VECTOR SIMILARITY MEASURES FOR SIMPLIFIED NEUTROSOPHIC HESITANT FUZZY SET

Journal of Inequalities and Special Functions ISSN: 2217-4303, URL: http://ilirias.com/jiasf Volume 7 Issue 4(2016), Pages 176-194.

VECTOR SIMILARITY MEASURES FOR SIMPLIFIED NEUTROSOPHIC HESITANT FUZZY SET AND THEIR APPLICATIONS TAHIR MAHMOOD, JUN YE AND QAISAR KHAN

Abstract. In this article we present three similarity measures between simplified neutrosophic hesitant fuzzy sets, which contain the concept of single valued neutrosophic hesitant fuzzy sets and interval valued neutrosophic hesitant fuzzy sets, based on the extension of Jaccard similarity measure, Dice similarity measure and Cosine similarity in the vector space. Then based on these three defined similarity measures we present a multiple attribute decision making method to solve the simplified neutrosophic hesitant fuzzy multiple criteria decision making problem, in which the evaluated values of the alternative with respect to the criteria is represented by simplified neutrosophic hesitant fuzzy elements. Further we applied the proposed similarity measures to pattern recognition. At the end a numerical examples are discussed to show the effectiveness of the proposed similarity measures.

1. Introduction The evidence provided by people to examine issues is frequently fuzzy and crisp, g and the fuzzy set (F S) has been verified to be very important in multi-criteria g decision making. F S theory was first presented by L. A. Zadeh in 1965[49] seeing degrees of membership to progressively asses the membership of elements in a set. g After the introduction of F S, atanassov[1] presented the concept of intuitionistic gS) by considering membership and non-membership degrees of an fuzzy set (IF gS to interval valued intuitionistic element to the set. Atanassov et al. extended IF gS a lot of similarity measures ^ fuzzy set (IV IF S)[2]. After the introduction of IF are proposed by many researchers as the extension of the similarity measures for g F S, as a similarity measure is an important tool for determining similarity between gS which is the first to objects. Li and cheng[20] presented similarity measure for IF gS applied them to pattern recognition. S.K De et al. gave some application of IF g to medical daignosis[19]. Ye[36] presented cosine similarity measure for IF S and applied it to medical diagnosis and pattern recognition. Ye[37, 38] also presented 2010 Mathematics Subject Classification. 90B50, 91B06, 91B10, 62C99.

Key words and phrases. Vector similarity measure; Jaccard similarity measure; Dice similarity measure; Cosine similarity measure; hesitant fuzzy sets; interval valued hesitant fuzzy sets; Single valued neutrosophic hesitant fuzzy sets; interval neutrosophic hesitant fuzzy sets; multi-criteria decision making; pattern recognition. c 2016 Universiteti i Prishtin¨ es, Prishtin¨ e, Kosov¨ e.

Submitted May 1, 2016. Published November 17, 2016.

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VECTOR SIMILARITY MEASURES FOR SNHFS

177

^ cosine similarity measure and vector similarity measures for IV IF S and trapezoidal ] IF N and applied them to multiple criteria decision making problem. g Neutrosophic set( N S) was first presented by F. Smarandache[25, 26] which g gS, IV g ^ simplifies the concept of crisp set, F S, IF IF S, and so on. In N S its indeterminacy is computed unambiguously and its truth-membership, indeterminacyg membership and falsity-membership are signified freely. N S simplifies all the above g defined sets from philosophical point of view. Several similarity measures for N S has been defined by Broumi. S. et al. [3]. Broumi. S. et al. [6, 7, 8, 9, 10, 11] single valued neutrosophic graph, Bipolar Single Valued Neutrosophic Graphs, Isolated Single Valued Neutrosophic Graphs,Bipolar Single Valued Neutrosophic Graph Theory, Single Valued Neutrosophic Graphs: Degree, Order and Size, and An Introduction to Bipolar Single Valued Neutrosophic Graph Theory. Majumdar g g et al.[21] presented similarity and entropy for N S. N S is difficult to apply in real life ^ and engineering problem. The concept of single valued neutrosophic sets SV N Ss was introduced for the first time by F. Smarandachein 1998 [25]. Wang et al. ^ [30, 31] introduced the concept of interval neutrosophic sets ( IN Ss) and provided ^ ^ ^ ^ basic operation and properties of SV N Ss and IN Ss. The SV N Ss and IN Ss g ^ is a subclass of N S. Sahin R. et al.[24] defined subsethood for SV N Ss. Zhang ] et al [50] also defined some operation for IN S. Ye[39, 40, 41, 42] defined corre^ lation coefficient, improved correlation,entropy and similarity measure for SV NS ] and IN S and gave their application in multiple attribute decision making problems (M^ CDM ). Broumi. S. et al. [4, 5] presented correlation and cosine similarity ] measure for IN S. Ye[43] presented the concept of simplified neutrosophic sets ( ^ ^ ] SN Ss) which contained the concept of SV N S and IN S and defined some basic ^ operation for SN Ss and applied them to multi-criteria decision making problem. ^ Ye [44, 45] also defined vector similarity measures for SN Ss and improved cosine ^ similarity measures for SN Ss. Peng et al[18] find some drawbacks of the operation ]S defined by Ye and presented new operations for SN ]S. for SN g ^ Hesitant fuzzy set ( HF S) is another generalization of F S proposed by Torra ^ and Narukawa[28, 29]. The advantege of HF S is that it authorizations the membership degree of an element to a given set with a limited different values, which ^ can ascend in a group decision making problem. Recently a HF S has acknowledged more and more consideration since its presence. Xia and Xu [32, 33, 34, 35] ^ presented hesitant fuzzy information, similarity measures for HF S and correlation ^ ^ for HF S.Chen et al[13] presented some correlation coefficient for HF S and applied them to clustering analysis. Ye[46] presented vector similarity measures for ^ HF S. Chen et al.[12] further extended HFS to interval valued hesitant fuzzy set. ^ ( IV^ HF S) Farhadinia. B [15, 16] defined information measures for HF S, and ^ ^ IV HF S. and distance measures for high order HF S. ^S) was presented by Zhu[51, 52] which is the Dual hesitant fuzzy sets (DHF g gS and fuzzy multi sets are special cases of DHF ^ ^S. generalization of HF S, F S, IF ^ Recently Su. Z[27] presented similarity measure for DHF Ss and applied it to

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^S was presented by Ye[47] and applied pattern recognition.Correlation of DHF ^ ^ them to M CDM under DHF information. As mentioned above that hesitancy ^ is the most common problem in decision making for which HF S is a suitable means by allowing several possible values for an element to a set. However in ^ HF S they consider only one truth-membership function and it cannot express this problem with a few different values assigned by truth-membership hesitant degree, indeterminacy-membership degree, and falsity-membership degrees due to doubts of ^S they consider two functions that is membership decision makers.and also in DHF and non-membership functions and can not consider indeterminacy-membership function. To overawed this problem and the decision maker can get more information Ye [48] presented the concept of single valued neutrosophic hesitant fuzzy sets (SV^ N HF ) and defined some basic operations, aggregation operators and applied it to M^ CDM under SV^ N HF .environment.

Lui

P. [22] presented the concept of in^ terval neutrosophic hesitant fuzzy set IN HF and defined some basic operations, aggregation operators and applied them to multicriteria decision making problem.

Since vector similarity measures played an important role in decision making, so ^ we present vector similarity measures for SN HF and then give its applications in M^ CDM and pattern recognition.

The rest of the article is organized as follows. In section 2 we defined some basic definitions related to our work. In section 3 we defined vector similarity measures for simplified neutrosophic hesitant fuzzy sets. In section 4 we apply the proposed similarity measures to multi attribute decision making problem under simplified neutrosophic environment. in section 5 we apply the proposed similarity measures to pattern recognition. At the end we give some numerical examples to show the effectiveness of the proposed similarity measures, conclusion and references are given.

2. Preliminaries In this section we define some basic definitions about hesitant fuzzy sets, interval valued hesitant fuzzy set, single valued neutrosophic hesitant fuzzy set, interval neutrosophic hesitant fuzzy sets and the vector similarity measures, such as Jaccard similarity measure, Dice similarity measure, Cosine similarity measure. ¯ be a fixed set, a hesitant fuzzy set M ˆ on U ¯ is defined Definition 2.1. [28, 29]Let U ¯ in terms of a function fMˆ (a) that when applied to U returns a finite subsets of [0, 1].

Hesitant fuzzy set is mathematically represented as, ˆ = {ha, f ˆ (a)|a ∈ U ¯ i}, M M Where fMˆ (a) is a set of some different values in [0, 1], denoting the possible ¯ to M ˆ. membership degrees of the element a ∈ U ¯ be a fixed set, an interval valued hesitant fuzzy set D ˆ on Definition 2.2. [12]Let U ¯ ¯ U is defined in terms of a function fMˆ (a) that when applied to U returns a finite set of subintervals of [0, 1]. An interval valued hesitant fuzzy set is mathematically represented as,

VECTOR SIMILARITY MEASURES FOR SNHFS

179

ˆ = {ha, f ˆ (a)|a ∈ U ¯ i}, D D Where fDˆ (a) is a set of some different values in [0, 1], denoting the possible ¯ to D. ˆ membership degrees of the element a ∈ U ¯ be a non empty fixed set. A single valued neutrosophic Definition 2.3. [48]Let U ˆ is the structure of the form: hesitant fuzzy set N ˆ = {ha, t˘(a), ˘ı(a), f˘(a)i} N In which t˘(a), ˘ı(a) and f˘(a) are three sets of some values in the real unit interval [0, 1], representing the possible truth-membership hesitant degrees, indeterminacymembership hesitant degrees and falsity-membership hesitant degree of the element ¯ to the set N ˆ , respectively, with the condition that 0 ≤ ϕ, χ, ψ ≤ 1 and a ∈ U 0 ≤ ϕ+ +χ+ +ψ + ≤ 3, where ϕ ∈ t˘(a), χ ∈ ˘ı(a), ψ ∈ f˘(a), ϕ+ = ∪ϕ∈t˘ max{ϕ}, χ+ = ∪χ∈ei max{χ}, ψ + = ∪ψ∈fe max{ψ}. ¯ be a non empty fixed set. A interval neutrosophic hesiDefinition 2.4. [22]Let U ˆ tant fuzzy set H is structure of the form ˆ = {ha, t˘(a), ˘ı(a), f˘(a)i} H In which t˘(a), ˘ı(a) and f˘(a) are three sets of some interval values in the set of real unit interval [0, 1], representing the possible truth-membership hesitant degrees, indeterminacy-membership hesitant degrees and falsity-membership hesitant degree ¯ to the set H, ˆ respectively, with the condition that ϕ = of the element a ∈ U l u l u [ϕ , ϕ ] ∈ t˘(a), χ = [χ , χ ] ∈ ˘ı(a), ψ = [ψ l , ψ u ] ∈ f˘(a) ⊆ [0, 1] and 0 ≤ sup ϕ+ + sup χ+ + sup ψ + ≤ 3, ϕ+ = ∪ϕ∈t˘ max{ϕ}, χ+ = ∪χ∈ei max{χ}, ψ + = ∪ψ∈fe max{ψ}.

2.1. Vector similarity measures. Let A = (c1 , c2 , ..., cm ) and B = (e1 , e2 , ..., em ) be two vectors of length is m where all the coordinates are positive. The jaccard similarity measure of these two vectors [17] is given by m P ck ek A.B k=1 ˜ B) = J(A, = .

(2.1) m m m P P P ||A||22 + ||B||22 − A.B c2k + e2k − ck e k k=1

k=1

k=1

m P

ck ek is the inner product of the vectors A and B and ||A||2 = s s m m P P 2 2 c and ||B|| = e2 are the Euclidean norms of A and B. Where A.B =

k=1

k=1

k=1

Then the Dice similarity measure [14] is defined as follows: 2

˜ D(A, B) =

2A.B = P m ||A||22 + ||B||22

k=1

m P

ck ek

k=1

c2k

+

m P k=1

(2.2) e2k

and the Cosine similarity measure[23] is defined as follows:

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T. MAHMOOD, J. YE AND Q. KHAN

m P

˜ C(A, B) =

ck ek A.B = s k=1 s ||A||2 .||B||2 m m P P c2k . e2k k=1

(2.3)

k=1

The Cosine similarity measure is nothing just the cosine angle between two vectors. the above formulas are same in the sense each take the value in the unit interval [0, 1]. The jaccard and dice and cosine similarity measures are respectively undefined if ck = ek = 0, f or k = 1, 2...m and ck = 0 or ek = 0, k = 1, 2...m. Now we assume that the cosine similarity measure equal to zero when ck = 0 or ek = 0, k = 1, 2...m.

These vector similarity measure satisfy the following properties for the vectors A and B. (1) J(A, B) (2) 0 (3) J(A, B)

=

J(B, A), D(A, B) = D(B, A), and C(A, B) = C(B, A)

≤ J(A, B), D(A, B), C(A, B) ≤ 1 =

D(A, B) = C(A, B) = 1 if A = B

3. Vector Similarity Measure For Simplified Neutrosophic hesitant fuzzy Sets If the simplified neutrosophic hesitant fuzzy set contains finite set of single points, that is single valued neutrosophic hesitant fuzzy set. Then the Jaccard, Dice and Cosine similarity measures are defined as follows: e and B e be two SV \ b = {u1 , u2 , ..., um }, Definition 3.1. Let A N HF Ss on the universal set U e b e respectively denoted by A = {huj , τAe(uj ), ιAe(uj ), κAe(uj )|uj ∈ U } and B = {huj , τBe (uj ), ιBe (uj ), κBe (uj )|uj ∈ b } for all j = 1, 2, ..., m.Then we define the jaccard,Dice and cosine similarity meaU e and B e as follows: sures for SV \ N HF Ss A

lej P i=1

e B) e = 1 JbSV\ (A, N HF S m

+

lej P i=1

¯ιAσ(i) (˜ uj )¯ιBσ(i) (¯ uj ) e e

lej P

κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e i=1 i2 P i2 P lej lej h

m X j=1

τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + e e

lej h P τ¯Aσ(i) (˜ uj ) e

+

i=1 lej h P

i=1

¯ιAσ(i) (˜ uj ) e

+

i=1

h i2 κ ¯ Aσ(i) (˜ uj ) e

i2 P i2 P i2 lej h lej h τ¯Bσ(i) (˜ uj ) + ¯ιBσ(i) (˜ uj ) + κ ¯ Bσ(i) (˜ uj ) − + e e e i=1 i=1 i=1   lej lej P P (˜ uj )¯ τBσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj )¯ιBσ(i) (˜ uj )+  e e e e  i=1 τ¯Aσ(i) i=1   lej   P κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e i=1

(1)

VECTOR SIMILARITY MEASURES FOR SNHFS

lej P

  2 

i=1

lej P

 (˜ u ) (˜ u )¯ ι ¯ιAσ(i) j  j Bσ(i) e e i=1  le  Pj (˜ uj ) (˜ uj )¯ κBσ(i) + κ ¯ Aσ(i) e e

(˜ uj ) + (˜ uj )¯ τBσ(i) τ¯Aσ(i) e e

m

X b \ (A, e B) e = 1 D SV N HF S m j=1

181

i=1

i2 P i2 P i2 lej h lej h lej h P τ¯Aσ(i) (˜ u ) ¯ ι (˜ u ) κ ¯ (˜ u ) + + j j j e e e Aσ(i) Aσ(i)

i=1 lej P

+

i=1

i=1

i=1

h i2 P i2 P i2 lej h lej h τ¯Bσ(i) (˜ uj ) + ¯ιBσ(i) (˜ uj ) + κ ¯ Bσ(i) (˜ uj ) e e e i=1

i=1

(2)

e B) e = 1 b \ (A, C SV N HF S m

m X j=1

lej P

lej lej P P κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj )¯ιBσ(i) (˜ uj ) + e e e e e e i=1 i=1 i=1 s i2 P i2 P i2 lej h lej h lej h P τ¯Aσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj ) e e e i=1 i=1 si=1 i2 P i2 P i2 lej h lej h lej h P + τ¯Aσ(i) (˜ u ) + + ¯ ι (˜ u ) κ ¯ (˜ u ) j j j e e e Aσ(i) Aσ(i) i=1

i=1

i=1

(3) Here lej = max(le(¯ τAe(˜ uj ), ¯ιAe(˜ uj ), κ ¯ Ae(˜ uj )), le(¯ τBe (˜ uj ), ¯ιBe (˜ uj ), κ ¯ Be (˜ uj ))) for all u ˜j ∈ b , where le(¯ U τAe(˜ uj ), ¯ιAe(˜ uj ), κ ¯ Ae(˜ uj )) and le(¯ τBe (˜ uj ), ¯ιBe (˜ uj ), κ ¯ Be (˜ uj )), respectively represents the number of values in τ¯Ae(˜ uj ), ¯ιAe(˜ uj ), κ ¯ Ae(˜ uj ) and τ¯Be (˜ uj ), ¯ιBe (˜ uj ), κ ¯ Be (˜ uj ).

When the number of values in τ¯Ae(˜ uj ), ¯ιAe(˜ uj ), κ ¯ Ae(˜ uj ) and τ¯Be (˜ uj ), ¯ιBe (˜ uj ), κ ¯ Be (˜ uj ). are not equal that is le(¯ τAe(˜ uj ), ¯ιAe(˜ uj ), κ ¯ Ae(˜ uj )) 6= le(¯ τBe (˜ uj ), ¯ιBe (˜ uj ), κ ¯ Be (˜ uj )). Then the pessimistic add the minimum value while the optimistic add the maximum value.

This depend on the decision makers that was successfully applied for hesitant fuzzy sets by [34].

The above defined three vector similarity measures satisfy the condition defined in Ye[46]. e B) e = e A), e C b \ (A, e B) e =D b \ (B, e A), e D b \ (A, e B) e = Jb \ (B, (A1 ) JbSV\ (A, SV N HF S SV N HF S SV N HF S SV N HF S N HF S e A). e b \ (B, C SV N HF S e B), e D b \ (A, e B), e C b \ (A, e B) e ≤ 1.

(A2 ) 0 ≤ JbSV\ (A, N HF S SV N HF S SV N HF S b e e b e e b e B) e =< 1, 0, 0 > (A3 ) JSV\ (A, B) = DSV\ (A, B) = CSV\ (A, N HF S N HF S N HF S e=B e if f A Proof. (A1) obviously it is true.

(A2) obviously the property is true due to the inequality x ˜2 + y˜2 ≥ 2˜ xy˜ for eq 1 and eq 2 and the cosine value for eq 3. e=B e then τ¯ e(˜ (A3) When A ¯Be (˜ uj ), ¯ιAe(˜ uj ) = ¯ιBe (˜ uj ) and κ ¯ Ae(˜ uj ) = κ ¯ Be (˜ uj ) A uj ) = τ b , j = 1, 2, ..., m. So there are for each u ˜j ∈ U e B) e = 1, D b \ (A, e B) e = 1, and C b SV\ e e JbSV\ (A, N HF S (A, B) = 1.

N HF S SV N HF S b = In real life problem, the elements u ˜j (j = 1, 2, ..., m) in a universal set U Tb {˜ u1 , u ˜2 , ..., u ˜j } have different weights. Let $ b = ($1 , $2 , ..., $j ) be the weight vecm P tor of u ˜j (j = 1, 2, ..., m) with $j ≥ 0, j = 1, 2, ..., m, and $j = 1. Therefore we j=1

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T. MAHMOOD, J. YE AND Q. KHAN

extend the above three similarity measures to weighted vector similarity measures for SV \ N HF Ss. e and B e as follows: The weighted Jaccard similarity measure for SV \ N HF Ss A

e B) e = JbSV\ (A, N HF S

m X

lej P

$j

i=1

τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + e e lej h P

j=1

+

i=1 lej h P i=1

lej P i=1

τ¯Aσ(i) (˜ uj ) e

τ¯Bσ(i) (˜ uj ) e

lej P i=1

i2

i2

τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + e e

¯ιAσ(i) (˜ uj )¯ιBσ(i) (˜ uj ) + e e

lej P i=1

i=1

κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e

i2 P i2 lej h lej h P ¯ιAσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj ) e e

+ +

lej P

i=1 lej h P

i=1

i=1

i=1

i2 P i2 lej h ¯ιBσ(i) (˜ uj ) + κ ¯ Bσ(i) (˜ uj ) − e e

¯ιAσ(i) (˜ uj )¯ιBσ(i) (˜ uj ) + e e

lej P i=1

! κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e

(4) e and B e as follows: The Dice similarity measure for SV \ N HF Ss A

lej P

  2  b SV\ e e D N HF S (A, B) =

m X

i=1

lej P

 ¯ιAσ(i) (˜ u )¯ ι (˜ u ) j Bσ(i) j  e e i=1  le  Pj + κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e

τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + e e i=1

$j

lej h P

j=1

i=1 lej P

+

i=1

i2 P i2 P i2 lej h lej h τ¯Aσ(i) (˜ u ) + + ¯ ι (˜ u ) κ ¯ (˜ u ) j j j e e e Aσ(i) Aσ(i) i=1

h

i=1

i2 P i2 P i2 lej h lej h τ¯Bσ(i) (˜ uj ) + ¯ιBσ(i) (˜ uj ) + κ ¯ Bσ(i) (˜ uj ) e e e i=1

i=1

(5)

e and B e as follows: The Cosine similarity measure for SV \ N HF Ss A

lej lej P P τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj )¯ιBσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e e e e e i=1 i=1 i=1 b \ (A, e B) e =   C $ s j SV N HF S i2 i2 P i2 P lej h lej h lej h j=1 P  ¯ιAσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )  τ¯Aσ(i) (˜ uj ) + e e e m X

lej P

i=1

s 

lej h P i=1

i=1

τ¯Aσ(i) (˜ uj ) e

i2

i=1

 i2 P i2 lej h lej h P + ¯ιAσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )  e e i=1

i=1

(6) If the SN HF S contain a set of interval values instead of a set of single points.

Then

VECTOR SIMILARITY MEASURES FOR SNHFS

lej P i=1 lej P

L L (˜ uj ) + (˜ uj )¯ τBσ(i) τ¯Aσ(i) e e

+

i=1 lej P

m

X e B) e = 1 JbSV\ (A, N HF S 2m j=1

¯ιL (˜ uj )¯ιL (˜ uj ) + e e Aσ(i) Bσ(i)

κ ¯L (˜ uj )¯ κL (˜ uj ) + e e Aσ(i) Bσ(i)

i=1

lej P i=1 lej P i=1 lej P i=1

183

U U τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) e e

¯ιU (˜ uj )¯ιU (˜ uj )+ e e Aσ(i) Bσ(i) κ ¯U (˜ uj )¯ κU (˜ uj ) e e Aσ(i) Bσ(i)

i2 P i2 P i2 lej h lej h lej h P L U L τ¯Aσ(i) (˜ u ) τ ¯ (˜ u ) ¯ ι (˜ u ) + + + j j j e e e Aσ(i) Aσ(i)

i=1

i=1

i=1

i2 P i2 P i2 lej h lej h lej h P L U (˜ u ) ¯ιL + (˜ u ) κ ¯ + (˜ u ) κ ¯ + j j j e e e Aσ(i) Aσ(i) Aσ(i)

i=1 lej P

h

i=1 lej h P

i2

L (˜ uj ) τ¯Bσ(i) e

+

i=1 lej P

i=1

i=1

i2 P i2 h lej h U L (˜ u ) + (˜ u ) + τ¯Bσ(i) ¯ ι j j e e Bσ(i) i=1

i2 P i2 P i2 lej h lej h L U (˜ u ) + (˜ u ) + (˜ u ) − ¯ιU κ ¯ κ ¯ j j j e e e Bσ(i) Bσ(i) Bσ(i) i=1 i=1 i=1   lej lej P P U U L L (˜ uj )  (˜ uj )¯ τBσ(i) (˜ uj ) + τ¯Aσ(i) uj )¯ τBσ(i) τ¯ e (˜  e e e   i=1 Aσ(i) i=1   lej lej P P   U U L L  + (˜ uj )+  (˜ uj )¯ιBσ(i) (˜ uj ) + ¯ιAσ(i) uj )¯ιBσ(i) ¯ι e (˜ e e e   i=1 Aσ(i) i=1   lej lej   P L P U U L (˜ uj ) (˜ uj )¯ κBσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) κ ¯ Aσ(i) e e e e i=1

i=1

(7) The dice similarity measure for IN\ HF Ss



m

X b \ (A, e B) e = 1 D SV N HF S 4m j=1

lej P

L L (˜ uj ) (˜ uj )¯ τBσ(i) τ¯Aσ(i) e e

lej P

!

U U (˜ uj ) (˜ uj )¯ τBσ(i) τ¯Aσ(i) e e

+   i=1 i=1  !  le lej j P L P  ¯ιAσ(i) (˜ uj )¯ιL (˜ uj ) + ¯ιU (¯ uj )¯ιU (˜ uj ) 2 + e e e e Bσ(i) Aσ(i) Bσ(i)  i=1 i=1  !  lej lej P P  L L U U + κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) e e e e i=1

lej P

h

i=1 lej h P i=1 lej P

i=1

L τ¯Aσ(i) (˜ uj ) e

¯ιL (˜ uj ) e Aσ(i)

i2

i2

+

lej P i=1

i2 h i2 P lej h L U ¯ ι (˜ u ) + τ¯Aσ(i) (˜ u ) + j j e e Aσ(i) i=1

i2 P i2 lej h lej h P U + κ ¯L (˜ u ) + κ ¯ (˜ u ) + j j e e Aσ(i) Aσ(i) i=1

i=1

h i2 P i2 P i2 lej h lej h L U L τ¯Bσ(i) (˜ u ) + τ ¯ (˜ u ) + ¯ ι (˜ u ) + j j j e e e Bσ(i) Bσ(i)

i=1 lej P

i=1

i=1

i=1

h i2 P i2 P i2 lej h lej h L U ¯ιU (˜ u ) + κ (u ) + κ (u ) j j j e e e Bσ(i) Bσ(i) Bσ(i) i=1

i=1

(8)

          

184

T. MAHMOOD, J. YE AND Q. KHAN

The cosine similarity measure for IN HF Ss.

lej P

m

X b \ (A, e B) e = 1 C SV N HF S 2m j=1

i=1 lej P i=1

L L (˜ uj ) + (˜ uj )¯ τBσ(i) τ¯Aσ(i) e e

lej P i=1 lej P

U U (˜ uj ) + (˜ uj )¯ τBσ(i) τ¯Aσ(i) e e

lej P i=1 lej P

(˜ uj )+ (˜ uj )¯ιL ¯ιL e e Bσ(i) Aσ(i)

¯ιU (˜ uj )¯ιU (˜ uj ) + κ ¯L (˜ uj )¯ κL (˜ uj ) + κ ¯U (˜ uj )¯ κU (˜ uj ) e e e e e e Aσ(i) Bσ(i) Aσ(i) Bσ(i) Aσ(i) Bσ(i) i=1 i=1 v  u P h i2 P i2 P i2 lej h lej h u lej L U L τ¯ e (˜ uj ) + τ¯Aσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj ) +  u e e u i=1 Aσ(i)  i=1 i=1 u lej h i2 P i2 P i2  lej h lej h t P L  L U ¯ιAσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj ) + e e e i=1 i=1  v i=1 u P i2 P i2 P i2 h lej h lej h u lej L U uj ) + (˜ uj ) + (˜ uj ) +  τ¯ e (˜ τ¯Bσ(i) ¯ιL u e e Bσ(i)  u i=1 Bσ(i) i=1 i=1 u lej h i2 P i2 P i2  lej h lej h t P U  ¯ιBσ(i) (˜ uj ) + κ ¯L (˜ uj ) + κ ¯U (˜ uj ) e e e Bσ(i) Bσ(i) i=1

i=1

i=1

(9) The weighted Jaccard similarity measures for IN HF S

lej P i=1 lej P

e B) e = d (A, W J SV\ N HF S

m X j=1

i=1 lej P

$j

i=1 lej h P i=1

L L τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + e e

(˜ uj )¯ιL (˜ uj ) + ¯ιL e e Aσ(i) Bσ(i)

lej P i=1 lej P

i2

+

lej h P i=1

¯ιU (˜ uj )¯ιU (˜ uj )+ e e Aσ(i) Bσ(i)

i=1 lej P

κ ¯L (˜ uj )¯ κL (˜ uj ) + e e Aσ(i) Bσ(i)

L (˜ uj ) τ¯Aσ(i) e

U U τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj )+ e e

i=1

κ ¯U (˜ uj )¯ κU (˜ uj ) e e Aσ(i) Bσ(i)

U (˜ uj ) τ¯Aσ(i) e

i2

+

i2 lej h P (˜ u ) + ¯ιL j e Aσ(i)

i=1

i2 P i2 P i2 lej h lej h lej h P L U ¯ιL + κ ¯ + κ ¯ + (˜ u ) (˜ u ) (˜ u ) j j j e e e Aσ(i) Aσ(i) Aσ(i)

i=1 lej P

i=1

i=1

i=1

h i2 P i2 P i2 lej h lej h L U τ¯Bσ(i) (˜ uj ) + τ¯Bσ(i) (˜ uj ) + ¯ιL (˜ uj ) + e e e Bσ(i) i=1

i=1

i2 P i2 P i2 lej h lej h lej h P L U ¯ιU (˜ u ) + κ ¯ (˜ u ) + κ ¯ (˜ u ) − j j j e e e Bσ(i) Bσ(i) Bσ(i) i=1 i=1 i=1   lej lej P P L L U U τ¯ e (˜ uj )¯ τBσ(i) (˜ uj ) + τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj )+   e e e   i=1 Aσ(i) i=1   lej lej P P   L U U   ¯ιL (˜ u )¯ ι (˜ u ) + ¯ ι (˜ u )¯ ι (˜ u ) j j j j e e e e Bσ(i) Aσ(i) Bσ(i)  i=1 Aσ(i)  i=1   le le j j   P L P U L U + κ ¯ Aσ(i) (˜ u )¯ κ (˜ u ) + κ ¯ (˜ u )¯ κ (˜ u ) j j j j e e e e Bσ(i) Aσ(i) Bσ(i) i=1

i=1

(10) The weighted Dice similarity measure

VECTOR SIMILARITY MEASURES FOR SNHFS



d e B) e = W DSV\ (A, N HF S

m X

lej P

185

!

lej P

L L U U τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj ) + τ¯Aσ(i) (˜ uj )¯ τBσ(i) (˜ uj )  e e e e  i=1 i=1  !  lej lej P P  L L U U ¯ιAσ(i) (˜ uj )¯ιBσ(i) (˜ uj ) + ¯ιAσ(i) (¯ uj )¯ιBσ(i) (˜ uj ) 2 + e e e e  i=1 i=1  !  lej lej P P  L L U U κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )¯ κBσ(i) (˜ uj ) + e e e e i=1

$j

lej h P

j=1

i=1

L τ¯Aσ(i) (˜ uj ) e

          

i=1

i2

+

lej h P i=1

U τ¯Aσ(i) (˜ uj ) e

i2

+

i2 lej h P ¯ιL (˜ u ) + j e Aσ(i)

i=1

i2 P i2 P i2 lej h lej h lej h P L U ¯ιL (˜ u ) + κ ¯ (˜ u ) κ ¯ (˜ u ) + + j j j e e e Aσ(i) Aσ(i) Aσ(i)

i=1 lej P

i=1

i=1

i2 P i2 P i2 h lej h lej h L U (˜ uj ) + (˜ uj ) + (˜ u ) + τ¯Bσ(i) τ¯Bσ(i) ¯ιL j e e e Bσ(i)

i=1 lej P

i=1

i=1

i=1

i2 P i2 P i2 h lej h lej h L U (˜ u ) + (u ) + (u ) ¯ιU κ κ j j j e e e Bσ(i) Bσ(i) Bσ(i) i=1

i=1

(11) The weighted Cosine measure lej P

e B) e = b \ (A, C SV N HF S

m X j=1

i=1 lej P

$j

i=1

L L (˜ uj )¯ τBσ(i) (˜ uj ) + τ¯Aσ(i) e e

lej P i=1 lej P

U U (˜ uj )¯ τBσ(i) (˜ uj ) + τ¯Aσ(i) e e

lej P i=1 lej P

(˜ uj )¯ιL (˜ uj )+ ¯ιL e e Aσ(i) Bσ(i)

(˜ uj )¯ κU κ ¯U (˜ uj )¯ κL (˜ uj ) + κ ¯L (˜ uj )¯ιU (˜ uj ) + ¯ιU (˜ uj ) e e e e e e Aσ(i) Bσ(i) Aσ(i) Bσ(i) Aσ(i) Bσ(i) i=1 i=1  v u P i2 P i2 P i2 h lej h lej h u lej L U L (˜ u ) τ ¯ (˜ u ) ¯ ι (˜ u ) τ¯Aσ(i) + + +  u j j j e e Aσ(i) Aσ(i)  u i=1 e i=1 i=1 u lej h i2 P i2 P i2  lej h lej h t P L  ¯ιAσ(i) (˜ uj ) + (˜ uj ) + (˜ uj ) + κ ¯L κ ¯U e e e Aσ(i) Aσ(i) i=1 i=1 v i=1  u P i2 P i2 P i2 h lej h lej h u lej L U L uj ) + (˜ uj ) + (˜ uj ) +  τ¯ e (˜ τ¯Bσ(i) ¯ιBσ(i) u e e u i=1 Bσ(i)  i=1 i=1 u lej h i2 P i2 P i2  lej h lej h t P U  ¯ιBσ(i) (˜ uj ) + κ ¯L (˜ uj ) + κ ¯U (˜ uj ) e e e Bσ(i) Bσ(i) i=1

i=1

i=1

(12) 4. Multi attribute decision making With single valued neutrosophic hesitant information In this section we present a multi-criteria decision making method adapted from Ye[46] to utilize the vector similarity measures for SV \ N HF Ss and IN\ HF Ss with \ \ SV N HF and IN HF information respectively. \ For a M\ CDM problem with SV\ N HF and IN HF information respectively, let ˇ ˇ ˇ ˇ A = {A1 , A2 , ..., An } be a set of alternatives and C˘ = {C˘1 , C˘2 , ..., C˘m } be a set of attributes. If for the alternatives Aˇj (f or j = 1, 2, ..., n) the decision makers provide several values under the criteria C˘i (f or i = 1, 2, ..., m), then each value \ is considered as a SV\ N HF element and IN HF element (¯ τji , ¯ιji , κ ¯ ji )( f or j = 1, 2, ..., n, i = 1, 2, ..., m).

186

T. MAHMOOD, J. YE AND Q. KHAN

\ Therefore we can stimulate a SV\ N HF and IN HF decision matrix respectively ˜ = (¯ D τji , ¯ιji , κ ¯ ji )n×m , where (¯ τji , ¯ιji , κ ¯ ji )( f or j = 1, 2, ..., n, i = 1, 2, ..., m).is in \ the form of SV\ N HF elements or IN HF elements.

For the selection of best alternatives in multiple-criteria decision making environments,the concept of ideal point has been used in the decision set. Although in real world the ideal point does not exists. The ideal point deliver a suitable theoretical hypothesis to evaluate alternatives. Therefore we define each value \ in each ideal SV\ N HF or IN HF element (¯ τj∗ , ¯ι∗j , κ ¯ ∗j ) for the ideal alternative ∗ ˘ as τ¯∗ ι∗jσ(k) = 0, κ ¯ ∗jσ(k) = 0, or τ¯jσ(k) = ¯ ∗j )i|C˘i ∈ C} A˘∗ = {hC˘i , (¯ τj∗ , ¯ι∗j , κ jσ(k) = 1, ¯ ∗ ∗ [1, 1], ¯ιjσ(k) = [0, 0], κ ¯ jσ(k) = [0, 0], f or k = 1, 2, ..., lei , where lei is the number of values or interval values in (¯ τji , ¯ιji , κ ¯ ji )( f or j = 1, 2, ..., n, i = 1, 2, ..., m).

For the different importance of each criteria the weighting vector criteria is Pof m given as $ ˇ = ($1 , $2 , ..., $m )T , where $i ≥ 0, i = 1, 2, ..., m, and i=1 $i = 1.

Then we exploit the three weighted vector similarity measures for SV \ N HF Ss \ or IN\ HF Ss for M\ CDM under SV\ N HF or IN HF information, which can be defined as follows: Step 1. Calculate one of the three vector similarity measures between the alternative Aˇj (f or j = 1, 2, ..., n) and the ideal alternative A˘∗ by using one of the three formulas:

lej P i=1

˘∗

d \ (Aˇj , A ) = $J SV N HF

m X

τ¯Aσ(i) (˜ uj )¯ τA∗˘∗ σ(i) (˜ uj ) + e +

lej P i=1

$i

lej h P

j=1

+

i=1 lej P i=1

lej P i=1

¯ιAσ(i) (˜ uj )¯ι∗A˘∗ σ(i) (˜ uj ) e

κ ¯ Aσ(i) (˜ uj )¯ κ∗A˘∗ σ(i) (˜ uj ) e

i2 i2 P lej h ¯ ι (˜ u ) τ¯Aσ(i) (˜ u ) + j j e e Aσ(i) i=1

h

i2 P i2 lej h κ ¯ Aσ(i) (˜ uj ) + τ¯A∗˘∗ σ(i) (˜ uj ) e i=1

i2 P i2 lej h lej h P + ¯ι∗A˘∗ σ(i) (˜ uj ) + κ ¯ ∗A˘∗ σ(i) (˜ uj ) − i=1



lej P

  

i=1

τ¯Aσ(i) (˜ uj )¯ τA∗˘∗ σ(i) (˜ uj ) e +

lej P i=1

+

i=1 lej P

i=1

¯ιAσ(i) (˜ uj )¯ι∗A˘∗ σ(i) (˜ uj ) e

κ ¯ Aσ(i) (˜ uj )¯ κ∗A˘∗ σ(i) (˜ uj ) e (13)

   

VECTOR SIMILARITY MEASURES FOR SNHFS

d Aˇj , A˘∗ ) = $D(

m X



lej P

 2 

i=1

τ¯Aσ(i) (˜ uj )¯ τA∗˘∗ σ(i) (˜ uj ) e +

lej P i=1

$i

lej h P

j=1

+

i=1 lej P i=1

+

lej P i=1

187

¯ιAσ(i) (˜ uj )¯ι∗A˘∗ σ(i) (˜ uj ) e

κ ¯ Aσ(i) (˜ uj )¯ κ∗A˘∗ σ(i) (˜ uj ) e

   

i2 P i2 lej h τ¯Aσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj ) e e i=1

h

i2 P i2 lej h κ ¯ Aσ(i) (˜ uj ) + uj ) τ¯A∗˘∗ σ(i) (˜ e i=1

i2 P i2 lej h lej h P + ¯ι∗A˘∗ σ(i) (˜ uj ) + κ ¯ ∗A˘∗ σ(i) (˜ uj ) i=1

i=1

(14)

lej lej P P τ¯Aσ(i) (˜ uj )¯ τA∗˘∗ σ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj )¯ι∗A˘∗ σ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj )¯ κ∗A˘∗ σ(i) (˜ uj ) e e e i=1 i=1 i=1 ∗ d Aˇj , A˘ ) = s $C( $i i2 P i2 P i2 lej h lej h lej h P j=1 τ¯Aσ(i) (˜ uj ) + ¯ιAσ(i) (˜ uj ) + κ ¯ Aσ(i) (˜ uj ) e e e i=1 i=1 s i=1 i2 P i2 P i2 lej h lej h lej h P τ¯A∗˘∗ σ(i) (˜ uj ) + ¯ι∗A˘∗ σ(i) (˜ uj ) + κ ¯ ∗A˘∗ σ(i) (˜ uj ) m X

lej P

i=1

i=1

i=1

(15) We use the following formulas under theSV\ N HF information. \ Under IN HF information, we use the following formulas:

188

T. MAHMOOD, J. YE AND Q. KHAN

lej P i=1 lej P

L τ¯Aσ(i) (˜ uj )¯ τA∗L uj ) + ˘∗ σ(i) (˜ e

+

d \ (A, e B) e = $J IN HF S

m X

i=1 lej P

$i

j=1

i=1 lej h P i=1

∗L ¯ιL (˜ uj )¯ιA uj ) + ˘∗ σ(i) (˜ e Aσ(i)

∗L κ ¯L (˜ uj )¯ κA uj ) + ˘∗ σ(i) (˜ e Aσ(i)

L (˜ uj ) τ¯Aσ(i) e

i2

+

lej P

U τ¯Aσ(i) (˜ uj )¯ τA∗U uj ) ˘∗ σ(i) (˜ e

i=1 lej P i=1 lej P i=1

¯ιU (˜ uj )¯ι∗U uj )+ ˘∗ σ(i) (˜ e A Aσ(i) ∗U κ ¯U (˜ uj )¯ κA uj ) ˘∗ σ(i) (˜ e Aσ(i)

i2 P i2 lej h lej h P U L (˜ u ) + (˜ u ) + τ¯Aσ(i) ¯ ι j j e e Aσ(i)

i=1

i=1

i2 P i2 P i2 lej h lej h lej h P L U ¯ιL (˜ u ) κ ¯ (˜ u ) κ ¯ (˜ u ) + + + j j j e e e Aσ(i) Aσ(i) Aσ(i)

i=1

i=1

i=1

i2 P i2 i2 P lej h lej h lej h P τ¯A∗U uj ) + ¯ι∗L uj ) + uj ) + τ¯A∗L ˘∗ σ(i) (˜ ˘∗ σ(i) (˜ ˘∗ σ(i) (˜ A

i=1 lej P

i=1

i=1

h i2 P i2 P i2 lej h lej h ∗U ∗L ∗U ¯ιA (˜ u ) + κ ¯ (˜ u ) + (˜ u ) − κ ¯ j j j ∗ ∗ ˘ σ(i) ˘ σ(i) e A Bσ(i) i=1 i=1 i=1   lej lej P P ∗L ∗U L U uj )¯ τA˘∗ σ(i) (˜ uj ) + (˜ uj )¯ τA˘∗ σ(i) (˜ uj )  τ¯ e (˜ τ¯Aσ(i)  e   i=1 Aσ(i) i=1   lej lej P P   ∗L U ∗U   + ¯ιL (˜ u )¯ ι (˜ u ) + ¯ ι (˜ u )¯ ι (˜ u )+ j j j j ˘∗ σ(i) ˘∗ σ(i) e e A A Aσ(i)   i=1 Aσ(i) i=1   lej le j   P L P U ∗L ∗U κ ¯ Aσ(i) (˜ u )¯ κ (˜ u ) + κ ¯ (˜ u )¯ κ (˜ u ) j j j j ∗ ∗ ˘ ˘ e e A σ(i) A σ(i) Aσ(i) i=1

i=1

(16)

lej P



d \ (A, e B) e = $D IN HF S

m X j=1

      2      

$i

L τ¯Aσ(i) (˜ uj )¯ τA∗L uj ) ˘∗ σ(i) (˜ e

i=1 lej P

+

i=1 lej P i=1

+

lej P

 U (˜ uj )¯ τA∗U uj )+ τ¯Aσ(i) ˘∗ σ(i) (˜ e

i=1 lej P

¯ιL (˜ uj )¯ι∗L uj ) + ˘∗ σ(i) (˜ e A Aσ(i)

∗U ¯ιU (˜ uj )¯ιA uj ) ˘∗ σ(i) (˜ e Aσ(i)

i=1 lej P

∗L κ ¯L (˜ uj )¯ κA uj ) + ˘∗ σ(i) (˜ e Aσ(i)

i=1

∗U κ ¯U (˜ uj )¯ κA uj ) ˘∗ σ(i) (˜ e Aσ(i)

            

i2 i2 P i2 P lej h lej h lej h P L L U ¯ ι (˜ u ) + τ¯Aσ(i) (˜ u ) + τ ¯ (˜ u ) + j j j e e e Aσ(i) Aσ(i)

i=1

i=1

i=1

i2 P i2 P i2 lej h lej h lej h P L U ¯ιL (˜ u ) + κ ¯ (˜ u ) + κ ¯ (˜ u ) + j j j e e e Aσ(i) Aσ(i) Aσ(i)

i=1

lej h P i=1 lej P i=1

τ¯A∗L uj ) ˘∗ σ(i) (˜

i2

+

i=1 lej h P i=1

τ¯A∗U uj ) ˘∗ σ(i) (˜

i2

i=1 lej P

+

i=1

h i2 ∗L ¯ιA (˜ u ) + j ˘∗ σ(i)

h i2 P i2 P i2 lej h lej h ∗U ∗L ¯ιA uj ) + κ ¯A uj ) + κ ¯ ∗U (˜ u ) j ˘∗ σ(i) (˜ ˘∗ σ(i) (˜ e Bσ(i) i=1

i=1

(17)

VECTOR SIMILARITY MEASURES FOR SNHFS

lej P

L Ï&#x201E;¯AÏ&#x192;(i) (Ë&#x153; uj )¯ Ï&#x201E;Aâ&#x2C6;&#x2014;L uj ) + Ë&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; e

i=1 lej P

2

d \ (A, e B) e = $C IN HF S

m X j=1

+ $i

i=1 lej P

lej P

U Ï&#x201E;¯AÏ&#x192;(i) (Ë&#x153; uj )¯ Ï&#x201E;Aâ&#x2C6;&#x2014;U uj )+ Ë&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; e

i=1 lej P

¯ιL (Ë&#x153; uj )¯ιâ&#x2C6;&#x2014;L uj ) + Ë&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; e A AÏ&#x192;(i)

189

i=1 lej P

¯ιU (Ë&#x153; uj )¯ιâ&#x2C6;&#x2014;U uj ) Ë&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; e A AÏ&#x192;(i)

â&#x2C6;&#x2014;L â&#x2C6;&#x2014;U κ ¯L (Ë&#x153; uj )¯ κA uj ) + κ ¯U (Ë&#x153; uj )¯ κA uj ) Ë&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; Ë&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; e e AÏ&#x192;(i) AÏ&#x192;(i) i=1  v u lej h i2 P i2 lej h u P L U u Ï&#x201E; Ë&#x153; (Ë&#x153; u ) + Ï&#x201E; ¯ (Ë&#x153; u ) +  j j e e AÏ&#x192;(i)  u i=1 AÏ&#x192;(i) i=1 u le h i2 P i2  lej h  u Pj L  u ¯ιL (Ë&#x153; u ) + ¯ ι (Ë&#x153; u ) j j  u e e AÏ&#x192;(i)  u i=1 AÏ&#x192;(i) i=1 u i i h h le le 2  2 Pj U Pj L  t (Ë&#x153; uj ) κ ¯ AÏ&#x192;(i) (Ë&#x153; uj ) + + κ ¯ AÏ&#x192;(i) e e i=1 i=1 v  u lej h i2 P i2 lej h u P â&#x2C6;&#x2014;L â&#x2C6;&#x2014;U u  uj ) + Ï&#x201E;¯AË&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; uj ) u i=1 Ï&#x201E;¯AË&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153;  i=1 u  h i h i le le j j u  2 2 P â&#x2C6;&#x2014;U u + P ¯ιâ&#x2C6;&#x2014;L  (Ë&#x153; u ) + ¯ ι (Ë&#x153; u ) j j â&#x2C6;&#x2014; â&#x2C6;&#x2014; u  Ë&#x2DC; Ë&#x2DC; A Ï&#x192;(i) u i=1 A Ï&#x192;(i)  i=1 u h i i h lej lej 2 2  P P t  â&#x2C6;&#x2014;L â&#x2C6;&#x2014;U + κ ¯ AË&#x2DC;â&#x2C6;&#x2014; Ï&#x192;(i) (Ë&#x153; uj ) + (Ë&#x153; uj ) κ ¯ BÏ&#x192;(i) e

i=1

i=1

i=1

(18) 4.1. Practical example. This example is adopted from Ye [46] is used as the demonstration of the effectiveness of the proposed decision making method in real life problem.

For SVNH information There is an investment company, which wants to invest some money in the best option. There is a panel with four possible alternatives to invest the money: (1) A1 is a car company (2) A2 is a food company , (3) A4 is a computers company, (4) A4 is an arms company. The following three criterions on which the investment company must take decision are (1) C1 is the risk, (2) C2 is the growth, (3) C3 is the environmental impact. The weight vector of the criteria is given as (0.35, 0.25, 0.4)T the four possible alternative The decision matrix for attributes and alternative the data is represented by SV \ N HF N        Ë&#x2DC; D=      

AË&#x153;1 AË&#x153;2 AË&#x153;3 AË&#x153;4

 CË&#x153;1 CË&#x153;2 CË&#x153;3

 {0.3, 0.4}, {0.2, 0.3}, {0.4, 0.5}, {0.1, 0.3}, {0.1, 0.2}, {0.3, 0.4},   {0.5, 0.6} {0.4, 0.6} {0.6, 0.7}

  {0.9, 1}, {0.05, 0.07}, {0.8, 0.9}, {0.1, 0.2}, {0.7, 0.9}, {0.1, 0.2},   {0.01, 0.02} {0.1, 0.2} {0.01, 0.02}

  {0.5, 0.6}, {0.2, 0.3}, {0.5, 0.7}, {0.1, 0.2}, {0.5, 0.7}, {0.1, 0.2},   {0.1, 0.2} {0.2, 0.3} {0.2, 0.3}

  {0.9, 1}, {0.1, 0.2}, {0.8, 0.9}, {0.1, 0.2}, {0.4, 0.5}, {0.2, 0.4}, {0.15, 0.25} {0.1, 0.2} {0.3, 0.5}

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Table 1: The values obtained through the defined weighted vector similarity measures for SV\ N HF S W JSV N HF (A˜∗ , Aˆk ) W DSV\ (A˜∗ , Aˆk ) W CSV\ (A˜∗ , Aˆk ) N HF N HF A˜1 0.2254 0.3384 0.4099 A˜2 0.9420 0.9762 0.8752 ˜ A3 0.68125 0.81120 0.8812 A˜4 0.73040 0.8205 0.8395 Ranking order A˜2 A˜4 A˜3 A˜1 A˜2 A˜4 A˜3 A˜1 , A˜3 A˜2 A˜4 A˜1 The values obtained in table 1 shows that the Jaccard and Dice similarity meae2 is the best alternative while the Cosine simisures have the same result that is A e3 is the best alternative. From table 1 we conclude that larity measure shows that A the Jaccard and Dice similarity measure are better to use in multi-criteria decision making.

For INS information Now we consider the above example for IN HF Ss, The decision matrix for attributes and alternative the data is represented by IN\ HF N   C˜1 C˜2 C˜3          {[0.3, 0.4], [0.4, 0.5]},   {[0.4, 0.5], [0.5, 0.6]},   {[0.1, 0.2], [0.2, 0.3]},      A˜1 {[0.2, 0.3], [0.3, 0.4]}, {[0.1, 0.2], [0.3, 0.4]}, {[0.3, 0.4], [0.4, 0.5]},             {[0.5, 0.6], [0.6, 0.7]}   {[0.4, 0.5], [0.5, 0.6]}   {[0.6, 0.7], [0.7, 0.8]}    {[0.8, 0.9], [0.9, 1]}, {[0.7, 0.8], [0.9, 1]},    {[0.8, 0.9], [0.9, 1]},         A˜2 {[0.01, 0.02], [0.02, 0.03]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.05, 0.15], [0.15, 0.25]},         ˘   D=   {[0.01, 0.02], [0.02, 0.03]}   {[0.1, 0.2], [0.1, 0.2]}   {[0.01, 0.02], [0.05, 0.09]}     {[0.5, 0.6], [0.6, 0.7]},   {[0.4, 0.5], [0.5, 0.7]},   {[0.4, 0.5], [0.5, 0.7]},      A˜3 {[0.2, 0.3], [0.3, 0.4]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.1, 0.2], [0.2, 0.3]},              {[0.1, 0.2], [0.2, 0.3]}   {[0.2, 0.3], [0.3, 0.4]}   {[0.2, 0.3], [0.3, 0.4]}    {[0.8, 0.9], [0.9, 1]},    {[0.7, 0.8], [0.9, 1]},   {[0.4, 0.5], [0.5, 0.6]},      A˜4 {[0.1, 0.2], [0.2, 0.3]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.5]},       {[0.15, 0.25], [0.25, 0.35]} {[0.1, 0.2], [0.2, 0.3]} {[0.3, 0.4], [0.4, 0.5]} Table 2: The values obtained through using weighted vector similarity measures for IN\ HF N W JSV N HF () W DSV N HF () W JSV N HF () A˜1 0.2825 0.4238 0.4357 A˜2 0.9437 0.9708 0.9765 A˜3 0.6100 0.7568 0.7297 A˜4 0.7171 0.8197 0.8337 Ranking order A˜2 A˜4 A˜3 A˜1 A˜2 A˜4 A˜3 A˜1 , A˜2 A˜4 A˜3 A˜1 e2 .

The values obtained from table 2 shows that the best alternatives is A 5. Pattern recognition In this section we applied the above defined vector similarity measures to pattern recognition. The method use here is adapted from Ye[36].

Example 5.1. We are given three sample pattern A˜1 , A˜2 , and A˜3 respectively which e = {a1 , a2 , a3 }. are represented by the following SV \ N HF Ss in the finite universe U

VECTOR SIMILARITY MEASURES FOR SNHFS

191

˘ Our aim is to classify Q ˘ in one of the We are also given an unknown pattern Q. given pattern. The principle of recognizing pattern is that of the maximum degree of similarity between SV \ N HF Ss is described by n o e ek , Q) ˜ N = arg M ax1≤k≤3 C ( A SV^ N HF S A˜1

{ha1 , {0.6, 0.7}, {0.2, 0.3}, {0.3, 0.4}i, ha2 , {0.4, 0.5}, {0.3, 0.4}, {0.4, 0.5}i,

=

ha3 , {0.3, 0.4}, {0.2, 0.4}, {0.5, 0.7}i} A˜2

= {ha1 , {0.4, 0.5}, {0.3, 0.5}, {0.5, 0.6}i, ha2 , {0.6, 0.7}, {0.2, 0.3}, {0.3, 0.4}i,

A˜3

= {ha1 , {0.2, 0.3}, {0.3, 0.4}, {0.6, 0.7}i, ha2 , {0.4, 0.5}, {0.3, 0.4}, {0.4, 0.5}i,

ha3 , {0.5, 0.6}, {0.2, 0.3}, {0.2, 0.4}i} ha3 , {0.6, 0.7}, {0.2, 0.3}, {0.3, 0.4}i} ˘ = {ha1 , {0.1, 0.2}, {0.3, 0.4}, {0.7, 0.8}i, ha2 , {0.2, 0.3}, {0.3, 0.4}, {0.6, 0.7}i, Q ha3 , {0.4, 0.5}, {0.3, 0.5}, {0.5, 0.6}i} ˘ = 0.3501, (A˜2 , Q) ˘ = 0.4317, (A ˜ , Q) ˘ = 0.4723 (A˜1 , Q) 3 e3 according to the One can observe that the pattern should be classified in A pattern recognition principle. This result is the same obtained by Ye[36]. \ For INHFS \ we use the same example defined above just the patExample 5.2. For INHFS \ terns are represented by INHFSs. A˜1

=

{ha1 , {[0.4, 0.5], [0.6, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.4]}i, ha2 , {[0.3, 0.4], [0.4, 0.5]}, {[0.3, 0.4], [0.3, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}i, ha3 , {[0.2, 0.3], [0.3, 0.4]}, {[0.1, 0.3], [0.2, 0.4]}, {[0.4, 0.6], [0.5, 0.7]}i}

A˜2

= {ha1 , {[0.3, 0.4], [0.4, 0.5]}, {[0.2, 0.3], [0.4, 0.5]}, {[0.3, 0.5], [0.4, 0.6]}i, ha2 , {[0.4, 0.6], [0.5, 0.7]}, {[0.2, 0.3], [0.3, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}i, ha3 , {[0.4, 0.5], [0.6, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.1, 0.3], [0.2, 0.4]}i}

A˜3

=

{ha1 , {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.4]}, {[0.5, 0.6], [0.6, 0.7]}i, ha2 , {[0.3, 0.4], [0.4, 0.5]}, {[0.2, 0.3], [0.3, 0.4]}, {[0.3, 0.4], [0.4, 0.5]}i, ha3 , {[0.5, 0.6], [0.6, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.4]}i}

˘ = {ha1 , {[0.1, 0.3], [0.2, 0.4]}, {[0.2, 0.3], [0.3, 0.4]}, {[0.6, 0.7], [0.7, 0.8]}i, Q ha2 , {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.4], [0.4, 0.5]}, {[0.5, 0.7], [0.6, 0.8]}i, ha3 , {[0.2, 0.4], [0.4, 0.6]}, {[0.3, 0.4], [0.4, 0.5]}, {[0.4, 0.5], [0.5, 0.6]}i} ˘ = 0.1954, (A˜2 , Q) ˘ = 0.1960, (A ˜ , Q) ˘ = 0.2150 (A˜1 , Q) 3 e3 according to the One can observe that the pattern should be classified in A pattern recognition principle. This result is the same obtained by Ye[36].

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6. Discussion As simplified neutrosophic hesitant fuzzy set is an important extension of the g gS), Single valued neutrosophic set(SV ^ Fuzzy set(F S),Intuitionistic fuzzy set(IF N S), ] ^ interval neutrosophic set (IN S),Hesitant fuzzy set(HF S) and Dual hesitant fuzzy ^S), single valued neutrosophic hesitant fuzzy set (SV^ set(DHF N HF S), interval ^ neutrosophic hesitant fuzzy set (IN HF S). As mentioned above that hesitancy ^ is the most common problem in decision making for which HF S is a suitable means by allowing sevres possible values for an element to a set. However in ^ HF S they consider only one truth-membership function and it cannot express this problem with a few different values assigned by truth-membership hesitant degree, indeterminacy-membership degree, and falsity-membership degrees due to doubts of ^S they consider two functions that is membership decision makers.and also in DHF and non-membership functions and can not consider indeterminacy-membership function. Therefore simplified neutrosophic sets is the more suitable for decision making and can handle incomplete, inconsistence and indeterminate information which occur in decision making. The other advantege of the simplified neutrosophic set is that it contain the concept of single valued neutrosophic hesitant fuzzy sets and as well as interval neutrosophic hesitant fuzzy sets. As comparing to other sets the simplified neutrosophic hesitant fuzzy set contain more information and the decision makers can get more information to take their decision.

7. Conclusion \ As simplified neutrosophic hesitant fuzzy set (SN HF S) is a new extension which g gS), Single valconsists of the concept of Fuzzy set(F S),Intuitionistic fuzzy set(IF ^ ] ued neutrosophic set(SV N S), interval neutrosophic set (IN S), Hesitant fuzzy ^ ^ set(HF S) and Dual hesitant fuzzy set(DHF S), single valued neutrosophic set ^ (SV^ N HF S), interval neutrosophic hesitant fuzzy set (IN HF S). In this article we defined vector similarity measures for simplified neutrosophic hesitant fuzzy set and applied them to multi-criteria decision making and pattern recognition. We also observe that the Jaccard and Dice similarity measures are better to apply in multi-criteria decision making problem then the Cosine similarity measure. In future we shall apply the above distance measures to medical diagnosis and also defined for Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset.

Compliance with Ethical Structure We declare that we have no commercial or associative interest that represents a conflict of interest in connection with work submitted.

Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.

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Tahir Mahmood Department of Mathematics and Statistics, International Islamic University, 44000Islamabad, Pakistan E-mail address: tahirbakht@yahoo.com Jun Ye Department of Electrical and Information Engineering, Shoaxing University, China E-mail address: yehjun@aliyun.com Qaisar Khan Department of Mathematics and Statistics, International Islamic University, 44000Islamabad, Pakistan E-mail address: qaisarkhan421@gmail.com

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