A Generalized Multi-Attribute Group Decision Making with Intuitionistic Fuzzy Set

A Generalized Multi-Attribute GroupDecision Making with IntuitionisticFuzzy Set

Zhifu Tao, Huayou Chen, Weiyuan Zou, Ligang Zhouand Jinpei Liu

Abstract The aim of this paper is to introduce a new decision making model calledthe generalized multi-attribute group decision making (GMAGDM), which provides avery general form that includes multi-attribute decision making (MADM) and multi-attribute group decision making (MAGDM) as two special cases. A GMAGDM underintuitionistic fuzzy environment is proposed. The relation between intuitionistic fuzzyset and hesitant fuzzy set is utilized. Then we propose a hesitant fuzzy TOPSIS methodfor the solution of the mentioned problem. Finally, a numerical example is given toillustrate the flexibility and validity of the proposed approach.

Keywords Generalized multi-attribute group decision making � MADM �MAGDM � Intuitionistic fuzzy set � Hesitant fuzzy TOPSIS

1 Introduction

Many applications involve the selection or ordering of a group of alternativesbased upon their satisfaction to a collection of criteria or a number of experts,which is normally shown as a MADM problem or a MAGDM process. MAGDM

Z. Tao � H. Chen (&) � W. Zou � L. ZhouSchool of Mathematical Science, Anhui University, Hefei, Anhui 230601, Chinae-mail: [email protected]

Z. Taoe-mail: [email protected]

W. Zoue-mail: [email protected]

L. Zhoue-mail: [email protected]

J. LiuSchool of Business, Anhui University, Hefei, Anhui 230601, Chinae-mail: [email protected]

Z. Yin et al. (eds.), Proceedings of The Eighth International Conferenceon Bio-Inspired Computing: Theories and Applications (BIC-TA), 2013,Advances in Intelligent Systems and Computing 212,DOI: 10.1007/978-3-642-37502-6_8, � Springer-Verlag Berlin Heidelberg 2013

63

consists of multiple decision makers (DMs) interacting to reach a final decision,while MADM can be seen as a special condition of MAGDM where there is onlyone DM. For the sake of comprehensive and as commonly applied in the decisionmaking literatures, the case of MAGDM model is considered in this paper. Thecase of MADM model is a straightforward simplicity of this methodology.

Dozens of studies have shown that there are several aspects to attract people’sattention. They are the formats of decision information, the construction ofaggregation methods and the application of the model in real problems.

Classically in MAGDM models, much of the early work focus on individualmember preference as the legitimate inputs for group choice. Traditionally, thepreference information provided by DMs is represented in the same format. But inpractical group decision making applications, due to different cultural and edu-cational backgrounds of DMs, it’s necessary to consider different preference for-mats in MAGDM. Commonly used formats include: preference orderings, utilityvalues, multiplicative preference relations, fuzzy preference relations and lin-guistic variables, etc. [1–6]. Totally, these types of preference information areshown with scalars. However, in real-world applications, the available informationis often uncertain and fuzzy and thus, it’s necessary to use another approach that isable to assess the uncertainty and vagueness in decision making. Zadeh [7] firstlyintroduced the fuzzy set to illustrate this fuzziness, which minimized the differencebetween the model and real decision process. Since then, some other forms offuzzy decision information are proposed to illustrate different kinds of uncertainty,including the interval valued fuzzy sets (IVFs) [8], the intuitionistic fuzzy set (IFs)[9], the interval valued intuitionistic fuzzy set (IVIFs) [10], the hesitant fuzzy set(HFS) [11], etc. Fuzzy information has widely been taken into consideration todeal with imprecision of information in decision making.

In order to construct information aggregation approach for decision. In Yager[12], Yager introduced the definition of ordered weighted averaging (OWA)operator, which was contributed to cut down the negative affection of certainoperation in decision making process. Related operator theories have been widelystudied and applied in decision making field ([13–20]). In the meanwhile, someother ways to utilize decision information have also been developed, such asentropy/cross entropy method, programming methods, TOPSIS theory, ELECTREI-IV, IS and TRI theory, Promethee method, etc. [19, 21–23].

When to the application of decision making theory in real-world, most crucialissues are to choose a right one from several alternatives. It can be divided aseconomic decision making, political decision making, etc. while combine thedecision making process with actual application backgrounds [13–19, 21–27].

However, in the real process of decision making problems, a normal and specialcondition is that several DMs are often faced with different numbers of attributes.For example, a project needs three departments to decide whether it should beimplemented, the three departments are technology department (Responsible forthe assessment of technology and environment), design department (Check theeconomical benefits, technical feasibility and social affection of the project) andadministration (Give evaluation to all properties include economic, environment,

64 Z. Tao et al.

technical and social). Then this is a special decision making problem, but it has notbeen researched before.

The aim of this paper is to propose the generalized multi-attribute groupdecision making method in which the attributes provided to DMs are not the same.We can obtain a generalization that includes normal MADM and MAGDM as itsspecial cases. The rest of the paper is organized as follows: The following sectionprovides some basic concepts of fuzzy set theory as preliminaries. In Sect. 3, theconcept of this new type of multi-attribute group decision making process isproposed. Section 4 mainly gives the method to deal with such kind of decisionmaking problem. The application of the proposed decision making method isaddressed in Sect. 5. Finally, some conclusions are presented at the end.

2 Preliminaries

This section mainly provides with basic concepts for further discussions.

Definition 2.1 An intuitionistic fuzzy set, IFS for short, A on the universe ofdiscourse U is defined as an object of the form A = {\x,uA(x),v A(x) [ |X 2U},where uA(x)and v A(x) represent the degrees of membership and non-membershipof the element x to the set A, respectively, satisfying 0 � uA(x) ? v A(x) � 1.

As another extension and generalization of traditional fuzzy sets, hesitant fuzzyset was proposed by Torra and Narukawa [11]; it permitted the membership degreeof an element to a set to be presented as several possible values between 0 and 1,which can be shown as follows:

Definition 2.2 Let X be a fixed set, a hesitant fuzzy set (HFS for short) on X is interms of a function that when applied to X returns a subset of [0, 1], which can berepresented as E = {\x,hE(x) [ |x 2X}, where hE(x) is a set of values in [0, 1],denoting the possible membership degree of the element x 2 X to the set E. Forconvince, we call hE(x) a hesitant fuzzy element (HFE).

Torra [11] showed that IFS could be transformed to be a HFS, which can beexpressed as follows:

Definition 2.3 Given an IFS represented by{\ x,uA(x),v A(x)}, the definition ofcorresponding HFE is straightforward: h(x) = [uA(x),1-v A(x)] if uA(x) 6¼1-v A(x).

3 Generalized Multi-Attribute Group Decision Makingwith Intuitionistic Fuzzy Numbers

3.1 Concept of GMAGDM

The concept of generalized multi-attribute group decision making (GMAGDM)can be defined as follows:

A Generalized Multi-Attribute Group Decision Making 65

Definition 3.1 Let x1; x2; � � � ; xmf g are m alternatives, e1; e2; � � � ; elf g are l deci-sion makers, and Ai ¼ a1; a2; � � � ; amif g is the corresponding attribute set of expertei, where Ai \ Aj i 6¼ j; i; j 2 1; 2; � � � ; lf gð Þ could be an empty set or a non-empty

set andSl

i¼1Ai ¼ A(A ¼ a1; a2; � � � ; anf g is attribute set, and Ai;Aj are subsets of

A. Then, a generalized multi-attribute group decision making is aggregating thedecision information of all alternatives under the attribute set Ai of the expertei; i ¼ 1; 2; � � � ; l.

Note 1 The GMAGDM degenerates to be a MADM when Ai \ Aj ¼ ;.

Actually, Ai \ Aj ¼ ; means that all the alternatives are judged by l decisionmakers under different attributes, which is the same with MADM process whilethinking about the weights of decision makers.

Note 2 The GMAGDM degenerates to be a MAGDM when Ai = Aj.

In fact, Ai = Aj illustrates that the attributes provided to each decision makerare the same, which is the same with the MAGDM problem.

Note 3 Definition 3.1 and note 1, 2 indicate that the MADM and MAGDM aretwo special cases of the GMAGDM.

In this paper, we mainly discuss a GMAGDM problem under intuitionisticenvironment.

3.2 Hesitant Fuzzy TOPSIS for GMAGDMwith Intuitionistic Fuzzy Numbers

On the basis of relation between IVFs and HFs and fuzzy TOPSIS theory, we nowdevelop a simple and straightforward method for GMAGDM with intuitionisticfuzzy decision information. The method is shown as follows:

Step 1. Construct aggregated intuitionistic fuzzy decision matrix based on theopinions of decision makers.

For a GMAGDM problem, let X¼ x1; � � � ; xmf g be a finite set of alternatives,and E¼ e1; � � � ; elf g be the set of DMs/experts. Given attribute set of i-th DM/expert is Ai i ¼ 1; 2; � � � ; lð Þ, where Ai;Aj i 6¼ j; i; j 2 1; 2; � � � ; lf gð Þ are not requiredto be the same. The decision makers ej j ¼ 1; 2; � � � ; lð Þ provides their judgment ofalternatives under attribute set Aj, which can be described as Dj ¼ di;KAj

, where

KAj is subscript set of attributes in Aj, and di;KAjare intuitionistic fuzzy numbers.

66 Z. Tao et al.

Step 2. Construct aggregated hesitant fuzzy decision matrix.

Collect the values of alternatives under each attribute according to

Dj j ¼ 1; 2; � � � ; lð Þ; denoted as d 2ð Þij ¼ dij i ¼ 1; 2; � � � ;m; j ¼ 1; 2; � � � ;ð

n 2Tl

k¼1KAkÞ, here dij is the intuitionistic value of alternative xi i ¼ 1; 2; � � � ;mð Þ

under attribute aj j ¼ 1; 2; � � � ; nð Þ. As a result, there is no less than one value of analternative under an attribute, and the values are intuitionistic fuzzy numbers.

Therefore, change d 2ð Þij to be HFSs h

d 2ð Þij

based on Definition 3.1, and then the

GMAGDM problem with intuitionistic fuzzy numbers comes to be a normalMADM with hesitant fuzzy attribute values.

Step 3. Obtain hesitant fuzzy positive-ideal and negative-ideal solution.

Extend the hesitant fuzzy numbers under the same attribute so that they havethe same length and choose the positive and negative hesitant fuzzy numbers ofeach attribute based on Definition 2.3, denoted as

hþ¼ max1� i�m

hd 2ð Þ

i1; max

1� i�mh

d 2ð Þi2; � � � ; max

1� i�mh

d 2ð Þin

� �

; h�¼ min1� i�m

hd 2ð Þ

i1; min

1� i�mh

d 2ð Þi2; � � � ; min

1� i�mh

d 2ð Þin

� �

:

Step 4. Calculate the separation measures.

In order to measure separation between HFSs, Xu and Xia [28] proposed theextensions of the well-known distance measures such as Hamming distance,Euclidean distance, and Hausdorff metric under hesitant fuzzy environment, In thispage, we use normalized Hamming distance [28]

dhnh M;Nð Þ ¼ 1n

Xn

i¼1

1lxi

Xlxi

j¼1

hr jð ÞM xið Þ�hr jð Þ

N xið Þ��

��

" #

:

where hM; hN are two HFSs, hr jð ÞM ; hr jð Þ

N are reorders of the elements in hM; hN .Compute the distance between each alternative and the positive and negative

hesitant fuzzy numbers, denoted as d xi; hþð Þ and d xi; h�ð Þ, respectively.

Step 5. Calculate the relative closeness coefficient to the ideal solution.

For each alternative, determine a ratio R equal to the distance to the negativehesitant fuzzy number divided by the sum of the distance to the negative hesitantfuzzy number and the distance to the positive hesitant fuzzy number,

R xið Þ ¼ d xi; h�ð Þ= d xi; h

þð Þþd xi; h�ð Þð Þ:


Step 6. Rank the alternatives.

After the relative closeness coefficient of each alternative is determined,alternatives are ranked according to descending order of the ratio in Step 5.

4 Numerical Example

A company desires to select a new information system. Four alternatives xi haveremained in the candidate list after preliminary selection. Three departments ej areresponsible to make the final decision; they are Technology Department (TD),Evaluation Department (ED) and Administration Department (AD). Totally thereare four attributes need to be considered: costs of investment (a1); contribution tothe company (a2); effort to transform from current system (a3); and outsourcingsoftware developer reliability (a4).A1 = {a1,a3}, A2 = {a2,a3,a4},A3 = {a1,a2,a3,a4} are three departments’ attribute set, respectively. The expertsevaluate the software packages with respect to their corresponding attribute sets,and construct the following fuzzy decision matrices Dj(j = 1,2,3) (See Table 1).

Assume that the weights of the experts are the same because they are experts intheir own fields; we utilize the former approach to select the software packages:

Step 1. Through the description of the problem, decision matrices Dj(j = 1,2,3)are the corresponding intuitionistic fuzzy numbers in Table 1.

Step 2. Collect the values of alternatives, d 2ð Þij can be described as:

d 2ð Þi1 ¼

0:5; 0:4; 0:1h i 0:4; 0:5; 0:1h i0:4; 0:5; 0:1h i 0:4; 0:6; 0:0h i0:8; 0:2; 0:0h i 0:7; 0:3; 0:0h i0:5; 0:3; 0:2h i 0:7; 0:2; 0:1h i

0

BB@

1

CCA;

d 2ð Þi2 ¼

0:5; 0:5; 0:0h i 0:3; 0:5; 0:2h i 0:5; 0:4; 0:1h i0:6; 0:4; 0:0h i 0:5; 0:3; 0:2h i 0:7; 0:3; 0:0h i0:7; 0:3; 0:0h i 0:5; 0:2; 0:3h i 0:6; 0:4; 0:0h i0:6; 0:2; 0:2h i 0:7; 0:3; 0:0h i 0:5; 0:3; 0:2h i

0

BB@

1

CCA;

Table 1 Intuitionistic fuzzy decision matrix (unit: 10-1)

Decision matrix D1 Decision matrix D2 Decision matrix D3

a1 a2 a2 a3 a4 a1 a2 a3 a4

x1 (5,4,1) (5,5,0) (3,5,2) (5,2,3) (4,5,1) (4,5,1) (5,4,1) (6,2,2) (3,5,2)x2 (4,5,1) (6,4,0) (5,3,2) (2,6,2) (6,4,0) (4,6,0) (7,3,0) (3,5,2) (5,5,0)x3 (8,2,0) (7,3,0) (5,2,3) (4,4,2) (4,6,0) (7,3,0) (6,4,0) (3,5,2) (6,2,2)x4 (5,3,2) (6,2,2) (7,3,0) (4,2,4) (7,1,2) (7,2,1) (5,3,2) (9,1,0) (6,4,0)

68 Z. Tao et al.

d 2ð Þi3 ¼


0

BB@

1

CCA;

d 2ð Þi4 ¼


0

BB@

1

CCA

Then change them to be hesitant fuzzy values and shows as follows:

hd 2ð Þ

i1¼

0:4; 0:5; 0:6ð Þ0:4; 0:5ð Þ0:7; 0:8ð Þ

0:5; 0:7; 0:8ð Þ

0

BBB@

1

CCCA; h

d 2ð Þi2¼

0:3; 0:5; 0:6ð Þ0:5; 0:6; 0:7ð Þ

0:5; 0:6; 0:7; 0:8ð Þ0:5; 0:6; 0:7; 0:8ð Þ

0

BBB@

1

CCCA;

hd 2ð Þ

i3¼

0:5; 0:6; 0:8ð Þ0:2; 0:3; 0:4; 0:5ð Þ0:3; 0:4; 0:5; 0:6ð Þ

0:4; 0:8; 0:9ð Þ

0

BBB@

1

CCCA; h

d 2ð Þi4¼

0:3; 0:4; 0:5ð Þ0:5; 0:6ð Þ

0:4; 0:6; 0:8ð Þ0:6; 0:7; 0:9ð Þ

0

BBB@

1

CCCA:

Step 3. Extend hesitant fuzzy numbers under each attribute and indentify thepositive and negative hesitant fuzzy set, denoted as follows:

d+ = ((0.7,0.8,0.8),(0.5,0.6,0.7,0.8),(0.4,0.8,0.9,0.9),(0.6,0.7,0.9));d- = ((0.4,0.5,0.5),(0.3,0.5,0.6,0.6),(0.2,0.3,0.4,0.5),(0.3,0.4,0.5)).

Step 4. Compute the distance between each alternative and separation measures.

d(x1,h-) = 0.0896, d(x1,h+) = 0.2063; d(x2,h-) = 0.0729, d(x2,h+) = 0.2229;d(x3,h-) = 0.1875, d(x3,h+) = 0.1083; d(x4,h-) = 0.2708, d(x4,h+) = 0.2958.

Step 5. For each alternative, indentify the ratio R, and the result is,

R(x1) = 0.3028, R(x2) = 0.2465, R(x3) = 0.6338, R(x4) = 0.9155.

Step 6. Rank the software according to the result in Step5, the order should be:x4 � x3 � x1 � x2:

where‘‘�’’ denotes ‘‘be superior to’’. Therefore, x4 is the best alternative.By comparison with the traditional information aggregation approach, the

proposed GMAGDM model presents some advantages:

• A lot of computation among intuitionistic fuzzy numbers and possible loss ofinformation are avoided.

• The application of hesitant fuzzy set is established, through which one can alsofound the relation of two fuzzy sets in real, but not just stay in the written.


5 Conclusions

In this paper, we introduced the GMAGDM model, which can be regarded as ageneralization of normal MADM and MAGDM where attribute sets provided forDMs are not required to be the same. The characteristic of IVFs and HFs wereutilized, and the hesitant fuzzy TOPSIS theory was introduced and applied in thesolving of GMAGDM. Numerical example illustrated that this method had someadvantages and we also proposed a new type of group decision making.

Acknowledgments The work is supported by National Science Foundation of China (Grant No.71071002), the academic innovation team of Anhui University (KJTD001B, SKTD007B), thefoundation for the young scholar of Anhui University (Grant No. 2009QN022B), and theundergraduate students’ innovative project of Anhui University (cxcy2012002).

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A Generalized Multi-Attribute Group Decision Making with Intuitionistic Fuzzy Set

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