This paper focuses on multiattribute group decision-making problems with interval-valued intuitionistic fuzzy values (IVIFVs) and develops a consensus reaching model with minimum adjustments to improve the consensus among decision-makers (DMs). To check the consensus, a consensus index is introduced by measuring the distance between each decision matrix and the collective one. For the group decision-making with unacceptable consensus, Consensus Rule 1 and Consensus Rule 2 are, respectively, proposed by minimizing adjustment amounts of individual decision matrices. According to these two consensus rules, two algorithms are devised to help DMs reach acceptable consensus. Moreover, the convergences of algorithms are proved. To determine weights of attributes, an interval-valued intuitionistic fuzzy program is constructed by maximizing comprehensive values of alternatives. Finally, alternatives are ranked based on their comprehensive values. Thereby, a novel method is proposed to solve MAGDM with IVIFVs. At length, a numerical example is examined to illustrate the effectiveness of the proposed method.

1. Introduction

Multiattribute group decision-making (MAGDM), where several decision-makers (DMs) evaluate a finite set of alternatives with respect to multiple attributes and select a best one, has been applied in many fields, such as supply chain management, risk investment, and industry engineering. In classical MAGDM, attribute values are usually represented as real numbers. However, due to the ambiguity of human thinking and the lack of decision information as well as time, DMs are unable to express their opinions with real numbers precisely. After Zadeh [1] introduced the fuzzy set (FS), more and more DMs used fuzzy sets to describe their opinions [24]. Nevertheless, FS characterizes the fuzziness only by the membership degree. Later, Atanassov [5] extended the FS and introduced the intuitionistic fuzzy set (IFS) which uses real number to express the membership, nonmembership, and hesitancy of an alternative on a given set. In 1989, Atanassov and Gargov [6] generalized IFS to the interval-valued IFS (IVIFS), in which the membership, nonmembership, and hesitancy are represented by intervals. In recent years, MAGDM with interval-valued intuitionistic fuzzy values (IVIFVs) has received widely attentions [716].

In group decision-making with IVIFVs, different DMs come from various research fields and have varying perceptions, attitudes, and motivations. Thus, they may have different preferences for the same decision problem and provide distinct opinions on alternatives. Thereby, inconsistency among DMs’ opinions is inevitable. If individual opinions are aggregated directly without consensus, final decision results may be unable to represent the opinion of this group and the group decision-making may be meaningless. Hence, the consensus process, which helps DMs bring their opinions closer, is one of key issues in MAGDM to reach a collective decision result accepted by most DMs. As so far, pools of methods [1725] have been developed to reach consensus. A popular method, proposed by Dong et al. [21], aids DMs in reaching consensus by minimizing adjustments between original individual decision matrices and adjusted ones. Later, this method is improved and extended to different environments. For example, Zhang et al. [19] improved this method by measuring adjustment with distances and the number of adjusted elements in real number scenario. Wu et al. [22] and Zhang et al. [23] extended method [21] to linguistic distribution context or incomplete linguistic distribution context and, respectively, constructed two different feedback mechanisms based consensus models with minimum adjustment cost. Subsequently, Zhang et al. [24] and Dong et al. [25], respectively, developed a minimum adjustment distance consensus rule and a minimum number of adjusted simple terms consensus model under hesitant fuzzy linguistic environment. In fact, consensus models with minimum adjustments have been successfully applied to many contexts, such as social network, opinion dynamics, and dishonest MAGDM contexts [2628].

Although the consensus model with minimum adjustment has been widely used in GDM, its application on GDM problems with IVIFVs has not appeared. Meanwhile, the research on the consensus of GDM with IVIFVs is very few. Only Zhang and Xu [16] and Cheng [10] discussed this issue. Zhang and Xu [16] presented a consensus index based on dominant relations between alternatives. Employing similarity degrees between individual preference vectors and the group one, Cheng [10] presented another consensus index and proposed an iterative approach to improving the consensus. For enriching the study on the consensus model in the IVIF context, this paper presents a new consensus feedback mechanism by generalizing the consensus model with minimum adjustment [19] to IVIF environment.

After reaching the consensus in MAGDM with IVIFVs, the next key issue is how to determine weights of attributes, which plays a significant role while aggregating individual opinions into a collective one. To determine attributes’ weights, different mathematical programs are constructed [8, 9, 11, 15]. For example, by maximizing comprehensive values of alternatives, Wan et al. [15] and Hajiagha et al. [11] constructed distinct mathematical models to determine attribute weights. The difference between them is that the former is an interval program, while the latter is an evolving linear program. By maximizing the weighted scores of alternatives, Chen and Huang [9] built a linear programming model to obtain attributes’ weights. Chen [8] set up a nonlinear program to derive attributes’ weights by maximizing inclusion-based closeness coefficients of alternatives. Finally, alternatives are ranked by distinct decision methods or aggregation methods, such as the plant growth simulation method [14], the inclusion-based TOPSIS method [8], the permutation method [7], the extended ELECTREE [12, 29], and IVIF power Heronian aggregation operators [13].

Although previous studies are effective for solving MAGDM with IVIFVs, there are still some limitations as follows:

Most existing methods [79, 1115] ignored the consensus before integrating individual opinions. Despite of methods [10, 16] discussing the consensus, method [16] only introduced a consensus index but did not provide any approach to improving the consensus. Although method [10] designed an algorithm for reaching consensus, the convergence of this algorithm is not proved. In fact, this algorithm sometimes is unable to help DMs reach the predefined level of the consensus, which is verified in Section 6.2.3.

Some methods [1214, 16] assigned attributes’ weights in advance, and this may result in the subjective randomness. Although methods [8, 9, 15, 29] determined attributes’ weights objectively by constructing and solving mathematical programs, the determined weights are real numbers. Considering advantages of IVIFSs over real numbers mentioned before, it is more suitable that attributes’ weights are represented by IVIFVs.

Due to that attribute values of alternatives are IVIFVs, it is reasonable that comprehensive values of alternatives should be IVIFVs, too. Thus, the decision information supplied by DMs can be retained as much as possible. However, comprehensive values of alternatives derived by methods [8, 12, 14, 15, 29] are real numbers or intervals. This may lead to the lost or distortion of decision information to some extent.

To make up above limitations, this paper discusses the consensus of MAGDM with IVIFVs. A consensus index is introduced to check the degree of the consensus among DMs. To improve the consensus, Consensus Rule 1 and Consensus Rule 2 are presented by minimizing adjustment amounts of original individual decision matrices. The difference between these two consensus rules is that Consensus Rule 1 is to minimize the distances between original decision matrices and adjusted ones, while Consensus Rule 2 is to minimize the number of adjusted elements in original matrices. Subsequently, maximizing comprehensive values of alternatives with IVIFVs, an IVIF program is constructed and solved to determine attributes’ weights objectively. Finally, comprehensive values of alternatives are generated and alternatives are ranked.

Compared with existing methods, the proposed method has following prominent characteristics:

Before aggregating individual decision matrices, the consensus among DMs is considered. A simple index is introduced to measure the consensus degree among DMs and two consensus rules are presented to help DMs reach an acceptable consensus. Furthermore, the convergences of these two rules are proved explicitly. Thus, it is guaranteed that the consensus among DMs can achieve predefined consensus degree for any MAGDM with IVIFVs.

For determining attributes’ weights, an IVIF program is built and a new solving approach is provided. First, DMs assign attributes’ weights in the form of IVIFVs. Afterwards, accurate attributes’ weights are objectively determined by solving the built IVIF program. Thus, not only the activeness of DMs is explored, but also the objectiveness of attributes’ weights is ensured.

The comprehensive values of alternatives obtained by the proposed method are in the form of IVIFVs, which is consistent with the form of attribute values provided by DMs. Thus, the decision information may be retained as much as possible. Therefore, the decision results based on comprehensive values may be more reasonable.

The remainder of this paper is organized as follows: Section 2 reviews some definitions of IVIFSs and describes MAGDM problems with IVIFVs. Section 3 introduces a consensus index for measuring the degree of consensus among DMs and defines two types of adjustment amounts used in the consensus reaching process. Section 4 presents two consensus rules for reaching consensus. In Section 5, a multiobjective interval intuitionistic fuzzy program is constructed and solved to determine attributes’ weights objectively. At the end of this section, a novel method is developed to solve MAGDM problems with IVIFVs. Section 6 provides a numerical example to show the application of the proposed method. The paper ends with some conclusions in Section 7.

2. Preliminaries

To facilitate subsequent analyses, this section reviews some definitions related to IVIFSs and describes MAGDM problems with IVIFVs.

2.1. Interval-Valued Intuitionistic Fuzzy Set

Definition 1 (see [6]). An interval-valued intuitionistic fuzzy set in is defined aswhere and and for any . The intervals and represent the membership degree and nonmembership degree of element to the IVIFS , respectively. Denote and . Therefore, the IVIFS can be equivalently expressed aswhere , , and for any .

In addition, is called the hesitancy index of element , where and . The pair is called an IVIFV and simply denoted by , where , , and .

Definition 2 (see [6]). Let and be two IVIFVs, and then we stipulate(1) if and only if and .(2) if and only if and .(3);(4).

Definition 3 (see [30]). Let be an IVIFV. Thenare called the score function and accuracy function of the IVIFV , respectively.

Definition 4 (see [30]). Let and be two IVIFVs. Then,
If , then ;
If , then (i)If , then ;(ii)If , then .

Definition 5 (see [30]). Let and be two IVIFVs. Then, the Manhattan distance between and is defined as

According to Definition 5, the Manhattan distance between two IVIF matrices is defined as Definition 6.

Definition 6. Let be two IVIF matrices whose elements are IVIFVs, where . The Manhattan distance between and is defined as

Definition 7. Let be a collection of IVIFVs. Ifwhere is an associated weight vector of , satisfying and , then IVIFWM is called an IVIF weight mean operator of dimension . Particularly, when , the IVIFWM is called an IVIF mean operator.

2.2. Multiattribute Group Decision-Making Problems with IVIFVs

For the sake of convenience, letting , and , the MAGDM problem concerned in this paper is described as follows.

Let be a discrete set of alternatives. Let be a finite set of attributes. Assume the weight vector of attributes is , where are IVIFVs. Let be a set of DMs whose weight vector is with and . Suppose that DM provides an IVIF decision matrix , where represents the performance of the alternative with respective to the attribute supplied by DM .

For solving the above MAGDM problems with IVIFVs, two processes, the consensus process and the selection process, are necessary. The consensus process aims to reach a high degree of consensus among DMs, which guarantees that the final decision results obtained in the selection process is accepted by most DMs. The selection process is to obtain the final decision results based on individual decision matrices. In the consensus process, this paper focuses on how to measure the degree of consensus among DMs and how to reach an acceptable consensus degree. In the selection process, this paper proposes an IVIF program based method for ranking alternatives.

3. Consensus Index and Adjustment Amounts in MAGDM with IVIFVs

This section introduces a distance-based consensus index to measure the degree of consensus among DMs and defines an acceptable consensus. If the consensus degree among DMs does not reach the defined acceptable consensus, it is necessary to adjust original individual decision matrices to improve the consensus. In this process, how to measure adjustment amounts of adjusted matrices from original decision matrices is an interesting topic. As for this topic, this section defines two different types of adjustment amounts.

3.1. Consensus Index in MAGDM with IVIFVs

The consensus index for GDM is often introduced by measuring proximity degree between the individual performance and the group performance. The distance function is a popular tool for measuring the proximity degree. Therefore, according to Definition 6, this subsection introduces a consensus index for GDM with IVIFVs.

Let be individual decision matrices and be the collective one obtained by aggregating individual decision matrices, where and . The consensus index is introduced by three levels.

Level 1. Consensus degrees of alternatives on attributes: The consensus degree of DM on alternative with respective to attribute is computed as

Level 2. Consensus degrees on alternatives: The consensus degree of DM on alternative is computed as

Level 3. Consensus degree on decision matrices: The consensus degree of DM is computed as

Thus, the consensus index is defined as

Plugging (8)-(10) into (11), (11) can be written as

It is shown from (12) that full consensus is reached if . Otherwise, the smaller the consensus index , the higher the consensus among DMs.

Definition 8. Let be a predefined threshold. If , the group is called acceptable consensus. Otherwise, the group is called unacceptable consensus.

3.2. Adjustment Amounts in MAGDM with IVIFVs

For the group with unacceptable consensus, it is necessary to adjust until they reach acceptable consensus. Denote the adjusted matrices by , where and . For convenience, let be the set of original decision matrices and be the set of adjusted decision matrices. As we know, the smaller the adjustment amounts of the set from , the more the decision information adjusted matrices retain. Thereby, how to measure adjustment amounts is an important issue. In [1820], distances between original matrices and adjusted ones were applied to measure the adjustment amounts. According to Definition 6, this paper presents a type of Manhattan distance-based adjustment amount as

The adjustment amount in (13) describes the average deviation of all adjusted matrices from their original matrices. The smaller , the closer adjusted matrices are to their corresponding original matrices and hence the more decision information adjusted matrices preserves.

Sometimes, DMs hope to use the number of adjusted elements in original matrices as a measure of the adjustment amount. The less the number of adjusted elements, the smaller the adjustment amount. In this case, another measure for the adjustment amount of from is proposed aswhere , , , and , respectively, indicate the numbers of the adjusted , , , and , i.e.,

In (14), adjustment amount counts the total number of adjusted elements in the consensus reaching process. If , all elements of all original decision matrices are not adjusted, i.e., for any . The larger , the more elements of original matrices being adjusted and, namely, the more adjustment amount.

For preserving the decision information as much as possible, it is sensible to minimize the adjustment amount while reaching consensus. Bearing this idea in mind and employing above two different adjustment amounts, we propose two consensus rules for reaching consensus in the sequel.

4. Two Consensus Rules with Minimum Adjustment Amounts

By minimizing two types of adjustment amounts mentioned in Section 3, respectively, this section develops two consensus rules to reach consensus. Consensus Rule 1 is to minimize the Manhattan distance-based adjustment amount , while Consensus Rule 2 is to minimize the total number of adjusted elements (i.e., ).

4.1. Consensus Rule 1 for Reaching Consensus

Due to the fact that the adjusted matrices are considered as final decision matrices, it is natural that the collective matrix should be obtained by aggregating matrices with IVIFWA operator in (7). Denote the obtained collective matrix by , where . Therefore, it yields thatwhere is the vector of DMs’ weights.

In reaching consensus process, the group should be required to be acceptable consensus, i.e.,where .

In addition, to guarantee that adjusted matrices are IVIF matrices, where , one has

Accordingly, Consensus Rule 1 is built by minimizing the Manhattan distance-based adjustment amount under such constraints described by (19)-(21), i.e.,where and are decision variables.

Plugging (13), (19) and (20) into (22), (22) can be rewritten as

Obviously, (23) is a nonlinear programming model. To solve this model, we can transform it into a linear programming model. Supposing , , , , , , , , , , , , , , , and , then it induces that , , , , , , , , , , , , , , , and . Employing these deviation variables, (23) can be converted as

Equation (24) is a linear programming model and can be easily solved by popular software, such as Lingo and Matlab. Thus, the optimal adjusted individual decision matrices, denoted by , are obtained.

4.2. Consensus Rule 2 for Reaching Consensus

Different from Consensus Rule 1, Consensus Rule 2 aims to minimize the total number of adjusted elements in all individual decision matrices. From (14), the objective function of Consensus Rule 2 is described aswhere , , , and are binary variables described by (15), (16), (17), and (18).

Rest constraints of Consensus Rule 2 are similar to those of Consensus Rule 1. Thus, combining constraints into (15), (16), (17), and (18), Consensus Rule 2 is built aswhere , , , , , , , , , , and are decision variables. It is clear that (26) is a mixed 0-1 programming model.

To solve (26), the key issue is how to handle the last constraints. Let us first analyze binary variables . From (18), one has if . Hence, (18) is equivalent to when . When , variables are taken as 0 or 1 in the equation . If we take , the value of the objective function in (26) is larger than that with . Therefore, variables should be taken as 0 when . Consequently, (18) can be rewritten as regardless of or . Similarly, (15), (16), and (17) can be also rewritten as , , and , respectively.

According to the above analyses, (26) can be converted into another mixed 0-1 program, i.e.,

To facilitate the solving process of (27), Lemma 9 is introduced.

Lemma 9 (see [31]). If a constraint in a mixed 0-1 programming contains a product of a binary variable with a linear term , where are variables with finite bounds, this product can be replaced by a new variable together with the following linear constraints:

Theorem 10. Equation (27) can be equivalently transformed into the following mixed 0-1 linear program:

Proof. Letting and , then we have and . Similarly, suppose that , , , , , and . Thus, the constraints, , can be equivalently converted as