TOPSIS method for multi-attribute group decision. making under single-valued neutrosophic environment

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1 Manuscript Click here to download Manuscript: NCAA.pbspbcg.tex Neural Computing and Applications manuscript No. (will be inserted by the editor) TOPSIS method for multi-attribute group decision making under single-valued neutrosophic environment Pranab biswas Surapati pramanik Bibhas C. Giri. Received: date / Accepted: date Abstract A single-valued neutrosophic set is a special case of neutrosophic set. It has been proposed as a generalization of crisp sets, fuzzy sets, and intuitionistic fuzzy sets in order to deal with incomplete information. In this paper, a new approach for multi-attribute group decision making problems is P. Biswas ( ) 1 Department of Mathematics, Jadavpur University, Kolkata 000, West Bengal, India. Tel.: +1 paldam0@gmail.com S. Pramanik Department of Mathematics, Nandalal Ghosh B.T College, Panpur-, West bengal, India,Tel.: sura pati@yahoo.co.in B.C. Giri Department of Mathematics, Jadavpur University, Kolkata 000, West Bengal, India. Tel.: +1 bcgiri.umath@gmail.com

2 Pranab biswas, Surapati pramanik Bibhas C. Giri. proposed by extending the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to single-valued neutrosophic environment. Ratings of alternative with respect to each criterion are considered as single-valued neutrosophic set that reflect the decision makers opinion based on the provided information. Neutrosophic set characterized by three independent degrees namely truth-membership degree (T), indeterminacy-membership degree (I), and falsity-membership degree (F) which is more capable to catch up incomplete information. Single-valued neutrosophic set based weighted averaging operator is used to aggregate the individual decision maker s opinion into one common opinion for rating the importance of criteria and alternatives. Finally, an illustrative example is provided in order to demonstrate its applicability and effectiveness of the proposed approach. Keywords Fuzzy set Intuitionistic fuzzy set Multi-attribute group decision making Neutrosophic set Single-valued neutrosophic set TOPSIS 1 Introduction Multiple attribute decision making (MADM) problems with quantitative or qualitative attribute values have broad applications in the area of operation research, management science, urban planning, natural science and military affairs, etc. The attribute values of MADM problems cannot be expressed always with crisp numbers because of ambiguity and complexity of attribute. In classical MADM methods, such as Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) developed by Hwang and Yoon [1], PROMETHEE

3 TOPSIS method for multi-attribute group decision making [], VIKOR [], ELECTRE [], the weight of each attribute and ratings of alternative are presented by crisp numbers. However, in real world, decision maker may prefer to evaluate attributes by using linguistic variables rather than exact values because of partial knowledge about the attribute and lack of information processing capabilities of the problem domain. In such situation, a preference information of alternatives provided by the decision makers may be vague, imprecise or incomplete. Fuzzy set [] introduced by Zadeh, is one of such tool that utilizes impreciseness in a mathematical form. MADM problem with imprecise information can be modeled quite well by using fuzzy set theory into the field of decision making. Chen [] extended the TOPSIS method for solving multi-criteria decision making problems in fuzzy environment. However,fuzzysetcanonlyfocusonthemembershipdegreeofvagueparametersor events. It fails to handle non-membership degree and indeterminacy degree of imprecise parameters. In 1, Atanassov [] introduced intuitionistic fuzzy set (IFS) characterized by membership and non-membership degrees simultaneously. Boran et al. [] extended the TOPSIS method for multi-criteria intuitionistic decision making problem. Pramanik and Mukhopadhyaya [] studied teacher selection in intuitionistic fuzzy environment. However in IFSs, sum of membership degree and non-membership degree of a vague parameter is less than unity. Therefore, a certain amount of incomplete information or indeterminacy arises in an intuitionistic fuzzy set. It cannot handle all types of uncertainties successfully in different real physical problems such as problems involving indeterminate information.

4 Pranab biswas, Surapati pramanik Bibhas C. Giri. Smarandache [] first introduced the concept of neutrosophic set (NS) from philosophical point of view to handle indeterminate or inconsistent information usually exist in real situation. A neutrosophic set is characterized by a truth membership degree, an indeterminacy membership degree and a falsity membership degree independently. An important feature of NS is that every element of the universe has not only a certain degree of truth (T) but also a falsity degree (F) and indeterminacy degree (I). This set is a generalization of crisp, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, intervalvalued intuitionistic fuzzy set etc. NS is difficult to apply directly in real engineering and scientific applications. In order to deal with difficulties, Wang et al. [] introduced a subclass of NS called single-valued neutrosophic set (SVNS) characterized by truth membership degree, an indeterminacy membership degree and a falsity membership degree. SVNS can be applied quite well in real scientific and engineering fields to handle the uncertainty, imprecise, incomplete, and inconsistent information. Ye [] studied multi-criteria decisionmaking problem by using the weighted correlation coefficient of SVNSs. Ye [] also developed single-valued neutrosophic cross entropy for multi-criteria decision-making problems. Biswas et al. [] proposed an entropy based grey relational analysis method for solving a multi attribute decision making problem under SVNSs. Biswas et al. [] also developed a new methodology for solving SVNSs based MADM with unknown weight information. Zhang et al. [1] studied multi-criteria decision making problems under interval neutrosophic sets information. Ye [1] further discussed multi-criteria decision

5 TOPSIS method for multi-attribute group decision making making problem by using aggregation operators for simplified neutrosophic sets. Chi and Liu [1] discussed an extended TOPSIS method for interval neutrosophic set based MADM problems. The obective of this paper is to extend the concept of TOPSIS method for multi-attribute group decision making (MAGDM) problems into MAGDM with SVNS information. The information provided by different domain experts in MAGDM problems about alternative and attribute values takes the form of single valued neutrosophic set. In a group decision making process, neutrosophic weighted averaging operator needs to be used to aggregate all the decision makers opinions into a single opinion for rating the selected alternatives. The remaining part of this paper is organized as follows: section briefly introduces some preliminaries relating to neutrosophic set and the basics of single-valued neutrosophic set. In section, basics of TOPSIS method are discussed. Section is devoted to discuss TOPSIS method for MADM under simplified neutrosophic environment. In section, an illustrative example is provided to show the effectiveness of the proposed approach. Finally, section presents the concluding remarks. Preliminaries of neutrosophic sets and single valued neutrosophic set In this section, some basic definitions of neutrosophic set defined by Smrandache [] have been provided to develop the paper.

6 Pranab biswas, Surapati pramanik Bibhas C. Giri..1 Neutrosophic Set Neutrosophic set is originated from neutrosophy, a new branch of philosophy which reflects the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra []. Definition 1 Let X be a universal space of points (obects), with a generic element of X denoted by x. A neutrosophic set N X is characterized by a truth-membership function T N (x), an indeterminacy membership function I N (x) and a falsity-membership function F N (x). T N (x), I N (x) and F N (x) are realstandardornon-standardsubsetsof[0,1 + ],sothatallthreeneutrosophic components T N (x) [0,1 + ], I N (x) [0,1 + ] and F N (x) [0,1 + ]. The set I N (x) may represent not only indeterminacy but also vagueness, uncertainty, imprecision, error, contradiction, undefined, unknown, incompleteness, redundancy etc, [1,0]. In order to catch up vague information, an indeterminacy membership degree can be split into sub-components, such as contradiction, uncertainty, unknown, etc, [1]. It should be noted that the sum of three independent membership degrees T N (x), I N (x) and F N (x) have no restriction such that [] 0 T N (x)+i N (x)+f N (x) + Definition The complement of neutrosophic set A is denoted by A c and is defined as TA c(x) =1+ T A (x), IA c (x) =1+ I A (x), and FA c(x) =1+ F A (x) for all x X

7 TOPSIS method for multi-attribute group decision making Definition A neutrosophic set A is contained in other neutrosophic set B, i.e. A B if and only if inf T A (x) inf T B (x), supt A (x) supt B (x), inf I A (x) inf I B (x), supi A (x) supi B (x), inf F A (x) inf F B (x), supf A (x) supf B (x), for all x in X.. Single-valued neutrosophic Set Single-valued neutrosophic set is a special case of neutrosophic set. It can be used in real scientific and engineering applications. In the following sections some basic definitions, operations, and properties regarding single valued neutrosophic sets [] are provided. Definition Let X be a universal space of points (obects), with a generic element of X denoted by x. A single-valued neutrosophic set (SVNS) Ñ X is characterized by a truth membership function TÑ(x), an indeterminacy membership function IÑ(x), and a falsity membership function FÑ(x) with TÑ(x), IÑ(x), FÑ(x) [0, 1] for all x X. It should be noted that for SVNS Ñ, the relation 0 TÑ(x)+IÑ(x)+FÑ(x) for x X holds good. When X is continuous a SVNS Ñ can be written as Ñ = TÑ(x),IÑ(x),FÑ(x) x, x X. x When X is discrete a SVNS Ñ can be written as Ñ = TÑ(x),IÑ(x),FÑ(x) x, x X. x

8 Pranab biswas, Surapati pramanik Bibhas C. Giri. SVNS has the following pattern: Ñ={(x TÑ(x),IÑ(x),FÑ(x) ) x X}. Thus finite SVNS can be presented by the ordered tetrads: Ñ = {(x 1 TÑ(x 1 ),IÑ(x 1 ),FÑ(x 1 ) ),...,(x M TÑ(x M ),IÑ(x M ),FÑ(x M ) )} x X. For convenience, a SVNS Ñ = {(x T Ñ (x),i Ñ (x),f Ñ (x) ) x X} is denoted by the simplified symbol Ñ = TÑ(x),IÑ(x),FÑ(x) for any x X. The following definitions are defined in [] for SVNSs à and B as follows: 1. à B if and only if TÃ(x) T B(x), IÃ(x) I B(x), FÃ(x) F B(x) for any x X.. à = B if and only if à B and B à for any x X.. à c = {(x FÃ(x),1 IÃ(x),TÃ(x) ) x X}. Wang et al. [] defined the following operations for two SVNS à and B as follows: 1. à B= max(tã(x),t B(x)),min(IÃ(x),I B(x)),min(FÃ(x),F B(x)) for any x X.. à B= min(tã(x),t B(x)),max(IÃ(x),I B(x)),max(FÃ(x),F B(x)) for any x X.. à B = TÃ(x).T B(x),IÃ(x)+I B(x) IÃ(x).I B(x),FÃ(x)+F B(x) FÃ(x).F B(x) for any x X.. Distance between two SVNSs Maumder and Samanta[] studied similarity and entropy measure by incorporating euclidean distances of neutrosophic sets.

9 TOPSIS method for multi-attribute group decision making Definition (Euclidean distance) Let à = {(x 1 TÃ(x 1 ),IÃ(x 1 ),FÃ(x 1 ),...,(x n TÃ(x n ),IÃ(x n ),FÃ(x n ) } and B= {(x 1 T B(x 1 ),I B(x 1 ),F B(x 1 ),...,(x n T B(x n ),I B(x n ),F B(x n ) } be two SVNSs for x i X, where i = 1,,...,n. Then the Euclidean distance between two SVNSs à and B can be defined as follows: D Eucl (Ã, B) n (TÃ(x 1 ) T B(x 1 )) +(IÃ(x 1 ) I B(x 1 )) = i=1 +(FÃ(x 1 ) F B(x 1 )) and the normalized Euclidean distance between two SVNSs à and B can be defined as follows: DEucl N (Ã, B) 1 n (TÃ(x 1 ) T B(x 1 )) +(IÃ(x 1 ) I B(x 1 )) = () n i=1 +(FÃ(x 1 ) F B(x 1 )) Definition (Deneutrosophication of SVNS) Deneutrosophication of SVNS Ñ can be defined as a process of mapping Ñ into a single crisp output ψ X i.e. f : Ñ ψ for x X. If Ñ is discrete set then the vector of tetrads Ñ = {(x T Ñ (x),i Ñ (x),f Ñ (x) ) x X} is reduced to a single scalar quantity ψ X by deneutrosophication. The obtained scalar quantity ψ X best represents the aggregate distribution of three membership degrees of neutrosophic element TÑ(x),IÑ(x),FÑ(x). TOPSIS TOPSIS method is used to determine the best alternative from the concepts of the compromise solution. The best compromise solution should have the (1)

10 Pranab biswas, Surapati pramanik Bibhas C. Giri. shortest Euclidean distance from the ideal solution and the farthest Euclidean distance from the negative ideal solution. The procedures of TOPSIS can be described as follows. Let A= {A i i = 1,,...,m} be the set of alternatives, C= {C = 1,,...,n}bethesetofcriteriaandD={d i i = 1,,...,m : = 1,,...,n} be the performance ratings with the criteria weight vector W= {w = 1,,...,n}. TOPSIS method is presented with these following steps: Step 1. Normalization the decision matrix The normalized value d N i is calculated as follows: For benefit criteria (larger the better), d i =(d i d )/(d+ d ), where d + = max i(x i ) and d = min i(x i ) or setting d + is the aspired or desired level and d is the worst level. For cost criteria (smaller the better), d i =(d d i)/(d d+ ). Step. Calculation of weighted normalized decision matrix In the weighted normalized decision matrix, the modified ratings are calculated as the following way: v i = w v i for i = 1,,...,m and = 1,,...,n. () where w is the weight of the -th criteria such that w 0 for = 1,,...,n and n =1 w = 1.

11 TOPSIS method for multi-attribute group decision making Step. Determination of the positive and the negative ideal solutions The positive ideal solution (PIS) and the negative ideal solution(nis) are derived as follows: and PIS = A + = { v 1 +,v+,...v+ n,} () {( ) ( ) } = maxv i J 1, minv i J = 1,,...,n NIS = A = { v1,v,...v n, } () {( ) ( ) } = minv i J 1, maxv i J = 1,,...,n where J 1 and J are the benefit and cost type criteria respectively. Step. Calculate the separation measures for each alternative from the PIS and the NIS. The separation values for the PIS can be measured by using the n-dimensional Euclidean distance which is given as: D + n ( i = vi v + ) i = 1,,...m. () =1 Similarly, separation values for the NIS is Di n ( = vi v ) i = 1,,...m. () =1

12 Pranab biswas, Surapati pramanik Bibhas C. Giri. Step. Calculation of the relative closeness coefficient to the positive ideal solution. The relative closeness coefficient for the alternative A i with respect to A + is D i C i = D + i +D i for i = 1,,...m. () Step. Ranking the alternatives According to relative closeness coefficient to the ideal alternative, larger value of C i indicates the better alternative A i. TOPSIS Method for multi-attribute decision making with single-valued neutrosophic information. Consider a multi-attribute decision-making problem with m alternatives and n attributes. Let A = (A 1, A,..., A m ) be a discrete set of alternatives, and C = (C 1, C,..., C n ) be the set of attributes. The rating provided by the decision maker describes the performance of alternative A i against attribute C.LetusalsoassumethatW ={w 1,w...,w n }betheweightvectorassigned for the attributes C 1, C,..., C n by the decision makers. The values associated with the alternatives for MADM problems can be presented in the following

13 TOPSIS method for multi-attribute group decision making decision matrix C 1 C... C n A 1 d d... d 1n A d 1 d... d n D = d i m n = () A m d 1m d m... d mn Step 1. Determination of the most important attribute Generally, there are many criteria or attributes in decision-making problems, wheresomeofthem areimportant andothersmaynot be soimportant. Soit is crucial to select the proper criteria or attributes for decision-making situation. The most important attributes may be chosen with the help of expert opinions or by some others method that are technically sound. Step. Construction of decision matrix with SVNSs It is assumed that the rating of each alternative with respect to each attribute is expressed as SVNS for MADM problem.the neutrosophic values associated with the alternatives for MADM problems can be represented in the following

14 Pranab biswas, Surapati pramanik Bibhas C. Giri. decision matrix: DÑ = d s i = T m n i,i i,f i m n () = C 1 C... C n A 1 T,I,F T,I,F... T 1n,I 1n,F 1n A T 1,I 1,F 1 T,I,F... T n,i n,f n A m T m1,i m1,f m1 T m,i m,f m... T mn,i mn,f mn () In the matrix DÑ = T i,i i,f i m n, T i,i i and F i denote the degree of truth membership value, indeterminacy membership value and falsity membership value of alternative A i with respect to attribute C satisfying the following properties under the single-valued neutrosophic environment: 1. 0 T i 1; 0 I i 1; 0 F i T i +I i +F i for i = 1,,..., n and = 1,,..., m. The ratings of each alternative over the attributes are best illustrated by the neutrosophic cube [] proposed by Dezert in 00. The vertices of neutrosophic cube are (0,0,0), (1,0,0),(1,0,1),(0,0,1),(0,1,0),(1,1,0),(1,1,1) and (0,1,1). The area of ratings in neutrosophic cube are classified in three categories namely,1. highly acceptable neutrosophic ratings,. tolerable neutrosophic rating and. unacceptable neutrosophic ratings. Definition (Highly acceptable neutrosophic ratings) The subcube(λ) of a neutrosophic cube( ) (i.e. Λ ) represents the area

15 TOPSIS method for multi-attribute group decision making of highly acceptable neutrosophic ratings (U)for decision making. Vertices of Λ are defined with these eight points (0.,0,0),(1,0,0),(1,0,0.), (0.,0,0.), (0.,0,0.),(1,0,0.),(1,0.,0.)and(0.,0.,0.).U includesalltheratingsof alternative considered with the above average truth membership degree, below average indeterminacy degree and below average falsity membership degree for multi-attribute decision making. Therefore, U has a great contribution in decision making process and can be defined as U = T i,i i,f i where 0. < T i < 1, 0 < I i < 0. and 0 < F i < 0.. for i=1,,...,m and =1,,...,n. Definition (Unacceptable neutrosophic ratings) The area Γ of unacceptable neutrosophic ratings V is defined by the ratings which are characterized by 0% membership degree, 0% indeterminacy degree and 0% falsity membership degree. Thus the set of unacceptable ratings V can be considered as the set of all ratings whose truth membership value is zero. V = T i,i i,f i where T i = 0, 0 < I i 1 and 0 < F i 1. for i=1,,...,m and =1,,...,n. Consideration of V should be avoided in decision making process. Definition (Tolerable neutrosophic ratings) Excluding the area of highly acceptable ratings and unacceptable ratings from a neutrosophic cube, tolerable neutrosophic rating area Θ (= Λ Γ)can be determined. The tolerable neutrosophic rating (Z) considered with below average truth membership degree, above average indeterminacy degree and above average falsity

16 Pranab biswas, Surapati pramanik Bibhas C. Giri. membership degree are taken in decision making process. Z can be defined by the following expression Z = T i,i i,f i where 0 < T i < 0., 0. < I i < 1 and 0. < F i < 1. for i=1,,...,m and =1,,...,n. Definition Fuzzification of SVNS Ñ = {(x T Ñ (x),i Ñ (x),f Ñ (x) ) x X} can be defined as a process of mapping Ñ into fuzzy set F = {x µ F(x) x X} i.e. f : Ñ F for x X. The representative fuzzy membership degree µ F(x) [0, 1] 1 of the vector tetrads {(x TÑ(x),IÑ(x),FÑ(x) ) x X} is defined from the concept of neutrosophic cube. It can be obtained by determining the root mean square of 1 TÑ(x),IÑ(x) and FÑ(x) for all x X. Therefore the equivalent fuzzy membership degree is as: 1 {(1 TÑ(x)) +IÑ(x) +FÑ(x) }/, for x U Z µ F(x) = 0, for x V () Step. Determination of the weights of decision makers Let us assume that the group of p decision makers having their own decision weights. Thus the importance of the decision makers in a committee may not be equal to each other. Let us assume that the importance of each decision maker is considered with linguistic variables and expressed it by neutrosophic numbers.

17 TOPSIS method for multi-attribute group decision making 1 Let E k = T k,i k,f k be a neutrosophic number for the rating of k th decision maker. Then,according to Eq.() the weight of the k th decision maker can be written as : ψ k = and p k=1 ψ k = 1 1 {(1 TÑ(x)) +IÑ(x) +FÑ(x) }/ p k=1( 1 {(1 TÑ(x)) +IÑ(x) +FÑ(x) }/ ) () Step. Construction of aggregated single-valued neutrosophic decision matrix based on decision makers assessments Let D (k) = (d (k) i ) m n be the single-valued neutrosophic decision matrix of the k th decision makerand Ψ= (ψ 1,ψ...,ψ p ) T be the weight vectorof decision maker such that each ψ k [0,1]. In the group decision making process, all the individual assessments need to be fused into a group opinion to make an aggregated neutrosophic decision matrix. This aggregated matrix can be obtained by using neutrosophic aggregation operator proposed by Ye[1] for SVNS as follows: D = (d i ) m n where, ( ) d i = SVNSWA Ψ d (1) i,d() i,...,d(p) i = ψ 1 d (1) i ψ d () i ψ (p) d (p) i p ( ) ψk p ( = 1, k=1 1 T (p) i k=1 I (p) i ) ψk p (, k=1 F (p) i ) ψk ()

18 Pranab biswas, Surapati pramanik Bibhas C. Giri. Therefore the aggregated neutrosophic decision matrix is defined as follows: D = d i m n = T i,i i,f i m n () = C 1 C... C n A 1 T,I,F T,I,F... T 1n,I 1n,F 1n A T 1,I 1,F 1 T,I,F... T n,i n,f n A m T m1,i m1,f m1 T m,i m,f m... T mn,i mn,f mn (1) where, d i = T i,i i,f i is the aggregated element of neutrosophic decision matrix D for i = 1,,...m and = 1,,...n. Step. Determination of the attribute weights In the decision-making process, decision makers may feel that all attributes are not equal importance. Thus every decision maker may have their very own opinion regarding attribute weights. To obtain the grouped opinion of the chosen attribute, all the decision makers opinions for the importance of each attribute need to be aggregated. Let w k =(w (1),w ()...,w k ) be the neutrosophic number(nn) assigned to the attribute C by the k th decision maker. Then the combined weight W=(w 1,w...,w n ) of the attribute can be

19 TOPSIS method for multi-attribute group decision making 1 determined by using neutrosophic aggregation operator [1]. ( ) w = SVNWA Ψ w (1),w (),...,w (p) = ψ 1 w (1) ψ w () ψ (p) w (p) p ( ) ψk p ( ) ψk p ( = 1,, k=1 1 T (p) where, w = T,I,F for = 1,,...n. k=1 I (p) k=1 F (p) ) ψk (1) W = [w 1,w 1,...,w n ] (1) Step. Aggregation of the weighted neutrosophic decision matrix In this section, the obtained weights of attribute and aggregated neutrosophic decision matrix need to be further fused to make the aggregated weighted neutrosophic decision matrix. Then, the aggregated weighted neutrosophic decision matrix can be defined by using the multiplication properties between two neutrosophic sets as follows: D W = D w = d w i m n = T w i,iw i,fw i m n = C 1 C... C n A 1 T w1,iw1,fw1 Tw,Iw,Fw A T w1 1,Iw1 1,Fw1 1 Tw,Iw,Fw A m T w1 m1,iw1 m1,fw1 m1 Tw m,iw m,fw m... Twn 1n,Iwn 1n,Fwn 1n... Twn n,iwn n,fwn n... Twn mn,imn,f wn wn mn (1) (0)

20 Pranab biswas, Surapati pramanik Bibhas C. Giri. Here, d w i = T w i,iw i,fw i is an element of the aggregated weighted neutrosophic decision matrix D w for i = 1,,...,m and = 1,,...,n. Step. Determination of the relative positive ideal solution (RPIS) and the relative negative ideal solution (RNIS) for SVNSs Let DÑ= d w i = T m n i,i i,f i m n be a SVNS based decision matrix, where, T i, I i and F i are the membership degree, indeterminacy degree and non-membership degree of evaluation for the attribute C with respect to the alternative A i. In practical, two types of attributes namely, benefit type attribute and cost type attribute exist in multi-attribute decision making problem. Definition Let J 1 and J be the benefit type attribute and cost type attribute respectively. Q + Ñ is the relative neutrosophic positive ideal solution (RNPIS) and Q is the relativeneutrosophic negativeideal solution (RNNIS). Ñ Then Q + can be defined as follows: Ñ Q + Ñ = [ d w 1 +,d w +,...,d w n+ ] (1) where, d w+ = T w+,i w+,f w+ for = 1,,...,n. T w + = I w+ = F w + = {( max{t w i } J 1 i ) {( min i {I w i } J 1 {( min i {F w i } J 1 ), (, ) (, ( )} min{t w i } J i )} max{i w i } J i max{f w i } J i )} () () ()

21 TOPSIS method for multi-attribute group decision making 1 and Q Ñ can be defined by Q Ñ = [ d w 1,d w,...,d w n ] () where, d w = T w,i w,f w for = 1,,...,n. T w = I w = F w = {( ) ( min{t i } J 1 i, {( ) {( max{i w i } J 1 i max{f w i } J 1 i, ), max{t w i } J i ( min i {I w i } J ( min i {F w i } J )} )} )} () () () Step. Determination of the distance measure of each alternative from the RNPIS and the RNNIS for SVNSs Similar to Eq.(), the normalized Euclidean distance measure of each alternative T w i,iw i,fw i from the RNPIS T w +,I w+,f w+ for i = 1,,...,m and = 1,,...,n can be written as follows: D i+ ( w Eu d i,dw + ) = 1 n ( w n T i (x ) T w =1 + (x ) ) ( w + I i (x ) I w+ (x ) ) + ( F w i (x ) F w similarly, the normalized Euclidean distance measure of each alternative w T i,iw i,fw i from the RNNIS T w,i w,f w can be written as: D i ( w Eu d i,dw ) 1 = n ( w n T i (x ) T w =1 + (x ) ) () (x ) ) ( w + I i (x ) I w (x ) ) + ( F w i (x ) F w (x ) ) (0)

22 Pranab biswas, Surapati pramanik Bibhas C. Giri. Step. Determination of the relative closeness co-efficient to the neutrosophic ideal solution for SVNSs The relative closeness coefficient of each alternative A i with respect to the neutrosophic positive ideal solution Q + is defined as follows: Ñ ( w d where, 0 C i 1. C i = D i Eu D i+ ( w Eu d i,dw Step. Ranking the alternatives i,dw ) ) +D i Eu ( w d i,dw ) () According to the relative closeness co-efficient values larger the values of C i reflects the better alternative A i for i = 1,,...,m. Numerical example Let us suppose that a group of four decision makers (DM 1,DM, DM, DM ) intend to select the most suitable tablet from the four initially chosen tablet (A 1,A,A,A ) by considering six attributes namely: Features C 1, Hardware C, Display C, Communication C, Affordable Price C, Customer care C. Based on the proposed approach discussed in section, the considered problem is solved by the following steps: Step 1. Determination of the weights of decision makers. The importance of four decision makers in a selection committee may not be equal to each other according their status. Their decision powers are considered

23 TOPSIS method for multi-attribute group decision making as linguistic terms expressed in Table -1. The importance of each decision Table 1 Linguistic Terms for Rating Attributes and Decision Makers Linguistic Terms SVNNs Very Good / Very Important (VG/VI) 0.0, 0., 0. Good / Important(G /I) 0.0, 0.0, 0. Fair / Medium(F/M) 0., 0., 0. Bad / Unimportant (B / UI) 0., 0., 0.0 Very Bad / Very Unimportant (VB/VUI) 0., 0.0, 0.0 maker expressed by linguistic term with its corresponding SVNN shown in Table-. The weight of decision maker is determined with the help of Eq.() as follows: 1 ( )/ ψ 1 = ( 0.0/ 0./ 0./ ) = 0. 0./ Similarly, other three weights of decision ψ = 0., ψ = 0.1 and ψ = 0. can be obtained. Thus the weight vector of the four decision maker is: Ψ = (0., 0., 0.1, 0.) () Table Importance of Decision makers expressed with SVNNs. DM-1 DM- DM- DM- LT VI I MI I W 0.0, 0., , 0.0, 0. 0., 0., , 0.0, 0.

24 Pranab biswas, Surapati pramanik Bibhas C. Giri. Table Linguistic terms for rating the candidates with SVNNs. Linguistic terms SVNNs Extremely Good/High (EG/EH) 1.00, 0.00, 0.00 Very Good/High (VG/VH) 0.0, 0., 0.0 Good/High (G/H) 0.0, 0.0, 0. Medium Good/High (MG/MH) 0., 0., 0.0 Medium/Fair (M/F) 0., 0., 0. Medium Bad/Medium Law (MB/ML) 0., 0., 0. Bad/Law (G/L) 0.0, 0., 0.0 Very Bad/Low (VB/VL) 0., 0., 0.0 Very Very Bad/low (VVB/VVL) 0.0, 0.0, 0. Step-. Construction of the aggregated neutrosophic decision matrix based on the assessments of decision makers. The linguistic term along with SVNNs is defined in Table- to rate each alternative with respect to each attribute. The assessment values of each alternative A i (i = 1,,,) with respect to each attribute C ( = 1,,,,,) provided by four decision makers are listed in Table-. Then the aggregated neutrosophic decision matrix can be obtained by fusing all the decision makers opinion with the help of aggregation operator [1] as in Table-: Step-. Determine the weights of attribute. The linguistic terms shown in Table-1 are used to evaluate each attribute. The importance of each attribute for every decision maker are rated with linguistic terms shown in Table-. Four decision makers opinions need to be aggregated

25 TOPSIS method for multi-attribute group decision making Table Assessments of alternatives and attribute weights given by four decision makers. Alternatives (A i ) Decision Makers C 1 C C C C C A 1 DM-1 VG G G G G VG DM- VG VG G G G VG DM- G VG G G VG G DM- G G G G G G A DM-1 M G M G G M DM- G MG G G MG G DM- G M G G M M DM- M G M G M M A DM-1 VG VG G G VG VG DM- G VG VG G G VG DM- VG G G MG G MG DM- VG VG G G MG G A DM-1 M VG G G VG M DM- M M G G M G DM- G G G G M VG DM- G M M G G VG Weights DM-1 VI VI I M I I DM- I VI I I M M DM- M I M M I M DM- M VI M I VI I

26 Pranab biswas, Surapati pramanik Bibhas C. Giri. Table Aggregated neutrosophic decision matrix C 1 C C A 1 0.,0.1, ,0.1, ,0.00,0.1 A 0.,0.,0. 0.,0.,0.1 0.,0.,0. A 0.0,0.0,0.0 0.,0.,0.0 0.,0.1,0. A 0.,0.,0. 0.,0.,0.1 0.,0.,0.10 C C C A 1 0.0,0.,0. 0.,0.1,0. 0.,0.1,0.01 A 0.,0.,0.0 0.,0.,0. 0.,0.,0. () A 0.,0.1, ,0.1,0. 0.,0.1,0.0 A 0.,0., ,0.0, ,0.1,0. to determine the combined weight of each attribute.the fused attribute weight vector is determined by using Eq.(1) as follows: 0., 0., 0.1, 0., 0., 0., W = 0., 0., 0.1, 0., 0., 0., () 0., 0.00, 0.10, 0.00, 0., 0. Step-. Construction of the aggregated weighted neutrosophic decision matrix After obtaining the combined weights of attribute and the ratings of alternative, the aggregated weighted neutrosophic decision matrix can be obtained by using Eq.(1) as shown in Table-.

27 TOPSIS method for multi-attribute group decision making Table Aggregated weighted neutrosophic decision matrix. C 1 C C A 1 0.,0.,0.0 0.,0.,0. 0.,0.1,0.0 A 0.,0.1,0. 0.,0.,0. 0.0,0.,0. A 0.,0.,0. 0.,0.,0. 0.,0.,0. A 0.,0.1,0. 0.,0.,0.0 0.,0.,0. C C C A 1 0.,0.1,0. 0.,0.,0.1 0.,0.1,0.0 A 0.,0.,0. 0.,0.,0.0 0.,0.,0. () A 0.,0.,0. 0.,0.1,0. 0.,0.1,0. A 0.,0.,0. 0.,0., ,0.,0. Step-. Determination of the neutrosophic relative positive ideal solution and the neutrosophic relative negative ideal solution. The NRPIS can be calculated from the aggregated weighted decision matrix on the basis of attribute types i.e. benefit type or cost type by using Eq. (1) as 0., 0., 0., 0., 0., 0., Q + = Ñ 0., 0., 0., 0., 0., 0., () 0., 0., 0., 0., 0.1, 0.0

28 Pranab biswas, Surapati pramanik Bibhas C. Giri. where, d w 1 + = T1 w+,i1 w+,f1 w+ is calculated as T1 w I1 w F1 w + = max{0.,0.,0.,0.} = 0. + = min{0.,0.1,0.,0.1} = 0. + = min{0.0,0.,0.,0.} = 0. and others. Similarly,the NRNIS can be calculated from aggregated weighted decision matrix on the basis of attribute types i.e. benefit type or cost type by using Eq. () as 0., 0.1, 0., 0., 0., 0., Q = Ñ 0.0, 0., 0., 0., 0.1, 0., () 0., 0., 0.0, 0., 0., 0. where, d w 1 = T1 w,i1 w,f1 w is calculated as T1 w I1 w F1 w = min{0.,0.,0.,0.} = 0. = max{0.,0.1,0.,0.1} = 0.1 = max{0.0,0.,0.,0.} = 0. and other components are similarly calculated. Step-. Determination of the distance measure of each alternative from the RNPIS and the RNNIS and relative closeness co-efficient Normalized Euclidean distance measures defined in Eq.() and Eq. (0) are used to determine the distances of each alternative from the RNPIS and the

29 TOPSIS method for multi-attribute group decision making Table Distance measure and relative closeness co-efficient of each alternative. Alternatives (A i ) D i+ Eucl D i Eucl C i A A A A RNNIS. With these distances relative closeness co-efficient is calculated by using Eq. (). These results are listed in Table-. Step-. Ranking the alternatives According to the values of relative closeness coefficient of each alternative shown in Table-, the ranking order of four alternatives is A A 1 A A. Thus A is the best alternative tablet to buy. Conclusions This paper is devoted to present a new TOPSIS based approach for MAGDM under simplified neutrosophic environment. In the evaluation process, the ratings of each alternative with respect to each attribute are given as linguistic variables characterized by single valued neutrosophic numbers. Neutrosophic aggregation operator is used to aggregate all the opinions of decision makers. Neutrosophic positive ideal and neutrosophic negative ideal solution are defined from aggregated weighted decision matrix. Euclidean distance measure

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