Cotangent Similarity Measures of Vague Multi Sets

International Journal of Pure and Applied Mathematics Volume 120 No. 7 2018, 155-163 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/

S. Cicily Flora1 and I. Arockiarani2 1,2Department of Mathematics Nirmala College for Women, Coimbatore-18 Email: [email protected]

Abstract This paper throws light on a simpliﬁed technique to solve the selection procedure of candidates using vague multi set. The new approach is the enhanced form of cotangent similarity measures on vague multi sets which is a model of multi criteria group decision making. AMS Subject Classiﬁcation: 3B52, 90B50, 94D05. Key Words and Phrases: Vague Multi sets, Cotan- gent similarity measures, Decision making.

1 Introduction

The theory of sets, one of the most powerful tools in modern mathematics is usually considered to have begun with Georg Cantor[1845- 1918]. Considering the uncertainity factor, Zadeh [7] introduced Fuzzy sets in 1965. If repeated occurrences of any object are al- lowed in a set, then the mathematical structure is called multi sets [1]. As a generalizations of multi set,Yager [6] introduced fuzzy multi set. An element of a fuzzy multi set can occur more than once with possibly the same or diﬀerent membership values. The concept of Intuitionistic fuzzy multi set was introduced by [3]. Sim- ilarity measure is a very important tool to determine the degree of similarity between two objects. Using the Cotangent function, a new similarity measure was proposed by wang et al [5]. Later a

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new fuzzy cotangent similarity measures was introduced by Tian Maoying [4]. In this paper we have introduced some cotangent similarity measures for vague multi sets and derived some of their properties. A decision making method based on this similarity measure is con- structed.

2 Preliminaries

Deﬁnition 1. [6] Let X be a nonempty set. A Fuzzy Multiset (FMS) A drawn from X is characterized by a function, ’count membership’ of A denoted by CMA such that CMA : X Q where Q is the set of all crisp multi sets drawn from the unit→ interval [0, 1]. Then for any x X , the value CMA(x) is a crisp multiset drawn from [0, 1]. ∈

Definition 2. [2] A vague set A in the universe of discourse U is a pair [tA,fA] where tA : U [0, 1],fA : U [0, 1] , are the mappings (called truth membership→ function and→ false membership function respectively) where tA(x) is a lower bound of the grade of membership of x derived from the evidence for x and fA(x) is a lower bound on the negation of x derived from the evidence against x and t (x)+ f (x) 1 for all x U A A ≤ ∈ Definition 3. [2] The interval [t (x), 1 f (x)] is called the A − A vague value of x in A , and it is denoted by VA(x). Thatis VA(x)= [t (x), 1 f (x)] A − A Definition 4. [2] Let X be a non-empty set and the Vague set A and B in the form A = x,t (x), 1 f (x) : x X ,B = { A − A ∈ } x,t (x), 1 f (x) : x X . Then { B − B ∈ } (i) A B if and only if t (x) t (x) and 1 f (x) 1 f (x) ⊆ A ≤ B − A ≤ − B (ii) A B= max t (x),t (x) and max 1 f (x), 1 f (x) ∪ { A B } { − A − B } (iii) A B= min t (x),t (x) and min 1 f (x), 1 f (x) ∩ { A B } { − A − B } (iv) A = x,f (x), 1 t (x) : x X . { A − A ∈ } for all x X. ∈

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3 Vague Multi Sets

Definition 5. Let X be a nonempty set. A vague multi sets (VMS) A in X is characterized by two functions namely count truth membership function tC and count false membership function fC such that t : X Q and f : X Q where Q is the set of all C → C → crisp multi sets in [0, 1]. Hence, for any x X, tC (x) is the crisp multi sets from [0, 1], whose truth membership∈ sequence is defined 1 2 j as (tA(x),tA(x), ...... tA(x)) and the corresponding false membership 1 2 j sequence is defined as (1 fA(x), 1 fA(x),..., 1 fA(x)) such that i i − − − 0 tA(x)+ fA(x) 1, x X and i =1, 2, 3, . . . , j . Therefore, A ≤ ≤ ∀ ∈ 1 2 P 1 VMS is given by A = < x, (tA(x),tA(x),...,tA(x)), (1 fA(x), 1 f 2 (x), ..., 1 f P (x)){> /x X . − − A − A ∈ } Definition 6. The cardinality of the truth membership function tC (x) and false membership function 1 fC (x) is the length of a element x in a VMS A denoted as ζ, defined− as ζ = t (x) = | C | 1 fC (x) . If A, B, C are the VMS defined on X , then their cardinality| − |ζ = Max ζ(A),ζ(B),ζ(c) . { } 4 Cotangent Similarity Measure For Vague Multi Sets

j j j j Let N = x, [tN (xi), 1 fN (xi)] and P = x, [tP (xi), 1 fP (xi)] be two vague multisets.− Then we deﬁne the cotangent− similarity measures between N and P as follows

ξ 1 1 n π COT (N, P )= cot VMS ξ n 4 j=1 j=1 j j j j π t (xi) t (xi) (1 f (xi)) (1 f (xi)) + N − P ∨ − N − − P (1) 4 where the symbol is the maximum operator. This similarity measure satisfy the∨ axiomatic requirements of similarity measure.

Theorem 7. Let X be the non-empty set. Then for two vague multisets N and P over X, the cotangent similarity measure COTV MS(N, P ) between N and P satisﬁes the following properties:

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(i) 0 COT (N, P ) 1 ≤ V MS ≤

(ii) COTV MS(N, P )=1 if and only if N = P

(iii) COTV MS(N, P )= COTV MS(P,N)

(iv) If N P R for N,P,R, are VMS then COTV MS(N,R) COT⊂ (N,⊂ P ) and COT (N,R) COT (P,R). ≤ V MS V MS ≤ V MS Proof: (i) Obvious. (ii) For any two VMS N and P , then the following relations hold tj (x )= tj (x ), 1 f j (x )=1 f j (x ). Hence tj (x ) tj (x ) = N i P i − N i − P i N i − P i 0 and (1 f j (x )) (1 f j (x )) = 0, thus COT (N, P ) = − N i − − P i VMS 0. Conversely, if COT (N, P )=1, then tj (x ) tj (x ) = VMS N i − P i 0 and (1 f j (x )) (1 f j (x )) = 0, since tan(0) = 0 So we − N i − − P i can write tj (x )= tj (x ), 1 f j (x )=1 f j (x ). Hence N = P . N i P i N i P i − − j (iii) COTVMS(N, P ) = COTVMS(P,N). It is obvious that tN (xi) j j j j j − tP (xi) = tP (xi) tN (xi) and (1 fN (xi)) (1 fP (xi)) = (1 f j (x )) (1 f−j (x )). But tj (−x ) tj (x−) =−tj (x ) tj (x −) P i − − N i N i − P i P i − N i and (1 f j (x )) (1 f j (x )) = (1 f j (x )) (1 f j (x )) . N i P i P i N i Hence from− (1) COT− − (N, P ) = COT− (P,N−). − VMS VMS j j j j (iv) If N P R, then tN (xi) tP (xi) tR(xi), (1 fN (xi)) j ⊂ ⊂ j ≤ ≤ − ≤ (1 fP (xi)) (1 fR(xi)). Then we have the following inequalities tj−(x ) tj (≤x ) − tj (x ) tj (x ) , tj (x ) tj (x ) tj (x ) N i − P i ≤ N i − R i N i − P i ≤ N i tj (x ) , (1 f j (x )) (1 f j (x )) (1 f j (x )) (1 f j (x )) , − R i − N i − − P i ≤ − N i − − R i (1 f j (x )) (1 f j (x )) 1 f j (x ) 1 f j (x) . N i P i N i R i Thus− COT − (N,R− ) COT≤ −(N, P ) and− COT− (N,R) VMS ≤ VMS VMS ≤ COTVMS(P,R), since cotangent function is decreasing in the in- π π terval , . 4 2 5 Decision Making under Vague Multi sets based on Cotangent Similarity measure

STEP 1: Determine the relation between experts and attributes.

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The relation between Experts Ei(i = 1, 2,...,m) and their attributes Aj(j =1, 2,...,n) in VMS can be presented as follows: Table 1. A1 A2 ... An 1 1 1 1 1 1 [t11, 1 f11] , [t12, 1 f12] , [t1n, 1 f1n] , 2 − 2 2 − 2 2 − 2 [t11, 1 f11] , [t12, 1 f12] , [t1n, 1 f1n] , E1  −.   −.  ...  −.   .   .   .   .   .   .   tj , 1 f j   tj , 1 f j ,   tj , 1 f j ,  11 − 11 12 − 12 1n − 1n  [t1 , 1 f 1 ] ,   [t1 , 1 f 1 ] ,   [t1 , 1 f 1 ] ,   21 21   22 22   2n 2n   [t2 , 1 − f 2 ] ,   [t2 , 1 − f 2 ] ,   [t2 , 1 − f 2 ] ,   21 − 21   22 − 22   2n − 2n  E2 ......  .   .   .   j j   j j   j j  t21, 1 f21 t22, 1 f22 , t2n, 1 f2n , . .− −. . −. .  .   .  .  .        [t1 , 1 f 1 ] , [t1 , 1 f 1 ] , [t1 , 1 f 1 ] , m1 − m1 m2 − m2 mn mn [t2 , 1 f 2 ] , [t2 , 1 f 2 ] , [t2 , 1 − f 2 ] ,  m1 m1   m2 m2   mn mn  Em −. −. ... −.  .   .   .   .   .   .   tj , 1 f j   tj , 1 f j ,   [tj , 1 f j ] ,  m1 − m1 m2 − m2 mn − mn       STEP 2 : Determine  the relation between  attributes (Aj)  and candidates(Ct) The relation between attributes Aj(j =1, 2,...n) and candidates Ct(t =1, 2,...k) in terms of single valued vague sets is presented in table 2

Table 2

C1 C2 ... Ck A [t , 1 f ] [t , 1 f ] ... [t , 1 f ] 1 11 − 11 12 − 12 1k − 1k A2 [t21, 1 f21] [t22, 1 f22] ... [t2k, 1 f2k] . .− .− . .− . . . . . A [t , 1 f ] [t , 1 f ] ... [t , 1 f ] n n1 − n1 n2 − n2 nk − nk STEP 3 : Determine the relation between Experts and candidates using the equation (1), the cotangent similarity measure COTV MS(N, P ) between the table 1 and table 2 is presented in table 3.

Table 3

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C1 C2 ... Ck E1 α11 α12 ... α1k E2 α21 α22 ... α2k ...... Em αm1 αm2 ... αmk The average of each column is presented as follows: m αit ηt = m where t =1, 2,...K i=1 STEP 4: Ranking the Candidates Ranking of candidates based on the descending order avarage of each column ηt of the table 3. Highest value of correlation measure reﬂects the best candidate. STEP 5: End.

6 Example on Candidate selection

Suppose that the human resources department of a company de- sires to hire a competent. A selection board has been formed con- sisting of three experts (E1,E2,E3). After primary selection four candidates (C1,C2,C3,C4) appear before the board. ﬁve attributes (criteria) (emotional steadiness(A1); oral communication skill (A2); education experience (A3); work experience (A4); personality and self conﬁdence (A5)) obtained from experts opinions are the param- eters for selection. Selection process is divided into three phases represented by vague multi sets. The relation between experts and attributes has been presented in the table 4 Table 4: The relation between experts and criteria

Q A1 A2 A3 A4 A5 [0.6, 0.8] [0.3, 0.7] [0.6, 0.7] [0.4, 0.6] [0.1, 0.6] E1 [0.3, 0.6] [0.6, 0.9] [0.2, 0.7] [0.6, 0.8] [0.5, 0.6] [0.2, 0.7] [0.4, 0.4] [0.0, 0.8] [0.5, 0.6] [0.5, 0.8] [0.3, 0.4] [0.2, 0.8] [0.4, 0.7] [0.4, 0.6] [0.5, 0.8] E2 [0.2, 0.7] [0.1, 0.7] [0.1, 0.6] [0.1, 0.4] [0.4, 0.5] [0.4, 0.7] [0.3, 0.7] [0.5, 0.8] [0.2, 0.7] [0.1, 0.2] [0.4, 0.7] [0.2, 0.6] [0.2, 0.7] [0.4, 0.8] [0.2, 0.7] E3 [0.5, 0.8] [0.1, 0.3] [0.3, 0.9] [0.5, 0.7] [0.4, 0.6] [0.2, 0.7] [0.3, 0.5] [0.1, 0.6] [0.4, 0.6] [0.2, 0.8]

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The relation between attributes Aj(j = 1, 2,...n) and candidates (alternatives) Ct(t =1, 2,...K) in terms of single valued vague sets is presented in table 5. Table 5: The relation between criterion and candidates

C1 C2 C3 C4 A1 [0.3, 0.5] [0.4, 0.8] [0.2, 0.5] [0.4, 0.9] A2 [0.2, 0.6] [0.6, 0.8] [0.3, 0.7] [0.5, 0.8] A3 [0.4, 0.8] [0.3, 0.5] [0.2, 0.4] [0.6, 0.7] A4 [0.3, 0.7] [0.6, 0.9] [0.4, 0.7] [0.2, 0.8] A5 [0.4, 0.5] [0.2, 0.7] [0.5, 0.8] [0.6, 0.8]

Using equation 1, the cotangent similarity measure between table 4 and table 5 is presented in table 6 Table 6: The cotangent similarity measure between experts and candidates

C1 C2 C3 C4 E1 0.7013 0.6845 0.7079 0.6547 E2 0.7740 0.5988 0.7245 0.6395 E3 0.7569 0.7053 0.6947 0.5988 ηt 0.7741 0.6629 0.7090 0.631

The merit list on the basis of average has been shown in table 7 Table 7: Merit list Rank Candidates 1 C1 2 C3 3 C2 4 C4

From the merit list, it is observed that C1 is the most eligible candidates for this post.

CONCLUSION: In this paper, we have proposed a vague multi set cotangent similarity measure in a decision making problem. The concept presented in this paper can be extended to the other decision making problems involving vague multi sets.

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References

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[3] T.K. Shinoj, Sunil Jacob John, Intuitionistic fuzzy multi sets, International Journal of Engineering Science and Innovative Technology, 2(6), (2013), 1-24.

[4] Tian Maoying, A new fuzzy similarity based on cotangent Functions for medical diagnosis, Advanced Modelling and Op- timization, 15(2), 2013.

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