A Consensus Reaching Model with Minimum Adjustments in Interval-Valued Intuitionistic MAGDM

This paper focuses onmultiattribute group decision-making problemswith 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 helpDMs 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.


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 [2][3][4].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 [7][8][9][10][11][12][13][14][15][16].
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 [17][18][19][20][21][22][23][24][25] 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 2 Mathematical Problems in Engineering 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 [26][27][28].
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: (1) Most existing methods [7][8][9][11][12][13][14][15] 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.
(2) Some methods [12][13][14]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.
(3) 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: (1) 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.
(2) 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.
(3) 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.

Preliminaries
To facilitate subsequent analyses, this section reviews some definitions related to IVIFSs and describes MAGDM problems with IVIFVs.
Definition 3 (see [30]) are called the score function and accuracy function of the IVIFV α, respectively.
Let  = { 1 ,  2 , ...,   } be a discrete set of alternatives.Let  = { 1 ,  2 , ...,   } be a finite set of attributes.Assume the weight vector of attributes is ω = (ω 1 , ω2 , ..., ω ) 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.

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.

Consensus Index in MAGDM with
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 R1 = R2 = ⋅ ⋅ ⋅ = R = R .Otherwise, the smaller the consensus index CI( R1 , R2 , ⋅ ⋅ ⋅ , R ), the higher the consensus among DMs.

Adjustment Amounts in MAGDM with
IVIFVs.For the group { R1 , R2 , ⋅ ⋅ ⋅ , R } with unacceptable consensus, it is necessary to adjust R1 , R2 , ⋅ ⋅ ⋅ , R until they reach acceptable consensus.Denote the adjusted matrices by R1 , R2 , ⋅ ⋅ ⋅ , R , where R = (r  ) × and r ).For convenience, let R = { R1 , R2 , ⋅ ⋅ ⋅ , R } be the set of original decision matrices and R = { R1 , R2 , ⋅ ⋅ ⋅ , R } be the set of adjusted decision matrices.As we know, the smaller the adjustment amounts of the set R from R, the more the decision information adjusted matrices retain.Thereby, how to measure adjustment amounts is an important issue.In [18][19][20], 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 AM( R, R) in ( 13) describes the average deviation of all adjusted matrices from their original matrices.The smaller AM( R, R) is, 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 R from R is proposed as where  −  ,  +  ,  −  , and  +  , respectively, indicate the numbers of the adjusted  −  ,  +  , V −  , and V +  , i.e., In ( 14), adjustment amount NAM( R, R) counts the total number of adjusted elements in the consensus reaching process.If NAM( R, R) = 0, all elements of all original decision matrices R ( ∈ ) are not adjusted, i.e., R = R for any  ∈ .The larger NAM( R, R), 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.

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 AM( R, R), while Consensus Rule 2 is to minimize the total number of adjusted elements (i.e., NAM( R, R)).

Consensus Rule 1 for Reaching Consensus.
Due to the fact that the adjusted matrices R ( ∈ ) are considered as final decision matrices, it is natural that the collective matrix should be obtained by aggregating matrices R ( ∈ ) with IVIFWA operator in (7).Denote the obtained collective matrix by R = (r  ) × , where r where  = ( 1 ,  2 , ⋅ ⋅ ⋅ ,   ) T is the vector of DMs' weights.
In reaching consensus process, the group { R1 , R2 , ⋅ ⋅ ⋅ , R } should be required to be acceptable consensus, i.e., where In addition, to guarantee that adjusted matrices R = (r  ) × ( ∈ ) are IVIF matrices, where r = Mathematical Problems in Engineering Accordingly, Consensus Rule 1 is built by minimizing the Manhattan distance-based adjustment amount AM( R, R) under such constraints described by ( 19)-( 21), i.e., min AM ( R, R) where r and r 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 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 R *  = (r *  ) × ( ∈ ), are obtained.

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 as where  −  ,  +  ,  −  , and  +  are binary variables described by ( 15), ( 16), (17), and (18).
Step  (24).Remark 12.In Step 3 * , the parameter  0 is the predefined threshold of the acceptable consensus, while the parameter  is a critical value that is used to decide whether Consensus Rule 1 or Consensus Rule 2 is applied to improve the consensus among DMs.In addition, the parameter  is more than  0 .
)  ) <  0 , then the consensus is acceptable consensus.Otherwise, the consensus is unacceptable and needs to be improved.While improving the consensus, if From Theorems 13 and 14, it is concluded that Algorithm I is convergent.It is easily understand that Algorithm II is also convergent because it is a combination of Consensus Rule 1 and Rule 2.

An Interval-Valued Intuitionistic Fuzzy Program Based Method for Solving GDM with IVIFVs
This section proposes an IVIF program based method for solving GDM with IVIFVs.To derive attributes' weights, an IVIF program is built.By transforming this IVIF program into a multiobjective interval program, a solving approach is proposed.The collective decision matrix is derived by integrating individual decision matrices with the IVIFWAM operator.Finally, combining the consensus reaching process described in Section 4, a novel method is presented to solve GDM with IVIFVs.

Interval Objective Program.
As a preparation for solving the IVIF program which will be constructed to determine attributes' weights, this subsection reviews an approach to solving an interval objective program.Ishibuchi and Tanaka [32] presented a popular solving approach by transforming an interval objective programming into an equivalent multiobjective programming.
) . ( where  * = ( * 1 ,  * 2 , ⋅ ⋅ ⋅ ,  *  ) T is the accurate weight vector of attributes and satisfies As we know, the larger the comprehensive value r of alternative   , the better the alternative   is.Thus, a mathematical program is built by maximizing comprehensive values, i.e., max r ( ∈ ) As comprehensive values r ( ∈ ) are IVIFVs, this program is called an IVIF program.According to Definition 2, to maximize r ( = 1, 2, ⋅ ⋅ ⋅ , ), it is necessary to maximize their memberships and minimize their nonmemberships.Hence, (35) can be equivalently transformed into the following multiobjective interval program: Employing the solving approach in Section 5.1, (36) can be converted as Equation ( 37) can be further transformed as By the min-max summation method, (38) can be converted into the following program: Plugging ( 34) into ( 39), ( 39) can be rewritten as min  ..
Solving (40), the weight vector of attributes  * is determined.

A Novel Method for
Solving GDM with IVIFVs.According to above analyses, a novel method for solving GDM with IVIFVs is generalized below.
Step 1.Each DM establishes his/her individual decision matrix R = (r  ) × with IVIFVs and provides the vector ω of attribute weights with IVIFVs.
In general, there are cost attributes and benefit attributes.Denote the set of cost attributes by  1 and the set of benefit attributes by  2 .Individual decision matrices often are normalized by the following formula: Step 3. Check the consensus among individual decision matrices R ( ∈ ) by ( 12) and (19).If the group is acceptable consensus, go to Step 3. Otherwise, go to the next step.
Step 4. Reach consensus by Algorithm I or II and derive the final adjusted individual decision matrices R  ( ∈ ) and the final collective decision matrix R  .
Step 7. Rank alternatives based on Definition 4.
Figure 1 depicts the above decision-making process.

A Practical Example of a Cloud Service Provider Selection and Comparative Analyses
To illustrate applications and advantages of the proposed method, this section provides a cloud service provider selection example and conducts comparative analyses.

A practical Example of a Cloud Service Provider Selection.
Cloud computing [33,34] is a kind of computing paradigms based on the internet.By this paradigm, shared hardware and software resources can be provided to computers on demand.Cloud computing has many advantages, such as flexibility, business agility, and pay-as-you-go cost structure.Many enterprises with limited financial and human resources can apply cloud computing to deliver their business services and products online to extend their business markets.Thus, the selection of cloud service provider becomes a key issue for enterprises.
Guilin FeiYu Electronic Technology limited company (Feiyu for short) is devoted to the development and sales of electronic products.To increase the running speed of their servers, Feiyu tries to seek a cloud service provider.After the market research and preliminary screening, five providers are selected and be further evaluated, including Ali Cloud ( 1 ), Tencent Cloud ( 2 ), Field Could ( 3 ), American Cloud ( 4 ), and Wangsu Technology limited company ( 5 ).Three decision-makers   ( = 1, 2, 3) are invited to evaluate these five alternatives with respect to four attributes, including performance ( 1 ), payment ( 2 ), security ( 3 ), and scalability ( 4 ).Suppose the vector of DMs' weights is  = (0.
In what follows, we will apply the proposed method to solve this example.
Steps -.Normalized individual decision matrices are shown as Tables 1-3.
Step .Check the consensus of individual IVIF decision matrices.

Comparative Analyses.
To further show advantages of the proposed method, some comparative analyses are conducted in this subsection.

IR and DR Rules.
In consensus reaching process, Identification Rule (IR) and Direction Rule (DR) are two Step '.While improving the consensus degree, one element is modified each time.To complete the step, four substeps are needed.

}.
Step 3'-3.Seek the maximum consensus degree of DM   0 on alternative   0 with respect to attributes and denoted by CI () Step 3'-4.Modify the element r() Set  =  + 1 and go to Step 2.

Consensus Rule 4
Step ".While improving the consensus of the group Set  =  + 1 and go to Step 2.

Comparative Analysis with IR and DR Consensus Rules.
For showing merits of Consensus Rule 1 and Consensus Rule 2 proposed in this paper, the example in Section 6.1 is solved by Consensus Rules 1-4 with different values of the parameter , respectively.Suppose the threshold  0 = 0.1.Comparative results are represented as Tables 9 and 10.Table 9 shows distances between original and adjusted individual decision matrices by Consensus Rules 1-4.Table 10 demonstrates iterative rounds for reaching consensus by Consensus rules 1-4.
From Tables 9 and 10, merits of Consensus Rule 1 and 2 proposed in this paper are outlined as follows: (1) In the consensus reaching process, Consensus Rules 1 and 2 are able to retain more decision information provided by DMs compared with Consensus rules 3 and 4. Observing total distances between original decision matrices and their corresponding adjusted ones in Table 9, the distances obtained by Consensus Rules 1 and 2 are obviously less than those obtained by Consensus Rules 3 and 4. As we know, the less the distance, the closer the adjusted decision matrix is to the original one and more decision information is preserved.At this point, Consensus Rules 1 and 2 retain more decision information and are better than Consensus Rules 3 and 4. In addition, as the distances obtained by the Consensus Rule 4 are very large, most decision information may be lost while reaching consensus by this rule.
(2) Compared with Consensus Rule 3, Consensus Rules 1 and 2 need much less iterative rounds for reaching consensus under any values of parameter  ∈ [0,1], which can be verified by Table 10.In other words, Consensus Rules 1 and 2 are time-saving.Although Consensus Rule 4 also needs fewer rounds, it is not sensible enough to select this rule for

Figure 1 :
Figure 1: The decision-making process of the MAGDM with IVIFVs.
Consensus degrees on alternatives: The consensus degree of DM   on alternative   is computed as −  ,  +  ,  −  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 1 programming contains a product of a binary variable  with a linear Mathematical Problems in Engineering term ∑  =1     , where   ( = 1, 2, ⋅ ⋅ ⋅ , ) are variables with finite bounds, this product can be replaced by a new variable  together with the following linear constraints: , the optimal value and the adjusted individ- ), the final collective IFPR R  , and the number of the iterations .

Table 4 :
Temporary interval-valued intuitionistic fuzzy collective decision matrix R .

Table 6 :
Final adjusted interval-valued intuitionistic fuzzy decision matrix R 2 .

Table 8 :
Final interval-valued intuitionistic fuzzy collective decision matrix R .

Table 9 :
Total distances between original decision matrices and adjusted ones by Consensus Rules 1-4.DR rule is used to direct DMs in how to modify their opinions of alternatives on attributes.Before conducting comparative analysis, IR and DR are extended into the IVIF environment.Thus, another two consensus rules, called Consensus Rule 3 and Consensus Rule 4, are generated, respectively.By replacing Step 3 in Algorithm I with Step 3' and Step 3", respectively, Consensus Rule 3 and Consensus Rule 4 can be obtained and described as follows: