Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—6

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Pacific Science Review A: Natural Science and Engineering

journal homepage: www.elsevier.com/locate/PSRA

Generalised multi-fuzzy soft set and its application in decision making

Asit Dey*, Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721102, India

ARTICLE INFO ABSTRACT

The aim of this paper is to introduce the concept of generalised multi-fuzzy soft set. We also define several generalised multi-fuzzy soft set operations and study their properties, and applications of generalised multi-fuzzy soft sets in decision making problem were presented.

Copyright © 2015, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article history: Available online xxx

Keywords:

Soft sets

Fuzzy soft sets

Multi-fuzzy sets

Multi-fuzzy soft sets

Generalised multi-fuzzy soft sets

Decision making

1. Introduction

Algebraic structures play a prominent role in mathematics with applications in such disciplines as theoretical physics, computer science, coding theory, and topological spaces. Researchers should continue to review abstract algebraic concepts and results in the broader framework of the fuzzy setting. One structure that's mathematical applications are most extensively discussed is lattice theory.

In real life, problems in economics, engineering, social sciences, and medical science do not always involve crisp data. We frequently cannot use traditional mathematical methods because of various uncertainties presented in these applied problems. To overcome these uncertainties, theories like fuzzy set, intuitionistic fuzzy set, rough set, and bipolar fuzzy set have been produced. However, each theory possesses inherent difficulties. Molodtsov initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties, which is free from the above limitations.

Maji et al. [14] offered the first practical application of soft sets in decision making problems. These authors introduced and studied the more generalised concept of the fuzzy soft set, which is a combination of fuzzy and softs. Majumdar and Samanta [15] generalised the fuzzy soft set concept introduced by Maji et al. [13].

A new type of fuzzy set (multi-fuzzy set) was introduced by Sebastian and Ramakrishnan [19]. This set uses ordered sequences of membership function and provides a new method to represent problems not captured in other extensions of fuzzy set theory, such as pixel colour.

The notion of multi-fuzzy complex numbers and sets were first introduced by Dey and Pal [7]. These authors introduced multi-fuzzy complex nilpotent matrices over a distributive lattice. Recently, Yong et al. [23] proposed the concept of the multi-fuzzy soft set for its application to decision making, which is a more general fuzzy soft set.

The purpose of our paper is to generalise the concept of the multi-fuzzy soft set and overcome existing uncertainties related to this problem. The generalised multi-fuzzy soft set is more feasible because it accounts for uncertainty in the selection of a fuzzy soft set corresponding to each parameter value. Relations on generalised multi-fuzzy soft sets are defined and their properties are studied. A decision making problem is then solved to identify its practical applications. To facilitate our discussion, we first review some background on soft set, fuzzy soft set, multi-fuzzy set and multi-fuzzy soft set in Section 2. In Section 3, the concept of generalised multi-fuzzy soft set is introduced, and certain of its structural properties are studied. In Section 4, the multi-fuzzy soft set is used to analyse a decision making problem, and an algorithm is proposed. Final conclusions are presented in Section 5.

* Corresponding author. E-mail addresses: asitvu@gmail.com (A. Dey), mmpalvu@gmail.com (M. Pal). Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University

2. Preliminaries

Throughout this paper, U refers to an initial universal set; E is a set of parameters; andP(U) is the power set of U and A4E.

http://dx.doi.org/10.1016/j.psra.2015.12.006

2405-8823/Copyright © 2015, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

A. Dey, M. Pal / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—6

Definition 1 [16](Soft sets) A pair (F,A) is called a soft set over U, where F is a mapping given by F:A / P(U).

In other words, a soft set over U is a mapping from parameters to P(U). P(U) is not a set, but a parameterized family of subsets of U.

Example 1 Let U = {b-i,b2,b3,b4,b5} be a set of bikes under consideration. Let A = {e-[,e2,e3} be a set of parameters, where e1 = expensive, e2 = beautiful and e3 = good mileage. Suppose that F(eO = {b2,b4}, F(e2) = {biMfis}, F(e3) = {bi,b3}. The soft set (F,A) describes the "attractiveness of the bikes". F(e1) means "bikes (expensive)" whose function value is the set {b2,b4}, F(e2) means "bikes (beautiful)" whose function value is the set {b1,b4,bs} and F(e3) means "bikes (good mileage)" whose function value is the set {bi,b3}.

Definition 2 [13](Fuzzy soft sets) Let U) be all fuzzy subsets of U. A pair A) is called fuzzy soft set over U, where F is a mapping given by F : A/~(U).

Example 2 Consider Example l.The fuzzy soft set (F, A) can describe the "attractiveness of the bikes" under the fuzzy circumstances.

~(ei) = {bi/0.3, b2/0.8, b3/0.4, b4/0.7, bs/0.S}, F(e2) = {bi/0.7,62/0.2, b3/0.4, b4/0.8, bs/0.9}, ~(e3) = {bi/0.6, b2/0.4, b3/0.7, b4/0.2, bs/0.1}.

Definition 3 [19] (Multi-fuzzy sets) Let k be a positive integer. A multi-fuzzy set A in U is a set of ordered sequences A = {u/(mi (u), m2(u), ..., mk(u)) : u e U}, where mi ef>(U), i = 1,2,.. ,,k.

The function mA = (m1(u),m2(u), ...,mk(u)) is called the multi-membership function of multi-fuzzy set A, k is called a dimension of A. The set of all multi-fuzzy sets of dimension k in U is denoted by MkFS(U).

Note 1 A multi-fuzzy set of dimension 1 is a Zadeh's fuzzy set, and a multi-fuzzy set of dimension 2 with m1(u) + m2(u) < 1 is an Atanassov's intuitionistic fuzzy set.

Note 2 If ]Tk=1m[(u) < 1, for all ue U, then the multi-fuzzy set of dimension k is called a normalized multi-fuzzy set. If X]k=1 mi (u) = l > 1 for some u e U, we redefine the multi-membership degree (m1(u),m2(u),...,mk(u)) as 1 (m1 (u),m2(u), ...,mk(u)), then the non-normalized multi-fuzzy set can be changed into a normalized multi-fuzzy set.

Definition 4 [19] Let AeMkFS(U). If A = {u/(0,0,..., 0) : ueU}, then AF is called the null multi-fuzzy set of dimension k, denoted by ¿¡>k. If A = {u/(1,1, ..., 1) : ueU}, then A is called the null multi-fuzzy set of dimension k, denoted by 1Fk.

Example 3 Suppose a colour image is approximated by an m x n matrix of pixels. Let U be the set of all pixels of the colour image. For any pixel u in U, the membership values mr(u),mg(u),mb(u) are the normalized red value, green value and blue value of the pixel u, respectively. Therefore, the colour image can be approximated by the collection of pixels with the multi-membership function (mr(u),mg(u),mb(u)) and can be represented as a multi-fuzzy set A = {u/(mr(u),mg(u),mb(u)) : ueU}. In a two dimensional image, pixel colour cannot be characterized by a membership function of an ordinary fuzzy set. However, pixel colour can be characterized by a three dimensional membership function (mr(u),mg(u),mb(u)). In fact, a multi-fuzzy set can be understood as a more general fuzzy set using ordinary fuzzy sets as its building blocks.

Definition5 [19] Let A = {u/(m1(u), m2(u),..., mk(u)) : ueU} and B = {u/(n1(u),n2(u), ...,nk(u)) : ueU} be two multi-fuzzy sets of dimension k in U. We define the following relations and operations.

(1) AeB if and only if mi(u) < ni(u),VueU and 1 < i < k.

(2) A = ~ if and only if mi(u) = ni(u),VueU and 1 < i < k.

(3) AuB = {u/(m1(u)vn1 (u),m2(u)vn2(u), ...,mk(u)Wk(u)) : ueU}.

(4) AnB = {u/(m1 (u)a^1 (u),m2(u)An2(u), ...,mk(u)A^k(u)) : ueU}.

(5) Ac = {u/(m1(u),m2(u),...,mk(u)) : ueU}.

Definition 6 [23] (Multi-fuzzy soft sets) A pair A) is called a multi-fuzzy soft set of dimension k over U, where FF is a mapping given by A/MkFS(U).

A multi-fuzzy soft set is a mapping from parameters to MkFS(U). This set is a parameterized family of multi-fuzzy subsets of U. For ee A, FF(e) may be considered a set of e-approximate elements of the multi-fuzzy soft set (FF, A).

Example 4 Suppose that U = {c1,c2,c3,c4,cs} is the set of cell phones under consideration. A = {e1,e2,e3} is the set of parameters, where e1 stands for the parameter colour, which consists of red, green and blue, e2 stands for the parameter ingredient, which is made from plastic, liquid crystal and metal, and e3 stands for the parameter price, which can be high, medium and low. We define a multi-fuzzy soft set of dimension 3 as follows: ~(e1) = {c1/(0.4, 0.2,0.3), c2/(0.2,0.1,0.6), c3/(0.1, 0.3,0.4), c4/(0.3, 0.1, 0.3), cs/ (0.7,0.1,0.2 )},~(e2) = {c1/(0.1,0.2,0.6),c2/(0.3,0.2, 0.4),c3/(0.5, 0.3,0.1),c4/(0.6,0.1,0.3),cs /(0.6,0.2,0.1)}, F(ea) = {c1/(0.3,0.4, 0.1), c2/(0.4,0.1,0.2), 03/(0.2,0.2,0.5), €4/(0.7,0.1, 0.2), cs/(0.5, 0.2,0.3)}.

Definition 7 [23] Let A,B4E. Let (~,A) and (G,B) be two multi-fuzzy soft sets of dimension k over U. (FF, A) is said to be a multi-fuzzy soft subset of (G, B) if.

(1) A4B, and

(2) F(e)8G(e) for all eeA.

In this case, we write (F, A) 8 (G, B).

Definition 8 [23] (Null multi-fuzzy soft sets) A multi-fuzzy soft set (F, A) of dimension k over U is said to be null multi-fuzzy soft set, denoted by ¡>A if F(e) = Fk for all eeA.

Definition 9 [23] (Absolute multi-fuzzy soft sets) A multi-fuzzy soft set (F, A) of dimension k over U is said to be an absolute multi-fuzzy soft set, denoted by UA if F(e) = ik for all eeA.

3. Generalised multi-fuzzy soft sets

In this section, we introduce a modified definition of multi-fuzzy soft set.

Definition 10 Let U = {x-i,x2,...,xn} be the universal set of elements and E = {e-!,e2,...,en} be the universal set of parameters and A4E. Let (F,A) be a multi-fuzzy soft set of dimension k over U, where F is a mapping given by F : A/MkFS(U). Additionally, let m be a fuzzy subset of A i.e. m:A / I = [0,1]. Next, the pair, (Fm, A) is called a generalised multi-fuzzy soft set of dimension k over U, where Fm is a mapping given by Fm : A / MkFS(U) x I with Fm(e) = (JF(e), m(e)) and F(e)eMkFS(U).

Here, for each parameter ei, Fm(e,) = (^(e,), m(e,)) indicates the degrees of belongingness of the elements of U in FF(ei) and of the possibility of such belongingness. This parameter is represented by

m(ei).

Example 5 Suppose that U={c1,c2,c3,c4,cs} is the set of cell phones under consideration, A={e1,e2,e3} is the set of parameters, where e1 stands for the parameter colour, which consists of red, green and blue, e2 stands for the parameter ingredient, which is made from plastic, liquid crystal and metal, and e3 stands for the parameter price which can be high, medium and low. Let m:A / I = [0,1] be defined as m(e1) = 0.2, m(e2) = 0.3, m(e3) = 0.6. We define a generalised multi-fuzzy soft set of dimension 3 as follows:

~m(ei) = «Ci/(0.4, 0.2,0.3),c2/(0.2, 0.1,0.6),C3AO.1,0.3, 0.4),cA/ (0.3,0.1,0.3), C5/(0.7,0.1,0.2)},0.2), ~m(c2) = ({ci/(0.1, 0.2,0.6), C2/(0.3,0.2,0.4), C3/(0.5,0.3, 0.1), C4/(0.6,0.1, 0.3), C5/(0.6,

0.2,0.1)}, 0.3), ~m(e3) = ({C1/(0.3,0.4,0.1), c2/(0.4,0.1,0.2), C3/ (0.2,0.2,0.5),c4/ (0.7,0.1,0.2),c5/(0.5,0.2,0.3)}, 0.6). In matrix form, this can be expressed as

A. Dey, M. Pal / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—6

3.1. Operations on generalized multi-fuzzy soft sets

Definition 15 The complement of a generalised multi-fuzzy soft set (~m,A) of dimension k over U is denoted by (~m,A)c and is defined by (~m,A)c = (Fy,A) where jF> : A/MkFS(U) x I is a map-

(0.4, 0.2, 0.3) (0.1, 0.2, 0.6) .(0.3, 0.4, 0.l)

(0.2, 0.1, 0.6) (0.3, 0.2, 0.4)

(0.4,0.1,0.2)

(0.1, 0.3, 0.4) (0.5,0.3, 0.1) (0.2, 0.2, 0.5)

(0.3, 0.1, 0.3)

(0.6, 0.1, 0.3) (0.7,0.1,0.2)

(0.7, 0.1, 0.2) 0.21 (0.6, 0.2, 0.1) 0.3 (0.5, 0.2, 0.3) 0.6

where the ith row vector represents Fm(e,), the ith column vector represents ci, the last column represents the values of m and is called the membership matrix of

Definition 11 Let A,B4E. Let (Fm,A) and (Gd,B) be two generalised multi-fuzzy soft sets of dimension k over U. Now, (~m, A) is said to be a generalised multi-fuzzy soft subset of (Gd,B) if.

(1) (~, A)~ (G, B), and

(2) m(e) < 5(e) for all eeA.

Here, we write (Fm, A)F (Gd, B).

Example 6 Let U=(c1,c2,c3,c4,c5), A=(e1,e2} and B = {e1,e2,e3}. m:A / I = [0,1] be defined as m(e1) = 0.4, m(e2) = 0.3 and 5:B / I = [0,1] be defined as 5(e1) = 0.6,5(e2) = 0.3,5(e3) = 0.5 Let's

ping given by ~mc(e) = ((~(e))c, mc(e)) for all eeA. Clearly, (~mc)c is the same as Fm i.e. generalised multi-fuzzy soft complement is

involute in nature. Suppose fF(m, A) and U(m, A) are generalised multi-

fuzzy soft sets of dimension k over U, then (Fkm,A))c = U^) and k

(m,A) •

(U (m,A))c =

Example 7 Consider Example 5, we have (~m,A)c as follows:~mc (e1) = ({c1/(0.6,0.8,0.7), c2/(0.8,0.9,0.4),c3/(0.9, 0.7,0.6), c4/(0.7,0.9,0.7), c5/(0.3,0.9,0.8)}, 0.8), F^ (e2) = ({1/ (0.9,0.8,0.4), c2/(0.7,0.8,0.6), c3/(0.5,0.7,0.9), c4/(0.4,0.9,0.7), c5/(0.4,0.8,0.9)}, 0.7), Fm (e3) = ({1/(0.7,0.6,0.9), c2/(0.6,0.9, 0.8), c3/(0.8,0.8,0.5), c4/ (0.3,0.9,0.8), c5/(0.5,0.8,0.7)}, 0.4). In matrix form this can be expressed as.

(0.6, 0.8, 0.7) (0.9, 0.8, 0.4)

(0.7,0.6, 0.9)

(0.8, 0.9, 0.4) (0.7, 0.8, 0.6) (0.6, 0.9, 0.8)

(0.9, 0.7, 0.6) (0.5, 0.7, 0.9) (0.8, 0.8, 0.5)

(0.7, 0.9, 0.7) (0.4, 0.9, 0.7) (0.3, 0.9, 0.8)

(0.3, 0.9, 0.8) (0.4, 0.8, 0.9) (0.5, 0.8, 0.7)

0.8 0.7 0.4

suppose (~m, A) and (Gd, B) be two generalised multi-fuzzy soft sets of dimension 3 over U, defined as follows:

Fm(e1) = ({c1/(0.4,0.2,0.3), c2/(0.2,0.1,0.6), c3/(0.1,0.3,0.4), c4/(0.3,0.1,0.3), c5/(0.7,0.1,0.2)}, 0.4), Fv(e2) = ({1/(0.1,0.2, 0.6), c2/(0.3,0.2,0.4), c3/(0.5,0.3,0.1), c4/(0.6,0.1, 0.3), c5/(0.6, 0.2,0.1)}, 0.3), Gd (e1) = ({c1/(0.5,0.2,0.6), c2/(0.2,0.4,0.8), c3 /(0.1, 0.4,0.4), c4/(0.5,0.4,0.4), c5/(0.7,0.5,0.2)}, 0.6), Gd(e2) = ({c1/(0.3,0.2,0.9), c2/(0.4,0.4,0.4), c3/(0.6,0.4,0.2), c4/(0.6,0.1, 0.3), c5/(0.6,0.4,0.2)}, 0.3), Gd(e3) = ({c1/(0.3,0.4,0.1), c2/(0.4, 0.1, 0.2), c3/(0.2,0.2,0.5 ), c4/(0.7,0.1, 0.2), c5/(0.5,0.2,0.3)}, 0.5).

Therefore, (~m,A) ~(Gd,B).

Definition 12 Let A,B4E. Let (~m,A) and (Gd,B) be two generalised multi-fuzzy soft sets of dimension k over U. (~m, A) and (Gd, B) are said to be a generalised multi-fuzzy soft equal if (~m, A) is a multi-fuzzy soft subset of (Gd, B) and (Gd, B) is a multi-fuzzy soft subset of (~m, A).

In this case, we write (~m, A) y (Gd, B).

Definition 13 A generalised multi-fuzzy soft set (~m,A) of dimension k over U is said to be a generalised null multi-fuzzy soft set, denoted by ¿£>km A), if Fm(e) = (~(e), m(e)) = (Fk, 0) for all eeA.

Definition 14 A generalised multi-fuzzy soft set (~m,A) of dimension k over U is said to be a generalised absolute multi-fuzzy soft set, denoted by U^ A), if ~m(e) = (~(e), m(e)) = (Ik, 1) for all eeA.

Now, we present the notion of AND and OR operations on two generalised multi-fuzzy soft sets, as follows.

Definition 16 If (~m,A) and (Gd,B) are two generalised multi-fuzzy soft sets of dimension k over U, the (~m,A) and (Gd, B), denoted by fa, A)a(Gs, B) is defined by (~m, A)a(Gs, B) = (H~, A x B),

where H (a, b)= ~(a) nG(b) and ~(a, b) = m(«)Ad(b), for all a,beAxB.

Definition 17 If (~m, A) and (Gd, B) are two generalised multi-fuzzy soft sets of dimension k over U, the (~m, A) OR (Gd, B), denoted by (~m,A)v(Ga,B) is defined by (~m,A)v(Gd,B) = (6~,A x B),

where O (a, b)=~(a)uG (b) and ~(a, b) = m(a)v5(b), for all a,beAxB.

Example 8 Let U = {c1,c2,c3}, A = {e1,e2} and B = {e2,e3,e4}. m:A / I = [0,1] be defined as m(e1) = 0.4, m(e2) = 0.3 and 5:B / I = [0,1] be defined as. 5(e2) = 0.6,5(e3) = 0.2,5(e4) = 0.5. Let's suppose (~m, A) and (Gd, B) be two generalised multi-fuzzy soft sets of dimension 3 over U, defined as follows: ~m(e1) = ({ c1 / (0.4,0.7,0.5), c2/(0.2,0.1, 0.8), c3/(0.1, 0.3,0.8)}, 0.4), Fm(e2) = ({c1/(0.6,0.2,0.6), c2/(0.3,0.9,0.4), c3/(0.5,0.3,0.4)}, 0.3), Gs (e2) = ({c1/ (0.3,0.2,0.9),c2/(0.4,0.4, 0.4),c3/(0.6,0.4,0.2)},0.2), Gd(e3) = ({c1 /(0.5,0.2,0.6), c2/(0.2,0.4,0.8), c3/(0.1,0.4,0.4)},

A Dey, M. Pal / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—6

0.6), Gd (e„) = ({Ci/(0.6,0.4,0.9), C2/(0.4,0.1,0.5), C3/(0.2,0.2,

0.5)}, 0.5).Then, we have (Fm,A)A(Ga,B) — (H, ,A x B) and (Fm,A)v (Gd , B) — (O_,A x B) as follows:

H~(e1, ej) = ({c1/(0.3,0.2,0.5), c2/(0.2,0.1,0.4), c3/(0.1,0.3, 0.2)}, 0.2), H-(e1, e3) = ({c1 /(0.4,0.2,0.5), c2/(0.2,0.1,0.8), c3/

(0.1,0.3,0.4)}, 0.4),

(ei, e4) = ({ci/(0.4,0.4,0.5), c2/(0.2,

0.1,0.5), C3/(0.1,0.2,0.5)},0.4),H~(e2, e2) = ({C1/(0.3, 0.2,0.6),C2/

Definition 19 The intersection of two generalised multi-fuzzy soft sets (Fm,A) and (Gd ,B) of dimension k over U with AnBsf is the generalised multi-fuzzy soft set (Hh, C), where C — AnB, and VeeC, H(e) — F(e)nG(e) and ,(e) — m(e)Ad(e). We write (Fm, A)ri(Ga, B) — (H, , C).

Example 10 Considering Example 8, we have

(Fm,A)ri(Ga, B) — (H, , C), where C—{e2} and H,(e2) — ({c1/(0.3,0.2,

(0.3,0.4,0.4), c3/(0.5,0.3,0.2)}, 0.2), H.(e2, e3) = ({c1/(0.5,0.2, 0.6), C2/(0.3,0.4,0.4), 03/(0.5,0.3,0.2)}, 0.2).

0.6), c2/(0.2,0.4,0.4), c3/(0.1,0.3,0.4)}, 0.3), H~(e2, e4) = ({c1/ (0.6,0.2,0.6), c2/(0.3,0.1,0.4), c3/(0.2,0.2,0.4)}, 0.3), and O~ (e1, e2) — ({c1 /(0.4,0.7,0.9), c2/(0.4,0.4,0.8), c3/(0.6,0.4,0.8)}, 0.4), Oh(e1, e3) = ({c1/(0.5,0.7,0.6), c2/(0.2,0.4,0.8), c3/(0.1, 0.4,0.8)},0.6), O~(e1,e4) = ( {c1/(0.6,0.7,0.9),c2/(0.4,0.1, 0.8), c3/(0.2,0.3,0.8)}, 0.5), Oh(e2, e2 ) = ({c1/(0.6,0.2,0.9), c2/(0.4, 0.9,0.4), c3/(0.6,0.4,0.4)}, 0.3), O~ (e2, e3) = ({c1/(0.6,0.2,0.6), c2/(0.3,0.9,0.8),c3/(0.5,0.4,0.4)},0.6), O^,e4) = ({c1/(0.6,0.4, 0.9), c2/(0.4,0.9,0.5), C3/(0.5,0.3,0.5)}, 0.5).

Theorem 1: Let (~m,A) and (Gd , B) be two generalised multi-fuzzy soft sets of dimension k over U. Then

(1) ((Fm,A)A(Gd,B))c = (Fm,A)cv(Gd,B)c.

(2) ((Fm,A)v(Gd,B))c = (Fm,A)cA(Gd,B)c.

The following results are trivial.

Theorem 2: Let (~m, A) and (Gd, B) are two multi-fuzzy soft sets of dimension k over U. Then

(1) (Fm,A)U(Fm,A) = A),

(2) (Fm,A)u(Fm,A) = A),

(3) (Fm,A)UFFkm, A) - (Fm,A),

(4) (~m, A)n<, a) — Ffm, A),

(5) (~m, A)UUjm, A) — U^ A) ,

(6) (~m , A)nUkm, A) —(Fm, A),

(7) (~m,A)U(Gd, B) — (Gd, B)U(~m,A),

(8) (~m,A)n(Gd, B) — (Gd , B)n(Fm,A).

Proof.

(1) Let (~m,A)A(Gd , B) = (H_ ,A x B).

4. Application of generalised multi-fuzzy soft sets in decision making

Therefore, ((Fm ,A)A(Gd , B))c — ,A x B)c — (Huc ,A x B), where

Uc(a ,b) — (mAd)c(a ,b) — mc(a)vdc(b) for all a,beAxB.

Now, (~m,A)cv(Gd,B)c — (^mc ,A)v(GiF,B) — (^f,A x B), where t(a ,b) — mc(a)vdc(b) — hc(a ,b) for all a,beAxB. Therefore,

H ~c (a ,b)

— (H c(a ,b), Uc (a ,b))

— ((F(a)nG (b))c(a ,b) , Uc(a ,b))

— ((Fic(a)uGc(b))(a ,b) , Uc(a ,b))'

— (O (a ,b) , t(a ,b))

— (Of , A x B)

Hence, ((~m,A)A(Gd,B))c — (Fm ,A)cv(Gd ,B)c.

(2) By similar arguments, the results can be proved.

Definition 18 The union of two generalised multi-fuzzy soft sets (~m , A) and (Gd , B) of dimension k over U is the generalised multi-

H (e) =

h(e) = <

(H h , C) if if if

where eeA - B , eeB - A, ee AnB.

C = AuB, and VeeC,

if if 'if

eeB - A , Here, we write (~m , A)û(Gd, B) = eeAnB.

soft set

8 F(e) G (e) 8 F(e)uG(e) < m(e) ^ d(e) [ m(e)vd(e)

(H, , C).

Example 9 Considering (Fm , A)U(G d , B) — (HH, , C), where H,(e1) — ({c1 /(0.4,0.7 ,0.5) , c2/ (0.2 ,0.1,0.8) ,c3/(0.1,0.3 ,0.8)} ,0.4), H,(e2) — ({c1/(0.6 ,0.2 ,0.9), c2/(0.4, 0.9 ,0.4) ,c3/(0.6,0.4,0.4)},0.3), H, (e3) — ({c1/ (0.5 ,0.2 , 0.6) , c2/(0.2 ,0.4,0.8) , c3/ (0.1,0.4,0.4)} , 0.6), H,(e4) — ({c1/ (0.6 , 0.4 , 0.9) , c2/(0.4 , 0.1, 0.5) , c3/ (0.2 , 0.2 , 0.5)} , 0.5).

Example 8,

Roy et al. [14] presented an object identification algorithm based on fuzzy soft set theory. Ref. [12] modified Roy's algorithm to compare choice values of different objects (i.e. higher choice value) [10]. Feng et al. proposed that the concept of choice values is designed for crisp soft sets and is unfit to solve decision making problems involving fuzzy soft sets. Yong Yang et al. [23] presented a novel approach to multi-fuzzy soft set based decision making problems using Feng's algorithm. We use the possibility of belongingness to solve a decision making problem in the following section. This problem is based on the concept of the generalised multi-fuzzy soft set.

Let U—{x1,x2,^,xn} be the universal set of elements, E— {e-i,e2,...,en} be the universal set of parameters and A4E. Let (Fd1, A) be a generalised multi-fuzzy soft set of dimension k over U. For each eeA Fd, (e) — ({x1/(m1(x1),m2(x1),...,Mkfa)),x2/(m1(x2)Lm2 (x2) ,... ,Mk(x2)) ,...,xn/(m1(xn),m2(xn), ... ,mk(xn))} , d1(e)) — (~(e) , d1(e)), where ~(e) is a normalized multi-fuzzy soft set of dimension k over U.

~(e) =

M1(X2) M2(X2) \M1(Xn) M2 (Xn)

mk (x1)\ mk (X2)

mk (xn)/

~d1 (e) =

//m1 (X1) m2 (X1 ) m1 (X2) m2(x2)

mk(x1)\ \ mk(X2) ,d1(e)

\m1 (xn) m2(xn)

mk(xn)/

Suppose Wd2(e)((w1,w2, .„,wk)' relative weight of the parameter e.

d2(e))k=1w( = 1), be the

A. Dey, M. Pal / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—6 5

Table 1

Tabular representation of the induced generalised fuzzy soft set A~d d = (fd1d2, A).

U e1wd2(e1 )=((0.3,0.4,0.3)T,0.38) e2W"2(e2)=((0.5,0.4,0.1)T,0.37) e3W"2(e3)=((0.2,0.3,0.5)T,0.39)

m1 (0.293,0.1710) (0.299,0.1591) (0.296,0.1794)

m2 (0.334,0.1710) (0.409,0.1591) (0.248,0.1794)

m3 (0.270,0.1710) (0.362,0.1591) (0.236,0.1794)

m4 (0.323,0.1710) (0.268,0.1591) (0.315,0.1794)

m5 (0.302,0.1710) (0.369,0.1591) (0.320,0.1794)

Table 2

Tabular representation of the mid-level soft set L (A~d d ; mid) of A~d d with choice values.

U e1d1(e1)d2(e1)=0.1710 e2d1(e2)d2(e2)=0.1591 e3"1(e3)"2(e3)=0.1794 Choice value

m1 0 0 1 c1 = 1

m2 1 1 0 c2 = 2

m3 0 1 0 c3 = 1

m4 1 0 1 c4 = 2

m5 0 1 1 c5 = 2

Table 3

Tabular representation of the mid-level soft set L (A~ weighted choice values.

mid) of Af with

U e1 e2 e3 Weighted choice value

m1 0 0 1 c2 = 0.1794

m2 1 1 0 c2 = 0.3301

m3 0 1 0 c2 = 0.1591

m4 1 0 1 c2 = 0.3504

m5 0 1 1 c2 = 0.3385

Now, we define an induced generalized fuzzy soft set as:

( W1 \ 1

( ,mt(xt) M2(xt) ... mk(xt)\ mt(x2) m2(x2) ■ ■• mk(x2)

V mt(xn) m2(xn) ... mk(xn)) \wkj

, d1(e)d2(e)

( / £ (x1)-

\ \ ,d1(e)d2(e)

Thus, if wd2 (e) is given, we can change a generalised multi-fuzzy soft set ~d1 (e) into an induced generalised fuzzy soft set~d1d2 (e).

We shall now use the induced generalised fuzzy soft set ~d1d2 (e) to make a decision.

In the following section, we present an algorithm to select an optimal decision.

4.1. An algorithm

Input: The generalised multi fuzzy soft set (~d1, A) and relative weight (w(ei),d2(ei)) of parameter ei.

Output: The product xk for some k.

Step 1. A generalised multi fuzzy soft set (~d1, A) and relative weight (w(ei),d2(ei)) of parameter ei is taken as input.

Step 2. Change the multi-fuzzy soft set (~, A) into the normalized multi-fuzzy soft set.

Step 3. Compute the induced generalised fuzzy soft set

AFd,d2 = (Fd,d2 , A).

Step 4. Choose the mid-level decision rule for decision making. Step 5. Compute the mid-level soft set L (A~ ; mid). Step 6. Present the mid-level soft set L (A~ 12; mid) in tabular form and compute the choice value ci of xi, ci.12

Step 7. Compute max ci.

Step 8. If maximum attained at only one value (e.g., j), then the optimal decision is to select xj.

Step 9. If maximum attained for multiple values of i, then compute the weighted choice vales cj of xj for all j and present the weighted choice vales cj in tabular form.

Step 10. Compute maximum weighted choice value

ck = max {cj}. j

Step 11. If k has only one value, then the optimal decision is to select xk.

Step 12. If k has more than one value, then any one of xk may be an optimal decision.

4.2. Experimental analysis

Let U = {m1,m2,m3,m4,m5} be the universe consisting of five types of cell phones. Let A = {e1,e2,e3} be the parameters set. Here, e1 stands for the parameter colour, which consists of red, green and blue; e2 stands for the parameter ingredient, which is made from plastic, liquid crystal and metal; and e3 stands for the parameter price which can be high, medium and low. Suppose that Fd, (e1) = ({m1/(0.34,0.32,0.21), m2/(0.40,0.37,0.22), m3/(0.51, 0.12,0.23), m4/(0.41, 0.41,0.12),m5/(0.35,0.35,0.19)},0.45),Fd1 (e2) = ({m1/(0.24,0.35,0.39),m2/(0.57, 0.28,0.12),m3/(0.35, 0.41, 0.23), m4/(0.25,0.27,0.35), m5/(0.18,0.67,0.11)}, 0.43),~d1 (e3) = ({m1/(0.43,0.25,0.27),m2/(0.25,0.26,0.24), m3/(0.79,0.11,0.09), m4/(0.15,0.45,0.30), m5/(0.35,0.35,0.29)}, 0.46). Suppose a customer would like to select a cell phone depending on its performance and the customer has imposed the following weights for the parameters in A: for the parameter colour, wd2 (e1) = ((0.3,0.4,0.3)T,0.38), for the parameter ingredient, wd2(e2) = ((0.5,0.4,0.1)T,0.37), and for the parameter price, wd2(e3) = ((0.2,0.3,0.5)T,0.39). Thus, we have induced generalised fuzzy soft set A~ =(^d]d2,A) with its tabular representation as in Table 1. 1112

A. Dey, M. Pal / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—6

Here, we shall consider the mid-level decision rule. Therefore, midDu d,d2 — {(e1,0.3044), (e2 ,0.3414) , (e3 , 0.2830)}.

Thus, we shall obtain the mid-level soft set L (A~ ; mid) of

Aud d with choice values with tabular representations, as in Table 2.

From Table 2, the choice values c2 — c4 — c5 — 2 > c1 — c3 — 1. Therefore, cj — max{c1, c2 , c3, c4 , c5} — c2 or c4 or c5. Thus, the maximum is attained for multiple values, and we have to compute the weighted choice value cj of mj for all j. Now from Table 3, we have,

(1) weighted choice value for m1 — c1 — (0 x 0.1710) +(0 x 0.1591) + (1 x 0.1794) — 0.1794.

(2) weighted choice value for m2 — c2 — (1 x 0.1710)+ (1 x 0.1591) + (0 x 0.1794) — 0.3301.

(3) weighted choice value for m3 — c2 — (0 x 0.1710) +(1 x 0.1591) + (0 x 0.1794) — 0.1591.

(4) weighted choice value for m4 — c4 — (1 x 0.1710) +(0 x 0.1591) + (1 x 0.1794) — 0.3504.

(5) weighted choice value for m5 — c5 —(0 x 0.1710) + (1 x 0.1591) + (1 x 0.1794) — 0.3385.

Therefore, the maximum weighted choice value — max

{c^ c2, c3 ,c4 , c5} — c4.

Because the maximum weighted choice value was attained for the weighted choice value was c4 of m4. Hence, the customer should select m4 as the best cell phone after specifying weights for different parameters.

5. Conclusions

The purpose of this paper was to generalise the concept of multi-fuzzy soft set and obtain a new approach for managing

uncertainties. The definition of the generalised multi-fuzzy soft set is more realistic because it involves uncertainty in the selection of a fuzzy soft set corresponding to each parameter value. Generalised multi-fuzzy soft set relations are defined, their properties are studied and a decision making problem is solved to illustrate their applications. To extend this work, one could study the topological structure for generalised multi-fuzzy soft sets. We hope that our work enhances the understanding of generalised multi-fuzzy soft sets for future researchers.

Acknowledgements

Financial support provided by the Council of Scientific and Industrial Research, New Delhi, India (Sanction no. 09/599(0054)/ 2013-EMR-I) is thankfully acknowledged.

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