Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations

Expert Systems with Applications 39 (2012) 2415–2420

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations Guiqing Zhang a, Yucheng Dong a,⇑, Yinfeng Xu a,b a b

Management School, Xi’an Jiaotong University, Xi’an 710049, PR China State Key Lab for Manufacturing Systems Engineering, Xi’an 710049, PR China

a r t i c l e

i n f o

Keywords: Group decision making Fuzzy preference relations Consistency Optimization

a b s t r a c t The consistency measure is a vital basis for group decision making (GDM) based on fuzzy preference relations, and includes two subproblems: individual consistency and consensus consistency. This paper proposes linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense), and maximizes the consistency level of fuzzy preference relations in calculating the missing values of incomplete fuzzy preference relations. Linear optimization models can be solved in very little computational time using readily available softwares. Therefore, the results in this paper are also of simplicity and convenience for the application of consistent fuzzy preference relations in GDM problems.  2011 Published by Elsevier Ltd.

1. Introduction Fuzzy preference relations are widely used in group decision making (GDM) models (Chiclana, Herrera, & Herrera-Viedma, 1998; Dong, Li, & Xu, 2008; Dong, Xu, & Yu, 2009; Fan, Xiao, & Hu, 2004; Fedrizzi & Brunelli, 2010; Fodor & Roubens, 1994; Tanino, 1984). The consistency problem is a vital basis in GDM using fuzzy preference relations. Generally, the problem of consistency itself includes two subproblems (Herrera-Viedma, Herrera, Chiclana, & Luque, 2004): (1) when can a decision maker, considered individually, be said to be consistent and, (2) when can a whole group of decision makers be considered consistent. In this paper, we call the first subproblem individual consistency, and the second subproblem consensus consistency. Individual consistency of fuzzy preference relations is related to rationality, which is associated with the transitivity property. Types of transitivity for fuzzy preference relations (i.e., reciprocal relations) have been widely devised (Cavallo & D’Apuzzo, 2009; Chiclana, Herrera-Viedma, Alonso, & Herrera, 2009; De Baets & De ⇑ Corresponding author. E-mail addresses: [email protected] (G. Zhang), [email protected] (Y. Dong), [email protected] (Y. Xu). 0957-4174/$ - see front matter  2011 Published by Elsevier Ltd. doi:10.1016/j.eswa.2011.08.090

Meyer, 2005; De Baets, De Meyer, De Schuymer, & Jenei, 2006). Additive transitivity is equivalent to the consistency property of multiplicative preference relations (Saaty, 1980) used in the analytic hierarchy process, and is widely used to character the individual consistency of fuzzy preference relations. Herrera-Viedma et al. (2004) propose a method to construct an additive consistent fuzzy preference relation from n  1 preference values. Herrera-Viedma, Chiclana, Herrera, and Alonso (2007) propose the consistency index based on additive transitivity to evaluate the individual consistency level of fuzzy preference relations. Ma, Fan, Jiang, Mao, and Ma (2006) and Xu and Da (2003) study some approaches to improve the individual consistency level of fuzzy preference relations. Alonso et al. (2008), Fedrizzi and Giove (2007), Herrera-Viedma et al. (2007) and Xu (2004) discuss the management of incomplete fuzzy preference relations in GDM. As applications of individual consistency of fuzzy preference relations, Chang and Chen (2009) study the cooperative learning in E-learning, Wang and Chang (2007) propose a method to predict the chance of the successful knowledge management implementation, and Wang and Chen (2007) discuss the partnership selection problem. The consensus model is an important aspect in GDM (Ben-Arieh & Easton, 2007; Ben-Arieh, Easton, & Evans, 2009; Chen & Cheng, 2009; Dong, Xu, Li, & Feng, 2010; Herrera, Herrera-Viedma, & Verdegay, 1996; Hsiao, Lin, & Chang, 2008; Kacprzyk, 1986; Kacprzyk, Fedrizzi, & Nurmi, 1992; Khorshid, 2010; Zhang, Dong, Xu, & Li, in press). Classically, consensus is defined as the full and unanimous agreement of all the decision makers regarding all

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the possible alternatives. However, some researchers consider that complete agreement is not necessary in real life. This has led to the use of the consensus consistency measure, which also is called ‘‘soft’’ consensus degree. Bordogna, Fedrizzi, and Pasi (1997) propose a linguistic model for GDM based on the ordered weighted averaging operator. Herrera-Viedma, Herrera, and Chiclana (2002) present a consensus model for fuzzy GDM problems with different preference structures. Herrera-Viedma, Alonso, Chiclana, and Herrera (2007) further develop a consensus model to deal with GDM with incomplete fuzzy preference relations. Chiclana et al. (2008) present a framework for integrating individual consistency into consensus model. By using Chiclana et al.’s consensus framework, Dong, Zhang, Hong, and Xu (2010) propose two AHP consensus models. An excellent survey of consensus models can be found in Cabrerizo, Pérez, and Herrera-Viedma (2010). The main purpose of this paper is to provide linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. In constructing individual consistency and reaching consensus, our proposal optimally preserves original preference information (in Manhattan distance sense). When calculating the missing values of incomplete fuzzy preference relations, our proposal maximizes the consistency level of fuzzy preference relations. This paper is organized as follows. In Section 2, we introduce some preliminary knowledge. In Section 3, we present linear optimization models for solving consistency issues of fuzzy preference relations. In Section 4, illustrative examples are provided. Concluding remarks are included in Section 5.

2. Consistency indexes

2.2. Consensus consistency Consider a GDM problem using fuzzy preference relations. Let D = {d1, d2, . . . , dm} be the set of decision makers, and {F(1), F(2), . . . , F(m)} be the fuzzy preference relations provided by m   ðkÞ decision makers dk (k = 1, 2, . . . , m), where F ðkÞ ¼ fij ; ðk ¼ 1; nn

2; . . . ; m; i; j ¼ 1; 2; . . . ; nÞ. In general, the computation of the consensus level among the decision makers is done by measuring the distance between their preference values. Chiclana et al.      ðrÞ ðrÞ ðtÞ ðtÞ  (2008) used the function s fij ; fij ¼ 1  fij  fij  to measure the similarity of the preference values of two decision makers, dr and dt, on a pair of alternatives, Ai and Aj. The computation of consensus consistency level, presented in Chiclana et al. (2008), is carried out as follows: (1) For each pair of decision makers r and t(r 6 t), a similarity matrix is calculated

  SMrt ¼ smrtij      ðrÞ ðrÞ ðtÞ ðtÞ  with smrtij ¼ s fij ; fij ¼ 1  fij  fij ; i; j ¼ 1; . . . ; n: (2) A consensus matrix, CM = (cmij), is obtained by aggregating all similarity matrices using arithmetic mean:

cmij ¼

2.1. Individual consistency Let X = {A1, A2, . . . , An} be a finite set of alternatives. When a decision maker makes pairwise comparisons using the values in [0, 1], he/she can construct a fuzzy preference relation F = (fij)nn, where fij + fji = 1 and 0 6 fij 6 1, to represent his/her own opinion on X. Transitive properties are very important concept to character consistent fuzzy preference relations. Some transitive properties of fuzzy preference relations can be described as follows (Cavallo & D’Apuzzo, 2009; Chiclana et al., 2009; De Baets & De Meyer, 2005; De Baets et al., 2006; Tanino, 1984): (1) Weak stochastic transitivity. fij P 0.5, fjk P 0.5 ) fik P 0.5 "i, j, k. (2) Strong stochastic transitivity (or restricted maximum transitivity). fij P 0.5, fjk P 0.5 ) fik P max(fij, fik) "i, j, k. (3) Additive transitivity. fij + fjk  fik = 0.5 "i, j, k. (4) Multiplicative transitivity (or product rule). "i, j, k : fij, fjk, fki R {0, 1} ) fijfjkfki = fikfkjfji. Additive transitivity is most commonly used to character consistency of fuzzy preference relations in the existing literatures. Herrera-Viedma et al. (2007) propose the consistency index based on additive transitivity to evaluate the individual consistency level (CL) of a fuzzy preference relation F:

2 CLðFÞ ¼ 1  3nðn  1Þðn  2Þ

threshold ðCLÞ for CL(F). If CLðFÞ P CL, we conclude that F is of acceptable consistency; otherwise, we conclude that F is of unacceptable consistency.

n X

n X

jfij þ fjk  fik  0:5j:

ð1Þ

i;k¼1;i–k j¼1;j–i;k

If CL(F) = 1, then the preference relation F is consistent, otherwise, the higher CL(F) the more consistent. According to the actual situation, the decision makers may establish the consistency

m X m X 2 smrtij : mðm  1Þ tPr r¼1

Obviously, cmij = cmji("i, j). (3) Consensus consistency Level (CCL) among {F(1), . . . , F(m)} is defined as follows:

CCLfF ð1Þ ; . . . ; F ðmÞ g ¼

n n X X 1 cmij nðn  1Þ i¼1 j¼1;j–i

2 nmðm  1Þðn  1Þ  n n m X m  X X X  ðrÞ ðtÞ   fij  fij :

¼1

ð2Þ

i¼1 j¼1;j–i tPr r¼1

If CCL{F(1), . . . , F(m)} = 1, then the decision makers are of full consensus, otherwise, the higher CCL{F(1), . . . , F(m)} the more consensus. According to the actual situation, the decision makers also may establish the thresholds ðCCLÞ for CCL{F(1), . . . , F(m)}. If CCLfF ð1Þ ; . . . ; F ðmÞ g P CCL, we conclude that {F(1), . . . , F(m)} are of acceptable consensus; otherwise, we conclude that {F(1), . . . , F(m)} are of unacceptable consensus. 3. Linear optimization models Before proposing the linear optimization models, we introduce the concept on distance metrics. A standard set of metrics is the p-norms (k kp) for p P 1. When using different p, we obtain different distance metrics. If p = 1, then the p-norm is equivalent to the Manhattan or rectilinear distance. This paper uses Manhattan distance to measure the distance between two fuzzy preference relations. Let E = (eij)nn and F = (fij)nn be two fuzzy preference relations. The Manhattan distance between E and F is

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dðE; FÞ ¼

n X n 1 X jeij  fij j: 2 n i¼1 j¼1

ð3Þ

n X n 1 X dij 2 n j¼1 i¼1

min

subject to fij P 0;

ð13Þ

i; j ¼ 1; 2; . . . ; n

3.1. Constructing consistency of individual fuzzy preference relations

fij þ fji ¼ 1;

Consistency construction issue of individual fuzzy preference relations is presented in Herrera-Viedma et al. (2004). We denote K F n ¼ fF ¼ ðfij Þnn : fij P 0; fij þ fji ¼ 1; i; j ¼ 1; 2; . . . ; ng as the set of n  n fuzzy preference relations, and M F n ¼ fF ¼ ðfij Þnn : fij P 0; fij þfji ¼ 1; fij þ fjc  fic ¼ 0:5; i; j; c ¼ 1; 2; . . . ; ng as the set of n  n individual consistent fuzzy preference relations. Let F = (fij)nn be a fuzzy preference relation. The main task of constructing the individual consistent fuzzy preference relation of F = (fij)nn is to find a fuzzy preference relation F ¼ ðfij Þnn 2 M F n . In order to preserve the information in F as much as possible, we hope that the distance measure between F and F is minimal, namely

aijc ¼ fij þ fjc  fic  0:5;

min dðF; FÞ:

ð4Þ

F2M F n

We use two transformed decision variables: yij ¼ fij  fij and zij = jyijj. In this way, model (4) is transformed into a linear optimization model P1: n X n 1 X zij 2 n j¼1 i¼1

min

subject to fij P 0;

ð5Þ

i; j ¼ 1; 2; . . . ; n

fij þ fji ¼ 1;

i; j ¼ 1; 2; . . . ; n;

fij þ fjc  fic ¼ 0:5; yij ¼ fij  fij ; yij 6 zij ;

i; j; c ¼ 1; 2; . . . ; n;

i; j ¼ 1; 2; . . . ; n;

i; j ¼ 1; 2; . . . ; n;

 yij 6 zij ;

i; j ¼ 1; 2; . . . ; n:

ð6Þ ð7Þ ð8Þ ð9Þ ð10Þ ð11Þ

In P1, constraints (6)–(8) guarantee that F 2 M F n and constraints (9)–(11) enforce that zij P jyij j ¼ jfij  fij j. According to the objective function (i.e., (5)), we find that any feasible solutions with zij > jyijj are not the optimal solution to P1. Thus, constraints (9)–(11) guarantee that zij P jyij j ¼ jfij  fij j. Clearly, any n  n consistent fuzzy preference relations satisfy all the constraints of P1, and therefore represent feasible solutions. A closed bounded feasible region for P1 would satisfy the assumption of Weierstrass theorem. This could be achieved, for instance, by introducing an upper bound for zij. Since yij ¼ fij  fij , a suitable inequality that doesn’t affect the optimal solution could be zij 6 2, i, j = 1, 2, . . . , n. According to Weierstrass theorem, the optimal solution to P1 exists. In general, it is hard to obtain consistency of fuzzy preference relations, especially when the number of alternatives is large. Let N F n ¼ fF : F 2 K F n ; CLðFÞ P CLg be the set of n  n fuzzy preference relations with acceptably individual consistency. Furthermore, we present an optimization model to construct the acceptable consistency fuzzy preference relation of F = (fij)nn. The main task of the proposed model is to find a fuzzy preference relation F ¼ ðfij Þnn 2 N F n . In order to preserve the information in F as much as possible, we hope that the distance measure between F and F is minimal, namely

min dðF; FÞ:

ð12Þ

F2N F n

We use four transformed decision variables: aijc ¼ fij þ fjc  fic  0:5; bijc ¼ jaijc j; cij ¼ fij  fij and dij = jcijj. In this way, model (12) is transformed into a linear optimization model P2:

aijc 6 bijc ;

i; j ¼ 1; 2; . . . ; n;

ð15Þ

i; j; k ¼ 1; 2; . . . ; n;

i; j ¼ 1; 2; . . . ; n;

 aijc 6 bijc ;

cij ¼ fij  fij ;

ð18Þ bijc 6 1  CL;

ð19Þ

i;c¼1;i–c j¼1;j–i;c

i; j ¼ 1; 2; . . . ; n;

ð20Þ

i; j ¼ 1; 2; . . . ; n;

 cij 6 dij ;

ð16Þ ð17Þ

i; j ¼ 1; 2; . . . ; n; n n X X

2 3nðn  1Þðn  2Þ

cij 6 dij ;

ð14Þ

ð21Þ

i; j ¼ 1; 2; . . . ; n:

ð22Þ

In P2, constraints (16)–(18) guarantee that bijc ¼ jaijc j ¼ jfij þ fjc  fic  0:5j, constraints (14), (15), (19) guarantee that F 2 N F n , and constraints (20)–(22) guarantee that dij ¼ jcij j ¼ jfij  fij j. Similar to P1, P2 has optimum solutions. Obviously, when setting CL ¼ 1; P2 reduces to P1. 3.2. Consensus model Consensus problem is a hot topic in group decision making. In particular, Chiclana et al. (2008) presented a framework for integrating individual consistency into consensus model. This framework is composed of two processes: individual consistency control process and consensus reaching process. Here, we propose a linear optimization model for reaching consensus under Chiclana et al.’ framework.   ðkÞ Let F ðkÞ ¼ fij ðk ¼ 1; 2; . . . ; mÞ be a group of fuzzy prefermn ence relations with unacceptable consensus. We find that the key   ðkÞ task of reaching consensus among F ðkÞ ¼ fij ðk ¼ 1; 2; . . . ; mÞ nn

under Chiclana et al.’ framework is to find a group of individual   ðkÞ ðk ¼ 1; 2; . . . ; mÞ with fuzzy preference relations F ðkÞ ¼ fij nn

acceptable consensus and acceptably individual consistency. In or  ðkÞ der to preserve the information in F ðkÞ ¼ fij ðk ¼ 1; 2; . . . ; mÞ mn

as much as possible, we hope that the distance measure between F(k) and F ðkÞ ðk ¼ 1; 2; . . . mÞ is minimal, namely

min F ðkÞ

m X

dðF ðkÞ ; F ðkÞ Þ;

ð23Þ

k¼1

i.e.,

min ðkÞ

fij

m X n X n   1 X  ðkÞ ðkÞ  fij  fij :

n2 m

k¼1 i¼1

ð24Þ

j¼1

  ðkÞ At the same time, F ðkÞ ¼ fij

nn

ðk ¼ 1; 2; . . . ; mÞ has acceptably

individual consistency level,

CLðF ðkÞ Þ P CI;

k ¼ 1; 2; . . . m;

ð25Þ

i.e., n X 2 3nðn  1Þðn  2Þ i;c¼1;i–c

6 1  CL;

n

m

j¼1;j–i;c

   ðkÞ  ðkÞ ðkÞ fij þ fjc  fic  0:5

k ¼ 1; 2; . . . m

ð26Þ

and has the acceptable consensus level, that is

n o CCL F ð1Þ ; . . . ; F ðmÞ P CCL;

ð27Þ

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G. Zhang et al. / Expert Systems with Applications 39 (2012) 2415–2420

i.e.,

3.3. Management of incomplete fuzzy preference relation

2 nmðm  1Þðn  1Þ

 n n m X m  X X X  ðrÞ ðtÞ  fij  fij  6 1  CCL:

ð28Þ

i¼1 j¼1;j–i tPr r¼1

In this way, an optimization model to reaching consensus is constructed as follows: 8  m P n P n  P  ðkÞ ðkÞ  > > min n21m fij  fij  > > > ðkÞ k¼1 i¼1 j¼1 > > > fij > <   n n P P  ðkÞ ðkÞ ðkÞ  2 s:t: 3nðn1Þðn2Þ fij þ fjc  fic  0:5 6 1  CL k ¼ 1;2;...m > > i;c¼1;i–c j¼1;j–i;c > > >  > n n m  P P P P >  ðrÞ ðtÞ  > 2 > t P rm fij  fij  6 1  CCL: : nmðm1Þðn1Þ

ð29Þ

r¼1

i¼1 j¼1;j–i

ðkÞ

ðkÞ

ðkÞ

We use six transformed decision variables: aijck ¼ fij þ fjc  fic ðrÞ

ðtÞ

ðkÞ

ðkÞ

0:5; bijck ¼ jaijck j; cijrt ¼ fij  fij ; dijrt ¼ jcijrt j; eijk ¼ fij  fij

and

gijk = jeijkj. In this way, model (29) is transformed into the following linear optimization model P3:

min

m X n X n 1 X g n2 m k¼1 i¼1 j¼1 ijk

ð31Þ

¼ 1; i;j ¼ 1;2;...;n; k ¼ 1;2;...m;

ð32Þ

ðtÞ

ð38Þ

 cijrt 6 dijrt i;j ¼ 1;2;...n; r;t ¼ 1;2;...;m;

ð39Þ

dijrt 6 1  CCL;

ð45Þ

i; j ¼ 1; 2; . . . ; n

fij0 þ fji0 ¼ 1;

ð46Þ

i; j ¼ 1; 2; . . . ; n;

aijc ¼ fij0 þ fjc0  fic0  0:5; aijc 6 bijc ;

cijrt 6 dijrt i;j ¼ 1;2;...n; r;t ¼ 1;2;...;m;

ðkÞ ðkÞ eijk ¼ fij  fij ;

subject to fij0 P 0;

ð36Þ ð37Þ

2 nmðm  1Þðn  1Þ

bijc

fij0 ¼ fij ; for f ij – null;

cijrt ¼ fij  fij ; i;j ¼ 1;2;...n; r;t ¼ 1;2;...;m;

n n m X m X X X

n X

i;c¼1;i–c j¼1;j–i;c

ð35Þ

n X

2 bijck 6 1  CL; k ¼ 1;2;...m; 3nðn  1Þðn  2Þ i;c¼1;i–c j¼1;j–i;c ðrÞ

ð33Þ ð34Þ

 aijck 6 bijck i;j;c ¼ 1;2;...;n; k ¼ 1;2;...m; n X

n X

min

aijck 6 bijck i;j;c ¼ 1;2;...;n; k ¼ 1;2;...m;

j¼1;j–i;c

We use two transformed decision variables: aijc ¼ fij0 þ fjc0  fic0  0:5 and bijc = jaijcj, In this way, model (44) is transformed into the following linear programming model P4:

P 0; i;j ¼ 1;2;...;n; k ¼ 1;2;...m

i;j;c ¼ 1;2;...;n; k ¼ 1;2;...m;

  n P  0  fij þ fjc0  fic0  0:5

ð44Þ

ðkÞ ðkÞ fij þ fji

ðkÞ ðkÞ ðkÞ aijck ¼ fij þ fjc  fic  0:5

8 n P 2 > > CLðF 0 Þ ¼ 1  3nðn1Þðn2Þ > max 0 > f > i;c¼1;i–c > < ij s:t: fij0 P 0 i; j ¼ 1; 2 . . . ; n > > > fij0 þ fij0 ¼ 1 i; j ¼ 1; 2 . . . ; n; > > > : fij0 ¼ fij for f ij – null:

ð30Þ

ðkÞ fij

subject to

When some of the elements in F = (fij)nn cannot be given by the decision maker, which we denote by null, we call F the incomplete fuzzy preference relation. Some researchers focused on calculating the missing values of incomplete fuzzy preference relations (i.e., management of incomplete fuzzy preference relations). We argue that the key task of calculating the missing values of F is to find a complete fuzzy preference relations F 0 ¼ ðfij0 Þnn with fij0 ¼ fij for non-null entries of F. Fedrizzi and Giove (2007) proposed to calculate the missing values of F by maximizing the consistent level of F0 . Inspired by this idea, we present an optimization model to calculate the missing values of incomplete fuzzy preference relations, based on the consistency level defined in Herrera-Viedma et al. (2007):

ð48Þ i; j; k ¼ 1; 2; . . . ; n;

i; j ¼ 1; 2; . . . ; n;

 aijc 6 bijc ;

ð47Þ

i; j ¼ 1; 2; . . . ; n:

ð49Þ ð50Þ ð51Þ

In P4, constraints (46) and (47) guarantee that F ðkÞ 2 K F n , and constraints (49)–(51) guarantee that bijc ¼ jaijc j ¼ jfij0 þ fjc0  fic0  0:5j. Similar to P1, P4 has optimum solutions.

ð40Þ 4. Numerical examples

i¼1 j¼1;j–i tPr r¼1

i;j ¼ 1;2;...;n; k ¼ 1;2;...m;

ð41Þ

eijk 6 g ijk ; i;j ¼ 1;2;...;n; k ¼ 1;2;...m;  eijk 6 g ijk ; i;j ¼ 1;2;...;n; k ¼ 1;2;...m:

ð42Þ

In this section, three numerical examples are provided to demonstrate the proposed models (P1, P2, P3 and P4).

ð43Þ

4.1. Example 1

ðkÞ

In P3, constraints (31) and (32) guarantee that F 2 K F n , con  ðkÞ ðkÞ ðkÞ straints (33)–(35) guarantee that bijck ¼ jaijck j ¼ fij þ fjc  fic 0:5j, constraint (36) guarantees F ðkÞ 2 N F n , constraints (37)–(39)    ðrÞ ðtÞ  guarantee that dijrt ¼ jcijrt j ¼ fij  fij , constraint (40) guarantees CCLfF ð1Þ ; . . . ; F ðmÞ g P CCL, and constraints (41)–(43) guarantee that    ðkÞ ðkÞ  g ijk ¼ jeijk j ¼ fij  fij . Similar to P1, P3 has optimum solutions. Obviously, when setting m = 1, P3 reduces to P2. Remark. This paper uses Manhattan distance to measure the information loss of original fuzzy preference relations. The proposed optimization model (i.e., P1, P2 and P3) can optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense). When using different distance metrics, we will obtain different models.

We consider the following fuzzy preference relation:

0

0:5

0:59 0:67 0:93

1

B 0:41 0:5 0:59 0:62 C C B F¼B C: @ 0:33 0:41 0:5 0:61 A 0:07

0:38 0:39 0:5

We use model P1 to construct consistent fuzzy preference relations of F. The corresponding consistent fuzzy preference relation is

1 0:5 0:59 0:6754 0:7854 B 0:41 0:5 0:5854 0:6954 C C B F¼B C: @ 0:3246 0:4146 0:5 0:61 A 0

0:2146 0:3046 0:39

0:5

We find dðF; FÞ ¼ 0:0288. When setting CL ¼ 0:95, we use model P2 to construct the fuzzy preference relations with acceptable consistency:

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G. Zhang et al. / Expert Systems with Applications 39 (2012) 2415–2420

0

0

1

0:5 0:59 0:6736 0:8776 B 0:41 0:5 0:5847 0:6387 C B C F ¼B C; @ 0:3264 0:4153 0:5 0:61 A 0:1224 0:3613 0:39

0:5

F

ð4Þ

0:5 B B 0:75 B ¼B B 0:65 @

We also set CL ¼ 0:98 and construct the fuzzy preference relations with acceptable consistency: 0 1 0:5 0:59 0:6732 0:8159 B 0:41 0:5 0:5854 0:6682 C B C F0 ¼ B C; CLðF 0 Þ ¼ 0:98; dðF;F 0 Þ ¼ 0:0213: @ 0:3268 0:4146 0:5 0:61 A 0:1841 0:3318 0:39

0:5

We

4.2. Example 2 We consider the example presented in Chiclana et al. (2008). In Chiclana et al.’s example, there are four decision makers providing the following fuzzy preference relations on a set of four alternatives:

0:5 0:2 0:6 0:4

1

C B B 0:8 0:5 0:9 0:7 C C F ð1Þ ¼ B B 0:4 0:1 0:5 0:3 C; A @ 0:6 0:3 0:7 0:5

F ð2Þ

0:5 B 0:3 B ¼B @ 0:1 0:5 0

0:7 0:9 0:5

0:5

1

0:7

0:5 0:5

0:5 0:2 0:6 0:4

1

0:8 C C C: 0:5 A 0:5

1

C B B 0:8 0:5 0:9 0:7 C C F ð1Þ ¼ B B 0:4 0:1 0:5 0:3 C; A @ 0:6 0:3 0:7 0:5 0

F

ð2Þ

1 0:5 0:475 0:575 0:525 B 0:525 0:5 0:6 0:7 C B C ¼B C; @ 0:425 0:4 0:5 0:6 A 0:475 0:3 0

0:5

0:4

0:37 0:5 0:42

B 0:63 0:5 B F ð3Þ ¼ B @ 0:5 0:4 0:58 0:6

0:5 1

0:6 0:4 C C C; 0:5 0:3 A 0:7 0:5

that

CLðF ð1Þ Þ ¼ 1; CLðF ð2Þ Þ ¼ 0:95; CLðF ð3Þ Þ ¼ 0:95;

We also consider the fuzzy preference relation presented in Fedrizzi and Giove (2007):

0:5

B B 0:5 B B B 0:5 B G¼B B 0:1845 B B B 0:5 @

0:5

0:5

0:8155 0:5

0:3423

1

C 0:3423 C C C 0:3423 0:5 0:8662 0:75 0:3423 C C C: 0:1845 0:1338 0:5 0:25 0:25 C C C 0:5 0:25 0:75 0:5 0:25 C A

0:5

0:6577 0:8155 0:5

0:5

0:5

0:44

0:75 0:5

0:8155 0:5

0:3423

1

C 0:6577 0:8155 0:57 0:3423 C C C 0:3423 0:5 0:8662 0:75 0:3423 C C C: 0:1845 0:1338 0:5 0:25 0:44 C C C 0:43 0:25 0:75 0:5 0:25 C A

0:5

0:6577 0:6577 0:6577 0:56

We find that CL(F(1)) = 1, CL(F(2)) = 0.7667, CL(F(3)) = 0.65, CL(F(4)) = 0.8333 and CCL{F(1), F(2), F(3), F(4)} = 0.75. When respectively setting CL ¼ 0:95 and CCL ¼ 0:85, we use model P3 to reach consensus among {F(1), F(2), F(3), F(4)}. The adjusted fuzzy preference relations with acceptable consensus and acceptably individual consistency are

0

0:5

4.3. Example 3

B B 0:5 B B B 0:56 B 0 G ¼B B 0:1845 B B B 0:5 @

1

0:25 0:15 0:65 0:6

find

0

0:3 0:2 0:5

B 0:75 0:5 B F ð4Þ ¼ B @ 0:85 0:4 0:35 0:2

0:4

C 0:775 C C C: 0:525 C A

Let us assume that the decision maker is not able to perform the comparison for the comparison {A1, A3}, {A2, A5} and {A4, A6}. We respectively use model P4 to calculate the missing values of G. The corresponding complete fuzzy preference relation is

C B B 0:7 0:5 0:1 0:3 C C; F ð3Þ ¼ B B 0:5 0:9 0:5 0:25 C A @ 0:3 0:7 0:75 0:5 0

0:6

0:6577 0:6577 0:6577 0:75

0:5 0:6 0:7 C C C; 0:4 0:5 0:8 A

0:5 0:3 0:5

0:5

0:525

CLðF ð4Þ Þ ¼ 0:95 and CCLfF ð1Þ ; F ð2Þ ; F ð3Þ ; F ð4Þ g ¼ 0:85.

0

0

0:35

0:475 0:225 0:475 0:5

CLðFÞ ¼ 0:95; dðF;FÞ ¼ 0:01:

0

0:25

1

0:75 0:5

0

We find that CL(G ) = 0.9141. 5. Conclusions Based on the consistency definitions presented in Chiclana et al. (2008) and Herrera-Viedma et al. (2007), we propose a set of linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. Comparing the existing approaches, our proposal has two desired features: (1) it optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense), (2) it maximizes the consistency level of complete fuzzy preference relations in calculating the missing values of incomplete fuzzy preference relations. In general, linear optimization models can be solved in very little computational time using readily available softwares, such as Lingdo and Matlab. Therefore, the results in this paper are also of simplicity and convenience for the application of consistent fuzzy preference relations in GDM problems. Acknowledgement This research is supported by NSF of China under Grants 70801048 and 71171160.

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