Expert Systems with Applications 38 (2011) 6985–6993

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Expert Systems with Applications

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

A new linguistic MCDM method based on multiple-criterion data fusion

Yong Deng a,c,⇑,1, Felix T.S. Chan b,⇑,1, Ying Wu a, Dong Wang a

a School of Electronics and Information Technology, Shanghai Jiao Tong University, Shanghai 200030, Chinab Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kongc College of Computer and Information Sciences, Southwest University, Chongqing 400715, China

a r t i c l e i n f o

Keywords:MCDMDempster–Shafer evidence theoryFuzzy sets theory

0957-4174/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.eswa.2010.12.016

⇑ Corresponding authors. Address: School of Electrnology, Shanghai Jiao Tong University, Shanghai 2000

E-mail addresses: dengyong@sjtu.edu.cn, ydeng@sn@inet.polyu.edu.hk (F.T.S. Chan).

1 These authors contributed equally to this work.

a b s t r a c t

Multiple-criteria decision-making (MCDM) is concerned with the ranking of decision alternatives basedon preference judgements made on decision alternatives over a number of criteria. First, taking advantageof data fusion technology to comprehensively consider each criterion data is a reasonable idea to solvethe MCDM problem. Second, in order to efficiently handle uncertain information in the process of deci-sion making, some well developed mathematical tools, such as fuzzy sets theory and Dempster Shafertheory of evidence, are used to deal with MCDM. Based on the two main reasons above, a new fuzzy evi-dential MCDM method under uncertain environments is proposed. The rating of the criteria and theimportance weight of the criteria are given by experts’ judgments, represented by triangular fuzzy num-bers. Then, the weights are transformed into discounting coefficients and the ratings are transformed intobasic probability assignments. The final results can be obtained through the Dempster rule of combina-tion in a simple and straight way. A numerical example to select plant location is used to illustrate theefficiency of the proposed method.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Multi-criteria decision-making (MCDM) is one of the mostwidely used decision methodologies in science, business, and engi-neering worlds. MCDM methods aim at improving the quality ofdecisions by making the process more explicit, rational, and effi-cient. Due to the uncertain information in the process of decisionmaking, fuzzy sets theory (FST) is extensively used to deal withMCDM problems. On the other hand, it is well known that there al-ways exits conflicts between criteria in MCDM. For example, in thecase of plant location selection, if we pay more attention about itsexpansion possibility, we have to increase the investment cost.Thus, taking advantage of data fusion technology to comprehen-sively consider each criterion data is a reasonable idea to solveMCDM problem. Recently, many researches are inspired by theapplication of Dempster–Shafer evidence of theory (DSeT), one ofthe widely used mathematical tools in data fusion, to handle theMCDM problem (Bauer, 1997; Beynon, 2002, 2005a, 2005b;Beynon, Curry, and Morgan, 2000; Beynon, Cosker, and Marshall,2001; Yang and Sen, 1997; Yang and Xu, 2002) and risk analysis

ll rights reserved.

onics and Information Tech-30, China (Y. Deng).wu.edu.cn (Y. Deng), mffcha-

(Beynon, 2005a, 2005b; Demotier, Schon, and Denoeux, 2006;Kangas and Kangas, 2004; Sadiq, Kleiner, and Rajani, 2006; Sadiq,Kleiner, and Rajani, 2007; Sun, Srivastava, and Mock, 2006; Sii,Ruxton, and Wang, 2002).

In general, a multiple criteria decision-making (MCDM) prob-lem can be concisely expressed in matrix format as

ð1Þ

where A1,A2, . . . ,Am are possible alternative, C1,C2, . . . ,Cn are criteriawith which performance of alternatives are measured, xij is the rat-ing of alternative Ai with respect to criteria Cj. Due to the uncer-tainty, the decision maker prefers to give his opinions in linguisticitems. Hence, the rating rij of alternative Ai and the weights of thecriteria are assessed in linguistic terms represented as triangularfuzzy numbers.

In this paper, we proposed a fuzzy evidential model to deal withthe MCDM problem. The main idea is that the performance of eachcriterion can be fused to get a comprehensive evaluation. The newmethodology is taking advantage of two desirable properties. First,the present model can efficiently represent experts’ opinionsthrough fuzzy sets theory (FST). For example, the rating rij of

1.0

0.1 0.2 0.3 0.4

A

6986 Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993

alternative Ai and the weights of the criteria are assessed in linguis-tic terms given by the decision maker. Second, the performance ofthe criterion given by domain experts can be fused through DSeT inour proposed model. Hence, the final decision result takes theconsideration of all the criteria in MCDM in a comprehensive way.

This paper is arranged into five sections. First, in Section 2 webriefly introduce some preliminaries including FSS and DSeT. Theintent of the Section 3 is to propose our fuzzy evidential method.A numerical example on the evaluation of main battle tanks is usedto illustrate the feasibility of the proposed method in Section 4.Finally, a conclusion is made in Section 5.

A=(0.1 0.2 0.3 0.4)

Fig. 1. A normal fuzzy number.

2. Preliminaries

In this section, we introduce some background of mathematicstools – fuzzy sets theory (FST) and Dempster–Shafer theory of evi-dence (DSeT) – that are used in the proposed method.

2.1. Fuzzy set theory

2.1.1. Basic definition

Definition 2.1 (Fuzzy set). Let X be a universe of discourse, eA is afuzzy subset of X if for all x 2 X, there is a number leAðxÞ 2 ½0;1�assigned to represent the membership of x to eA, and leAðxÞ is calledthe membership of eA (Zimmermann, 1991).

Definition 2.2 (Fuzzy number). A fuzzy number eA is a normal andconvex fuzzy subset of X. Here, the ‘‘Normality’’ implies that(Zimmermann, 1991)

9x 2 R; _xleAðxÞ ¼ 1

and ‘‘Convex’’ means that

8x1 2 X; x2 2 X; 8a 2 ½0;1�;leAðax1 þ ð1� aÞx2ÞP minðleAðx1Þ;leAðx2ÞÞ:

1.0

w

0.1 0.2 0.3 0.4

B=(0.1 0.2 0.3 0.4 w)

B

Fig. 2. A generalized fuzzy number.

Definition 2.3 (Generalized fuzzy number). A generalized fuzzynumber eA ¼ ða; b; c; d; wÞ is described as any fuzzy subset of thereal line R with membership function feA which has the followingproperties (Chen and Chen, 2003):

(a) feA is a continuous mapping from R to the closed interval[0,w], 0 6 w 6 1;

(b) feAðxÞ ¼ 0 for all x 2 (�1,a];(c) feA is strictly increasing on [a,b];(d) feAðxÞ ¼ w, for all x 2 [b,c], where w is a constant and

0 6 w 6 1;(e) feA is strictly decreasing on [c,d](f) feAðxÞ ¼ 0 for all x 2 (�1,a];

where 0 6 w 6 1, a, b, c and d are real numbers. A generalizedtrapezoidal fuzzy number can be defined as eA ¼ ða; b; c; d; wÞ,wherea 6 b 6 c 6 d, 0 6 w 6 1, and its membership function isdefined by

lAðxÞ ¼ feAðxÞ ¼0 x < aðx�aÞb�a a 6 x 6 b

w b 6 x 6 cðx�cÞd�c c 6 x 6 d

0 x > d

8>>>>>><>>>>>>:

ð2Þ

The generalized trapezoidal fuzzy number eA is plotted in Fig. 1. Ifw = 1, the generalized trapezoidal fuzzy number eA is called a normaltrapezoidal fuzzy number and often can be abbreviated aseA ¼ ða; b; c;dÞ, plotted in Fig. 2. If a = b andc = d, then eA is called acrisp or simple interval. Similarly, ifb = c, then eA is called a normaltriangular fuzzy number and if a = b = c = d, then eA becomes a realnumber. The membership function of the normal trapezoidal fuzzynumber eA ¼ ða; b; c;dÞ is defined by

lAðxÞ ¼ feAðxÞ ¼0 x < aðx�aÞb�a a 6 x 6 b

1 b 6 x 6 cðx�cÞd�c c 6 x 6 d

0 x > d

8>>>>>><>>>>>>:

ð3Þ

For example, Fig. 1 shows the normal fuzzy number A and Fig. 2shows the generalized fuzzy number B.

2.1.2. Linguistic variableThe concept of linguistic variable is very useful in dealing with

situations that are too complex or ill-defined to be reasonablydescribed in conventional quantitative expressions. Linguistic vari-ables are represented in words or sentences or artificial languages,where each linguistic value can be modeled by a fuzzy set (Zim-mermann, 1991). In this paper, the importance weights of variouscriteria and the ratings of qualitative criteria are considered as lin-guistic variables. For example, these linguistic variables can be ex-pressed in positive trapezoidal fuzzy numbers. Table 1 shows thelinguistic variables of the importance weight. Table 2 shows lin-guistic variables of the rating of criterion. It should be noticed thatthere are many different methods to represent linguistic items. Thekind of representing method used depends on the real applicationsystems and the domain experts’ opinions. Fig. 3 illustrate the se-ven trapezoidal fuzzy numbers of the ratings listed in Table 2.

Table 1Linguistic variables of the importance weight.

Linguistic variables for importance weights Fuzzy numbers

Very low (VL) (0,0,0.1,0.2)Low (L) (0.1,0.2,0.2,0.3)Medium low (ML) (0.2,0.3,0.4,0.5)Medium (M) (0.4,0.5,0.5,0.6)Medium high (MH) (0.5,0.6,0.7,0.8)High (H) (0.7,0.8,0.8,0.9)Very high (VH) (0.8,0.9,1,1)

Table 2Linguistic variables of the ratings.

Linguistic variables of the ratings Fuzzy numbers

Very poor (VP) (0,0,1,2)Poor (P) (1,2,2,3)Medium poor (MP) (2,3,4,5)Medium (M) (4,5,5,6)Medium good (MG) (5,6,7,8)Good (G) (7,8,8,9)Very good (VG) (8,9,10,10)

Fig. 3. A seven granular linguistic set about rating.

1

X

Fig. 4. Two overlapping fuzzy numbers.

Fig. 5. Two fuzzy numbers without overlapping area.

Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993 6987

2.1.3. Defuzzification of fuzzy numbersIn this paper, the mean value of fuzzy numbers is used as a

defuzzified value. For detailed information, please refer Zimmer-mann (1991).

Definition 2.4. Given trapezoidal fuzzy numbers eA ¼ ða1; a2; a3;

a4Þ, the mean value of the trapezoidal fuzzy numbers eA is definedas (Zimmermann, 1991)

PðeAÞ ¼ ða1 þ a2 þ a3 þ a4Þ4

: ð4Þ

For example, by applying Eq. (4), the mean value of linguistic itemof importance ‘‘Medium high’’ (MH) is

PðMHÞ ¼ ða1 þ a2 þ a3 þ a4Þ4

¼ ð0:5þ 0:6þ 0:7þ 0:8Þ4

¼ 6:5:

2.1.4. Similarity measure between fuzzy numbers

Similarity measure is an important concept, widely used in pat-tern recognition (Christopher, 2006). The similarity measure be-tween fuzzy numbers means the match degree between twofuzzy numbers. In our method, the BPA (The definition of BPAcan be seen in the following Section 2.2.) in Dempster–Shafer the-ory of evidence is determined by the similarity measure betweenfuzzy numbers. Hence, a good similarity measure is needed. Many

methods have been proposed to calculate the degree of similaritybetween fuzzy numbers (Chen, 1998; Chen & Chen, 2003; Deng,Shi, & Liu, 2004; Hsu & Chen, 1996; Liu, Xiong, & Deng, 2008).

Assume that there are two trapezoidal fuzzy numbers, whereeA ¼ ða1; a2; a3; a4Þ and eB ¼ ðb1; b2; b3; b4Þ, then the degree of simi-larity SðeA; eBÞ between the trapezoidal fuzzy numbers eA and eB canbe calculated as follows (Hsu & Chen, 1996):

SðeA; eBÞ ¼R

xðminfleAðxÞ;leBðxÞgÞdxRxðmaxfleAðxÞ;leBðxÞgÞdx

: ð5Þ

From Fig. 4, it can be seen that the more the interaction area of thetwo fuzzy numbers, the larger the value of SðeA; eBÞ. Several authorsargue about the limitation of area method (Chen & Chen, 2003;Deng et al., 2004; Liu et al., 2008). For example, as shown inFig. 5, the similarity measure between fuzzy numbers is 0. It maynot be suitable to some real situations. To overcome the drawbacksof the area method, center-of-gravity (CoG) method (Chen & Chen,2003) and radius-of-gravity (ROG) method (Deng et al., 2004) areproposed. How to select the reasonable similarity measure betweenfuzzy numbers is an open problem. In our opinion, the selection ofthe similarity measure depends on the real application environ-ments. For example, the area method to obtain similarity measureis more desirable than the other alternatives to get the basic prob-ability assignment (which is very important in the application ofDSeT) in our method. The reason is detailed in Section 2.2.

2.2. Dempster–Shafer theory of evidence

2.2.1. Basic definitionThe DSeT can be regarded as a general extension of the Bayesian

theory that can robustly deal with incomplete data. In addition tothis, DSeT offers a number of advantages, including the opportunityto assign measures of probability to focal elements, and allowingfor the attachment of probability to the frame of discernment. Inthis section, we briefly review the basic concepts of evidencetheory.

6988 Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993

The DSeT first defines a set of hypotheses H called frame of dis-cernment H = {H1,H2, . . . ,HN}. It is composed of N exhaustive andmutually exclusive hypotheses. The power set P(H) is composedof 2N propositions:

PðHÞ ¼ f;; fH1g; fH2g; . . . ; fHNg; fH1 [ H2g; fH1 [ H3g . . . ;Hg ð6Þ

where ; denotes an empty set. The subsets containing only oneelement are called singletons. A key concept in DST is the basicprobability assignment (BPA). The BPA for an element of H is sim-ilar to probability, but differs by the fact that the unit mass isdistributed among the elements of P(H), that is to say not onlyon the singletons in HN in H but also on composite hypothesestoo.

Definition 2.5 (Basic Probability Assignment, BPA). Given the frameof discernment H = {H1,H2, . . . ,HN}, A BPA function is defined by:

m : PðHÞ ! ½0;1� ð7Þ

and which satisfies the following conditions:

PA2PðHÞ

mðAÞ ¼ 1

mð;Þ ¼ 0ð8Þ

Definition 2.6 (Belief Function, Bel). Given the frame of discernmentH = {H1,H2, . . . ,HN} and the BPA defined in Definition 2.5, the belieffunction is defined by

BelðBÞ ¼Xn

i¼1

mðAiÞ; Ai # B ð9Þ

Definition 2.7 (Pluasibility Function, Pl). Given the frame of discern-ment H = {H1,H2, . . . ,HN} and the BPA defined in Definition 2.5, theplausibility function is defined by

BelðBÞ ¼Xn

i¼1

mðAiÞ; Ai \ B – ; ð10Þ

It can be seen that Bel and Pl some times can act as interval proba-bility (shown in Fig. 6). The Bel is the low probability and the Pl isthe up probability. The next example show the efficiency to dealwith uncertain information based on DSeT.

Example 2.1. Suppose that there are two types of balls in twoboxes. One type of ball is red and the other type of ball is green.All the balls in box A are red balls. However, box B contains bothred balls and green balls. In addition, we do not know the exactnumber of red balls and green balls. What’s more, we do not knowthe ratio of red balls and green balls in box B. What we know is thatthe probability of Box A being selected is 0.7 and the probability ofBox B being selected is 0.3. The question is that if we select a ball,what’s the probability of that ball being red?

Fig. 6. Interval probability of hypothesis A.

If we use the probability, the process may be complex to somedegree. However, if we use DSeT, we can easily obtain the followingresults:

M(red) = 0.7; M(red,green) = 0.3.Hence, Bel(red) = 0.7, Pl(red) = 0.7 + 0.3 = 1.

The final result can be shown that the probability is an interval[0.7,1].

Example 2.2. Suppose the frame of discernment is {a,b,c} and BPAis as follows M(a) = 0.7; M(b) = 0.2; M(c) = 0.1.

Then, the Bel and Pl can be calculated as follows:

Bel(a) = Pl(a) = M(a) = 0.7;Bel(b) = Pl(b) = M(b) = 0.2;

Bel(c) = Pl(c) = M(c) = 0.1;

As can be seen from above, if the BPA is assigned in singletons,the low probability is equal to up probability. In this situation,the interval probability is shortened as a point and the BPA isdegenerated as probability measure.

2.2.2. Dempster rule of combinationIn the case of imperfect data (uncertain, imprecise and incom-

plete), fusion is an interesting solution to obtain more relevantinformation. Evidence theory offers appropriate aggregation tools.From the basic belief assignment denoted by mi obtained for eachinformation source Si, it is possible to use a combination rule in or-der to provide combined masses synthesizing the knowledge of thedifferent sources. Dempster rule of combination (also calledorthogonal sum), denoted bym = m1 �m2, is the most commoncombination rule for two BPAs m1 and m2 to yield a new BPA:

mðAÞ ¼P

B\C¼Am1ðBÞm2ðCÞ1� k

ð11Þ

with

k ¼X

B\C¼;m1ðBÞm2ðCÞ ð12Þ

where k is a normalization constant, called conflict because it mea-sures the degree f conflict between m1 and m2, k = 0 corresponds tothe absence of conflict between m1 and m2, whereas k = 1 implies acomplete contradiction between m1 and m2. The belief functionresulting from the combination of J information sources SJ definedas

m ¼ m1 �m2 � � � �mj � � � �mJ ð13Þ

Example 2.3. Suppose the frame of discernment is H = {a,b} in atarget recognition system. There are two sensors to observe thetarget in the application systems. One sensor report is: M1{a} = 0.6;M1{H} = 0.4; the other sensor report is: M2{a} = 0.7; M2{H} = 0.3.

Using the Dempster rule of combination, the result can be listedas follows:

Mfag ¼ 0:88; M1fHg ¼ 0:12

Some interesting things should be pointed out from the above re-sult. On the one hand, the final result is that the target whoseBPA is 0.88, is greater than both two sensor reports. On the otherhand, the BPA of Unknown {a,b} (means that we just know it is a

Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993 6989

target, but we don’t know any more) decreases to 0.12, which islower than both two sensor reports. In other words, the uncertainmeasure decreases after the information fusion process based onDSeT.

Example 2.4. Suppose the frame of discernment is H = {a,b} in tar-get recognition systems. There are two sensors to observe the tar-get in the application systems. One sensor report is: M1{a} = 0.6;M1{b} = 0.4; the other sensor report is: M2{a} = 0.7; M2{b} = 0.3.

Using the Dempster rule of combination, the result can be listedas follows:

MðaÞ ¼ M1ðaÞ �M2ðaÞ1� ðM1ðaÞ �M2ðbÞ þM2ðaÞ �M1ðbÞÞ

¼ 0:6� 0:70:54

¼ 79

MðbÞ ¼ M1ðbÞ �M2ðbÞ1� ðM1ðaÞ �M2ðbÞ þM2ðaÞ �M1ðbÞÞ

¼ 0:4� 0:30:54

¼ 29

If we think of the first BPA as prior information, the prior informa-tion can be represented through prior distribution. Hence, usingBayesian method, the prior distribution can be updated as follows:

PðaÞ ¼ 0:6� 0:70:6� 0:7þ 0:4� 0:3

¼ 79

PðbÞ ¼ 0:4� 0:30:6� 0:7þ 0:4� 0:3

¼ 29

We can see that both methods have the same result. That is, if theBPA are assigned at singletons, the BAP can be degenerated as prob-ability function. In addition, the Dempster rule of combination isdegenerated as Bayesian updating method.

2.2.3. Discounted combinationIt should be noted that when evidence highly conflicts with

each other, the classical Dempster rule of combination is not effi-cient. For example, the famous numerical example is given by Za-deh (1986).

Example 2.5. Consider a situation in which we have two BPAs m1

and m2 as follows:

m1ðaÞ ¼ 0:99; m1ðbÞ ¼ 0:01m2ðbÞ ¼ 0:01; m2ðcÞ ¼ 0:99

Application of the Dempster rule yields

mðbÞ ¼ 0:00011� k

¼ 0:00011� 0:9999

¼ 1

Thus it can be seen that while m1 and m2 affords little support to b,the results afford complete support to b. This appears somewhatcounterintuitive. For the illogical aspects mentioned above, conflictmanagement in belief functions is a very important problem andhas been already studied in the past. Many methods are proposedto deal with the conflict evidence combination problem (Denget al., 2004; Guo, Shi, & Deng, 2006; Lefevre, 2002; Murphy,2000). One of the efficient methods is the use of discounting coeffi-cient. Discounting coefficient can be seen as reliability coefficients.It means the reliability of the information source. Given reliabilitycoefficients a, the next step is to incorporate them into the fusionprocess. To handle conflict between information sources, a dis-counting rule has been introduced in DSeT given as follows:

maðHÞ ¼ a�mðHÞ þ ð1� aÞ

maðAÞ ¼ a�mðAÞ; 8A � H and A – /: ð14Þ

Example 2.6. Consider a situation in which we have two BPAs m1

and m2 as follows:

m1ðaÞ ¼ 0:99; m1ðbÞ ¼ 0:01m2ðbÞ ¼ 0:01; m2ðcÞ ¼ 0:99

The discounting coefficient is 0.9 and 0.2, respectively.Application of the discounting coefficient equation in (14), the

discounted BPA can be shown as follows:

m0:91 ðaÞ ¼ 0:9� 0:99 ¼ 0:891

m0:91 ðbÞ ¼ 0:9� 0:01 ¼ 0:009

m0:91 ðHÞ ¼ 0� 0þ ð1� 0:9Þ ¼ 0:1

m0:12 ðbÞ ¼ 0:1� 0:01 ¼ 0:001

m0:12 ðcÞ ¼ 0:1� 0:99 ¼ 0:099

m0:12 ðHÞ ¼ 0� 0þ ð1� 0:1Þ ¼ 0:9

The final combination result is

mðaÞ ¼ 0:8812mðbÞ ¼ 0:0090mðcÞ ¼ 0:0109mðHÞ ¼ 0:0989

From this example, we can see that the result supports the hypoth-esis A. The main reason is that although the evidence highly con-flicts with each other, the result is more affected by the reliabilitysensor reports.From these examples listed above, we can see theflexibility and efficiency of DSeT. Generally speaking, comparedwith classical probability theory, DSeT can represent uncertaininformation in a more generalized manner. What’s more, the datacan be fused by the Dempster rule of combination without priorinformation, which is more preferable to Bayesian method in someuncertain environments. Finally, the reliability of each data sourcecan be evaluated by the discounting coefficient. In our paper, weproposed a fuzzy evidential method to analyze systems risk. Thepresented method is detailed in the following section.

2.2.4. Pignistic probability transformation (PPT)Beliefs manifest themselves at two levels – the credal level

(from credibility) where belief is entertained, and the pignistic levelwhere beliefs are used to make decisions. The term ‘‘pignistic’’ wasproposed by Smets (2000) and originates from the word pignus,meaning ‘bet’ in Latin. Pignistic probability is used for decision-making and uses Principle of Insufficient Reason to derive fromBPA. It represents a point estimate in a belief interval and can bedetermined as

betðAiÞ ¼X

Ai # Ak

mðAkÞjAkj

ð15Þ

Example 2.7. Consider a BPA as follows:

mðaÞ ¼ 0:2; mða; bÞ ¼ 0:2; mðb; cÞ ¼ 0:3; mða; b; cÞ ¼ 0:3

Then the result of PPT is

BelðaÞ ¼ mðaÞ þ 12

mða; bÞ þ 13

mða; b; cÞ ¼ 0:4

BelðbÞ ¼ 12

mða; bÞ þ 12

mðb; cÞ þ 13

mða; b; cÞ ¼ 0:35

BelðcÞ ¼ 12

mðb; cÞ þ 13

mða; b; cÞ ¼ 0:25

2.2.5. Determination of basic probability assignment

When data fusion is applied by the DSeT, one of the most impor-tant things is to determine the BPA. In this section, we propose a

6990 Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993

new method to generate BPA, which is suitable in the linguisticdecision making environment.

Suppose in a decision making environment, the linguistic itemsare given in Table 2 and illustrated in Fig. 7.

If the expert’s opinion is ‘‘Low’’, how can we transform it intoBPA?

Hence, using Eq. (5), the following similarity measures can beobtained:

SimLowfVery lowg ¼ 0:0233SimLowfLowg ¼ 1SimLowfFairly lowg ¼ 0:0577SimLowfLow;Very lowg ¼ 0:0303SimLowfLow; Fairly lowg ¼ 0:1333

SimLow{Very low} = 0.0233 means that the linguistic term ‘‘low’’which is represented as the trapezoid ‘‘low’’ has the little similaritymeasure to the linguistic term ‘‘very low’’. It should be pointedthat the hypothesis {Low,Very low}, the intersection parts between‘‘Low’’ and ‘‘Very low’’ can be represented as a generalized fuzzynumber.

Normalize the similarity measure to get the BPA of linguisticitem ‘‘LOW’’ as follows:

mLowfVery lowg ¼ 0:0187mLowfLowg ¼ 0:8035mLowfFairly lowg ¼ 0:0464mLowfLow;Very lowg ¼ 0:0243mLowfLow; Fairly lowg ¼ 0:1071

ð16Þ

Fig. 7. Generating BPA from the linguistic term low.

The Bes

Battle

Attack

Capability

Mobility

Capability

Challen

(UK

M1A1

(USA)

Fig. 8. The hierarchical structure of evalua

Here we should give some explanation about the reason whywe negative the ROG method but chose the area method to deter-mine similarity measure. As can be seen from Fig. 7, if the expertgive his opinion ‘‘LOW’’, the BPA assigned at {Very low}, {Low},{Fairly low}, {Low,Very low} and {Low,Fairly low} is acceptable.However, we cannot accept to assign BPA to {Very high}, {Veryhigh,high} if his opinion is ‘‘LOW’’. The main reason is that the lin-guistic items given in Table 1 have provided enough soft. If the ex-pert’s opinion is ‘‘LOW’’, he himself cannot agree that ‘‘High’’ isacceptable, otherwise he may use ‘‘Fairly high’’, which has someinteraction area with ‘‘High’’. As a result, though the linguisticitems themselves are not clear in their boundary, the degree iscrisp to some extend. It means that the BPA cannot assign to‘‘Good’’ given the linguistic item ‘‘Poor’’.

3. The proposed method

In this section, we use a main tank evaluation model to developour fuzzy evidential method step by step.

Step 1. Construct the decision hierarchy model.As can be shown in Eq. (1), the decision maker should constructa hierarchy model at the first step of decision making. The hier-archy model can be constructed according to the domainexperts’ knowledge. For example, when ranking the main battletanks, the decision hierarchy model can be shown in Fig. 8.In the above decision hierarchy model, the alternatives are inthe bottom. For example, three main battle tanks, namelyM1A1, Challenger 2 and Leopard 2 are evaluated in this situa-tion. The four criteria given by the domain experts are attackcapability, mobility capability, self-defense capability as wellas communication and command. The top of the decision hier-archy model is the decision result, namely the best main battletank in the evaluation.Step 2.According to the real data, whether quantitative or qualitative,the experts will give their opinions on the weight of each crite-rion and the rating of each criterion. Their opinions are repre-sented by the form of a linguistic item in Tables 2 and 3,respectively.The real data of three main battle tanks are shown in Table 3.Then, according to Table 3, the experts will use the linguisticvalues in Table 2 to give his rating of each criterion, shown inTable 4.

t Main

Tank

Self-defense

Capability Communication

and command

ger 2

)

Leopard 2

(Germany)

ting three types of main battle tanks.

Table 3Basic performance data for three types of main battle tank.

Item Type

M1A1 (USA) Challenger (UK) Leopard 2(Germany)

Armament 1 � 120 mmgun

1 � 120 mm L30gun 1 � 120 mmgun

2 � 7.62 mmMG

2 � 7.62 mm MG 2 � 7.62 mmMG3

1 � 12.7 mmMG

Ammunition 40 Up to 50 projectilestowage

42

1000 Positions (7.62 mm)4000

4750

11,400Smoke grenade

dischargers2 � 6 2 � 5 2 � 8

Power to weight ratio(hp/t)

27 10.2 25.12

Max. road speed 72 km 56 km/h 72 kmMax. range (km) 498 450 500Fording (m) 1.219 1.07 1Gradient (%) 60 60 60Vertical obstacles (m) 1.244 0.9 1.1Trench 2.743 2.43 3.00Armour protection Good Excellent FairAcclimatization Good Fair GoodCommunication Fair Fair FairScout Medium Medium Medium

TabThe

C

A

M

S

C

Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993 6991

Similarly, the importance of each criterion can also be givenusing the linguistic items in Table 1. The expert will give hisopinions listed in Table 5.Step 3. Aggregate the weights of the criterion and transformthem into a discounting coefficient.For each criteria, using the fuzzy mathematic operator to obtainthe average weights of the criterion. Let wjt = (ajt,bjt,cjt),j = 1,2, . . . ,n, t = 1,2, . . . ,k be the weight assigned by the deci-sion-maker Dk to criterion Cj. First, obtain the graded mean inte-gration representation of fuzzy numbers wjt = (ajt,bjt,cjt). Then,the aggregated importance weight Wj of criterion Cj assessedby the committee of k decision-makers can be evaluated as

Wj ¼Pk

t¼1wjt

kð17Þ

For example, according to the data in Table 5, three experts givetheir opinions about the importance of criterion ‘‘Attack’’ as Veryhigh (VH), High (H), and High (H), respectively. From Table 1, VH

le 4ratings of attribute performance for three types of main battle tanks and the correspondi

riteria Item Type

M1A1 (U

ttack Armament MGAmmunition VGSmoke grenade dischargers GMean (6.66,7.6

obility Power to weight ratio GMax. road speed GMax. range GPassing trench/ obstacle GMean (7.00,8.0

elf-defense Armour protection MGAcclimatization MGMean (5.00,6.0

ommunication and command Communication GScout MGMean (6.00,7.0

is (0.8,0.9,1.0,1.0) and H is (0.7,0.8,0.8,0.9). Then the aggregatedimportance weight can be obtained using Eq. (17) as follows:

Wattack ¼ð0:8;0:9;1:0;1:0Þ þ ð0:7;0:8;0:8;0:9Þ þ ð0:7;0:8;0:8;0:9Þ

3¼ ð0:73;0:83;0:86;0:93Þ

All the other aggregated importance weights can be obtained inthe same way and shown in Table 5.Defuzzify the aggregated importance of ‘‘Attack’’ into a crispnumber using Eq. (4).

Wcrispattack ¼

14ð0:73þ 0:83þ 0:86þ 0:93Þ ¼ 0:8375

All the other crisp numbers of aggregated importance weightscan be obtained in the same way and shown in Table 5.The relative can be easily obtained by dividing the crisp numberwith the maximum value among the crisp numbers.For example, the maximum crisp is 0.8755, hence, for the crite-rion ‘‘Attack’’, its relative importance is 0.8375/0.8755 = 0.9544.All the other relative importance of each criterion can be ob-tained in the same way and shown in Table 5. As can be seenfrom Table 5, the most important criterion is ‘‘Mobility’’, whilethe least important criterion is ‘‘Command and Communication’’.The relative importance will be used as the discounted coeffi-cient in the following fusion of multi-criteria data based onDempster rule of combination. The more the discounted coeffi-cient of the criterion, the more the effect by the criterion. Thus,the discounted coefficient of each criterion in our method plays asimilar role as the weight in classical MCDM to some degree.Step4. Aggregate the ratings of the criterion and transform theminto basic probability assignment.Similar to the aggregation of weights, the rating of each crite-rion given by different experts can also be obtained. Let rijt =(oijt,pijt,qijt), rijt 2 R+, i = 1,2, . . . ,m, j = 1,2, . . . ,n, t = 1,2, . . . ,k, bethe suitability rating assigned to alternative Ai by decision mak-ers Dt with respect to criteria Cj. Then, the aggregated ratingRij = (oij,pij,qij), of alternative Ai with respect to criteria Cj canbe obtained as

Rij ¼PK

t¼1rijt

Kð18Þ

For example, according to Table 4, the criterion ‘‘Attack’’ isdecomposed of three sub-criteria ‘‘Armament’’, ‘‘Ammunition’’and ‘‘Smoke grenade dischargers’’. To the M1A1, its ratings canbe evaluated as ‘‘Medium good (MG)’’, ‘‘Very good (VG)’’, and‘‘Good (G)’’, respectively. Then, the aggregated rating of thecriterion ‘‘Attack’’ of M1A1 is given as follows:

ng aggregation value.

SA) Challenger 2 (UK) Leopard 2 (Germany)

G GMG MGMG VG

6,8.33,9.00) (5.66,6.66,7.33,8.33) (6.66,7.66,8.33,9.00)

F GF GMG GMG MG

0,8.00,9.00) (4.50,5.50,6.00,7.00) (6.50,7.50,7.77,8.77)

G FF MG

0,7.00,8.00) (5.50,6.50,6.50,7.50) (4.50,5.50,6.00,7.00)

G GMG MG

0,7.50,8.50) (6.00,7.00,7.50,8.50) (6.00,7.00,7.50,8.50)

Table 5The linguistic importance weight of the criteria and its relative importance.

D1 D2 D3 Aggregated importance Crisp number Relative importance (discount coefficient)

Attack VH H H (0.73,0.83,0.86,0.93) 0.8375 0.9544Mobility VH H VH (0.76,0.86,0.93,0.96) 0.8775 1Self-defense M VH MH (0.56,0.66,0.73,0.80) 0.6875 0.7835Command and communication M M M (0.40,0.50,0.50,0.60) 0.5000 0.5698

Table 6Transform the aggregated ratings of each criterion into basic probability assignment.

m {MP} m {F} m {MP,F} m {MG} m {F,MG} m {G} m {MG,G} m {VG} m {G,VG}

Attack1 0 0 0 0.1720 0 0.5360 0.0745 0.1338 0.0838Attack2 0 0.0220 0 0.7423 0.0126 0.0835 0.1086 0.0174 0.0133Attack3 0 0.0220 0 0.7423 0.0126 0.0835 0.1088 0.0174 0.0133Mobility1 0 0 0 0.1020 0 0.6118 0.0765 0.1224 0.0874Mobility2 0.0303 0.3636 0.0242 0.4848 0.0970 0 0 0 0Mobility3 0 0 0 0.1801 0 0.3613 0.3613 0.0560 0.0412Self-defense1 0 0.1087 0 0.6522 0.0652 0.1087 0.0652 0 0Self-defense2 0 0.0599 0 0.7980 0.0299 0.0748 0.0374 0 0Self-defense3 0.0303 0.3636 0.0242 0.4848 0.0970 0 0 0 0CC1 0 0 0 0.4532 0 0.3569 0.1322 0.0330 0.0248CC2 0 0 0 0.4532 0 0.3569 0.1322 0.0330 0.0248CC3 0 0 0 0.4532 0 0.3569 0.1322 0.0330 0.0248

Table 7Discounted basic probability assignment of each criterion.

m {MP} m {F} m {MP,F} m {MG} m {F,MG} m {G} m {MG,G} m {VG} m {G,VG} m {H}

Attack1 0 0 0 0.1642 0 0.0711 0.5116 0.1277 0.0800 0.0456Attack2 0 0.0210 0 0.7085 0.0120 0.0797 0.1036 0.0166 0.0127 0.0456Attack3 0 0.0210 0 0.7085 0.0120 0.0797 0.1036 0.0166 0.0127 0.0456Mobility1 0 0 0 0.1020 0 0.6118 0.0765 0.1224 0.0874 0Mobility2 0.0303 0.3636 0.0242 0.4848 0.0970 0 0 0 0 0Mobility3 0 0 0 0.1801 0 0.3613 0.3613 0.0560 0.0412 0Self-defense1 0 0.0852 0 0.5110 0.0511 0.0852 0.0511 0 0 0.2165Self-defense2 0 0.0469 0 0.6252 0.0234 0.0586 0.0293 0 0 0.2165Self-defense3 0.0237 0.2849 0.0190 0.3798 0.0760 0 0 0 0 0.2165CC1 0 0 0 0.2582 0 0.2034 0.0753 0.0188 0.0141 0.4302CC2 0 0 0 0.2582 0 0.2034 0.0753 0.0188 0.0141 0.4302CC3 0 0 0 0.2582 0 0.2034 0.0753 0.0188 0.0141 0.4302

Table 8The multi-criteria data fusion results.

m {MP} m {F} m {MP,F} m {MG} m {F,MG} m {G} m {MG,G} m {VG} m {G,VG}

M1A1 (USA) 0 0 0 0.1746 0 0.7881 0.0069 0.0244 0.0060Challenger 2 (UK) 0.0004 0.0115 0.0003 0.9861 0.0017 0 0 0 0Leopard 2 (Germany) 0 0 0 0.8891 0 0.0858 0.0223 0.0019 0.0009

Table 9Results of pignistic probability transformation.

bet {MP} bet {F} bet {MG} bet {G} bet {VG}

M1A1 USA) 0 0 0.1780 0.7945 0.0274Challenger 2 (UK) 0.0005 0.0125 0.9870 0.0017 0Leopard 2 (Germany) 0 0 0.9002 0.0974 0.0023

6992 Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993

RM1A1Attack ¼

ð5;6;7;8Þ þ ð8;9;10;10Þ þ ð7;8;8;9Þ3

¼ ð6:66;7:66;8:33;9:00Þ

All the other aggregated ratings of the criterion can be obtainedin a same way. The results are listed in Table 4.Based on the method proposed in Section 2.2.3, the BPA of eachrating can be obtained and shown in Table 6.Step 5. Discountingthe BPA of ratings with the relative importance.Given the relative importance (discounted coefficient) in Table5 and the BPA in Table 6, the discounted BPA can be easilyobtained through Eq. (14). The results are listed in Table 7.Step 6. Combine the discounted data from each criterion basedon the Dempster rule.

Using the classical Dempster rule of combination, the data fromeach criteria can be fused. The fusion result of each alternativeis listed in Table 8.Step 7. Using the pignistic probability transformation to get thefinal ranking order.

Y. Deng et al. / Expert Systems with Applications 38 (2011) 6985–6993 6993

By the pignistic probability transformation in Eq. (15), theresults are shown in Table 8. The final ranking order isM1A1(USA) > Challenger (Germany) > Leopard (UK), which isthe same as the result presented in Cheng and Lin’s work(2002) (see Table 9).

4. Conclusions

Due to the uncertainity in decision-making, it is flexible to dealwith linguistic information in decision making under the frame-work of fuzzy sets theory and Dempster–Shafer evidence theory.In this paper, a fuzzy evidential method to handle MCDM problemis proposed. The new methodology represents experts’ opinions bythe use of fuzzy numbers. The linguistic weights can be trans-formed as the discounting coefficient. The ratings are used to gen-erate the BPA. Based on the discounting rule, the data from eachcriterion can be fused by the Dempster rule of combination. The fi-nal ranking order can be determined by the pignistic probabilitytransformation. A numerical example to select the best main battletank is used to illustrate the efficiency of the proposed method. Wewill apply it to the linguistic environment evaluation in the future.

Acknowledgements

The first author will appreciate the funding provided by theNational Natural Science Foundation of China, Grant Nos.60874105, 60904099, Program for New Century Excellent Talentsin University, Grant No. NCET-08-0345, Shanghai Rising-StarProgram Grant No. 09QA1402900, Chongqing Natural ScienceFoundation, Grant No. CSCT, 2010BA2003, Aviation ScienceFoundation, Grant Nos. 20090557004, 20095153022, the ChenxingScholarship Youth Found of Shanghai Jiao Tong University GrantNo. T241460612, Doctor Funding of Southwest University GrantNo. SWU110021, Leading Academic Discipline Project of ShanghaiMunicipal Education Commission Grant No. J50704.

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