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/ Cotangent Similarity Measures of Vague Multi Sets 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 sim- ilarity 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 mathe- matics 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 155 International Journal of Pure and Applied Mathematics Special Issue new fuzzy cotangent similarity measures was introduced by Tian Maoying [4]. In this paper we have introduced some cotangent similarity mea- sures 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 mem- bership’ 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. ∈ 156 International Journal of Pure and Applied Mathematics Special Issue 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 func- tion 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: 157 International Journal of Pure and Applied Mathematics Special Issue (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 at- tributes. 158 International Journal of Pure and Applied Mathematics Special Issue The relation between Experts Ei(i = 1, 2,...,m) and their at- tributes 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 , .