A consensus- and harmony-based feedback mechanism for multiple attribute group decision making with correlated intuitionistic fuzzy sets

1. Introduction

Multiple attribute group decision making (MAGDM) problems address decision situations in which a group of experts express preferences on multiple attributes and interact to derive a common solution (Ölçer and Odabaşi, 2005). During the past decades, significant work has been done to develop decision-making models for this issue. For example, recent proposals to deal with MAGDM can be found in the literature (Park and Kim, 1997; Wei et al., 2000; Cao and Wu, 2011). However, the MAGDM problems generally involve many conflicting aspects prevalent among its experts because different actors have diverse and conflicting value systems. It is preferable that the set of experts reach consensus before applying a selection process to derive the decision solution (Altuzarra et al., 2010; Wu and Cao, 2010; Wu et al., 2011).

Consequently, one key issue that needs to be addressed in an MAGDM problem is the evaluation of the consensus level of group experts. A comprehensive review of the literature (Herrera-Viedma et al., 2014) indicates the existence of different approaches to model consensus under group decision making (GDM). The GDM models usually have the following preference relations: the crisp numbers (Chiclana et al., 1998, 2007, 2009, 2013; Wu and Xu, 2012), interval numbers (Xu, 2004; Wu et al., 2009; Wu and Cao, 2010; Wu and Chiclana, 2014a), linguistic information (Herrera et al., 1996a, 2009; Zhou and Chen, 2013), fuzzy numbers (Bryson, 1997; Dong et al., 2008, 2010; Su et al., 2013), intuitionistic fuzzy numbers (Szmidt and Kacprzyk, 2003; Xu, 2007; Xu and Yager, 2009; Yu, 2014), and extended intuitionistic fuzzy numbers (Wu and Chiclana, 2012; Wu and Liu, 2012; Wu and Cao, 2013). However, these approaches are static in nature, that is, they do not include any type of feedback mechanism to produce rules or recommendations on how to increase consensus when it is unacceptably low. Cabrerizo et al. (2010) made a comprehensive comparison of different consensus measures, and then some interactive consensus models (Alonso et al., 2010; Dong et al., 2014; Herrera-Viedma et al., 2007; Wu and Chiclana, 2014b; Xu and Wu, 2013; Xu et al., 2013;) were developed for experts to change their opinion to a higher consensus level by a feedback mechanism. Furthermore, these interactive consensus models are extended into the following decision-making contexts: network (Pérez et al., 2010; Alonso et al., 2013), unbalanced fuzzy linguistic information (Cabrerizo et al., 2009, 2010), and discrete fuzzy linguistic information (Massanet et al., 2014). However, these interactive consensus models use the two comparison preference relations, therefore they are only suitable to deal with single-attribute GDM problems. Recently, Xu (2009) extended these models to the case of MAGDM problems. But, Xu's interactive consensus model contains a feedback mechanism that experts must change all of their opinions, if one expert has not reached the threshold value of consensus level. Later, Xu and Wu (2011) proposed an improved feedback mechanism in which every expert has to update only part of his/her opinion (some elements in the decision matrices).

Although these interactive consensus models are successful in investigating different kinds of consensus levels and proposing feedback mechanisms to provide advice to reach higher consensus degree, these models neglected a reality that if the recommended advice makes experts deviate from their original opinions, they may not accept these advice. Therefore, in nature, these approaches are forcing the discordant experts to change their opinions without considering whether they agree or not. However, in practice, it is up to the decision maker (DM) whether to implement the recommendations given to him/her (Eklund et al., 2008). A more reasonable policy should rest on this premise and, consequently, it would allow the DM to revisit his/her evaluations using appropriate and meaningful consensus information representation (Wu and Chiclana, 2014b). Also, Herrera-Viedma et al. (2014) recently pointed that some new decision rules should be taken into the consensus process of GDM. Inspired by these ideas, this article defines the concept of the harmony degree (HD) to determine the degree of maintaining the original information of expert. Then, the consensus level is defined as the proximity degree (PD) between an expert and other experts in the group. Particularly, the PD is computed on three levels: evaluation elements of alternatives, alternatives, and decision matrix. Combining the PD and HD, a three-dimensional (3D) feedback mechanism is proposed to show the consensus and harmony status of every expert with figures. Furthermore, a graphical simulation of future consensus and harmony status, if the recommended values were to be implemented, has been depicted (see Fig. 2(a)–(c) in the final numerical example). In the light of this extra visual information, a discordant expert can revisit his evaluations and make changes if considered appropriate to achieve a higher consensus level. Consequently, the novelty of our feedback mechanism is that it increases the consensus level of the set of experts and maintains, as much as possible, experts' original information simultaneously.

Considering that there may exist some degree of interactive characteristics in experts or interdependence in criteria (Grabisch, 1995; Grabisch et al., 2000; Torra, 2003; Tan, 2011), it could be interesting to investigate an interactive approach for MAGMD in which the preferences of experts are interactive and the attributes are interindependent. Because of the fuzziness in MAGDM, this article applies the intuitionistic triangular fuzzy numbers (ITFNs) to describe the evaluation elements (Zhang and Liu, 2010) that are an extension of triangular fuzzy sets (Zadeh, 1965) and intuitionistic fuzzy sets (Atanassov, 1986). After the group consensus level reaches a satisfied value, the individual decision matrices need to be aggregated into a collective matrix. Therefore, the aggregate operator is a key issue in GDM (Merigó and Casanovas, 2011; Merigó and Gil-Lafuente, 2011, 2013; Zhou et al., 2012a, 2012b). To deal with interactive characteristics between experts, a new PD-induced intuitionistic triangular fuzzy correlated averaging (PD-IITFCA) operator is investigated to compute the collective decision matrix. The PD-IITFCA operator associated weighting vector is derived using the PD of individual expert, providing a monotonic increasing mapping between the experts' consensus levels and their contribution weight in the collective decision matrix. Furthermore, this PD-IITFCA operator has the advantage that it does not require to assign the weight of each expert beforehand, while Pérez et al. (2013) assumed that the weights of experts are to be obtained directly. To rank the best alternative, an intuitionistic triangular fuzzy correlated averaging (ITFCA) operator is investigated to aggregate the evaluation elements of alternatives under correlative attributes, which can also be regarded as the generalization of these correlated aggregating operators in the literature (Wei and Zhao, 2012; Yang and Chen, 2012; Meng et al., 2013; Wu et al., 2013).

The rest of the article is organized as follows. Section 2. develops the score and accurate functions of ITFNs, then an order relation for ranking ITFNs is proposed. In Section 3., the definition of PD is proposed on three levels. Then, an interactive consensus model for MAGDM with ITFNs is covered with special attention paid to the design of the PD-based feedback mechanism. Section 4. proposes an approach for the selection process of MAGDM problems based on the PD-IITFCA and ITFCA operators. Section 5. gives an illustrative example to show the validity of our proposed method. Finally, Section 6. gives an analysis of the proposed consensus model highlighting the main differences with respect to existing consensus models in the literature, and then draws the conclusions.

2. Score and accurate functions for ranking ITFNs

Intuitionistic fuzzy sets (IFSs) were introduced by Atanassov (1986).

Definition 1. (IFS)An IFS A over a universe of discourse X is represented as

(1)

where , and For each , the numbers and are known as the degree of membership and degree of nonmembership of x to A, respectively.

To express more fuzzy information, Atanassov and Gargov (1989) introduced the definition of interval-valued IFSs (IVFSs), which was later generalized by Zhang and Liu (2010) with ITFNs. A prominent characteristic of ITFNs is that its membership values and nonmembership values are triangular fuzzy numbers (TFNs).

Definition 2. (ITFNs)An ITFN is denoted as . Its membership function can be given as

(2)

and its nonmembership function is

(3)

where , , and .

Let and be two ITFNs, and , then ITFNs have the following operational laws:

• 1. .
• 2. .
• 3. .
• 4. .

To rank TFNs, Liou and Wang (1992) proposed the following index.

Definition 3.Let is a collection of TFN with left membership function and right membership function . Suppose that is the inverse function of , and is the inverse function of , then ordering index is defined by

(4)

where is the optimism index reflecting the optimism degree of a decision maker. The larger η is, the more optimistic the decision maker is.

Suppose that , then expression 4 can be expressed as

(5)

Recall that TFNs are particular cases of ITFNs, we can apply this index to propose the score and accurate functions of ITFNs.

Definition 4. (Score function of ITFNs)Let be an ITFN. Then, a score function of ITFNs can be represented as follows:

(6)

where , the larger the value of , the more the degree of score of the ITFN . and are computed by expression 5, respectively.

Definition 5. (Accuracy function of ITFNs)An accuracy function of ITFNs can be represented as follows:

(7)

where , the larger the value of , the more the degree of accuracy of the ITFN . and are computed by expression 5, respectively.

3. Interactive consensus model for correlated MAGDM

The consensus process of GDM can be considered as an interactive methodology to obtain the high degree of consensus between the set of experts. Clearly, it is preferable that the experts achieve a high consensus level before applying the selection process. To do so, Xu (2009) proposed an interactive algorithm to reach consensus. If the expert does not reach the threshold value of consensus level, then the interactive algorithm (Xu, 2009) changes every original decision matrix

(8)

and

(9)

But this interactive consensus model has some drawbacks that force experts to change all of their opinions (Xu and Wu, 2011). This policy may not be acceptable to those experts with acceptable consensus level as it will make them deviate from their original opinions. A more reasonable policy of interactive consensus process is to increase the level of agreement and, at the same time, maintain the original information as much as possible. To achieve these “rational” criteria, this article introduces the definition of proximity degree (PD) to measure the actual consensus on three levels: evaluation elements of alternatives, alternatives, and decision matrix. If satisfies a minimum satisfaction threshold value , agreed by the group experts, then the selection process of GDM is carried out; otherwise a 3D feedback mechanism is proposed to identify the discordant experts, the discordant alternatives and the discordant evaluation elements at a fast speed. Also, this article will determine the degree of maintaining the original information of expert, named harmony degree (or HD). Combining the PD and HD, this feedback mechanism provides figures to show the consensus status of every expert. Furthermore, a graphical simulation of future consensus and harmony status, if the recommended values were to be implemented, is also presented.

In order to aggregate the individual decision matrix into a collective one, it is necessary to determine the weights associated to each expert. Because consensual information is considered more relevant or important than discordant information, we develop the PD-IITFCA operator that associates higher weights with more consensual information. This operator has the advantage that it does not need any weight information beforehand and assigns the experts' weights by their associated PDs. Furthermore, the PD-IITFCA operator aggregates individual opinions in such a way that more emphasis is given on higher consensus level. Finally, we develop the ITFCA operator to aggregate the correlative attribute values and derive the final priority of the alternatives.

These steps will be presented in more detail in the following subsections. A step-by-step example to illustrate the computation processes involved in each step is also provided.

3.1. The definition of PD with ITFNs

Based on the Hamming distance (Yang and Chiclana, 2012), we can define a distance function between ITFNs as follows:

Definition 7.Let and be two ITFNs, then we define the distance between and as follows:

(10)

Xu and Yager (2009) defined the similarity degree between two intuitionistic fuzzy numbers for consensus analysis in GDM. We extend it to the case of ITFNs.

Definition 8. (Similarity degree)Let and be two ITFNs, and be the complement of , then

(11)

is called the similarity degree of and , where .

The similarity degree ϑ has the following properties:

• 1. which means is more similar to than . Especially, , which means the identity of and .
• 2. which means is similar to than .
• 3. which means is more similar to than . Especially, , which means the complete dissimilarity of and .

In the following definition, we will apply the developed similarity to measure the PD for each expert.

Definition 9.Let be a decision matrix. If all are ITFNs, and

then we call an intuitionistic triangular fuzzy decision matrix.

To compute the PD between an expert and other experts in the group, we need to aggregate decision matrices into a collective matrix. However, in most cases, the weight information of experts is usually not known beforehand. Therefore, the arithmetic or geometric average operators are usually used for aggregating individual opinions (Bryson, 1997; Xu and Yager, 2009; Yu and Lai, 2011; Wu and Xu, 2012).

Definition 10.Let be a collection of intuitionistic triangular fuzzy decision matrices given by experts, respectively, then the average aggregated intuitionistic triangular fuzzy decision matrix is defined as:

where .

Once the average aggregated decision matrix is computed, we compute the PD for each expert at the three different levels of a relation:

• Level 1. PD on elements of alternatives. The PD of an expert to the group on the alternatives under attribute is
  (12)
• Level 2. PD on alternatives. The PD of an expert to the group on the alternative is
  (13)
• Level 3. PD on decision matrix. The PD of an expert to the group on decision matrix is
  (14)

The greater the value of , the greater the PD between intuitionistic triangular fuzzy decision matrices and . The PD has the following desirable properties.

Proposition 1.Let be t intuitionistic triangular fuzzy decision matrices, and be their averaged aggregated decision matrix, then

1. .
2. if and only if and are identical.

Proof 1.By expression 12, we have

then

2. Necessity. If , then , for all that is, , for all . Therefore, and are identical.

3. Sufficiency. If and are identical, then , for all that is, , for all . Therefore, .

The PD can be used to decide when the feedback mechanism should be applied to give advise to the experts, or when the consensus reaching process has to come to an end. When satisfies a minimum satisfaction threshold value the consensus reaching process ends, and the selection process is applied to achieve the solution of consensus.

Note 2.In GDM problems, it is rare to achieve complete consensus. Therefore, the threshold value . And in most cases, if more than half of experts can achieve consensus, the decision-making result may be acceptable. Thus, . Consequently, we have the threshold value . The bigger the threshold value, the more experts should change their preference relation.

3.2. Three-dimensional feedback mechanism

When at least one of the experts' consensus levels is below the fixed threshold value, a feedback mechanism is activated to generate personalized advice to those experts, which includes two steps: “identification of the evaluation values” that should be changed and “generation of advice.”

3.2.1 Identification of the evaluation values

To identify the evaluation values that are contributing less to the consensus, the following three steps of identification procedure that uses PD are carried out:

• Step 1. The experts with a PD lower than the threshold value γ are identified: .
• Step 2. For the identified experts, their alternatives with a lower than the satisfaction threshold γ are identified: .
• Step 3. Finally, the values to be changed are .

3.2.2 Generation of advice

The feedback mechanism generates personalized recommendation rules, that is, it will tell the experts which preference values they should change and also the new evaluation values to use in order to increase their consensus level. For all , the personalized recommendation rules are identified as follows:

• 1.

  If , the recommendation generated for expert is: “You should change your evaluation value for the alternatives under attribute , , to a value closer to .''

  

where is a parameter to control the degree of recommendation.

To determine the degree of maintaining the original information of expert before and after adopting the recommended advice, this article provides the definition of the HD as follows.

Definition 11.After accepting the recommended advice by feedback mechanism, the HD of expert can be as defined as

(15)

where . The bigger the , the less original opinion the expert changes.

Definition 12.The general HD (GHD) of experts can be as defined as

(16)

Note 3.. The bigger the GHD, the more original opinion is preserved in every feedback round. Therefore, a more reasonable policy of feedback mechanism is to arrive at the threshold value of consensus level and the maximum value of GHD simultaneously.

4. Selection process based on the PD-IITFCA and ITFCA operators

4.1. The PD-IITFCA operator for aggregating individual decision matrices

Let be the weights of the elements, where ϕ is a fuzzy measure. Wang and Klir (1992) gave the definition of ϕ as follows.

Definition 13.A fuzzy measure ϕ on set Y is a set function ϕ :

• 1. .
• 2. ; implies , for all .
• 3. , for all and , where .

Especially, if , then the condition (3) reduces to the axiom of additive measure:

If all the elements in Y are independent and we have

(17)

Combining the above definition and the well-known Choquet integral (Choquet, 1953), Xu (2009) developed some intuitionistic fuzzy aggregation operators that consider the importance of the elements or their ordered positions and reflect the correlations among the elements or their ordered positions simultaneously. In the following, we develop the PD-ITIFCA operator for aggregating decision-making matrices with ITFNS in which the weights of experts are correlative.

Definition 14.Let be a collection of ITFNs on X, a PD-IITFCA operator of dimension is a mapping PD-IITFCA: such that

(18)

where is a permutation of such that for all , that is, is the 2-tuple with the tth largest values in the set , and in is referred as the order inducing variable and as the ITFNs. for and .

By Definition 14, we get the following result using mathematical induction on n.

Theorem 1.Let be a collection of intuitionistic triangular fuzzy values on X and ϕ be a fuzzy measure on Y, then their aggregated value using the PD-TIFCA operator is also an ITFN, and

(19)

where for , and , and . , , and , respectively.

Proof.Clearly, the first result follows from Equation 18. In the following equation, we prove Equation 19 using mathematical induction on n. For , we have

and

According to the operation law (1) of ITFNs, we obtain

That is, for n = 2, Equation 19 holds. Suppose if for , Equation 19 holds, that is,

then, for , according to Definition 14, we have

That is, for , expression 19 always holds. Therefore, for all n, expression 19 always holds, which completes the proof of Theorem 1.

Before implementing the aggregation process, the correlated operator needs to assume a fuzzy measure ϕ in order to compute its correlated weights (Xu, 2010). However, in some real case, this supposition may not be met, that is, we have no information of experts' weights. In the case of PD-IITFCA operator, we propose to use the PD values associated with each expert, both as a weight associated with the argument and the order inducing values, which is similar to the method used determining the weighting vector associated with an induced ordered weighted averaging (IOWA) operator (Yager, 2003). Considering each comment in the aggregation consists of a triple : is the argument value to aggregate, is the importance weight value associated with , and is the order inducing value, the aggregation is

(20)

with

(21)

where , and σ is the permutation such that in is the lth largest value in the set . Q is a function: [0, 1] → [0, 1] such that and if then .

In our case, the ordering of the preference values is first induced by the ordering of experts from highest to lowest consensus, and the weights of the PD-IITFCA operator is obtained by applying Equation 21, which reduces to

(22)

where , , and σ is the permutation such that is the lth largest value in the set .

In an aggregation process, the weighting value of experts should be implemented in such a way that the effect from those experts who have less consensus is reduced. Supposing a concave function (Chiclana et al., 2007), we can verify that the higher the consensus of an expert the higher the weighting value of that expert in the aggregation, that is,

4.2. The exploitation of the alternatives

Definition 15.Let be a collection of intuitionistic triangular fuzzy values on X. Based on fuzzy measure, an ITFCA operator of dimension n is a mapping TIFCA: such that

(23)

where is a permutation of such that , , for and .

We can also obtain the following result for aggregating ITFNs using mathematical induction on n.

Theorem 2.Let be a collection of intuitionistic triangular fuzzy values on X, then their aggregated value using the ITFCA operator is also an intuitionistic triangular fuzzy value, and

(24)

where indicates a permutation on X such that , for , and , and . , , and ,respectively.

Proof.The proof of Theorem 2 is similar to the proof of Theorem 1 (omitted).

5. Numerical example

Example 1.A computer producing company wants to select the most appropriate green supplier for one of the key elements in its manufacturing process. After preevaluation, four suppliers have remained as alternatives for further evaluation. Three criteria are considered as follows: C₁, environmental costs; C₂, remanufacturing/reuse activity; C₃, hazardous waste management. Four experts from different departments organized this evaluation whose important degrees are not known beforehand. Procedure for the selection of most appropriate green supplier contains the following steps:

Step 1. According to Definition 10, all the decision matrices are synthesized into an aggregated decision matrix :

Step 2. Taking a minimum threshold value of , we calculate the PDs on the relation for each expert as follows:

Therefore, the feedback mechanism is activated to assist expert e₁.

Step 3. The following set is obtained:

Then the generation of advice is activated.

Step 4. Taking a value of , the recommendations for expert e₁ are as follows:

• You should change your evaluation value of alternative x₄ under attribute c₁ to a value closer to .
• You should change your evaluation value of alternative x₄ under attribute c₃ to a value closer to .

We can obtain the . Obviously, it is very close to 1, which means experts just need to make a little change of their original opinions. Therefore, expert e₁ may be glad to implement the changes in his evaluation values to a higher consensus levels. Furthermore, a visual graphical simulation of future consensus status after the recommended values were to be implemented is shown in Fig. 2(a)–(c).

Adopting Xu's feedback mechanism (Xu, 2009), we can calculate the , which is obviously lower than our . The reason for this is that Xu's feedback mechanism forces every expert to change the preference values regardless of whether they achieve the threshold value of consensus level shown in Fig. 3. In addition, Fig. 3(b) and (c) shows that all the alternatives and elements are changed. Obviously, our proposed feedback mechanism advises the discordant expert to change only his part evaluation values, and therefore it reduces the computation burden and can be easily accepted.

Step 5. Second consensus round: Assuming e₁ implements the values recommended above, he/she gives the new decision matrix as

and applying the above computation process, the new consensus levels would become , which are over the threshold value .

Step 6. Apply the new consensus levels : and assume the basic unit-interval monotonic (BUM) function , we calculate the correlated weights of experts by expression 22.

Then, by the PD-IITFCA operator 18, the collective decision matrix is computed as

Step 7. Thus, the exploitation of the alternatives would be activated.

Suppose that the fuzzy measure of attribute of as follows:

Utilizing the decision information given in matrix and the ITFCA operator, we derive the collective overall preference values of the alternative:

(25)

According to expression 6, we calculate the score functions as

Then

Thus, the best alternative is x₃.

6. Conclusions

Finally, the feedback mechanism of the proposed consensus model is designed following a top-to-bottom methodology, and therefore it identifies the discordant experts in the following order: (a) their decision matrices, (b) their alternatives, and (c) their evaluation elements. Then, the experts only need to update their discordant evaluation elements to consensus, that is, it preserves the original decision opinions and independence in maximum degree, and also reduces the computation burden with respect to other consensus method in the literature.