns-element ζ, ϒ( ζ ) is a non-empty fnselement of. If α N ϒ α ( ) contains only one non-empty fns-element ζ, then is a unique fixed point of ϒ. ζ Proof Observe that ϒ α ( ) ϒ α ( )foreachα N. Let ζ be an fns-element of such that ζ α N ϒ α ( ). That is, ζ α N ϒ α ( ). Consequently, ϒ( ζ ) ϒ( α N ϒ α ( )) α N ϒ α+ ( ) α N ϒ α ( )= ζ.sinceϒ( ζ ) is a non-empty fns-element of,wegetϒ( ζ )= ζ. Example 3.6 Let (, τ)beanfns-topological space and define ϒ : as ϒ( ζ )= ζ for all ζ,where φ and ζ represents the largest fns-element of or equivalently ζ ζ for each fns-element ζ.then α N ϒ α ( ) carries only one non-empty fns-element ζ.thus ζ is a unique fixed point of ϒ. Proposition 3.7 Let (, τ) be an fns-hausdorff topological space. Then each fnscompact set in is fns-closed in. Proof Let B be an fns-compact set in (, τ), we here show that B is fns-closed, that is, B c is fns-open. Let ζ c B,forevery ζ B,letU α, V α τ be such that U α V α = φ and ζ U α, ζ V α where α I.Since B is fns-compact, there exist ζ, ζ,..., ζ B such that B V α V α2 V αk.denoteu = U α U α2 U αk and V = V α V α2 V αk.then ζ U τ, U V = φ, which implies that ζ U c B.Thus B is fns-closed. Theorem 3.8 Let (, τ) be an fns-hausdorff topological space and ϒ : an fnscontinuous function such that: (a) for every non-empty fns-element ζ B, ϒ( ζ ) is a non-empty fns-element of B, (b) for every fns-closed set C B if ϒ( C )= C then C contains only one non-empty fns-element of B. Then there subsists a unique non-empty fns-element ζ B such that ϒ( ζ )= ζ. Proof Suppose an assembling of fns-subsets of B of the form Z = ϒ( B ), Z 2 = ϒ(Z ), Z 3 = ϒ(Z 2 ),...,Z α = ϒ(Z α )=ϒ α ( B )forα N. ItisclearthatZ α Z α for every α N. ByProposition3.7, foreveryα N, Z α is fns-closed. Using Proposition 3.3, it is clear that an fns-set D of the form D = α N Z α is non-empty. Observe that ϒ( D )=ϒ( α N ϒ α ( B )) α N ϒ α+ ( B ) α N ϒ α ( B )= D.Nextwewillprove that D = ϒ( D ). For this purpose, consider that there subsists ζ D such that ζ is not an fns-element of ϒ( D ). Put F α = ϒ ( ζ ) Z α.letusobservethatf α φ and F α F α for each α N.ByProposition3.3,thereexistsanon-emptyfns-element ζ ϒ ( ζ ) D and thus ζ = ϒ( ζ ) ϒ( D), a contradiction. Therefore, ϒ( D )= D. Hence the result follows using Proposition napplication of the fns-set to multi-criteria decision making Definition 4. ([32]) Wehaveamatrix,whererowsrepresentthepersonnamesp, p 2, p 3,...,p n and columns represent the parameters q, q 2, q 3,...,q m.theentriesε αβ are designed by ε αβ = a + b c, where a is the number premeditated as how many times T pα (q β ) exceeds

8 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 8 of 0 Table fns-data X Over population Wastage of energy Poor infrastructure Poor distribution system Major accidents and natural calamities Wars and attacks Over consumption ς (0.7, 0., 0.3) (0.8, 0.2, 0.3) (0.9, 0.2, 0.4) (0.9, 0.2, 0.) (0.8, 0., 0.2) (0.8, 0.3, 0.3) (0.9, 0.2, 0.2) ς 2 (0.3, 0.5, 0.7) (0.4, 0.5, 0.) (0.6, 0., 0.4) (0.4, 0.3, 0.) (0.6, 0., 0.4) (0.5, 0.2, 0.) (0.7, 0., 0.) ς 3 (0.4, 0.2, 0.) (0.7, 0.3, 0.2) (0.6, 0.5, 0.) (0.6, 0., 0.2) (0.8, 0.7, 0.6) (0.6, 0.5, 0.3) (0.4, 0., 0.) ς 4 (0.6, 0.2, 0.3) (0.5, 0.4, 0.4) (0.6, 0.5, 0.5) (0.5, 0.2, 0.) (0.4, 0., 0.2) (0.5, 0., 0.2) (0.2, 0., 0.) ς 5 (0.9, 0., 0.) (0.8, 0.2, 0.2) (0.7, 0., 0.2) (0.9, 0., 0.2) (0.9, 0.2, 0.) (0.8, 0., 0.3) (0.7, 0.2, 0.2) ς 6 (0.7, 0.2, 0.3) (0.8, 0.2, 0.2) (0.9, 0., 0.2) (0.6, 0., 0.2) (0.6, 0.2, 0.3) (0.6, 0., 0.4) (0.7, 0.2, 0.5) ς 7 (0.6, 0., 0.3) (0.7, 0., 0.2) (0.6, 0., 0.) (0.6, 0.3, 0.4) (0.5, 0., 0.2) (0.6, 0.2, 0.) (0.8, 0., 0.) ς 8 (0.4, 0.3, 0.2) (0.5, 0., 0.4) (0.3, 0., 0.) (0.4, 0.3, 0.2) (0.5, 0.2, 0.4) (0.4, 0.2, 0.3) (0.5, 0.5, 0.4) or equals T pγ (q β )forp α p γ p γ X, b is the number premeditated as how many times I pα (q β ) exceeds or equals I pγ (q β )forp α p γ p γ X and c is the number premeditated as how many times F pα (q β ) exceeds or equals F pγ (q β )forp α p γ p γ X. pplication: nenergydisasterisanysubstantialblockageintheprovisionofenergy resources to an economy. Energy is the supreme significant source of national power. ny country cannot attain financial success and military strength without passable sources of energy. Let X = {ς, ς 2, ς 3,...,ς m } be the universal set where the ς j, j =,2,3,...,m represent the possible solutions to reduce energy crises and R = {ϱ, ϱ 2, ϱ 3,...,ϱ n } be the assembling of alternatives or causes which increase this problem by the criteria of the fns-set. Here ϱ i, i =,2,3,...,n is for the alternatives or causes for the criteria of the fns-set. Now we demonstrate an algorithm for most suitable choice of an object. lgorithm: Input: step : Construct the table of given data in the form of the fns-sets. Output: step 2: Calculatethe averagevaluesbyusing fns-average operator, ( n i= k = T n ij i=, I ij, n n n i= F ) ij n for each i =,2,3,...,n and j =,2,3,...,m. step 3: Compute the comparison table C T by following Definition 4.. step 4: The maximum scored value should be more preferable from set X by using max j (C j ). Example 4.2 Suppose that a country is facing energy crisis problems. The authorities want to control this problem and create possible solutions to relieve it. greeing to a survey, experts find some main reasons of energy crises. They listed some possible solutions to curb this problem. We construct here a fns-model to find the order-wise possible solutions to curb this problem. The fns-information about the causes and possible solutions is given in Table. ThesetofpossiblesolutionsisrepresentedbyX = {ς, ς 2, ς 3, ς 4, ς 5, ς 6, ς 7, ς 8 },where: ς = Buy energy efficient products; ς 2 = Lighting controls; ς 3 = Energy simulation;

9 Riaz and Hashmi Fixed Point Theory and pplications (208) 208:7 Page 9 of 0 Table 2 fns-average table X veragevalues ς (0.828, 0.85, 0.257) ς 2 (0.5, 0.257, 0.27) ς 3 (0.585, 0.342, 0.228) ς 4 (0.47, 0.228, 0.257) ς 5 (0.84, 0.42, 0.85) ς 6 (0.7, 0.57, 0.3) ς 7 (0.628, 0.42, 0.2) ς 8 (0.428, 0.242, 0.285) Table 3 fns-comparison table C T Comparison values ς 7+3 4=6 ς =4 ς =8 ς = ς =7 ς = ς 7 4+ =4 ς = ς 4 = Common stand on climate change; ς 5 = Replacing thermal power fuel; ς 6 = Stand alone power projects; ς 7 = Use solar thermal; ς 8 = Perform energy audit. The set of some basic causes is given by R = {ϱ, ϱ 2, ϱ 3, ϱ 4, ϱ 5, ϱ 6, ϱ 7 },where: ϱ =Overpopulation; ϱ 2 = Wastage of energy; ϱ 3 =Poorinfrastructure; ϱ 4 = Poor dispersion scheme; ϱ 5 = Major fortuities and instinctive disasters; ϱ 6 =Warsandattacks; ϱ 7 = Over consumption; By applying the fns-average operator on Table the average values are given in Table 2. Comparison table for the above fns-set is calculated by using Definition 4. given in Table 3. The selection possibilities can be identified in the following order: ς 3 ς 5 ς ς 2 = ς 7 ς 4 = ς 6 ς 8. It can be easily seen from the above relation that the first three maximum resulting values are given by ς 3, ς 5 and ς, which shows that we should work on the energy simulation and should buy energy efficient products with replacing thermal power fuel. 5 Conclusion Fuzzy neutrosophic soft set theory has various applications in science and engineering, especially in the areas of neural networks, operations research, artificial intelligence and decision making. On this theme, we put forward the idea of fns-mappings which is based on an fns-element of an fns-set in the fns-topological space. We introduced the innovative

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