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International Journal of Fuzzy Logic and Intelligent Systems 2014; 14(3): 181-187

Published online September 30, 2014

https://doi.org/10.5391/IJFIS.2014.14.3.181

© The Korean Institute of Intelligent Systems

## Group Decision Making Using Intuitionistic Hesitant Fuzzy Sets

Ismat Beg1 and Tabasam Rashid2

1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan
2Department of Mathematics, University of Management and Technology, Lahore, Pakistan

Correspondence to :
Ismat Beg (ibeg@lahoreschool.edu.pk)

Received: March 17, 2014; Revised: September 20, 2014; Accepted: September 22, 2014

Dealing with uncertainty is always a challenging problem. Intuitionistic fuzzy sets was presented to manage situations in which experts have some membership and non-membership value to assess an alternative. Hesitant fuzzy sets was used to handle such situations in which experts hesitate between several possible membership values to assess an alternative. In this paper, the concept of intuitionistic hesitant fuzzy set is introduced to provide computational basis to manage the situations in which experts assess an alternative in possible membership values and non-membership values. Distance measure is defined between any two intuitionistic hesitant fuzzy elements. Fuzzy technique for order preference by similarity to ideal solution is developed for intuitionistic hesitant fuzzy set to solve multi-criteria decision making problem in group decision environment. An example is given to illustrate this technique.

Keywords: Hesitant fuzzy set,Intuitionistic fuzzy set,Multiple attribute group decisionmaking,Technique for order preference by similarity to ideal solution

Group decision making is the process of finding the best option among a set of feasible alternatives. Basic problem is how to aggregate several inputs into a single representative output [13]. Problems that are defined under uncertain situations are common in real world decision making. Zadeh [4] in his seminal paper introduced the notion of fuzzy sets to handle uncertainty. Bellman and Zadeh [5] used fuzzy sets in decision making for the solution of ambiguity in information obtained from human preferences. Recently, Dubois [6] presented a beautiful comparison about old and new techniques for fuzzy decision analysis. Atanassov [7, 8] introduced the concept of intuitionistic fuzzy sets (IFS) characterized by a membership function and a non-membership function, which is more suitable for dealing with fuzziness and uncertainty than the fuzzy set. The IFS is highly useful in depicting uncertainty and vagueness of an object, and thus can be used as a powerful tool to express data information under various different fuzzy environments which has attracted great attentions. Recently, the intuitionistic fuzzy set has been widely applied to decision making problems [912]. IFSs have been found to be a particularly useful tool to deal with vagueness. Torra [13] extended the concept of fuzzy sets to hesitant fuzzy sets. Hesitant fuzzy set theory tries to manage those situations where a set of values are possible in the definition process of the membership of an element. Group decision making problems are solved by using hesitant fuzzy sets and with aggregation operators in [13, 14, 15].

Hwang and Yoon [16] developed technique for order preference by similarity to ideal solution (TOPSIS) for multi-attribute/multi-criteria decision making (MADM/MCDM) problems. Shih et al. [17] addressed four advantages of TOPSIS: first is that a sound logic represents the rationale of human choice; secondly a scalar value considers the best and worst alternative simultaneously; third advantage is that it has a simple computation process and can be easily programmed and the last but not the least advantage is that it has ability of the performance measures of all alternatives on attributes to be visualized on a polyhedron, at least for any two dimensions. Fuzzy numbers are applied to establish a fuzzy TOPSIS [18].

The aim of this paper is to introduce the concept of intuitionistic hesitant fuzzy sets (IHFS) by merging the concept of IFS and HFS. It helps to manage those situations of uncertainty in which some values are possible as membership values of element as well as non-membership values of the same element. Additionally, we also develop fuzzy TOPSIS for IHFS. This article is organized as follows: In Section 2, we introduce the concept of IHFS and notion of distance between any two elements of IHFS. In Section 3, TOPSIS is constructed for IHFS. Then in Section 4, this fuzzy TOPSIS method is applied for the ranking of alternatives in an example, to demonstrate its feasibility. Conclusion is given in the last section.

### 2. Intuitionistic Hesitant Fuzzy Set

In this section, we introduce an extension of IFS to manage those situations in which several values are possible for the definition of a membership function and non-membership function. We propose the concept of IHFS by keeping in view the importance of IFS and HFS. IHFS is defined in terms of a function that returns a set of membership values and a set of non-membership values for each element in the domain.

### Definition 2.1

An IHFS on X are functions h and h′ that when applied to X return subsets of [0, 1], which can be represented as the following mathematical symbol:

E={(x,h(x),h(x))xX},

where h(x) and h′(x) are sets of some values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element xX to the set E with the conditions that max(h(x))+min(h′(x)) ≤ 1 and min(h(x))+max(h′(x)) ≤ 1. For convenience, (h(x), h′(x)) an intuitionistic hesitant fuzzy element (IHFE).

Examples of IHFS are given below where h(x) and h′(x) represent the possible membership and non-membership values of the set at x, respectively.

It is noted that the number of values in different IHFEs may be different, let lh(x) and lh′(x) be the number of values in h(x) and h′(x). In case values in an IHFE are out of order; we can arrange them in such a order, that IHFE (h, h′), let σ: (1, 2, ..., n) → (1, 2, ..., n) and ς: (1, 2, ...,m) → (1, 2, ...,m) be two permutations satisfying hσ(i)hσ(i+1), i = 1, 2, ..., lh–1 and hσ(i)hσ(i+1), j = 1, 2, ..., lh–1. We proposed that two IHFEs (h1, h1) and (h2, h2) have lh1 = lh2, lh1=lh2, h1σ(i) = h2σ(i) and h1σ(j)=h2σ(j) if and only if (h1,h1)=(h2,h2) for i = 1, 2, ..., lh1 and j=1,2,,lh1.

### Example 2.2

Consider an IHFS A given by

A = {(x1, (0.2, 0.3, 0.5, 0.6, 0.9), (0.01, 0.05, 0.1)), (x2, (0.1, 0.4, 0.7), (0.1, 0.15, 0.18, 0.2, 0.25))}.

Then

• max hA(x1) = max(0.2, 0.3, 0.5, 0.6, 0.9)=0.9;

• max hA(x1)=max(0.01,0.05,0.1)=0.1;

• max hA(x2) = max(0.1, 0.4, 0.7)=0.7;

• max hA(x2)=max(0.1,0.15,0.18,0.2,0.25)=0.25;

• min hA(x1) = min(0.2, 0.3, 0.5, 0.6, 0.9)=0.2;

• max hA(x1)=min(0.01,0.05,0.1)=0.01;

• min hA(x2) = min(0.1, 0.4, 0.7)=0.1;

• min hA(x2)=min(0.1,0.15,0.18,0.2,0.25)=0.1.

It is clear that max hA(x1)+min hA(x1)=0.9+0.01=0.911 and min hA(x1)+max hA(x1)=0.2+0.1=0.31, so (hA(x1),hA(x1)) is an IHFE. Similarly, (hA(x2),hA(x2))is an IHFE. Thus A is an IHFS.

### Definition 2.3

Let x and y be two IHFEs, such that x=(hx,hx)=((a1,a2,,an),(a1,a2,,an)) and y=(hy,hy)=((b1,b2,,bm),(b1,b2,,bm)) then distance ‘d’ between x and y is defined as

d(x,y)=max{maxaihx{minbihy(ai-bi)},maxbihy{minaihx(ai-bi)},maxaihx{minbihy(ai-bi)},maxbihy{minaihx(ai-bi)}}.

It is easy to show that this distance ‘d’ satisfies the following properties:

• 1) d(x, y) = 0 if and only if x = y;

• 2) d(x, y) = d(y, x).

In this section, we give construction of TOPSIS for IHFS. This TOPSIS is used for multi-criteria group decision making where the opinions about the criteria values are expressed in IHFS. We suppose that in this group decision making problem, E = {e1, e2, . . ., eK} is the set of the decision makers involved in the decision problem; A = {A1, A2, . . ., Am} is the set of the considered alternatives and C = {C1, C2, . . ., Cn} is the set of the criteria used for evaluating the alternatives.

• Step 1. Let X˜l=[xij=(HSijl,HSijl)]m×n be a fuzzy decision matrix for the MCDM problem where performance of alternative Ai with respect to decision maker el and criterion Cj is denoted as HSijl, in a group decision environment with K decision makers.

• Step 2. We calculate the one decision matrix X by aggregating the opinions of DMs (1, 2, · · ·, K);

X = [xij], where xij=((xxHSijland spijxsqijfor all l),(xxHSijland spijxsqijfor all l)) where

spij=min{minl=1K(max HSijl),maxl=1K(min HSijl)},sqij=max{minl=1K(max HSijl),maxl=1K(min HSijl)},spij=min{minl=1K(max HSijl),maxl=1K(min HSijl)}

and

sqij=max{minl=1K(max HSijl),maxl=1K(min HSijl)}.

Performance of alternative Ai with respect to criterion Cj is denoted as xij, in an aggregated matrix X.

• Step 3. Let Ωb be the collection of benefit criteria (i.e., the larger Cj, the greater preference) and Ωc be the collection of cost criteria (i.e., the smaller Cj, the greater preference). The IHFS positive-ideal solution (IHFS-PIS), denoted as A˜+=(V˜1+V˜2+V˜n+), and the IHFS negative-ideal solution (IHFS-NIS), denoted as A˜-=(V˜1-V˜2-V˜n-), are defined as follows:

A˜+=[(x,x)xHSijland xHSijliand (maxl=1K(maxi(min HSijl))x,xmaxl=1K(maxi(max HSijl)),minl=1K(mini(min HSijl))x,xminl=1K(mini(max HSijl)))jΩb,(x,x)xHSijland xHSijliand (minl=1K(mini(min HSijl))x,xminl=1K(mini(max HSijl)),maxl=1K(maxi(min HSijl))x,xmaxl=1K(maxi(max HSijl)))jΩc]i=1,2,,m,and j=1,2,,n.A˜+=(V˜1+V˜2+V˜n+)A˜-=[(x,x)xHSijland xHSijliand (maxl=1K(maxi(min HSijl))x,xmaxl=1K(maxi(max HSijl)),minl=1K(mini(min HSijl))x,xminl=1K(mini(max HSijl)))jΩc,(x,x)xHSijland xHSijliand (minl=1K(mini(min HSijl))x,xminl=1K(mini(max HSijl)),maxl=1K(maxi(min HSijl))x,xmaxl=1K(maxi(max HSijl)))jΩb]i=1,2,,m,and j=1,2,,n.A˜-=(V˜1-V˜2-V˜n-)

• Step 4. Construct positive ideal separation matrix (D+) and negative ideal separation matrix (D) which are defined as follows:

D+=[d(x11,V˜1+)++d(x1n,V˜n+)d(x21,V˜1+)++d(x2n,V˜n+)d(xm1,V˜1+)++d(xmn,V˜n+)]

and

D-=[d(x11,V˜1-)++d(x1n,V˜n-)d(x21,V˜1-)++d(x2n,V˜n-)d(xm1,V˜1-)++d(xmn,V˜n-)]

• Step 5. Calculate the relative closeness (RC) coefficient of each alternative to the ideal solution as follows:

RC(Ai)=Di-Di++Di-,i=1,2,,m,

where Di-=j=1nd(xij,V˜j-) and Di+=j=1nd(xij,V˜j+).

• Step 6. Rank all the alternatives Ai (i = 1, 2, . . ., m) according to the RC(Ai) coefficient, greater the value RC(Ai), better the alternative Ai.

Graphical representation of this proposed technique is given in Figure 1.

### 4. Illustrative Example

In this section, we give an example. We utilized the method proposed in Section 4 to get the most desirable alternative as well as rank the alternatives from the best to worst or vice versa. Consider five schools (School of Business and Economics (A1), School of Science and Technology (A2), School of Social Sciences and Humanities (A3), School of Communication and Cultural Studies (A4), School of Textile and Design (A5)) all in the same university. Management of the university want to manage the allocation of funds to these schools based on their performance. There are four criteria (expenses of school (C1), students in take per year in school (C2), publications from school (C3), covered area of the school (C4)) for assessing the performance of these five schools. These assessments are given by the nine members from board of directors.

• Step 1. There are five possible alternatives Ai (i = 1, 2, 3, 4, 5) are to be evaluated on the criteria Cj (j = 1, 2, 3, 4) using the IHFS by nine experts eK (K = 1, 2, . . ., 9), as listed in Tables 13.

• Step 2. The decision matrix constructed in Table 4 by utilize Tables 13.

• Step 3. For cost criteria C1, C4 and benefit criteria C2, C3 IHFS-PIS is as follows:

A+=[((0,0.1,0.2),(0.6,0.7))((0.6,0.7,0.8,0.9,1),(0))((0.7,0.8,0.9),(0,0.05))((0,0.1),(0.7,0.8))]

For cost criteria C1, C4 and benefit criteria C2, C3 IHFS-NIS is as follows:

A-=[((0.7,0.8,0.9),(0))((0.1,0.2),(0.6,0.65,0.7))((0,0.1),(0.7,0.8))((1),(0))]

• Step 4. Positive ideal separation matrix (D+):

D+=[0.4+0.2+0.6+0.40.3+0.6+0.4+0.70.35+0.6+0.25+0.10.7+0.6+0.8+0.70.8+0.6+0.7+0.9]D+=[1.621.32.83.0]

Negative ideal separation matrix (D):

D-=[0.5+0.6+0.3+0.70.6+0.5+0.65+0.50.5+0.5+0.6+0.90.1+0.4+0.2+0.50.3+0.3+0.2+0.3]D-=[2.12.252.51.21.1]

• Step 5.RC of each alternative to the ideal solutions:

• RC(A1) = 2.1/(1.6+2.1) = 0.5676;

• RC(A2) = 2.25/(2+2.25) = 0.5294;

• RC(A3) = 2.5/(1.3+2.5) = 0.6579;

• RC(A4) = 1.2/(2.8+1.2) = 0.3;

• RC(A5) = 1.1/(3+1.1) = 0.2683.

• Step 6. Rank all the alternatives Ai (i=1, 2, . . ., 5) according to the closeness coefficient RC(Ai):

• A3A1A2A4A5.

Thus the most desirable alternative is A3. So the rector will allocate the funds according to this ranking.

IHFS is the best way to deal with uncertainty when fuzzy set theory is not able to cope with the situation. Decision makers gave their opinions about the criteria of alternatives by IHFS. Multi-criteria analysis provides an effective frame work for the evaluation of alternatives. Fuzzy TOPSIS method is proposed for IHFS to solve multi-criteria decision-making problem in group decision environment. The RC coefficient has ranked the alternatives from the best to worst by considering the smallest distance from the PIS and also the largest distance from the NIS. In future, we plan to continue to study Choquet integral based TOPSIS for IHFS. Furthermore, algebraic operations for IHFS will also be develop.

No potential conflict of interest relevant to this article was reported.

Fig. 1.

Graphical structure of technique for order preference by similarity to ideal solution.

Table. 1.

Table 1. Decision matrix (1) with respect to experts 1, 2, 3, 4 (e1, e2, e3, e4).

C1
A1((0.5,0.6,0.8),(0.1,0.2))
A2((0.1,0.3),(0.3,0.4,0.5))
A3((0.5,0.7),(0.2,0.25))
A4((0.7,0.9),(0.05,0.1))
A5((1),(0))
C2
A1((0.6,0.8),(0.1,0.2))
A2((0.5,0.7,0.8),(0.1))
A3((0.5,0.6),(0.2,0.35))
A4((0.1,0.2),(0.6,0.7))
A5((0.1,0.3),(0.5,0.65))
C3
A1((0.1,0.3),(0.6,0.7))
A2((0.5,0.6),(0.1,0.3))
A3((0.7,0.9),(0.05,0.1))
A4((0.1,0.3),(0.6,0.7))
A5((0,0.2),(0.7,0.8))
C4
A1((0.1,0.3),(0.5,0.6))
A2((0.5,0.6),(0.2,0.3))
A3((0.1,0.2),(0.6,0.7))
A4((0.5,0.6,0.7),(0.2))
A5((0.4,0.7),(0.1,0.2))

Table. 2.

Table 2. Decision matrix (2) with respect to experts 5, 6, 7 (e5, e6, e7).

C1
A1((0.1,0.2),(0.5,0.6))
A2((0,0.2),(0.6,0.7))
A3((0.4,0.6),(0.1,0.3))
A4((0.6,1),(0))
A5((0.5,0.7),(0.1,0.2))
C2
A1((0.4,0.9),(0,0.1))
A2((0.1,0.3),(0.5,0.6))
A3((0.1,0.2),(0.6,0.7))
A4((0.4,0.7),(0.1,0.2))
A5((0.4,0.6),(0.2,0.3))
C3
A1((0,0.2),(0.6,0.7))
A2((0.4,0.5),(0.3,0.4))
A3((0.4,0.6),(0.3,0.4))
A4((0,0.1),(0.6,0.8))
A5((0,0.1),(0.7,0.8))
C4
A1((0.4,0.6),(0.1,0.3))
A2((0.6,1),(0))
A3((0,0.2),(0.5,0.7))
A4((0.5,0.7),(0.2,0.3))
A5((0.6,1),(0))

Table. 3.

Table 3. Decision matrix (3) with respect to experts 8, 9 (e8, e9).

C1
A1((0.4,0.6),(0.2,0.3))
A2((0.3,0.6),(0.2,0.3))
A3((0.1,0.3),(0.5,0.6))
A4((0.6,0.9),(0,0.1))
A5((0.5,0.6),(0.3,0.4))
C2
A1((0.6,1),(0))
A2((0.1,0.3),(0.6,0.7))
A3((0.6,0.9),(0,0.1))
A4((0.5,0.7),(0.1,0.2))
A5((0.1,0.3),(0.6,0.7))
C3
A1((0.3,0.5),(0.3,0.4))
A2((0.5,0.9),(0,0.05))
A3((0.3,0.7),(0.1,0.2))
A4((0,0.2,0.4),(0.4,0.5))
A5((0.2,0.4),(0.4,0.5))
C4
A1((0,0.3),(0.5,0.6))
A2((0.3,0.5),(0.3,0.4))
A3((0,0.1),(0.7,0.8))
A4((0.5,0.6,0.8),(0))
A5((1),(0))

Table. 4.

Table 4. Decision matrix (X).

C1
A1((0.2,0.4,0.5),(0.2,0.3,0.5))
A2((0.2,0.3),(0.3,0.4,0.5,0.6))
A3((0.3,0.4,0.5),(0.25,0.3,0.5))
A4((0.7,0.9),(0,0.05))
A5((0.6,0.7,1),(0,0.1,0.2,0.3))
C2
A1((0.6,0.8),(0,0.1))
A2((0.3,0.5),(0.1,0.5,0.6))
A3((0.2,0.5,0.6),(0.1,0.2,0.35,0.6))
A4((0.2,0.4,0.5),(0.2,0.6))
A5((0.3,0.4),(0.3,0.5,0.6))
C3
A1((0.2,0.3),(0.4,0.6))
A2((0.5),(0.05,0.1,0.3))
A3((0.6,0.7),(0.1,0.2,0.3))
A4((0.1),(0.5,0.6))
A5((0.1,0.2),(0.5,0.7))
C4
A1((0.3,0.4),(0.3,0.5))
A2((0.5,0.6),(0,0.2,0.3))
A3((0.1),(0.7))
A4((0.5,0.6,0.7),(0,0.2))
A5((0.7,1),(0,0.1))

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Ismat Beg is professor at Lahore School of Economics. He has vast experience of teaching and research. His field of interest and specialization is versatile in nature. It covers many areas of mathematics, economics, decision theory, computer science and engineering. He is a fellow of Pakistan Academy of Sciences.

Tabasam Rashid was born in Lahore (Pakistan) on 18 April 1985. He obtained his bachelor’s degree in Mathematics in 2008 at University of the Punjab, Lahore, Pakistan. In 2010, he obtained his M.S. in mathematics from National University of Computer and Emerging Sciences, Lahore Campus, Pakistan. He majored in fuzzy similarity measures in his master study. In 2010, he became a Ph.D. student at the Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Pakistan. His research focused on the multi-criteria decision making problems. He has published more than 10 papers in international refereed journals. Tabasam currently is a lecturer at University of Management and Technology. He is also a reviewer of several international journals.

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2014; 14(3): 181-187

Published online September 30, 2014 https://doi.org/10.5391/IJFIS.2014.14.3.181

Copyright © The Korean Institute of Intelligent Systems.

## Group Decision Making Using Intuitionistic Hesitant Fuzzy Sets

Ismat Beg1 and Tabasam Rashid2

1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan
2Department of Mathematics, University of Management and Technology, Lahore, Pakistan

Correspondence to:Ismat Beg (ibeg@lahoreschool.edu.pk)

Received: March 17, 2014; Revised: September 20, 2014; Accepted: September 22, 2014

### Abstract

Dealing with uncertainty is always a challenging problem. Intuitionistic fuzzy sets was presented to manage situations in which experts have some membership and non-membership value to assess an alternative. Hesitant fuzzy sets was used to handle such situations in which experts hesitate between several possible membership values to assess an alternative. In this paper, the concept of intuitionistic hesitant fuzzy set is introduced to provide computational basis to manage the situations in which experts assess an alternative in possible membership values and non-membership values. Distance measure is defined between any two intuitionistic hesitant fuzzy elements. Fuzzy technique for order preference by similarity to ideal solution is developed for intuitionistic hesitant fuzzy set to solve multi-criteria decision making problem in group decision environment. An example is given to illustrate this technique.

Keywords: Hesitant fuzzy set,Intuitionistic fuzzy set,Multiple attribute group decisionmaking,Technique for order preference by similarity to ideal solution

### 1. Introduction

Group decision making is the process of finding the best option among a set of feasible alternatives. Basic problem is how to aggregate several inputs into a single representative output [13]. Problems that are defined under uncertain situations are common in real world decision making. Zadeh [4] in his seminal paper introduced the notion of fuzzy sets to handle uncertainty. Bellman and Zadeh [5] used fuzzy sets in decision making for the solution of ambiguity in information obtained from human preferences. Recently, Dubois [6] presented a beautiful comparison about old and new techniques for fuzzy decision analysis. Atanassov [7, 8] introduced the concept of intuitionistic fuzzy sets (IFS) characterized by a membership function and a non-membership function, which is more suitable for dealing with fuzziness and uncertainty than the fuzzy set. The IFS is highly useful in depicting uncertainty and vagueness of an object, and thus can be used as a powerful tool to express data information under various different fuzzy environments which has attracted great attentions. Recently, the intuitionistic fuzzy set has been widely applied to decision making problems [912]. IFSs have been found to be a particularly useful tool to deal with vagueness. Torra [13] extended the concept of fuzzy sets to hesitant fuzzy sets. Hesitant fuzzy set theory tries to manage those situations where a set of values are possible in the definition process of the membership of an element. Group decision making problems are solved by using hesitant fuzzy sets and with aggregation operators in [13, 14, 15].

Hwang and Yoon [16] developed technique for order preference by similarity to ideal solution (TOPSIS) for multi-attribute/multi-criteria decision making (MADM/MCDM) problems. Shih et al. [17] addressed four advantages of TOPSIS: first is that a sound logic represents the rationale of human choice; secondly a scalar value considers the best and worst alternative simultaneously; third advantage is that it has a simple computation process and can be easily programmed and the last but not the least advantage is that it has ability of the performance measures of all alternatives on attributes to be visualized on a polyhedron, at least for any two dimensions. Fuzzy numbers are applied to establish a fuzzy TOPSIS [18].

The aim of this paper is to introduce the concept of intuitionistic hesitant fuzzy sets (IHFS) by merging the concept of IFS and HFS. It helps to manage those situations of uncertainty in which some values are possible as membership values of element as well as non-membership values of the same element. Additionally, we also develop fuzzy TOPSIS for IHFS. This article is organized as follows: In Section 2, we introduce the concept of IHFS and notion of distance between any two elements of IHFS. In Section 3, TOPSIS is constructed for IHFS. Then in Section 4, this fuzzy TOPSIS method is applied for the ranking of alternatives in an example, to demonstrate its feasibility. Conclusion is given in the last section.

### 2. Intuitionistic Hesitant Fuzzy Set

In this section, we introduce an extension of IFS to manage those situations in which several values are possible for the definition of a membership function and non-membership function. We propose the concept of IHFS by keeping in view the importance of IFS and HFS. IHFS is defined in terms of a function that returns a set of membership values and a set of non-membership values for each element in the domain.

### Definition 2.1

An IHFS on X are functions h and h′ that when applied to X return subsets of [0, 1], which can be represented as the following mathematical symbol:

$E={(x,h(x),h′(x))∣x∈X},$

where h(x) and h′(x) are sets of some values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element xX to the set E with the conditions that max(h(x))+min(h′(x)) ≤ 1 and min(h(x))+max(h′(x)) ≤ 1. For convenience, (h(x), h′(x)) an intuitionistic hesitant fuzzy element (IHFE).

Examples of IHFS are given below where h(x) and h′(x) represent the possible membership and non-membership values of the set at x, respectively.

It is noted that the number of values in different IHFEs may be different, let lh(x) and lh′(x) be the number of values in h(x) and h′(x). In case values in an IHFE are out of order; we can arrange them in such a order, that IHFE (h, h′), let σ: (1, 2, ..., n) → (1, 2, ..., n) and ς: (1, 2, ...,m) → (1, 2, ...,m) be two permutations satisfying hσ(i)hσ(i+1), i = 1, 2, ..., lh–1 and $hσ(i)′≤hσ(i+1)′$, j = 1, 2, ..., lh–1. We proposed that two IHFEs (h1, $h1′$) and (h2, $h2′$) have lh1 = lh2, $lh1′=lh2′$, h1σ(i) = h2σ(i) and $h1σ(j)′=h2σ(j)′$ if and only if $(h1,h1′)=(h2,h2′)$ for i = 1, 2, ..., lh1 and $j=1,2,…,lh1′$.

### Example 2.2

Consider an IHFS A given by

A = {(x1, (0.2, 0.3, 0.5, 0.6, 0.9), (0.01, 0.05, 0.1)), (x2, (0.1, 0.4, 0.7), (0.1, 0.15, 0.18, 0.2, 0.25))}.

Then

• max hA(x1) = max(0.2, 0.3, 0.5, 0.6, 0.9)=0.9;

• $max hA′(x1)=max(0.01,0.05,0.1)=0.1$;

• max hA(x2) = max(0.1, 0.4, 0.7)=0.7;

• $max hA′(x2)=max(0.1,0.15,0.18,0.2,0.25)=0.25$;

• min hA(x1) = min(0.2, 0.3, 0.5, 0.6, 0.9)=0.2;

• $max hA′(x1)=min(0.01,0.05,0.1)=0.01$;

• min hA(x2) = min(0.1, 0.4, 0.7)=0.1;

• $min hA′(x2)=min(0.1,0.15,0.18,0.2,0.25)=0.1$.

It is clear that $max hA(x1)+min hA′(x1)=0.9+0.01=0.91≤1$ and $min hA(x1)+max hA′(x1)=0.2+0.1=0.3≤1$, so $(hA(x1),hA′(x1))$ is an IHFE. Similarly, $(hA(x2),hA′(x2))$is an IHFE. Thus A is an IHFS.

### Definition 2.3

Let x and y be two IHFEs, such that $x=(hx,hx′)=((a1,a2,…,an),(a1′,a2′,…,an′′))$ and $y=(hy,hy′)=((b1,b2,…,bm),(b1′,b2′,…,bm′′))$ then distance ‘d’ between x and y is defined as

$d(x,y)=max {maxai∈hx {minbi∈hy(∣ai-bi∣)},maxbi∈hy {minai∈hx(∣ai-bi∣)},maxai′∈hx′ {minbi′∈hy′(∣ai′-bi′∣)},maxbi′∈hy′ {minai′∈hx′(∣ai′-bi′∣)}}.$

It is easy to show that this distance ‘d’ satisfies the following properties:

• 1) d(x, y) = 0 if and only if x = y;

• 2) d(x, y) = d(y, x).

### 3. TOPSIS for IHFS

In this section, we give construction of TOPSIS for IHFS. This TOPSIS is used for multi-criteria group decision making where the opinions about the criteria values are expressed in IHFS. We suppose that in this group decision making problem, E = {e1, e2, . . ., eK} is the set of the decision makers involved in the decision problem; A = {A1, A2, . . ., Am} is the set of the considered alternatives and C = {C1, C2, . . ., Cn} is the set of the criteria used for evaluating the alternatives.

• Step 1. Let $X˜l=[xij=(HSijl,HSij′l)]m×n$ be a fuzzy decision matrix for the MCDM problem where performance of alternative Ai with respect to decision maker el and criterion Cj is denoted as $HSijl$, in a group decision environment with K decision makers.

• Step 2. We calculate the one decision matrix X by aggregating the opinions of DMs (1, 2, · · ·, K);

X = [xij], where $xij=((x∣x∈HSijl and spij≤x≤sqij for all l),(x∣x∈HSij′l and spij′≤x≤sqij′ for all l))$ where

$spij=min {minl=1K(max HSijl),maxl=1K(min HSijl)},sqij=max {minl=1K(max HSijl),maxl=1K(min HSijl)},spij′=min {minl=1K(max HSij′l),maxl=1K(min HSij′l)}$

and

$sqij′=max {minl=1K(max HSij′l),maxl=1K(min HSij′l)}.$

Performance of alternative Ai with respect to criterion Cj is denoted as xij, in an aggregated matrix X.

• Step 3. Let Ωb be the collection of benefit criteria (i.e., the larger Cj, the greater preference) and Ωc be the collection of cost criteria (i.e., the smaller Cj, the greater preference). The IHFS positive-ideal solution (IHFS-PIS), denoted as $A˜+=(V˜1+V˜2+…V˜n+)$, and the IHFS negative-ideal solution (IHFS-NIS), denoted as $A˜-=(V˜1-V˜2-…V˜n-)$, are defined as follows:

$A˜+=[(x,x′)∣x∈HSijl and x′∈HSij′l∀iand (maxl=1K (maxi(min HSijl))≤x,x≤maxl=1K (maxi(max HSijl)),minl=1K (mini(min HSij′l))≤x′,x′≤minl=1K (mini(max HSij′l)))∣j∈Ωb,(x,x′)∣x∈HSijl and x′∈HSij′l∀iand (minl=1K (mini(min HSijl))≤x,x≤minl=1K (mini(max HSijl)),maxl=1K (maxi(min HSij′l))≤x′,x′≤maxl=1K (maxi(max HSij′l)))∣j∈Ωc]i=1,2,…,m, and j=1,2,…,n.$$A˜+=(V˜1+V˜2+…V˜n+)$$A˜-=[(x,x′)∣x∈HSijl and x′∈HSij′l∀iand (maxl=1K (maxi(min HSijl))≤x,x≤maxl=1K (maxi(max HSijl)),minl=1K (mini(min HSij′l))≤x′,x′≤minl=1K (mini(max HSij′l)))∣j∈Ωc,(x,x′)∣x∈HSijl and x′∈HSij′l∀iand (minl=1K (mini(min HSijl))≤x,x≤minl=1K (mini(max HSijl)),maxl=1K (maxi(min HSij′l))≤x′,x′≤maxl=1K (maxi(max HSij′l)))∣j∈Ωb]i=1,2,…,m, and j=1,2,…,n.$$A˜-=(V˜1-V˜2-…V˜n-)$

• Step 4. Construct positive ideal separation matrix (D+) and negative ideal separation matrix (D) which are defined as follows:

$D+=[d(x11,V˜1+)+⋯+d(x1n,V˜n+)d(x21,V˜1+)+⋯+d(x2n,V˜n+) ⋮ ⋮ ⋮d(xm1,V˜1+)+⋯+d(xmn,V˜n+)]$

and

$D-=[d(x11,V˜1-)+⋯+d(x1n,V˜n-)d(x21,V˜1-)+⋯+d(x2n,V˜n-) ⋮ ⋮ ⋮d(xm1,V˜1-)+⋯+d(xmn,V˜n-)]$

• Step 5. Calculate the relative closeness (RC) coefficient of each alternative to the ideal solution as follows:

$RC(Ai)=Di-Di++Di-,i=1,2,…,m,$

where $Di-=∑j=1nd(xij,V˜j-)$ and $Di+=∑j=1nd(xij,V˜j+)$.

• Step 6. Rank all the alternatives Ai (i = 1, 2, . . ., m) according to the RC(Ai) coefficient, greater the value RC(Ai), better the alternative Ai.

Graphical representation of this proposed technique is given in Figure 1.

### 4. Illustrative Example

In this section, we give an example. We utilized the method proposed in Section 4 to get the most desirable alternative as well as rank the alternatives from the best to worst or vice versa. Consider five schools (School of Business and Economics (A1), School of Science and Technology (A2), School of Social Sciences and Humanities (A3), School of Communication and Cultural Studies (A4), School of Textile and Design (A5)) all in the same university. Management of the university want to manage the allocation of funds to these schools based on their performance. There are four criteria (expenses of school (C1), students in take per year in school (C2), publications from school (C3), covered area of the school (C4)) for assessing the performance of these five schools. These assessments are given by the nine members from board of directors.

• Step 1. There are five possible alternatives Ai (i = 1, 2, 3, 4, 5) are to be evaluated on the criteria Cj (j = 1, 2, 3, 4) using the IHFS by nine experts eK (K = 1, 2, . . ., 9), as listed in Tables 13.

• Step 2. The decision matrix constructed in Table 4 by utilize Tables 13.

• Step 3. For cost criteria C1, C4 and benefit criteria C2, C3 IHFS-PIS is as follows:

$A+=[((0,0.1,0.2),(0.6,0.7))((0.6,0.7,0.8,0.9,1),(0))((0.7,0.8,0.9),(0,0.05))((0,0.1),(0.7,0.8))]$

For cost criteria C1, C4 and benefit criteria C2, C3 IHFS-NIS is as follows:

$A-=[((0.7,0.8,0.9),(0))((0.1,0.2),(0.6,0.65,0.7))((0,0.1),(0.7,0.8))((1),(0))]$

• Step 4. Positive ideal separation matrix (D+):

$D+=[0.4+0.2+0.6+0.40.3+0.6+0.4+0.70.35+0.6+0.25+0.10.7+0.6+0.8+0.70.8+0.6+0.7+0.9]D+=[1.621.32.83.0]$

Negative ideal separation matrix (D):

$D-=[0.5+0.6+0.3+0.70.6+0.5+0.65+0.50.5+0.5+0.6+0.90.1+0.4+0.2+0.50.3+0.3+0.2+0.3]D-=[2.12.252.51.21.1]$

• Step 5.RC of each alternative to the ideal solutions:

• RC(A1) = 2.1/(1.6+2.1) = 0.5676;

• RC(A2) = 2.25/(2+2.25) = 0.5294;

• RC(A3) = 2.5/(1.3+2.5) = 0.6579;

• RC(A4) = 1.2/(2.8+1.2) = 0.3;

• RC(A5) = 1.1/(3+1.1) = 0.2683.

• Step 6. Rank all the alternatives Ai (i=1, 2, . . ., 5) according to the closeness coefficient RC(Ai):

• A3A1A2A4A5.

Thus the most desirable alternative is A3. So the rector will allocate the funds according to this ranking.

### 5. Conclusions

IHFS is the best way to deal with uncertainty when fuzzy set theory is not able to cope with the situation. Decision makers gave their opinions about the criteria of alternatives by IHFS. Multi-criteria analysis provides an effective frame work for the evaluation of alternatives. Fuzzy TOPSIS method is proposed for IHFS to solve multi-criteria decision-making problem in group decision environment. The RC coefficient has ranked the alternatives from the best to worst by considering the smallest distance from the PIS and also the largest distance from the NIS. In future, we plan to continue to study Choquet integral based TOPSIS for IHFS. Furthermore, algebraic operations for IHFS will also be develop.

### Fig 1.

Figure 1.

Graphical structure of technique for order preference by similarity to ideal solution.

The International Journal of Fuzzy Logic and Intelligent Systems 2014; 14: 181-187https://doi.org/10.5391/IJFIS.2014.14.3.181

Decision matrix (1) with respect to experts 1, 2, 3, 4 (e1, e2, e3, e4).

C1
A1((0.5,0.6,0.8),(0.1,0.2))
A2((0.1,0.3),(0.3,0.4,0.5))
A3((0.5,0.7),(0.2,0.25))
A4((0.7,0.9),(0.05,0.1))
A5((1),(0))
C2
A1((0.6,0.8),(0.1,0.2))
A2((0.5,0.7,0.8),(0.1))
A3((0.5,0.6),(0.2,0.35))
A4((0.1,0.2),(0.6,0.7))
A5((0.1,0.3),(0.5,0.65))
C3
A1((0.1,0.3),(0.6,0.7))
A2((0.5,0.6),(0.1,0.3))
A3((0.7,0.9),(0.05,0.1))
A4((0.1,0.3),(0.6,0.7))
A5((0,0.2),(0.7,0.8))
C4
A1((0.1,0.3),(0.5,0.6))
A2((0.5,0.6),(0.2,0.3))
A3((0.1,0.2),(0.6,0.7))
A4((0.5,0.6,0.7),(0.2))
A5((0.4,0.7),(0.1,0.2))

Decision matrix (2) with respect to experts 5, 6, 7 (e5, e6, e7).

C1
A1((0.1,0.2),(0.5,0.6))
A2((0,0.2),(0.6,0.7))
A3((0.4,0.6),(0.1,0.3))
A4((0.6,1),(0))
A5((0.5,0.7),(0.1,0.2))
C2
A1((0.4,0.9),(0,0.1))
A2((0.1,0.3),(0.5,0.6))
A3((0.1,0.2),(0.6,0.7))
A4((0.4,0.7),(0.1,0.2))
A5((0.4,0.6),(0.2,0.3))
C3
A1((0,0.2),(0.6,0.7))
A2((0.4,0.5),(0.3,0.4))
A3((0.4,0.6),(0.3,0.4))
A4((0,0.1),(0.6,0.8))
A5((0,0.1),(0.7,0.8))
C4
A1((0.4,0.6),(0.1,0.3))
A2((0.6,1),(0))
A3((0,0.2),(0.5,0.7))
A4((0.5,0.7),(0.2,0.3))
A5((0.6,1),(0))

Decision matrix (3) with respect to experts 8, 9 (e8, e9).

C1
A1((0.4,0.6),(0.2,0.3))
A2((0.3,0.6),(0.2,0.3))
A3((0.1,0.3),(0.5,0.6))
A4((0.6,0.9),(0,0.1))
A5((0.5,0.6),(0.3,0.4))
C2
A1((0.6,1),(0))
A2((0.1,0.3),(0.6,0.7))
A3((0.6,0.9),(0,0.1))
A4((0.5,0.7),(0.1,0.2))
A5((0.1,0.3),(0.6,0.7))
C3
A1((0.3,0.5),(0.3,0.4))
A2((0.5,0.9),(0,0.05))
A3((0.3,0.7),(0.1,0.2))
A4((0,0.2,0.4),(0.4,0.5))
A5((0.2,0.4),(0.4,0.5))
C4
A1((0,0.3),(0.5,0.6))
A2((0.3,0.5),(0.3,0.4))
A3((0,0.1),(0.7,0.8))
A4((0.5,0.6,0.8),(0))
A5((1),(0))

Decision matrix (X).

C1
A1((0.2,0.4,0.5),(0.2,0.3,0.5))
A2((0.2,0.3),(0.3,0.4,0.5,0.6))
A3((0.3,0.4,0.5),(0.25,0.3,0.5))
A4((0.7,0.9),(0,0.05))
A5((0.6,0.7,1),(0,0.1,0.2,0.3))
C2
A1((0.6,0.8),(0,0.1))
A2((0.3,0.5),(0.1,0.5,0.6))
A3((0.2,0.5,0.6),(0.1,0.2,0.35,0.6))
A4((0.2,0.4,0.5),(0.2,0.6))
A5((0.3,0.4),(0.3,0.5,0.6))
C3
A1((0.2,0.3),(0.4,0.6))
A2((0.5),(0.05,0.1,0.3))
A3((0.6,0.7),(0.1,0.2,0.3))
A4((0.1),(0.5,0.6))
A5((0.1,0.2),(0.5,0.7))
C4
A1((0.3,0.4),(0.3,0.5))
A2((0.5,0.6),(0,0.2,0.3))
A3((0.1),(0.7))
A4((0.5,0.6,0.7),(0,0.2))
A5((0.7,1),(0,0.1))

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