An evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval-valued group consensus requirements

European Journal of Operational Research 223 (2012) 167–176

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

An evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval-valued group consensus requirements Chao Fu ⇑, Shanlin Yang School of Management, Hefei University of Technology, Box 270, Hefei 230009, Anhui, PR China Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei 230009, Anhui, PR China

a r t i c l e

i n f o

Article history: Received 13 July 2011 Accepted 29 May 2012 Available online 9 June 2012 Keywords: Decision analysis Multiple attribute group decision analysis Evidential reasoning based consensus model Missing assessment Feedback mechanism Nonlinear optimization

a b s t r a c t With the aim of modeling multiple attribute group decision analysis problems with group consensus (GC) requirements, a GC based evidential reasoning approach and further an attribute weight based feedback model are sequentially developed based on an evidential reasoning (ER) approach. In real situations, however, giving precise (crisp) assessments for alternatives is often too restrictive and difficult for experts, due to incompleteness or lack of information. Experts may also find it difficult to give appropriate assessments on specific attributes, due to limitation or lack of knowledge, experience and provided data about the problem domain. In this paper, an ER based consensus model (ERCM) is proposed to deal with these situations, in which experts’ assessments are interval-valued rather than precise. Correspondingly, predefined interval-valued GC (IGC) requirements need to be reached after group analysis and discussion within specified times. Also, the process of reaching IGC is accelerated by a feedback mechanism including identification rules at three levels, consisting of the attribute, alternative and global levels, and a suggestion rule. Particularly, recommendations on assessments in the suggestion rule are constructed based on recommendations on their lower and upper bounds detected by the identification rule at a specific level. A preferentially developed industry selection problem is solved by the ERCM to demonstrate its detailed implementation process, validity, and applicability. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction With the aim of generating commonly satisfactory solutions to multiple attribute group decision analysis (MAGDA) problems with group consensus (GC) requirements, a GC based evidential reasoning (GCER) approach (Fu and Yang, 2010) was proposed based on an evidential reasoning (ER) approach (Yang, 2001; Wang et al., 2006; Yang et al., 2006; Guo et al., 2007). Further, an attribute weight based feedback model (AWFM) was designed to accelerate convergence to GC. It can identify assessments damaging GC and yield recommendations on how to renew identified assessments (Fu and Yang, 2011), when GC has been gradually increased in its convergence process. Compared with other existing consensus based group decision analysis (GDA) approaches (e.g., Bordogna et al., 1997; Herrera et al., 1996; Herrera et al., 1997; Ben-Arieh and Chen, 2006; Dong et al., 2008; Dong et al., 2010a; HerreraViedma et al., 2005; Herrera-Viedma et al., 2007a; Mata et al., 2009; Cabrerizo et al., 2009; Szmidt and Kacprzyk, 2003; Choudhury et al., 2006; Fedrizzi et al., 1994), many practical factors, in⇑ Corresponding author at: School of Management, Hefei University of Technology, Box 270, Hefei 230009, Anhui, PR China. Tel.: +86 0551 2904930; fax: +86 0551 2905263. E-mail address: [email protected] (C. Fu). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.05.048

cluding experts’ utilities, subjective weights of experts, and the flexibility in GC and GC requirements, were considered in the GCER approach and the AWFM. In real situations, however, experts or decision makers may often feel too restrictive and difficult to give precise (crisp) assessments due to incompleteness or lack of information, which results in partial or total ignorance. To deal with these situations, experts are encouraged to give interval-valued assessments. For example, a consensus model was developed to solve GDA problems with interval fuzzy preference relations (Tapia Garcia et al., 2012). Although some ER extensions (e.g. Wang et al., 2006; Yang et al., 2006) deal with interval-valued assessments, they do not focus on solving MAGDA problems and further reaching consensus. Also, due to limitation or lack of knowledge, experience and provided data about the problem domain (Kim and Ahn, 1999; Kim et al., 1999; Herrera-Viedma et al., 2007a,b), some experts may find it difficult to give appropriate assessments on specific attributes (also called missing attributes) for specific or all alternatives. In this paper, based on the GCER approach and the AWFM, an ER based consensus model (ERCM) is developed to find commonly satisfactory solutions to MAGDA problems with interval-valued GC (IGC) requirements. In the ERCM, experts give interval-valued assessments, which are interval-valued belief structures (IBSs)

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(Denoeux, 1999; Wang et al., 2006, 2007), for problems. The ERCM deals with missing attributes and designs a feedback mechanism based on the AWFM to accelerate convergence to IGC. As stated in (Cabrerizo et al., 2010a), GDA approaches with feedback mechanism to accelerate convergence to consensus can be regarded as advanced methods to generate reasonable consensus based solutions, such as the approaches of Herrera-Viedma et al. (2005, 2007a) and Mata et al. (2009). Obviously, the ERCM is advanced. Similar to the GCER approach and the AWFM, the IGC is constructed at three levels, including the attribute, alternative and global levels. Based on the GC at the attribute level, a pair of nonlinear optimization problems is constructed to generate the IGC at the attribute level. The IGC at the alternative and global levels is constructed based on the one at the attribute level. In terms of the idea of reaching the maximal IGC, recommendations on missing attributes are yielded to help experts give their assessments after group analysis and discussion (GAD). Furthermore, a feedback mechanism, including identification rules at three levels, corresponding to the IGC, and a suggestion rule, is developed. By considering attribute weights, similar to the AWFM, the former identifies assessments damaging the lower and upper bounds of IGC. Particularly, the latter is designed to improve the lower and upper bounds of IGC after GAD simultaneously if necessary. After the predefined IGC is reached, the process of generating solutions to MAGDA problems in the GCER approach and the AWFM is extended to the context of interval-valued assessments by constructing and solving correspondingly nonlinear optimization problems. The rest of this paper is organized as follows. Section 2 presents the preliminaries related to the ERCM. Section 3 interprets the ERCM in detail. A preferentially developed industry selection problem is solved in Section 4 to demonstrate a detailed implementation process of the ERCM, its validity and applicability. Section 5 discusses the ERCM based on the problem in Section 4 and compares it with other existing methods. Finally, this paper is concluded in Section 6. 2. Preliminaries 2.1. The ER distributed modeling framework for MAGDA problems using interval-valued assessments Suppose a MAGDA problem includes T experts tj( j = 1, . . . , T) and a manager. The relative weights of T experts on the attribute ei for the alternative al are denoted by k(ei(al)) = (k1(ei(al)), k2 (ei(al)), . . . , kT(ei(al))) such that

0 6 kj ðei ðal ÞÞ 6 1 and

T X

kj ðei ðal ÞÞ ¼ 1:

ð1Þ

j¼1

All experts deal with a common multiple attribute decision analysis problem which has M alternatives al(l = 1, . . . , M), on the upper level attribute, referred to as a general attribute, and L lower level attributes ei(i = 1, . . . , L), called basic attributes. The relative weights of L basic attributes are denoted by w = (w1, w2, . . . , wL) such that

0 6 wi 6 1 and

L X wi ¼ 1:

ð2Þ

i¼1

Suppose Hn (n = 1, . . . , N) denotes a set of grades which forms the frame of discernment X = {H1, H2, . . . , HN}. M alternatives are assessed at L attributes using Hn(n = 1, . . . , N). Let Bðei ðal ÞÞ ¼ þ fðHn ; ½b n;i ðal Þ; bn;i ðal ÞÞ; n ¼ 1; . . . ; Ng denote the interval-valued distributed assessment vector on the attribute ei for the alternative al to the grade Hn with the interval belief degree of ½b n;i ðal Þ;  þ bþ n;i ðal Þ. The interval belief degree satisfies 0 6 bn;i ðal Þ 6 bn;i ðal Þ;



PN

  þ  n¼1 bn;i ðal Þ 6 1;bn;i ðal Þ 2 ½bn;i ðal Þ;bn;i ðal Þ;bX;i ðal Þ ¼ max

ðal ÞÞ; bþ X;i ðal Þ ¼ 1 

PN

 n¼1 bn;i ðal Þ; bX;i ðal Þ

0;1 

PN

þ 2 ½b X;i ðal Þ; bX;i ðal Þ and

þ n¼1 bn;i

PN

n¼1

þ bn;i ðal Þ þ bX;i ðal Þ ¼ 1, where bX,i(al) and ½b X;i ðal Þ; bX;i ðal Þ denote the belief degree assigned to X and its interval, respectively. If bX,i(al) = 0 always holds, then the assessment vector is complete; otherwise, it is incomplete. After M alternatives are all assessed on L basic attributes, an interval-valued belief decision matrix will be obtained, which is

Sg ¼ Bðei ðal ÞÞLM :

ð3Þ

Aided by the principle of utility equivalence (Yang, 2001), quantitative attributes can also be modeled as IBSs by using the defined assessment grades (Wang et al., 2006). The expert tj gives the interval-valued belief decision matrix jþ Sjg ¼ Bj ðei ðal ÞÞLM , where Bj ðei ðal ÞÞ ¼ fðHn ; ½bj n;i ðal Þ; bn;i ðal ÞÞ; n ¼ 1; PN j j . . . ; Ng such that n¼1 bn;i ðal Þ þ bX;i ðal Þ ¼ 1. All abbreviations in this paper include multiple attribute group decision analysis (MAGDA), group consensus (GC), group consensus based evidential reasoning (GCER), evidential reasoning (ER), group decision analysis (GDA), attribute weight based feedback model (AWFM), fuzzy preference relations (FPRs), evidential reasoning based consensus model (ERCM), belief structures (BSs), interval-valued belief structures (IBSs), interval-valued group consensus (IGC), interval-valued proximity measures (IPMs), group analysis and discussion (GAD), preferentially developed industry (PDI), processing industries (PIs) and group decision support system (GDSS). 3. The ERCM 3.1. The IGC at three levels To meet predefined IGC requirements, the IGC at the attribute, alternative and global levels should be constructed first. Based on the GC at the attribute level in the GCER approach and the AWFM, the IGC at the same level is defined as follows: Definition 1. Suppose the IGC on the attribute ei for the alternative al is denoted by [gc(ei(al)), gc+(ei(al))] such that gc(ei(al)) 2 [gc(ei(al)), gc+(ei(al))]. Then, the lower and upper bounds of IGC can be obtained by solving the following pair of optimization problems.

PT MIN=MAX gcðei ðal ÞÞ ¼

PT k¼1;k–j

cmðV j ðei ðal ÞÞ;V k ðei ðal ÞÞÞ

j¼1

T1

T

:

ð4Þ

  In Eq. (4), cm V j ðei ðal ÞÞ; V k ðei ðal ÞÞ measures the compatibility between the assessments of the experts tj and tk, which is defined based on Liu’s conflict measure (Liu, 2006) in Section A.2 in Appendix A in the Supplementary material. Based on cmðV j ðei ðal ÞÞ;  j PT k V ðei ðal ÞÞÞ; k¼1;k–j cm V ðei ðal ÞÞ; V k ðei ðal ÞÞ =T  1 denotes the overall compatibility of the assessment V j ðei ðal ÞÞ. The constraints of the above optimization problems and relevant explanations can be found in Section A.3 in Appendix A. The optimization problems on each attribute for each alternative are implemented in the Matlab environment. Consequently, gc(ei(al)) and gc+(ei(al)) can be obtained by the Matlab. Based on the IGC at the attribute level, the one at the alternative and global levels can be defined as follows. Definition 2. Suppose [gc(ei(al)),gc+(ei(al))](i = 1, . . . , L, l = 1, . . . , M) is obtained. Then, the IGC for the alternative al, [gc(al), gc+(al)](l = 1, . . . , M) such that gc(al) 2 [gc(al), gc+(al)], and the IGC for all alternatives [ggc, ggc+] such that ggc 2 [ggc, ggc+] are defined as

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gc ðal Þ ¼

L X

wi  gc ðei ðal ÞÞ and gcþ ðal Þ ¼

i¼1

and ggc ¼

M X

L X wi  gcþ ðei ðal ÞÞ;

ð5Þ

i¼1 M X gc ðal Þ=M and ggcþ ¼ gcþ ðal Þ=M:

l¼1

optimization problem, which yields the maximal gc+(esi(asl)) and the corresponding gc(esi(asl)). Also, gc(esi(asl)) reaches the maximum on the condition that the maximal gc+(esi(asl)) is reached.

ð6Þ

l¼1

Because the IGC on all attributes is exclusive of each other, the IGC at the alternative level can be defined as the weighted sum of the IGC on all attributes. Similarly, the IGC at the global level can be reasonably expressed as the mean of the IGC for all alternatives due to the same importance of alternatives and the exclusiveness of the IGC for all alternatives. Given an interval-valued threshold vector dI, two intervalvalued thresholds dIM and dIG instead of d, dM and dG in the GCER approach and the AWFM, the IGC at three levels is checked as described in the following.

PT MAX gcðesi ðasl ÞÞ ¼

PT k¼1;k–j

cmðV j ðesi ðasl ÞÞ;V k ðesi ðasl ÞÞÞ

j¼1

T1

:

T

ð10Þ

The constraints and the relevant proof of the optimization problem are presented in Section A.4 in Appendix A. Suppose n   o j j Hn ; ubn;si ðasl Þ ; X; ubX;si ðasl Þ ðj ¼ m þ 1; . . . ; TÞ denote the T–m BSs with which the maximum of gc(esi(asl)) is reached in the above optimization problem. They are distributed as recommendations to the T–m experts to help complete assessments after GAD. 3.3. The feedback mechanism

Definition 3. Suppose the interval-valued threshold vector þ  þ  þ dI ¼ ð½d 1 ; d1 ; . . . ; ½dL ; dL Þ such that 0 6 di 6 di 6 1ði ¼ 1; . . . ; LÞ, þ two interval-valued thresholds dIM ¼ ½d ; d  such that 0 6 d M M M 6  þ  þ dþ 6 1 and dI ¼ ½d ; d  such that 0 6 d 6 d 6 1, are given by G M G G G G the manager or the group. Then, the IGC is reached at the attribute level when Condition 1 is satisfied; it is reached at the alternative level when Condition 2 is satisfied; and it is reached at the global level when Condition 3 is satisfied. (1)

gc ðei ðal ÞÞ P di ;

gcþ ðei ðal ÞÞ P dþi ;

i ¼ 1; . . . ; L; l

¼ 1; . . . ; M; 

dM ;

(2)

gc ðal Þ P

(3)

ggc P dG ;

þ

gc ðal Þ P

ð7Þ dþM ;

l ¼ 1; . . . ; M;

ggcþ P dþG :

ð8Þ ð9Þ

Here, dI, dIM and dIG denote the IGC requirements at three levels, which should be satisfied before IGC-based solutions to MAGDA problems can be generated. The manager or the group can practically decide dI, dIM and dIG according to application constraints, historical setting, decision contexts, and their knowledge, experience, preference, and risk attitude. 3.2. Recommendations on missing attributes for experts When some experts cannot give effective assessments on missing attributes due to limitation or lack of knowledge, experience and provided data, the ERCM generates recommendations by reaching the maximal upper bounds of IGC and the corresponding lower bounds of IGC on missing attributes. This effectively avoids arbitrary assessments of experts. Then, the recommendations are distributed to the experts and all experts have a GAD to freely express opinions. After that, the experts independently give their assessments based on the recommendations according to their preference, which is related to their background, knowledge and experience, and the extent to which they agree with other experts who give effective assessments. This strategy of generating recommendations can accelerate convergence to IGC. For any missing attribute esi for the specific alternative asl, suppose only m(1 6 m < T) experts can give effective assessments. To reach the maximal gc+(esi(asl)) and the corresponding gc(esi(asl)) on the missing attribute esi for the alternative asl, recommendations to T–m experts can be decided by the following theorem. Theorem 1. Suppose m experts give assessments on the missing attribute esi for the alternative asl. Then, recommendations for the remaining T–m experts can be generated by solving the following

After specific experts give assessments on missing attributes after GAD based on Theorem 1, a feedback mechanism is developed to help experts improve IGC quickly. It consists of identification rules at three levels and a suggestion rule. The identification rule at a specific level is used to detect assessments damaging the lower bound, the upper bound or both of IGC. Moreover, the suggestion rule recommends specific experts for the way to renew their assessments comprehensively instead of the lower and upper bounds of their assessments separately. 3.3.1. The interval-valued proximity measure The identification rules in the ERCM will be similarly constructed according to the ones in the AWFM, so the compatibility between individual assessments and group assessment is necessarily measured. Differently, IPMs at three levels, including the attribute, alternative and global levels, instead of proximity measures at three levels in the AWFM, will be constructed in the ERCM. The IPM at the attribute level, denoted by [pmj(ei(al)), pmj+(ei(al))], is obtained by solving the following optimization problems, similar to the IGC at the same level.

MIN=MAX pmj ðei ðal ÞÞ ¼ cmðV j ðei ðal ÞÞ; Vðei ðal ÞÞÞ:

ð11Þ

The detailed optimization problems can be seen in Section A.5 in Appendix A. Corresponding to the IGC at the alternative and global levels, the IPMs at the alternative and global levels can be defined as

pmj ðal Þ ¼

L L X X wi  pmj ðei ðal ÞÞ and pmjþ ðal Þ ¼ wi  pmjþ ðei ðal ÞÞ; ð12Þ i¼1

i¼1

M M X X pmj ðal Þ=M and pmjþ ¼ pmjþ ðal Þ=M: and pm ¼ j

l¼1

ð13Þ

l¼1

Eqs. (12) and (13) are reasonable because the IPMs on all attributes and the IPMs for all alternatives are exclusive of each other, similar to the situation of the IGC at the alternative and global levels. 3.3.2. The identification rules Based on the identification rules in the AWFM (Fu and Yang, 2011), the ones in the ERCM are still constructed at the attribute, alternative and global levels. Differently, the lower and upper bounds of assessments damaging IGC are identified separately. Similar to (Mata et al., 2009; Fu and Yang, 2011), in the process of reaching IGC, the lower and upper bounds of the IGC are also at three states, which are very low, low and medium states. At different states, a different number of assessments damaging the lower and upper bounds of the IGC will be identified and renewed. Furthermore, on some important attributes with their associated weights more than or at least equal to the mean of the weights of all attributes, all experts are recommended to renew assessments.

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Fig. 1. The two sets of the ordinal relaxation of the identification rules at three levels.

As stated in (Fu and Yang, 2011), this will facilitate convergence to IGC and detection of underlying problems. þ þ  Two pairs of thresholds, q 1 and q1 , and q2 and q2 , are the parameters together with the IGC requirements to decide which states the lower and upper bounds of the IGC are at. They are decided by the manager or the group according to application constraints, historical setting, decision contexts, and their knowledge, experience, preference, and risk attitude, similar to the IGC requirements. Meanwhile, the IGC requirements are seen as references for the decision of the two pairs of thresholds. Based on the above analysis, the identification rules at three levels in the ERCM will be demonstrated in the following: (1) The identification rule at the attribute level. For the lower and upper bounds of the IGC at the attribute level, they can be only at very low and medium states due to the availability of only the IPM at the attribute level. When the lower and upper bounds of the IGC are at the very low state, i.e. þ þ gc ðei ðal ÞÞ 6 q 1 and gc ðei ðal ÞÞ 6 q1 , all experts are recommended to renew assessments to avoid that the process of reaching IGC is guided by some experts imposing their assessments. When the lower and upper bounds of the IGC are   increasingly at the medium state, i.e. q 1 < gc ðei ðal ÞÞ < di

þ þ and qþ 1 < gc ðei ðal ÞÞ < di , after several rounds of recommendation, GAD and assessment renewal, the identification of assessments to renew depends on the IPM at the attribute level. Because the lower and upper bounds of the IPM at the attribute level are dynamically changed after each GAD, PT PT j jþ j¼1 pm ðei ðal ÞÞ=T and j¼1 pm ðei ðal ÞÞ=T are selected as dynamic thresholds to control them. (2) The identification rule at the alternative level. Due to the availability of the IPMs at the attribute and alternative levels, the lower and upper bounds of the IGC at the alternative þ level can be at three states, i.e. gc ðal Þ 6 q 1 and gc ðal Þ 6 þ þ þ    þ q1 ; q1 < gc ðal Þ 6 q2 and q1 < gc ðal Þ 6 q2 , and q2 < þ þ þ gc ðal Þ < d M and q2 < gc ðal Þ < dM . Similar to the control of the lower and upper bounds of the IPM at the attribute level, dynamic thresholds are set to control the lower and upper bounds of the IGC at the attribute level and the IPMs at the attribute and alternative levels. The successive application of the IPMs at the attribute and alternative levels decreases the identified assessments on unimportant attributes when the lower and upper bounds of the IGC are at low and medium states, respectively. (3) The identification rule at the global level. The lower and upper bounds of the IGC at the global level can be still at three states. Similar to the identification rule at the alternative level, dynamic thresholds are still selected. Differently, the IPM at the global level is added and controlled. The successive application of the IPMs at three levels also decreases the identified assessments on unimportant attributes when the lower and upper bounds of the IGC are at low and medium states, respectively. On important attributes, i.e. wi P 1/L, all experts are recommended to renew assessments when the lower and upper bounds of the IGC at three levels are not at the very low state. The formulas of the identifications rules at three levels can be found in Section A.6 in Appendix A. The set of assessments to be renewed at different states identified at three levels are denoted by ASSATT VL ; ASSATT VLþ ; ASSATT M ; ASSATT Mþ ; ASSA j j Lþ M LT VL ;ASSALT VLþ ;ASSALT L ; ASSALT Mþ ;ASSGLOVL ; j ; ASSALT j ; ASSALT j j Lþ M ASSGLOVLþ ; ASSGLOL and ASSGLOMþ . j ; ASSGLOj ;ASSGLOj j

As a whole, due to the ordinal relaxation characteristic of the GC requirements at three levels (Fu and Yang, 2010), the lower and upper bonds of the IGC requirements have the same characteristic. This means that the identification rules at three levels have gradually become relaxed, that is, ASSATT VL / ASSALT VL /ASSGLOVL ;ASSATT VLþ / ASSALT VLþ /

Fig. 2. The procedure of the ERCM.

C. Fu, S. Yang / European Journal of Operational Research 223 (2012) 167–176

171

jþ j j (2) decrease bj n;i ðal Þ if ln;i ðal Þ ¼ bn;i ðal Þ–ln;i ðal Þ, or increase jþ jþ jþ j bn;i ðal Þ if ln;i ðal Þ ¼ bn;i ðal Þ–ln;i ðal Þ, respectively. jþ jþ The reason why the conditions of bj n;i ðal Þ–ln;i ðal Þ; bn;i ðal Þ– j jþ j l ln;i ðal Þ, and bn;i ðal Þ–ln;i ðal Þ are required in the above recommendations can also be known in Section A.7. Although the above recommendations seem reasonable, they may guide assessments of experts to the assessments with large intervals between lower and upper bounds of grades, or large ignorance intervals, or even the unnormalized assessments. Therefore, the suggestion rule is constructed based on the recommendations to help experts comprehensively renew assessments and preferably improve IGC. jþ j n;i ðal Þ; bn;i ðal Þ–

Fig. 3. The movement of the IGC for four PIs in four times of group assessments. VLþ

L

L

Lþ

Lþ

M

ASSGLO ; ASSALT j / ASSGLOj ; ASSALT j / ASSGLOj ; ASSATT j /ASSALT M / ASSGLOM , and ASSATT Mþ /ASSALT Mþ / ASSGLOMþ , j j j j j

where the notation ‘/’ means ‘stricter than’. Meanwhile, the fact that identified assessments to renew have gradually decreased along with the increase of the IGC decides the other ordinal relaxation characteristic, i.e. ASSATT VL / ASSATT M ; j

Definition 5. Suppose the expert tj is recommended to renew the lower and upper bounds of r grades, which are denoted by ns, ns+1, . . . , ns+r1, on the attribute ei for the alternative al according to Definition 4. When r = 1, the expert is recommended to comprehensively renew Bj(ei(al)) according to Situations (1)–(3); when r > 1, the expert can comprehensively renew Bj(ei(al)) according to Situations (4)–(5).

Theorem 2. Suppose fðHn ; lj n;i ðal ÞÞ; j ¼ 1; . . . ; T; n ¼ 1; . . . ; Ng and

(1) When ns = 1, if the expert is recommended to increase/ jþ decrease bj ns ;i ðal Þ or bns ;i ðal Þ, then he/she is recommended to j decrease/increase B (ei(al)). jþ (2) When 1 < ns < N, if bjþ ns 1;i ðal Þ > 0 and bns þ1;i ðal Þ > 0, then the j expert is recommended to renew B (ei(al)); if bjþ ns 1;i ðal Þ > 0; bjþ and the expert is recommended to ns þ1;i ðal Þ ¼ 0, jþ increase/decrease bj ns ;i ðal Þ or bns ;i ðal Þ, then he/she is recommended to increase/decrease Bj(ei(al)); if bjþ ns 1;i ðal Þ ¼ 0; bjþ ns þ1;i ðal Þ > 0, and the expert is recommended to increase/ jþ decrease bj ns ;i ðal Þ or bns ;i ðal Þ, then he/she is recommended to j decrease/increase B (ei(al)). (3) When ns = N, if the expert is recommended to increase/ jþ decrease bj ns ;i ðal Þ or bns ;i ðal Þ, then he/she is recommended to j increase/decrease B (ei(al)). (4) When there are different recommendations on the lower and upper bounds of r grades, if the expert is recommended

fðHn ; ljþ ða ÞÞ; j ¼ 1; . . . ; T; n ¼ 1; . . . ; Ng are two groups of BSs limited n;i l

jþ to increase/decrease bj nsþr1 ;i ðal Þ or bnsþr1 ;i ðal Þ, then he/she is

jþ in Bj ðei ðal ÞÞ ¼ fðHn ; ½bj n;i ðal Þ; bn;i ðal ÞÞ; j ¼ 1; . . . ; T; n ¼ 1; . . . ; Ng reach-

recommended to increase/decrease Bj(ei(al)). (5) When there exist identical recommendations on the lower and upper bounds of r grades, if ns+r1 < N, then the expert is recommended to renew Bj(ei(al)); otherwise, the expert is recommended to increase/decrease Bj(ei(al)) if he/she is jþ recommended to increase/decrease bj nsþr1 ;i ðal Þ or bnsþr1 ;i ðal Þ.

ASSATT VLþ /ASSATT Mþ ; ASSALT VL / ASSALT L /ASSALT M ; ASSALT VLþ / j j j Lþ VL L Mþ ASSALT j / ASS ALTj ; ASSGLO / ASSGLOj /ASSGLOM , and j ASSGLOVLþ / ASSGLOLþ / ASSGLOMþ . The two sets of the ordinal j j relaxation of the identification rules at three levels are shown in Fig. 1. Therefore, the constructed identification rules at three levels are a uniquely reasonable choice based on the IGC and the IPMs at three levels.

3.3.3. The suggestion rule A theorem about the increase trend of gc(ei(al)) and gc+(ei(al)) is given as follows:

ing gc(ei(al)) and gc+(ei(al)), respectively. Then, gc(ei(al)) and gc+(ei(al)) will increase when j j jþ (1) bj n;i ðal Þ is increased if ln;i ðal Þ ¼ bn;i ðal Þ, or bn;i ðal Þ is decreased jþ if lj ða Þ ¼ b ða Þ, and l l n;i n;i jþ j jþ (2) bj n;i ðal Þ is decreased if ln;i ðal Þ ¼ bn;i ðal Þ, or bn;i ðal Þ is increased jþ if ln;i ðal Þ ¼ bjþ ða Þ,respectively. l n;i This theorem is proved in Section A.7 in Appendix A. Based on Theorem 2, recommendations on renewing the lower and upper bounds of identified assessments are defined as follows:

Definition 4. Suppose the experts in the set T1 such that T1 # {1, . . . ,T} are identified to renew their assessments on the attribute ei for the alternative al according to identification rules mentioned in

Section

3.3.2.

Let

fðHn ; lj n;i ðal ÞÞ; n ¼ 1; . . . ; Ng

and

jþ n;i ðal ÞÞ; n

fðHn ; l

¼ 1; . . . ; Ng be two groups of BSs limited in n h i o jþ such that j 2 T1 Bj ðei ðal ÞÞ ¼ ðHn ; bj n;i ðal Þ; bn;i ðal Þ Þ; n ¼ 1; . . . ; N reaching gc(ei(al)) and gc+(ei(al)), respectively. Then, in order to improve gc(ei(al)) and gc+(ei(al)), the expert tj such that j 2 T1 is recommended to (1) increase bj n;i ðal Þ if

j jþ lj n;i ðal Þ ¼ bn;i ðal Þ–ln;i ðal Þ, or decrease jþ j jþ jþ bn;i ðal Þ if ln;i ðal Þ ¼ bn;i ðal Þ–ln;i ðal Þ, and

Relevant explanation about Definition 5 can be found in Section A.8 in Appendix A. If some experts give arbitrary assessments, then they will be recommended to renew assessments according to the identification and suggestion rules, because it is less possible that the assessments are compatible with other experts’ assessments. 3.4. The forming of group assessments for alternatives In the ERCM, the attribute based aggregation strategy in the GCER approach is employed to form group assessments for alternatives. Interval-valued assessments of experts are aggregated on attributes and further for alternatives. The aggregated group assessment on each attribute for each alternative is obtained by solving the following pair of optimization problems.

MIN=MAX bn;i ðal Þ ¼

T X j¼1

gjn;i  kj ðei ðal ÞÞði ¼ 1;. .. ;L;l ¼ 1; .. .; MÞ:

ð14Þ

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C. Fu, S. Yang / European Journal of Operational Research 223 (2012) 167–176 Table 1 The IGC for four PIs in four rounds. CYCLE

PI1

0 1 2 3

[0.3395, [0.4813, [0.5254, [0.5699,

0.4336] 0.5738] 0.621] 0.6574]

PI2

PI3

[0.436, 0.5383] [0.5193, 0.6112] [0.5621, 0.6603] [0.5621, 0.6603]

[0.4176, [0.4471, [0.5173, [0.5933,

The constraints of optimization problems are shown in Section A.9 in Appendix A. In the GDA process, experts freely give their assessments according to their preference and correspondingly their assessments are exclusive of each other. Therefore, their assessments can be aggregated according to Eq. (14). Furthermore, by solving optimization problems constructed by Wang et al. (Wang et al., 2006) based on Lefevre et al.’s method for combining conflicting BSs (Lefevre et al., 2002), the aggregated group assessment for each alternative can also be obtained. Relevant optimization problems are restated in Section A.10 in Appendix A. 3.5. The generation of the ranking order of alternatives Based on the aggregated group assessments on attributes for alternatives, the mini-max regret approach (MRA) (Wang et al., 2006), which is simply introduced in Section A.12 in Appendix A, can be applied to generate a ranking order of alternatives. Relevant optimization problems used to yield the maximum and minimum of expected utilities are restated in Section A.11 in Appendix A. When solutions to MAGDA problems with IGC requirements are only required to show the ranking order of alternatives, the maximum and minimum of expected utilities instead of group assessments for alternatives are necessary. However, the aggregated group assessments for alternatives are beneficial for the analysis about alternatives and solutions. 3.6. The procedure of the ERCM The procedure of the ERCM is shown in Fig. 2, which will be elaborated step by step. Step 1: Form a MAGDA problem. A manager selects T experts, identifies L basic attributes and their types (benefit or cost) and N assessment grades, and lists M alternatives to form a MAGDA problem. Step 2: Prepare for the ERCM in order to solve the MAGDA problem. The manager specifies MAXCYCLE, the maximum times of GAD to avoid the delayed convergence of collective solution and endless rounds of GAD (Bryson, 1996, 1997; Herrera-Viedma et al., 2002; Choudhury et al., 2006; Mata et al., 2009); sets CYCLE = 0, a cycle counter; decides the relative weights of L attributes; specifies a specific level, its corresponding interval-valued threshold vector or threshold, and two pairs of interval-valued þ þ  thresholds q 1 and q1 , and q2 and q2 for its corresponding identification rule; and decides swj(ei) (j = 1, . . . , T, i = 1, . . . , L) and c, which are demonstrated in Sections A.2 and A.5, respectively. Step 3: Collect experts’ interval-valued assessments and missing attributes. All experts independently give their interval-valued assessments for alternatives and their utilities of assessment grades. The manager collects their assessments

PI4 0.507] 0.5388] 0.6058] 0.6666]

[0.378, 0.4634] [0.4647, 0.5553] [0.5249, 0.6137] [0.5619, 0.6575]

and missing attributes. This can avoid arbitrary assessments of experts. Step 4: Generate recommendations on missing attributes. Generate recommendations to specific experts by reaching maximal upper bounds and correspondingly maximal lower bounds of IGC on missing attributes for specific or all alternatives. Step 5: Organize GAD and collect the experts’ assessments on missing attributes. The manager organizes GAD and collects the experts’ assessments on missing attributes for specific or all alternatives based on the recommendations according to the experts’ preference. Step 6: Decide whether the IGC at the specific level is reached. It is determined whether the IGC at the specific level is reached according to Definition 3. If so, go to Step 8. Otherwise, go to Step 7. Step 7: Generate recommendations, organize GAD, and collect experts’ renewed assessments. If CYCLE > MAXCYCLE, then go to Step 10. Otherwise, recommendations to specific experts on how to renew their assessments damaging the IGC at the specific level are generated according to the identification rule at the same level and the suggestion rule. The manager sets CYCLE = CYCLE + 1 and organizes GAD in order to eliminate the incompatibility among experts’ assessments. In GAD, experts are free to communicate with each other and never provided by suggestions from the manager on their assessments to reach the predefined IGC. Then, experts independently change or persist in assessments, which can avoid arbitrary assessments. After that, go to Step 6. Step 8: Form the aggregated group assessments for alternatives. Calculate the aggregated group assessments for alternatives, as mentioned in Section 3.4. The necessity of this step is decided by the real requirement of GDA. Step 9: Generate a ranking order of M alternatives. Generate a ranking order of M alternatives using the MRA (Wang et al., 2006) based on the maximum and minimum of expected utilities, which are obtained based on the aggregated group assessments on each attribute for each alternative, as stated in Section 3.5. Step 10: Finish the procedure. The manager checks whether CYCLE > MAXCYCLE holds. If so, a conclusion of no IGC-based solution for the MAGDA problem can be drawn. Otherwise, the optimum alternative, or the ranking order of M alternatives can be selected as a final solution to the MAGDA problem meeting predefined IGC requirements. 4. Illustrative example In this section, an industry selection problem will be solved by the ERCM as a real case to demonstrate its application to modeling a MAGDA problem with IGC requirements, its detailed implementation process, and its validity. The selected industry will be preferentially developed by the municipal government of a city in

C. Fu, S. Yang / European Journal of Operational Research 223 (2012) 167–176

Anhui province, China. Accordingly, the problem is called the PDI selection problem. A self-developed solving system developed in the Matlab environment is used to effectively and efficiently solve the PDI selection problem. 4.1. The description of the PDI selection problem Consider the PDI selection problem with four PIs including the food PI, the grease PI, the livestock PI and the aquatic product PI. This can be considered a MAGDA problem. A manager from the government and four experts from the development and reform committee, the agriculture committee, the environmental protection agency, and the research cooperator, depend on ten qualitative attributes shown in Table B.1 in Appendix B in the Supplementary material to solve this problem. In the GDA process, four experts always give assessments independently. The attributes are the gross product (GP), the growth speed (GS), the rationality of enterprise structure (RES), the rationality of product structure (RPS), the situation of brand construction (SBC), the situation of food safety (SFS), the current market-competitive advantage of product (CMAP), the future market space of product (FMSP), the energy-consumptionper-unit-output-value of scale enterprises (ESE), and the emission situation of industrial three wastes (ESITW). Ten attributes are exclusive of each other. The relative weights of ten attributes are specified by the manager as w = (0.09, 0.06, 0.07, 0.05, 0.1, 0.08, 0.2, 0.15, 0.08, 0.12). The manager owns relevant materials to generate ten attributes, so the weights specified by him are reasonable. Suppose four PIs are assessed by using the following set of assessment grades: Poor (P), Average (A), Good (G), VeryGood (V), and Excellent (E), say

X ¼ fHn ; n ¼ 1; . . . ; 5g ¼ fPoor; Av erage; Good; VeryGood; Excellentg ¼ fP; A; G; V; Eg: Ten attributes are all assessed by the above set of assessment grades. The manager wants to obtain an optimum PI which is an IGCbased optimum choice for four experts in order to ponder over their opinions. The manager specifies that MAXCYCLE is equal to 4, and the IGC is checked at the alternative level by a threshold interval þ dIM ¼ ½d M ; dM  ¼ ½0:55; 0:65, which is decided by the manager according to his knowledge and experience, and the consensus constraint specified by the municipal government of the city; sets CYCLE = 0, two pairs of thresholds for the identification rule þ þ  ½q 1 ; q1  ¼ ½0:3; 0:45 and ½q2 ; q2  ¼ ½0:45; 0:55 based on his knowledge and experience, and the reference of dIM; and decides c = 0.5, which means that four experts are a heterogeneous group. 4.2. Checking of the IGC at the alternative level To find the IGC-based solution to the PDI selection problem, the IGC at the alternative level must be reached first. Four experts independently give their utilities of Hn(n = 1, . . . ,5) using a probability assignment approach (Farquhar, 1984; Winston, 1994), as shown in Table B.2 in Appendix B. Due to the different background, knowledge and experience of four experts, their importance on each attribute is different, which is specified by the manager in a way employed in (Ölçer and Odabasßi, 2005), as shown in Table B.3 in Appendix B. Based on Theorem 1, recommendations on missing attributes in Table B.1 are generated. The recommendations and the corresponding assessments given by specific experts after GAD are shown in Table B.4 in Appendix B.

173

After missing attributes are handled, through solving the optimization problems in Eq. (4) and further calculating according to Eq (5), the IGC for four PIs can be obtained as shown in the first row in Table 1. Obviously, the required IGC is not reached because þ þ gc ðPIl Þ < d M and gc ðPI l Þ < dM ðl ¼ 1; . . . ; 4Þ hold. Accordingly, the manager necessarily organizes several rounds of GAD to reach the required IGC. 4.3. Reaching of the IGC at the alternative level Through face-to-face meetings, after three rounds of GAD organized by the manager when missing attributes are handled, i.e. CYCLE = 3, the required IGC is reached, as shown in Table 1 and plotted in Fig. 3. Also, the IGC-based solution to the PDI selection problem can be found because of CYCLE < MAXCYCLE. In addition, the IGC on ten attributes for four PIs in four rounds is shown in Tables B.5–B.8 in Appendix B, respectively, which shows the detailed variety of IGC on each attribute. After missing attributes are handled, i.e. CYCLE = 0, as shown in Table 1, except that gc(PI1) and gc+(PI1) are at the very low state, gc(PIl) and gc+ (PIl) (l = 2, 3, 4) are at the low state, due to þ þ    gc ðPI1 Þ 6 q and 1 ¼ 0:35; gc ðPI1 Þ6 q1 ¼ 0:45; q1 < gc ðPIl Þ 6 q2 ¼ 0:45 qþ1 < gcþ ðPIl Þ 6 qþ2 ¼ 0:55ðl ¼ 2; 3; 4Þ. When CYCLE = 1, except gc(PI3) and gc+(PI3) keeping the low state, gc(PIl) and gc+(PIl) (l = 1,  2, 4) are increased to be at the medium state, due to q 1 < gc  þ þ þ þ   ðPI3 Þ 6 q ; q < gc ðPI Þ 6 q ; q < gc ðPI Þ 6 d ¼ 0:55 and q < 3 l 2 2 M 1 2 2  + gcþ ðPIl Þ 6 dþ M ¼ 0:65ðl ¼ 1; 2; 4Þ. When CYCLE = 2, gc (PIl) and gc (PIl) (l = 1, 3, 4) are at the medium state except that gc(PI2) and gc+(PI2)  þ  þ satisfy the requirement, due to q 2 < gc ðPI l Þ < dM ; q2 < gc ðPIl Þ þ  þ  þ < dM ðl ¼ 1; 3; 4Þ; gc ðPI2 Þ P dM and gc ðPI2 Þ P dM . IPMs at three levels, after experts handle missing attributes, are shown in Table B.9 in Appendix B. According to the identification rule and Definition 4, recommendations on grades to specific experts can be generated based on Tables B.1, B.4, B.5 and B.9, as shown in Table B.10 in Appendix B. Further, recommendations on assessments can be obtained based on the ones on grades according to Definition 5, as shown in Table B.11 in Appendix B. In a similar way, recommendations on assessments after the last two rounds of GAD are obtained and shown in Tables B.12 and B.13 in Appendix B, respectively. After missing attributes are handled, four experts spend about 30, 20 and 15 min on considering the recommendations and renewing their assessments before and after three rounds of GAD, respectively. Furthermore, the time spent on three rounds of GAD is about 130, 70 and 30 min, respectively. All these are because the number of identification sets recommended to renew has gradually decreased, as shown in Tables B.11–B.13, and four experts have cognized the PDI selection problem profoundly, along with the increase of IGC in three rounds of GAD. In each round of GAD, all experts are encouraged to freely communicate and not provided by suggestions from the manager on their assessments to reach the required IGC. Recommendations generated by the identification and suggestion rules rather than the assessments of all experts are returned to some or all experts to effectively avoid herd behavior. The extent to which recommended experts change assessments based on the recommendations depends on the experts’ preference. This strategy can avoid arbitrary assessments of experts. In the following, we will give some examples in Tables B.11–B.13 to demonstrate convergence to IGC in the problem. (1) On the attribute e1 for PI1, when CYCLE = 1 and 2, it is found from Tables B.11 and B.12 that the expert t3 renews the assessment gradually as {(V,[0.1,0.2]), (E, [0.7,0.8]) ? {(G, [0.7,0.8]), (V, [0.1,0. 2]) ? {(G, [0.1,0.2]), (V, [0.7,0.8])}, but other three experts change assessments slightly. This means that the expert t3 is willing to change the assessment after

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Table 2 The expected utilities of four PIs and their ranking order. Expected utilities

PI1

PI2

PI3

PI4

Minimum expected utility Maximum expected utility Average expected utility Rank

0.4975 0.622 0.55975 2

0.5893 0.7095 0.6494 1

0.4455 0.5687 0.5071 4

0.463 0.6071 0.53505 3

free communication in two rounds of GAD because he thinks others’ opinions are reasonable. That is, if the expert t3 thinks his assessment is more reasonable than others’ assessments, then he will persist in his assessment or only change it slightly. (2) On the attribute e8 for PI1, when CYCLE = 1 and 3, the experts t1 and t4 change assessments slightly but the other experts greatly. This is because the experts t1 and t4 are more special on the attribute e8 so that their opinions in two rounds of GAD help alleviate the incompatibility among experts. Obviously, the renewed assessments of the experts t2 and t3 still depend on the experts’ preference. In addition, owing to dynamic thresholds to control the IPMs at the attribute and alternative levels and the IGC at the attribute level, four experts are not recommended to renew assessments on the attribute e8 for PI1, when CYCLE = 2. (3) On the attribute e10 for PI3, when CYCLE = 1, 2 and 3, expect the whole fluctuation of the assessment of the expert t4, other three experts change assessments gradually from a large range to a small range. On the one hand, this is due to four experts changing assessments on the important attribute e10. On the other hand, through free communication among experts in three rounds of GAD, the incompatibility induced by experts’ cognition for the problem has been gradually alleviated. In addition, although the fluctuation of the assessment of the expert t4, there is no significant difference between the renewed assessments when CYCLE = 1 and 3, which partly reflects the variation of experts’ assessments in three rounds of GAD and gradual convergence to IGC. (4) In three rounds of GAD, free communication among four experts and no guidance from the manager for convergence to IGC help them to find and cognize potential problems in GDA and further to find a valid solution to maximally reflecting the collective wisdom, as demonstrated in (Fu and Yang, 2010). Along with the increase of rounds of GAD, four experts have cognized the problem and alternatives more and more profoundly. Furthermore, after each round of GAD, they independently give their renewed assessments according to their preference. As a consequence, they may renew some assessments not recommended by the identification and suggestion rules when they consider it reasonable and necessary. For example, after the second round of GAD when missing attributes are handled, B2(e1(PI4)) is changed, as shown in Table B.12; and after the third round of GAD, B2(e6(PI1)), B2(e9(PI3)), B2(e3(PI4)), B3(e3(PI1)), B3(e6(PI4)), B4(e7(PI3)) and B4(e4(PI4)) are changed, as shown in Table B.13. 4.4. Obtaining of the IGC-based solution After three rounds of GAD when missing attributes are handled, through sequentially solving the optimization problems in Sections A.9 and A.10, the aggregated group assessments for four PIs are obtained, as shown in Table B.14 in Appendix B. Further, the expected utilities of each PI are obtained by solving the two optimization problems in Section A.11. Correspondingly, a ranking order of four PIs is generated using the MRA (Wang et al., 2006). Both of them are shown in Table 2. As a consequence, the

manager obtains the optimum PI, PI2 (the grease PI), which is the IGC-based optimum choice for four experts and meaningful on the mini-max regret criterion. 5. Discussions In this section, we will discuss two aspects related to the ERCM and compare it with other consensus methods in the ER context and other existing approaches considering missing attributes and feedback mechanism, which are two important contributions in the ERCM. (1) When some experts give BSs as assessments on missing attributes based on recommendations according to their preference after GAD, as shown in Table B.5, the situation of

j jþ jþ lj n;si ðasl Þ¼bn;si ðasl Þ¼bn;si ðasl Þ¼ ln;si ðasl Þðj2T m #fmþ1;...;TgÞ

certainly occurs. In the situation, the movement of bj n;si ðasl Þ  or bjþ n;si ðasl Þ will inevitably increase one of gc (ei(al)) and

gc+(ei(al)) and decrease the other, according to Definition 4. Therefore, experts may voluntarily renew the BSs when they freely communicate in several rounds of GAD rather than are passively recommended by the suggestion rule. (2) Because the identification rules respectively identify assessments damaging gc(ei(al)) and gc+(ei(al)), experts may only be recommended to renew lower or upper bounds of grades in assessments. If the experts renew the assessments according to Definition 4, then they may give the ones with large grade intervals or large ignorance intervals, or even unnormalized ones. Further, unilateral increase of gc+(ei(al)), which only increase the upper bound of grade or decrease the lower bound of grade in assessment, will inevitably increase the imprecision of assessments according to the imprecision measure of IBS, Du(B) = maxm2Bu(m)  minm2Bu(m), based P on the uncertainty measure uðmÞ ¼ £–A # X;mðAÞ>0 mðAÞ h i P P jA\Bj in log2 jAj  A # X; mðAÞ > 0mðAÞlog2 B # X;mðBÞ>0 mðBÞ jAj (Denoeux, 1999). Here, B and m denote some IBS and any BS limited in the IBS, respectively. Recommendations generated in Definition 5 focus on improving gc(ei(al)) and gc+(ei(al)) bilaterally, so they can prevent abnormal assessments mentioned above. In the ER context, the GCER approach (Fu and Yang, 2010) was proposed to consider many practical factors, including experts’ utilities, subjective weights of experts, the flexibility in GC and GC requirements, and the special design for MAGDA problems. Based on the GCER approach, the AWFM (Fu and Yang, 2011) was further developed to accelerate convergence to GC by a feedback mechanism consisting of the identification rules at the attribute, alternative and global levels, and a suggestion rule. Meanwhile, a consensus framework (Fu et al., 2012) was designed to accelerate convergence to GC by combining renewal of assessments with adjustment of relative weights of experts. Missing assessments were also handled in the framework. In the ERCM, however, IBSs rather than BSs denote assessments to make the situation more complex on the condition that the aforementioned factors are retained. Except generating recommendations on missing attributes by reaching the maximal gc(esi(asl)) on the condition that the maximal gc+(esi(asl)) is reached, assessments damaging IGC are identified and renewed to improve gc(ei(al)) and gc+(ei(al)) simultaneously, or improve one and keep the other. This will accelerate convergence to IGC. Meanwhile, assessments with large uncertainty or unnormalized assessments are effectively prevented. Particularly, adjustment of relative weights of experts is not considered in the ERCM.

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On the other hand, the two important contributions in the ERCM, missing attributes and feedback mechanism, have also been considered in some existing GDA methods. In the GDA approaches of Herrera-Viedma and his group, the additive transitivity property of FPRs, which can be seen as the parallel concept of Saaty’s consistency property for multiplicative preference relations (Herrera-Viedma et al., 2004), was used to develop an iterative procedure to estimate missing values of incomplete FPRs (Herrera-Viedma et al., 2007a,b). The procedure was further extended in an unbalanced fuzzy linguistic context (Cabrerizo et al., 2010b). Herrera-Viedma and his group pointed out that when a set of n  1 non-leading diagonal preferences was known, which meant each two alternatives were compared at least once, an incomplete (unbalanced) FPR could be completed. The estimation procedure focuses on making the resulting FPR more additive consistent, that is, it focuses on the self-consistency of expert’s assessments rather than the consensus among assessments of experts. In the meantime, due to the additive consistency requirement of FPR, individual consistency and consensus were measured and controlled simultaneously in the consensus models of Herrera-Viedma and his group. In the ERCM, however, experts give assessments on each attribute for each alternative, rather than compare any two alternatives on each attribute. It means that individual consistency is not a problem and correspondingly unnecessary to be measured and reached in the ERCM. Only when the required IGC is reached, can the ERCM generate the IGC-based solution. As a consequence, maximizing IGC can be considered an appropriate and feasible way to generate recommendations on missing attributes. The resulting recommendations will inevitably accelerate convergence to IGC if experts accept them. Furthermore, experts independently give their assessments after GAD based on the recommendations according to their preference. In GAD, free communication among experts will help them give authentic assessments rather than accept the recommendations straightforward. At least, the recommendation strategy on missing attributes in the ERCM avoids arbitrary assessments of experts and facilitates convergence to IGC to some extent, especially when most experts accept recommendations completely or almost completely after GAD. As for feedback mechanism, Herrera-Viedma and his group designed identification and suggestion rules in linguistic preference and fuzzy preference contexts to make experts’ assessments closer and avoid self-contradiction of expert’s assessments (HerreraViedma et al., 2005, 2007a). They also developed a model to generate adaptive recommendations to experts when consensus had been gradually increased (Mata et al., 2009). However, only the consensus at the preference relation level instead of three levels, including the pairs of alternatives, alternative, and preference relation levels, is considered, unlike the identification rules in the ERCM. In addition, their mechanisms could add GAD to help experts rationally renew assessments through free communication among experts. This will be beneficial for the detection of underlying problems and the avoidance of herd behavior in the iterative process, and the corresponding improvement of GDA quality. Particularly, states of lower and upper bounds of IGC are separately decided in the ERCM, which means that the states of lower and upper bounds of IGC may be different, partly due to the choice of þ þ  thresholds, q 1 and q1 , and q2 and q2 . Although this will cause that some assessments are identified to improve lower or upper bounds of IGC, and others to improve lower and upper bounds of IGC, experts can renew assessments to implement bilateral improvement of lower and upper bounds of IGC if they conform the suggestion rule in Definition 5. In the process of reaching the required IGC, lower and upper bounds of IGC may be moved from different states to the same states or from the same states to different states, so they have to be decided separately and improved wholly. It is nat-

175

ural that when one of lower and upper bounds of IGC is reached, the other may not be done. On the other hand, when experts renew assessments according to the suggestion rule in Definition 5, they should simultaneously consider the movement of lower and upper bounds of grades in assessments, to ensure the validity, normalization and meaningfulness of renewed assessments. In addition, the most discordant expert for two consecutive rounds of GAD will be recommended to modify opinions in Parreiras et al.’s consensus scheme (Parreiras et al., 2010). This may spend so many rounds of GAD, which hinders quick convergence to consensus. Further, experts’ utilities and the flexibility in consensus measures are not considered in the scheme. In Altuzarra et al.’s method (Altuzarra et al., 2010), based on pairwise comparison of alternatives, the most discrepant assessments are identified to revise, and different consensus paths are provided to reach consensus among experts. Only a unique criterion, however, is used to compare pairs of alternatives. Further, relative weights of experts are fixed rather than composed of subjective and objective parts, in which the objective part is related to GC, like the ERCM. Moreover, as stated in Section 4.3, experts may renew some assessments not recommended by the identification and suggestion rules after many rounds of GAD when they consider it reasonable. This may sound more reasonable than the strategy of automatically modifying the most discrepant assessments based on the aggregated group assessment in (Xu and Cai, 2011; Dong et al., 2010b; Wu and Xu, 2012), because experts decide their renewed assessments according to their own preference. 6. Conclusions This paper proposed an ERCM to model MAGDA problems with both quantitative and qualitative attributes, probabilistic uncertainties, especially and pivotally interval uncertainties and IGC requirements. Missing assessments were handled in the ERCM. The identification and suggestion rules were designed to identify assessments damaging IGC and generate recommendations to help accelerate convergence to IGC. Particularly, recommendations on assessments rather than on lower and upper bounds of grades in assessments were given to experts. This can effectively avoid assessments with large imprecision and unnormalized assessments. In the next step, a GDSS with a graphical interface will be constructed based on the ERCM, like existing GDSSs (Pérez et al., 2010; Alonso et al., 2010). Acknowledgements This research is supported by the National Natural Science Foundation of China (Nos. 71071045, 71131002 and 70925004), the Humanities and Social Science Foundation of Ministry of Education in China (No. 12YJC630046), the Natural Science Foundation of Anhui Province of China (No. 1208085QG130), and a grant from CityU (No. 7002700). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ejor.2012.05.048. References Alonso, S., Herrera-Viedma, E., Chiclana, F., Herrea, F., 2010. A web based consensus support system for group decision making problems and incomplete preferences. Information Sciences 180, 4477–4495.

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