# An Interactive Consensus Model for Group Decision-Making with Incomplete Multi-Granular 2-Tuple Fuzzy Linguistic Preference Relations

Authors: Yongming Song; Feng Yuan
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

As a result of the heterogeneity of decision-makers, it is more suitable to use multi-granular linguistic information to express assessing information over alternatives for decision-makers. Decision-makers may only provide partial preference information as the limitation of knowledge over the alternatives. Thus, the incomplete multi-granular 2-tuple fuzzy linguistic preference relations (IMGFLPRs) can be applied to manage the GDM problems in complex environments. In this paper, we propose a novel group decision-making (GDM) model based on mathematical programming with IMGFLPRs. It is more important that the proposed mathematical programming can directly deal with IMGFLPRs and does not need to be converted into a uniform form. By this means, we construct a consensus reaching process based on IMGFLPRs, considering simultaneously dynamic adjustment of experts’ weight. Finally, an emergency plan selection problem is solved by the proposed model, which shows the result of GDM is effectiveness.


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## Body

## 1. Introduction

With the development of society, decision-making environments are more and more complex and uncertain. Thus, group decision-making (GDM) is put forward and widely applied to integrate individual preference information on alternatives from multiple decision-makers. In some cases, it is difficult for decision-makers (DMs) to express their assessments about alternatives by exact numerical values. Therefore, utilizing linguistic information proposed by Zadeh [1] to represent their assessments may be more appropriate way for DMs. Besides, for linguistic information-based GDM problems, multiple DMs with different level of cognitive competence and knowledge may engage in the decision-making process, which usually implies the necessity of multiple linguistic scales (multi-granular linguistic term sets) to express properly different knowledge for different DMs [2]. Multi-granular fuzzy linguistic-based models have been frequently developed in GDM [2–9].Preference relation is a useful tool for DMs to provide their preferences information on alternatives, for example multiplicative preference relation [10], fuzzy preference relation (FPR) [11] and linguistic preference relation (LPR) [12], etc. Furthermore, it is well understood in the field of decision-making that an expert might not possess adequate levels of knowledge or information to address a given decision-making problem [13], then incomplete evaluations can occur in the complex environment. A number of incomplete preference relations have been developed in recent years [14–17].Consensus is a significant issue widely considered in GDM problems. The consensus assures the unity of opinions of DMs, which makes the result of decision-making accepted by most DMs. A number of methods with a range of preference relations have been proposed to manage the GDM based on consensus [2, 4, 8, 18–27]. Herrera-Viedma et al. [2] established an information feedback mechanism to provide advices with experts to adjust in order to come true a solution with high consistency and consensus levels. Wu and Xu [19] proposed a GDM model based on automatically adjusting consistency as well as consensus to achieve the acceptable predefined level. Liao et al. [24] proposed an improved consensus reaching mechanism in GDM based on multiplicative transitivity of intuitionistic fuzzy preference relation. Although a large number of models have already been proposed to manage GDM problems based on incomplete multi-granular 2-tuple fuzzy linguistic preference relations (IMGFLPRs) [4, 8], there are some drawbacks to the existing research. There is no consensus reaching process in both methods described above [4, 8], which makes the result of decision-making would not be accepted by most decision-makers. Therefore, an important goal of this paper is to establish a consensus reaching process to deal with IMGFLPRs in group decision-making. Besides, the previous studies [4, 8] need estimate unknown elements, which becomes burden on DMs due to the complexity of calculation.In light of the gaps identified above, we are devoted to presenting a consensus model with IMGFLPRs. First, a mathematical programming is established to get directly individual priority weight vector based on an equivalence theorem between FLPR and FPR. Following this, the collective preference vector is obtained by means of IOWA operator. After that, a consensus reaching algorithm is constructed based on direct consensus framework to obtain the collective priority weight vector with satisfactory level of consensus. In the process of consensus, the experts’ weight values are dynamically adjusted according the quality of information provided.The rest of this paper is organized as follows. Section2 reviews some basic knowledge of linguistic information and 2-tuple FLPR. In Section 3, we propose a novel GDM model with IMGFLPs. In Section 4, a problem of emergency decision is resolved by the proposed model. Section 5 carries out a comparison of our suggested method with previous related researches and summarizes the main contributions of this paper. Some concluding remarks are given in Section 6.


## 2. Preliminaries

In this section, we review the related knowledge about linguistic information, multi-granular linguistic information, two-tuple FLPR, and additive consistency of FLPR.


### 2.1. Linguistic Information

To facilitate the assessment using linguistic for DMs, a linguistic term set should be determined beforehand. The widely used linguistic term set has following characteristics: (a) the odd value with granularity; (b) its membership functions being symmetrical and uniformly distributed; (c) the midterm of linguistic term set representing ‘‘indifference’’, with the remainder of the linguistic terms symmetrically and uniformly being placed on either side of it. LetS=s0,s1,⋯,sg be a linguistic term set with odd granularity and g+1 is the granularity of the linguisticS. Moreover, the term set S should satisfy the following features [28]: (1) a negation operator: Negsi=sg-i; (2) an order: si≥sj, if i≥j.The 2-tuple linguistic expressive model was presented [28] and it may improve the accuracy of linguistic computational models.Definition 1 (see [28]).
Supposeβ∈0,g is the result of a symbolic aggregation operation in a linguistic term set S=s0,s1,⋯,sg. Then, the equivalent information to β in the 2-tuple is obtained by the following function:(1)Δ:0,g→S×-0.5,0.5Δβ=sii=roundβα=β-iα∈-0.5,0.5,where round is the rounding operation.Definition 2 (see [28]).
SupposeS=s0,s1,⋯,sg is a linguistic term set and si,α be a 2-tuple, then there exists a function Δ-1 which transforms a 2-tuple into its equivalent numerical value β∈0,g. The function Δ-1 is defined as follows:(2)Δ-1:S×-0.5,0.5→0,gΔ-1si,α=i+αAccording to Definitions1 and 2, a linguistic label can be transformed into a two-tuple linguistic by adding a zero as a symbolic translation, i.e., Δsi=si,0.In addition, the comparison of two-tuple linguistic was given in [28]. For two 2-tuples: si,α and sj,β, then(1)
ifi<j, then si,α is smaller thansj,β;(2)
ifi=j, then
(a)
ifα=β, then si,α and sj,β represent the same information;(b)
ifα<β, then si,α is smaller thansj,β.


### 2.2. Multi-Granular Linguistic Information

When multiple DMs are taking part in a decision-making concern, different experts showcase different levels of uncertainty regarding the items. It is quite natural that linguistic term sets with different cardinality could be used to provide their preferences over the alternatives. A decision-maker having the capacity to deliver precise information puts to use a finer granularity linguistic term set. Contrarily, the decision-maker is likely to select a coarse granularity linguistic term set [2].In this paper, we handle GDM problems based on multi-granular linguistic, i.e., GDM problems where the expertsekk=1,2,⋯,m may express their linguistic preference relations Pk=pijkn×n on the set of alternatives X=x1,x2,⋯,xn based on given multi-granular linguistic term sets. Let Sg1,Sg2,⋯,Sgr be the linguistic term sets to be used by the DMs, where Sgh=s0gh,s1gh,⋯,sgh-1gh is a linguistic term set with a granularity of gh, h∈1,2,⋯,r. Therefore, pijk∈Sghrepresents the preference of alternative xi over alternative xj for the expert ek based on the linguistic term set Sgh.


### 2.3. Two-Tuple FLPR and Additive Consistency of FLPR

Definition 3 (see [29]).
P = p i j n × n is called a 2-tuple FLPR; if the following conditions hold for all i,j,(3)Δ-1pij+Δ-1pji=g,Δ-1pii=g2.where S=s0,s1,⋯,sg is a given linguistic term set.
Moreover, Alonso et al. [29] established the additive consistency of 2-tuple FLPR as follows:(4)Δ-1pij=Δ-1pik+Δ-1pkj-Δ-1sg/2,0,∀i,j,k∈1,2,⋯,n


## 2.1. Linguistic Information

To facilitate the assessment using linguistic for DMs, a linguistic term set should be determined beforehand. The widely used linguistic term set has following characteristics: (a) the odd value with granularity; (b) its membership functions being symmetrical and uniformly distributed; (c) the midterm of linguistic term set representing ‘‘indifference’’, with the remainder of the linguistic terms symmetrically and uniformly being placed on either side of it. LetS=s0,s1,⋯,sg be a linguistic term set with odd granularity and g+1 is the granularity of the linguisticS. Moreover, the term set S should satisfy the following features [28]: (1) a negation operator: Negsi=sg-i; (2) an order: si≥sj, if i≥j.The 2-tuple linguistic expressive model was presented [28] and it may improve the accuracy of linguistic computational models.Definition 1 (see [28]).
Supposeβ∈0,g is the result of a symbolic aggregation operation in a linguistic term set S=s0,s1,⋯,sg. Then, the equivalent information to β in the 2-tuple is obtained by the following function:(1)Δ:0,g→S×-0.5,0.5Δβ=sii=roundβα=β-iα∈-0.5,0.5,where round is the rounding operation.Definition 2 (see [28]).
SupposeS=s0,s1,⋯,sg is a linguistic term set and si,α be a 2-tuple, then there exists a function Δ-1 which transforms a 2-tuple into its equivalent numerical value β∈0,g. The function Δ-1 is defined as follows:(2)Δ-1:S×-0.5,0.5→0,gΔ-1si,α=i+αAccording to Definitions1 and 2, a linguistic label can be transformed into a two-tuple linguistic by adding a zero as a symbolic translation, i.e., Δsi=si,0.In addition, the comparison of two-tuple linguistic was given in [28]. For two 2-tuples: si,α and sj,β, then(1)
ifi<j, then si,α is smaller thansj,β;(2)
ifi=j, then
(a)
ifα=β, then si,α and sj,β represent the same information;(b)
ifα<β, then si,α is smaller thansj,β.


## 2.2. Multi-Granular Linguistic Information

When multiple DMs are taking part in a decision-making concern, different experts showcase different levels of uncertainty regarding the items. It is quite natural that linguistic term sets with different cardinality could be used to provide their preferences over the alternatives. A decision-maker having the capacity to deliver precise information puts to use a finer granularity linguistic term set. Contrarily, the decision-maker is likely to select a coarse granularity linguistic term set [2].In this paper, we handle GDM problems based on multi-granular linguistic, i.e., GDM problems where the expertsekk=1,2,⋯,m may express their linguistic preference relations Pk=pijkn×n on the set of alternatives X=x1,x2,⋯,xn based on given multi-granular linguistic term sets. Let Sg1,Sg2,⋯,Sgr be the linguistic term sets to be used by the DMs, where Sgh=s0gh,s1gh,⋯,sgh-1gh is a linguistic term set with a granularity of gh, h∈1,2,⋯,r. Therefore, pijk∈Sghrepresents the preference of alternative xi over alternative xj for the expert ek based on the linguistic term set Sgh.


## 2.3. Two-Tuple FLPR and Additive Consistency of FLPR

Definition 3 (see [29]).
P = p i j n × n is called a 2-tuple FLPR; if the following conditions hold for all i,j,(3)Δ-1pij+Δ-1pji=g,Δ-1pii=g2.where S=s0,s1,⋯,sg is a given linguistic term set.
Moreover, Alonso et al. [29] established the additive consistency of 2-tuple FLPR as follows:(4)Δ-1pij=Δ-1pik+Δ-1pkj-Δ-1sg/2,0,∀i,j,k∈1,2,⋯,n


## 3. A Group Decision-Making Model with Incomplete Multi-Granular 2-Tuple FLPRs

In this section, a group decision-making model with IMGPLPRs is proposed based on mathematical programming. First, a mathematical programming is established to deal with IMGFLPRs and obtain the priority weight vector; second, I-IOWA operator of 2-tuple FLPR is introduced to obtain collective priority weight vector in Section3.2. Finally, a consensus reaching process algorithm is established with IMGFLPRs in Section 3.3.


### 3.1. A Mathematical Programming Model to Deal with IMGFLPR

In what follows, an equivalence property of additive consistency between a 2-tuple FLPR and its corresponding FPR [30] is reviewed.SupposeP=pijn×n is a 2-tuple FLPR and S=s0,s1,⋯,sg is a linguistic scale set. Let rij=Δ-1pij/g, then we get rij,rji∈0,1 and rij+rji=Δ-1pij/g+Δ-1pji/g=1. Hence, R=rijn×n is an FPR. Here, R=rijn×n is called the FPR matrix corresponding to the 2-tuple FLPR P=pijn×n. In the following, we review the equivalence property of consistency between P=pijn×n and R=rijn×n.Theorem 4 (see [30]).
P = p i j n × n is an additive consistent 2-tuple FLPR ⟺R=rijn×n is an additive consistent FPRRemark 5.
From the Theorem4, it can be seen that the additive consistent 2-tuple FLPR P=pijn×n can be obtained by its corresponding additive consistent FPR R=rijn×n.Based on the Theorem4, a mathematical programming is established to deal with IMGFLPR. Suppose P=pijn×n is an additive consistent 2-tuple FLPR and a normalized weight vector is ω=ω1,ω2,⋯,ωnT. Based on the relationship between ωi,ωj, and rij proposed by Xu et al. [31, 32], we can obtain(5)Δ-1pijg=0.5+n-12ωi-ωjEquation (5) can be transformed to(6)Δ-1pij=g2+gn-12ωi-ωj,If a 2-tuple FLPR is not additive consistency, then there does not exist a priority vector that satisfies (6). In practice, it is difficult to satisfy the perfect consistency. Then, the “Soft” consistency was proposed to express approximate consistency [33, 34]. Let Lijω=g/2+gn-1/2ωi-ωj-Δ-1pij; if Lijω=0, then we say that the satisfaction degree of the priorities equals one. Otherwise, the satisfaction degree should reduce for some deviation. In order to get the optimal weight vector, the following mathematical programming is established:(7)minεij=g2+gn-12ωi-ωj-Δ-1pijs.t.∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,For the sake of simplifying model (7), we present the following Theorem 6.Theorem 6.
For model (7), the following relationship is established:(8)g2+gn-12ωj-ωi-Δ-1pji=g2+gn-12ωi-ωj-Δ-1pij.Proof.
Based on the properties of the FLPR, we have(9)Δ-1pji=g-Δ-1pijThen,(10)g2+gn-12ωj-ωi-Δ-1pji=g2+gn-12ωj-ωi-g-Δ-1pji=g2+gn-12ωj-ωi-g+Δ-1pji=gn-12ωj-ωi-g2+Δ-1pji=g2+gn-12ωi-ωj-Δ-1pijwhich completes the proof of Theorem 6.Based on Theorem6, model (7) can be reduced to model (11):(11)minεij=g2+gn-12ωi-ωj-Δ-1pijs.t.∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,i<j,Moreover, model (11) can be transformed into the following mathematical programming:(12)minf=∑i=1n-1∑j=2,j>intijdij++mijdij-s.t.g2+gn-12ωi-ωj-Δ-1pij-tijdij++mijdij-=0dij+,dij-≥0∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,i<j,where dij+ and dij- are the positive and negative deviations with respect to the goal εij, respectively. tij and mij are the weights corresponding to dij+ and dij-, respectively.In general, we assume that all goalsεij(i,j=1,2,⋯,n, i<j) are fair and then tij=mij=1(i,j=1,2,⋯,n, i<j). Thus, model (12) can be translated into the following model: (13)minf=∑i=1n-1∑j=2,j>indij++dij-s.t.g2+gn-12ωi-ωj-Δ-1pij-dij++dij-=0dij+,dij-≥0∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,i<j,Based on positivedij+ and negative deviation dij- in model (13), the consistency level of P is defined as (14)clP=1-2∑i=1n-1∑j=2,j>indij++dij-gnn-1Remark 7.
Based on the above discussion, it is deserved to note that model (13) could serve a dual function: on the one hand, it could directly deal with incomplete multi-granular 2-tuple FLPR with different linguistic granular, and on the other hand it might also get the individual consistency level clP.In what follows, we apply model (13) to deal with Example 8.Example 8.
Suppose an incomplete FLPRP as follows:(15)P=s3,0s2,0xs2,0s4,0s3,0s1,0xxs5,0s3,0s4,0s4,0xs2,0s3,0,where S7={s07=verypoor,s17=poor, s27=slightlypoor,s37=fair, s47=slightlygood, s57=good,s67=verygood}.According to model (13), we obtain the following mathematical programming: (16)minf=d12++d12-+d14++d14-+d23++d23-+d34++d34-s.t.9ω1-9ω2-d12++d12-=-1,9ω1-9ω4-d14++d14-=-1,9ω2-9ω3-d23++d23-=-2,9ω3-9ω4-d34++d34-=1,d12+,d12-,d14+,d14-,d23+,d23-,d34+,d34-≥0,∑i=14ωi=1,ωi≥0,i=1,2,3,4By means of MATLAB, we can obtainω=0.1251,0.2082,0.4028,0.2638Tand clP=0.9722 based on d12+=0.2519,d14-=0.248,d23+=0.2486,d34+=0.2516,d12-=d14+=d23-=d34-=0.


### 3.2. IOWA Operator Based on Consistency Level

In order to obtain collective priority weight vector, the IOWA operator based on consistency level [35] is used.Individual priority weight vectorsωk=ω1k,ω2k,⋯,ωnkk=1,2,⋯,m can be combined into collective priority weight vector ωc=ω1c,ω2c,⋯,ωnc according to IOWA operator based on consistency level:(17)ωic=IOWAQcωi1,ωi2,⋯,ωim=Φωcl1,ωi1,cl2,ωi2,⋯,clm,ωim=∑τ=1muτωiτ(18)uτ=Q∑k=1τclστT-Q∑k=1τ-1clστTwhereT=∑k=1mclσk and clσk is the kth largest value incl1,cl2,⋯,clm.


### 3.3. Consensus Reaching Process Algorithm

As shown in [36, 37], the consensus reaching process algorithm based on direct approach has two desirable properties: (1) it can avoid internal inconsistency with multi-granular 2-tuple FLPRs, and (2) it is satisfied with the Pareto principle in social choice theory. Inspired by the direct approach [36, 37], we adopt the direct approach to manage consensus reaching process in this study.


#### 3.3.1. Consensus Index

Definition 9.
Letωk and ωc be shown earlier. The group consensus index (GCI) for ek is defined as follows:(19)GCIek=1-1n∑i=1nωik-ωic2,If GCIek=1k=1,2,⋯,m, all decision-makers agree with the group; otherwise, the larger GCIek is, the higher the consensus index is.


#### 3.3.2. Group Consensus Reaching Process

The goal of the feedback adjustment is to provide adjustment advice to help DMs improve the consensus level. In this study, an interaction feedback adjustment strategy is proposed based on the personal original preference information and the adjusted group preference information to regulate the personal preference information; that is to say, the DMs modify his/her preference to improve the group consensus level. According to the different granular linguistic structure, the interactive feedback adjustment mechanism should be considered.Forek∈Sgh=s0gh,s1gh,⋯,sgh-1gh, the adjustment proposals are given as follows:(20)pij,k¯∈minpij,k,pij,cgh,maxpij,k,pij,cgh,i<jpij,k¯=sgh-1/2,i=jpij,k¯=Δgh-1-Δ-1pji,k¯,i>jwhere pij,cgh=Δgh/2+ghn-1/2ωic-ωjc.Furthermore, based on the adjustment proposals, a group consensus reaching Algorithm1 is constructed as shown in Algorithm 1.Algorithm 1
Input:Individual FLPRs P1,P2,⋯,Pm, the experts’ weight vector u=u1,u2,⋯,umT, the thresholdsGCI¯ and the maximum iterations tmax≥1. Output:Adjusted FLPRs P1(t),P2(t),⋯,Pm(t), GCIPkt(k=1,2,⋯,m), Pct and the iterationst. Step  1 :Set t=0 and Pk0=Pkk=1,2,⋯,m. Step  2 : Obtain individual priority weight vector ω1t,ω2t,⋯,ωmt using the model (13) and then get the experts’ weight vectorut=u1t,u2t,⋯,umtTaccording to Eq. (18). Step  3 : Compute the collective priority weight vector ωct corresponding to ω 1 t , ω 2 t , ⋯ , ω m t by means of IOWA operator. Step  4 : Count the GCIPktk=1,2,⋯,m according to Eq. (19). If GCIPkt≥ G C I ¯ k = 1,2 , ⋯ , m or t≥tmax, go to Step 6; Or else, find the FLPR Pkt that satisfies G C I P k t < G C I ¯ and proceed to Step 5. Step  5 : Ascertain the position of the elements diτjτ,kt for expert ek satisfying GCIPkt< G C I ¯, where diτjτ,kt=maxi,jΔ-1pij,kt-Δ-1pij,ct, and then adjust expert ek’s FLPR according to Eq. (20). Sett=t+1 and go to Step 2. Step  6 : Let Pk¯=Pktk=1,2,⋯,m and ω∗=ωct. Output P1¯,P2¯,⋯,Pm¯, G C I P k ¯ k = 1,2 , ⋯ , m, ω∗=ωctand the iterations t.Remark 10.
ThresholdsGCI¯ provide a flexible option to control the decision process. Once thresholds GCI¯ are specified, Step 4 is implemented to determine which experts need to adjust their opinion and then Step 5 gives a specific scheme to make adjustments. After expert opinion Pkt is adjusted, optimization model (13) is reused to determine the new experts’ weights with those updated information. By iteratively adjusting the experts’ opinion and weights, the consensus level gradually is improved.


#### 3.3.3. A Step-by-Step Procedure of the GDM Model with IMGFLPRs

Below we shall provide a GDM model with IMGFLPRs based on mathematical programming. Assume that there aren alternatives x1,x2,⋯,xn and experts’ weight u=u1,u2,⋯,umT related to experts e1,e2,⋯,em. Pkk=1,2,⋯,m indicates the 2-tuple FLPRs given by experts e1,e2,⋯,em based on the given linguistic term sets Sg1,Sg2,⋯,Sgr. The flow chart of proposed GDM model is illustrated by Figure 1, and its procedure is specifically shown as follows.Figure 1
Procedure of the GDM model with incomplete multi-granular 2-tuple FLPRs.Step 1.
Every expert needs to provide only the most confident assessed linguistic terms between two alternatives by paired comparison method based on given linguistic scale setsSg1,Sg2,⋯,Sgr by means of their experience and expertise knowledge, where missing comparisons are denoted by “x”.Step 2.
Obtain the individual priority weight vectors for all expertse1,e2,⋯,em by means of model (13), respectively.Step 3.
Fuse individual priority weight vectors to collective priority weight vector using IOWA operator.Step 4.
Achieve group consensus with satisfying index by means of Algorithm1.Step 5.
Rank alternativesAii=1,2,⋯,n according to the final collective priority weight vector.Step 6.
End.


## 3.1. A Mathematical Programming Model to Deal with IMGFLPR

In what follows, an equivalence property of additive consistency between a 2-tuple FLPR and its corresponding FPR [30] is reviewed.SupposeP=pijn×n is a 2-tuple FLPR and S=s0,s1,⋯,sg is a linguistic scale set. Let rij=Δ-1pij/g, then we get rij,rji∈0,1 and rij+rji=Δ-1pij/g+Δ-1pji/g=1. Hence, R=rijn×n is an FPR. Here, R=rijn×n is called the FPR matrix corresponding to the 2-tuple FLPR P=pijn×n. In the following, we review the equivalence property of consistency between P=pijn×n and R=rijn×n.Theorem 4 (see [30]).
P = p i j n × n is an additive consistent 2-tuple FLPR ⟺R=rijn×n is an additive consistent FPRRemark 5.
From the Theorem4, it can be seen that the additive consistent 2-tuple FLPR P=pijn×n can be obtained by its corresponding additive consistent FPR R=rijn×n.Based on the Theorem4, a mathematical programming is established to deal with IMGFLPR. Suppose P=pijn×n is an additive consistent 2-tuple FLPR and a normalized weight vector is ω=ω1,ω2,⋯,ωnT. Based on the relationship between ωi,ωj, and rij proposed by Xu et al. [31, 32], we can obtain(5)Δ-1pijg=0.5+n-12ωi-ωjEquation (5) can be transformed to(6)Δ-1pij=g2+gn-12ωi-ωj,If a 2-tuple FLPR is not additive consistency, then there does not exist a priority vector that satisfies (6). In practice, it is difficult to satisfy the perfect consistency. Then, the “Soft” consistency was proposed to express approximate consistency [33, 34]. Let Lijω=g/2+gn-1/2ωi-ωj-Δ-1pij; if Lijω=0, then we say that the satisfaction degree of the priorities equals one. Otherwise, the satisfaction degree should reduce for some deviation. In order to get the optimal weight vector, the following mathematical programming is established:(7)minεij=g2+gn-12ωi-ωj-Δ-1pijs.t.∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,For the sake of simplifying model (7), we present the following Theorem 6.Theorem 6.
For model (7), the following relationship is established:(8)g2+gn-12ωj-ωi-Δ-1pji=g2+gn-12ωi-ωj-Δ-1pij.Proof.
Based on the properties of the FLPR, we have(9)Δ-1pji=g-Δ-1pijThen,(10)g2+gn-12ωj-ωi-Δ-1pji=g2+gn-12ωj-ωi-g-Δ-1pji=g2+gn-12ωj-ωi-g+Δ-1pji=gn-12ωj-ωi-g2+Δ-1pji=g2+gn-12ωi-ωj-Δ-1pijwhich completes the proof of Theorem 6.Based on Theorem6, model (7) can be reduced to model (11):(11)minεij=g2+gn-12ωi-ωj-Δ-1pijs.t.∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,i<j,Moreover, model (11) can be transformed into the following mathematical programming:(12)minf=∑i=1n-1∑j=2,j>intijdij++mijdij-s.t.g2+gn-12ωi-ωj-Δ-1pij-tijdij++mijdij-=0dij+,dij-≥0∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,i<j,where dij+ and dij- are the positive and negative deviations with respect to the goal εij, respectively. tij and mij are the weights corresponding to dij+ and dij-, respectively.In general, we assume that all goalsεij(i,j=1,2,⋯,n, i<j) are fair and then tij=mij=1(i,j=1,2,⋯,n, i<j). Thus, model (12) can be translated into the following model: (13)minf=∑i=1n-1∑j=2,j>indij++dij-s.t.g2+gn-12ωi-ωj-Δ-1pij-dij++dij-=0dij+,dij-≥0∑i=1nωi=1,ωi≥0i,j=1,2,⋯,n,i<j,Based on positivedij+ and negative deviation dij- in model (13), the consistency level of P is defined as (14)clP=1-2∑i=1n-1∑j=2,j>indij++dij-gnn-1Remark 7.
Based on the above discussion, it is deserved to note that model (13) could serve a dual function: on the one hand, it could directly deal with incomplete multi-granular 2-tuple FLPR with different linguistic granular, and on the other hand it might also get the individual consistency level clP.In what follows, we apply model (13) to deal with Example 8.Example 8.
Suppose an incomplete FLPRP as follows:(15)P=s3,0s2,0xs2,0s4,0s3,0s1,0xxs5,0s3,0s4,0s4,0xs2,0s3,0,where S7={s07=verypoor,s17=poor, s27=slightlypoor,s37=fair, s47=slightlygood, s57=good,s67=verygood}.According to model (13), we obtain the following mathematical programming: (16)minf=d12++d12-+d14++d14-+d23++d23-+d34++d34-s.t.9ω1-9ω2-d12++d12-=-1,9ω1-9ω4-d14++d14-=-1,9ω2-9ω3-d23++d23-=-2,9ω3-9ω4-d34++d34-=1,d12+,d12-,d14+,d14-,d23+,d23-,d34+,d34-≥0,∑i=14ωi=1,ωi≥0,i=1,2,3,4By means of MATLAB, we can obtainω=0.1251,0.2082,0.4028,0.2638Tand clP=0.9722 based on d12+=0.2519,d14-=0.248,d23+=0.2486,d34+=0.2516,d12-=d14+=d23-=d34-=0.


## 3.2. IOWA Operator Based on Consistency Level

In order to obtain collective priority weight vector, the IOWA operator based on consistency level [35] is used.Individual priority weight vectorsωk=ω1k,ω2k,⋯,ωnkk=1,2,⋯,m can be combined into collective priority weight vector ωc=ω1c,ω2c,⋯,ωnc according to IOWA operator based on consistency level:(17)ωic=IOWAQcωi1,ωi2,⋯,ωim=Φωcl1,ωi1,cl2,ωi2,⋯,clm,ωim=∑τ=1muτωiτ(18)uτ=Q∑k=1τclστT-Q∑k=1τ-1clστTwhereT=∑k=1mclσk and clσk is the kth largest value incl1,cl2,⋯,clm.


## 3.3. Consensus Reaching Process Algorithm

As shown in [36, 37], the consensus reaching process algorithm based on direct approach has two desirable properties: (1) it can avoid internal inconsistency with multi-granular 2-tuple FLPRs, and (2) it is satisfied with the Pareto principle in social choice theory. Inspired by the direct approach [36, 37], we adopt the direct approach to manage consensus reaching process in this study.


### 3.3.1. Consensus Index

Definition 9.
Letωk and ωc be shown earlier. The group consensus index (GCI) for ek is defined as follows:(19)GCIek=1-1n∑i=1nωik-ωic2,If GCIek=1k=1,2,⋯,m, all decision-makers agree with the group; otherwise, the larger GCIek is, the higher the consensus index is.


### 3.3.2. Group Consensus Reaching Process

The goal of the feedback adjustment is to provide adjustment advice to help DMs improve the consensus level. In this study, an interaction feedback adjustment strategy is proposed based on the personal original preference information and the adjusted group preference information to regulate the personal preference information; that is to say, the DMs modify his/her preference to improve the group consensus level. According to the different granular linguistic structure, the interactive feedback adjustment mechanism should be considered.Forek∈Sgh=s0gh,s1gh,⋯,sgh-1gh, the adjustment proposals are given as follows:(20)pij,k¯∈minpij,k,pij,cgh,maxpij,k,pij,cgh,i<jpij,k¯=sgh-1/2,i=jpij,k¯=Δgh-1-Δ-1pji,k¯,i>jwhere pij,cgh=Δgh/2+ghn-1/2ωic-ωjc.Furthermore, based on the adjustment proposals, a group consensus reaching Algorithm1 is constructed as shown in Algorithm 1.Algorithm 1
Input:Individual FLPRs P1,P2,⋯,Pm, the experts’ weight vector u=u1,u2,⋯,umT, the thresholdsGCI¯ and the maximum iterations tmax≥1. Output:Adjusted FLPRs P1(t),P2(t),⋯,Pm(t), GCIPkt(k=1,2,⋯,m), Pct and the iterationst. Step  1 :Set t=0 and Pk0=Pkk=1,2,⋯,m. Step  2 : Obtain individual priority weight vector ω1t,ω2t,⋯,ωmt using the model (13) and then get the experts’ weight vectorut=u1t,u2t,⋯,umtTaccording to Eq. (18). Step  3 : Compute the collective priority weight vector ωct corresponding to ω 1 t , ω 2 t , ⋯ , ω m t by means of IOWA operator. Step  4 : Count the GCIPktk=1,2,⋯,m according to Eq. (19). If GCIPkt≥ G C I ¯ k = 1,2 , ⋯ , m or t≥tmax, go to Step 6; Or else, find the FLPR Pkt that satisfies G C I P k t < G C I ¯ and proceed to Step 5. Step  5 : Ascertain the position of the elements diτjτ,kt for expert ek satisfying GCIPkt< G C I ¯, where diτjτ,kt=maxi,jΔ-1pij,kt-Δ-1pij,ct, and then adjust expert ek’s FLPR according to Eq. (20). Sett=t+1 and go to Step 2. Step  6 : Let Pk¯=Pktk=1,2,⋯,m and ω∗=ωct. Output P1¯,P2¯,⋯,Pm¯, G C I P k ¯ k = 1,2 , ⋯ , m, ω∗=ωctand the iterations t.Remark 10.
ThresholdsGCI¯ provide a flexible option to control the decision process. Once thresholds GCI¯ are specified, Step 4 is implemented to determine which experts need to adjust their opinion and then Step 5 gives a specific scheme to make adjustments. After expert opinion Pkt is adjusted, optimization model (13) is reused to determine the new experts’ weights with those updated information. By iteratively adjusting the experts’ opinion and weights, the consensus level gradually is improved.


### 3.3.3. A Step-by-Step Procedure of the GDM Model with IMGFLPRs

Below we shall provide a GDM model with IMGFLPRs based on mathematical programming. Assume that there aren alternatives x1,x2,⋯,xn and experts’ weight u=u1,u2,⋯,umT related to experts e1,e2,⋯,em. Pkk=1,2,⋯,m indicates the 2-tuple FLPRs given by experts e1,e2,⋯,em based on the given linguistic term sets Sg1,Sg2,⋯,Sgr. The flow chart of proposed GDM model is illustrated by Figure 1, and its procedure is specifically shown as follows.Figure 1
Procedure of the GDM model with incomplete multi-granular 2-tuple FLPRs.Step 1.
Every expert needs to provide only the most confident assessed linguistic terms between two alternatives by paired comparison method based on given linguistic scale setsSg1,Sg2,⋯,Sgr by means of their experience and expertise knowledge, where missing comparisons are denoted by “x”.Step 2.
Obtain the individual priority weight vectors for all expertse1,e2,⋯,em by means of model (13), respectively.Step 3.
Fuse individual priority weight vectors to collective priority weight vector using IOWA operator.Step 4.
Achieve group consensus with satisfying index by means of Algorithm1.Step 5.
Rank alternativesAii=1,2,⋯,n according to the final collective priority weight vector.Step 6.
End.


## 3.3.1. Consensus Index

Definition 9.
Letωk and ωc be shown earlier. The group consensus index (GCI) for ek is defined as follows:(19)GCIek=1-1n∑i=1nωik-ωic2,If GCIek=1k=1,2,⋯,m, all decision-makers agree with the group; otherwise, the larger GCIek is, the higher the consensus index is.


## 3.3.2. Group Consensus Reaching Process

The goal of the feedback adjustment is to provide adjustment advice to help DMs improve the consensus level. In this study, an interaction feedback adjustment strategy is proposed based on the personal original preference information and the adjusted group preference information to regulate the personal preference information; that is to say, the DMs modify his/her preference to improve the group consensus level. According to the different granular linguistic structure, the interactive feedback adjustment mechanism should be considered.Forek∈Sgh=s0gh,s1gh,⋯,sgh-1gh, the adjustment proposals are given as follows:(20)pij,k¯∈minpij,k,pij,cgh,maxpij,k,pij,cgh,i<jpij,k¯=sgh-1/2,i=jpij,k¯=Δgh-1-Δ-1pji,k¯,i>jwhere pij,cgh=Δgh/2+ghn-1/2ωic-ωjc.Furthermore, based on the adjustment proposals, a group consensus reaching Algorithm1 is constructed as shown in Algorithm 1.Algorithm 1
Input:Individual FLPRs P1,P2,⋯,Pm, the experts’ weight vector u=u1,u2,⋯,umT, the thresholdsGCI¯ and the maximum iterations tmax≥1. Output:Adjusted FLPRs P1(t),P2(t),⋯,Pm(t), GCIPkt(k=1,2,⋯,m), Pct and the iterationst. Step  1 :Set t=0 and Pk0=Pkk=1,2,⋯,m. Step  2 : Obtain individual priority weight vector ω1t,ω2t,⋯,ωmt using the model (13) and then get the experts’ weight vectorut=u1t,u2t,⋯,umtTaccording to Eq. (18). Step  3 : Compute the collective priority weight vector ωct corresponding to ω 1 t , ω 2 t , ⋯ , ω m t by means of IOWA operator. Step  4 : Count the GCIPktk=1,2,⋯,m according to Eq. (19). If GCIPkt≥ G C I ¯ k = 1,2 , ⋯ , m or t≥tmax, go to Step 6; Or else, find the FLPR Pkt that satisfies G C I P k t < G C I ¯ and proceed to Step 5. Step  5 : Ascertain the position of the elements diτjτ,kt for expert ek satisfying GCIPkt< G C I ¯, where diτjτ,kt=maxi,jΔ-1pij,kt-Δ-1pij,ct, and then adjust expert ek’s FLPR according to Eq. (20). Sett=t+1 and go to Step 2. Step  6 : Let Pk¯=Pktk=1,2,⋯,m and ω∗=ωct. Output P1¯,P2¯,⋯,Pm¯, G C I P k ¯ k = 1,2 , ⋯ , m, ω∗=ωctand the iterations t.Remark 10.
ThresholdsGCI¯ provide a flexible option to control the decision process. Once thresholds GCI¯ are specified, Step 4 is implemented to determine which experts need to adjust their opinion and then Step 5 gives a specific scheme to make adjustments. After expert opinion Pkt is adjusted, optimization model (13) is reused to determine the new experts’ weights with those updated information. By iteratively adjusting the experts’ opinion and weights, the consensus level gradually is improved.


## 3.3.3. A Step-by-Step Procedure of the GDM Model with IMGFLPRs

Below we shall provide a GDM model with IMGFLPRs based on mathematical programming. Assume that there aren alternatives x1,x2,⋯,xn and experts’ weight u=u1,u2,⋯,umT related to experts e1,e2,⋯,em. Pkk=1,2,⋯,m indicates the 2-tuple FLPRs given by experts e1,e2,⋯,em based on the given linguistic term sets Sg1,Sg2,⋯,Sgr. The flow chart of proposed GDM model is illustrated by Figure 1, and its procedure is specifically shown as follows.Figure 1
Procedure of the GDM model with incomplete multi-granular 2-tuple FLPRs.Step 1.
Every expert needs to provide only the most confident assessed linguistic terms between two alternatives by paired comparison method based on given linguistic scale setsSg1,Sg2,⋯,Sgr by means of their experience and expertise knowledge, where missing comparisons are denoted by “x”.Step 2.
Obtain the individual priority weight vectors for all expertse1,e2,⋯,em by means of model (13), respectively.Step 3.
Fuse individual priority weight vectors to collective priority weight vector using IOWA operator.Step 4.
Achieve group consensus with satisfying index by means of Algorithm1.Step 5.
Rank alternativesAii=1,2,⋯,n according to the final collective priority weight vector.Step 6.
End.


## 4. Case Study

In recent years, more and more disasters have taken place in the world. They have brought great influence and threat to people’s lives [4], for example, China’s Wenchuan Earthquake in 2008 and Japan’s Earthquake in 2011. The arrangement of the disaster relief work is an important link, how to ensure the earthquake materials supplies in the shortest possible time, the highest security, and the best economic way to transport goods demand point is very important. Here we give four different emergency decisions of the scheduling:(x1)
Airdrop relief supplies(x2)
Repairing damaged roads and transporting relief supplies by car(x3)
Repairing damaged rail and rail transport(x4)
Transporting a large number of people on foot.In order to choose the most suitable emergency rescue plan from the above four emergency rescue plans, in here, the emergency rescue alternativesX=x1,x2,x3,x4 will be evaluated by three experts E=e1,e2,e3. The linguistic term sets used by experts e1,e2,e3 are S5=s05,s15,⋯,s45, S7=s07,s17,⋯,s67, and S9=s09,s19,⋯,s89, respectively.Step 1.
Every expert gives his/her linguistic assessed values between every two alternatives by paired comparison method based on the given linguistic scale setsS5,S7,S9, respectively. In here, the experts only give the most confident comparisons based on their experience and expertise knowledge, where missing comparisons are denoted by “x”. Then, we obtain three incomplete multi-granular 2-tuple FLPRs Pkk=1,2,3 with three experts E=e1,e2,e3, respectively:(21)P1=s25,0s25,0s35,0s35,0s25,0s25,0xs35,0s15,0xs25,0xs15,0s15,0xs25,0,P2=s37,0s27,0s47,0s37,0s47,0s37,0s47,0s47,0s27,0s27,0s37,0s27,0s37,0s27,0s47,0s37,0,P3=s49,0s39,0xs59,0s59,0s49,0s59,0xxs39,0s49,0s69,0s39,0xs29,0s49,0.Step 2.
The individual priority weight vectors and consistency levels are obtained based onPkk=1,2,3 by model (13) and (14):(22)ω1=0.3333,0.3333,0.1667,0.1667T,d34-=4,andclP1=0.8333;ω2=0.25,0.3611,0.1389,0.25T,d23+=1,andclP2=0.9722;ω3=0.2395,0.3438,0.2813,0.1354T,d12-=0.251,d14+=0.2497,d23-=0.2494,d34-=0.2498,andclP3=0.9792.Step 3 (group consensus reaching process).
SinceclP1=0.8333, clP2=0.9722, clP3=0.9792, we have σ1=3,σ2=2,σ3=1. Then, we use the function Qx=x0.9 proposed by Xu et al. [35] in IOWA operator to determine experts’ weights as follows:(23)u3=Q0.9792∑i=13cli=Q0.3516=0.3903;u2=Q0.9792+0.9722∑i=13cli-Q0.3516=0.3359;u1=1-0.3903-0.3359=0.2738.On the basis of the original experts’ weight vector u0=0.2738,0.3359,0.3903T, the original collective preference vector is ωc0=0.2687,0.3467,0.2021,0.1825T and consensus indices for each expert are as follows:(24)GCIe1=0.9617,GCIe2=0.9523,GCIe3=0.9516.According to the practical problems, the experts agree to set upGCI¯=0.98. Then Algorithm 1 is applied to adjust the original FLPRs. Since GCIek<GCI¯k=1,2,3, we need to find the position of elements diτjτ,k0k=1,2,3, where diτjτ,kt=maxi,jΔ-1pij,kt-Δ-1pij,ct.With regard toP10, as d13,10=d31,10=maxi,jΔ-1pij,10-Δ-1pij,c0=0.6, d13,10,d31,10 need to be adjusted to d13,10=s3,-0.5,d31,10=s1,-0.5 according to SUG11.(25)P11=s25,0s25,0s35,-0.5s35,0s25,0s25,0xs35,0s25,-0.5xs25,0xs15,0s15,0xs25,0where(26)SUG11=s25,0,s25,0s25,-0.47,s25,0s25,0.4,s35,0s35,-0.48,s35,0s25,0,s25,0.47s25,0,s25,0xs35,0,s35,0s15,0,s25,-0.4xs25,0,s25,0xs15,0,s15,0.48s15,0,s15,0xs25,0,s25,0with sugij1=minpij1,pij1,c,maxpij1,pij1,c.With regard toP20and P30, the same procedures are implemented to obtain the adjusted FLPRs P21,P31 according to the SUG21, SUG31 as follows:(27)P21=s37,0s27,0s47,0s37,0s47,0s37,0s47,0s47,0s27,0s27,0s37,0s37,0s37,0s27,0s37,0s37,0,P31=s49,0s39,0xs59,0s59,0s49,0s59,0xxs39,0s49,0s59,0s39,0xs39,0s49,0where(28)SUG21=s37,0,s37,0s27,0,s27,0.3s47,-0.4,s47,0s37,0,s47,-0.22s47,-0.3,s47,0s37,0,s37,0s47,0,s47,0.3s47,0,s47,0.48s27,0,s27,0.4s27,-0.3,s27,0s37,0,s37,0s27,0,s37,0.18s27,0.22,s37,0s27,-0.48,s27,0s37,-0.18,s47,0s37,0,s37,0.SUG31=s49,0,s49,0s39,0,s39,0.06xs59,0,s59,0.03s59,-0.06,s59,0s49,0,s49,0s59,0,s69,-0.26xxs29,0.26,s39,0s49,0,s49,0s49,0.24,s69,0s39,-0.03,s39,0xs29,0,s49,-0.24s49,0,s49,0Let t=1, then go to Step 2.Going through 2 rounds of adjustment, Algorithm1 is terminated. On the whole, e1, e2, and e3 adjust their FLPRs 2, 2, and 1 times, respectively. In addition, the final group consensus index of every expert and final collective priority weight vector are as follows:(29)GCIP12=0.9857,GCIP22=0.9839,GCIP31=0.9814.w∗=wc2=0.2714,0.3235,0.2246,0.1805T.Furthermore, the adjustment variations of FLPRs for three experts are shown in Table1.Table 1
To compare the situation of FLPRs after adjusting for three experts.
Experts before the adjustment after the adjustment experte1 P 1 = s 2 5 , 0 s 2 5 , 0 s 3 5 , 0 s 3 5 , 0 s 2 5 , 0 s 2 5 , 0 x s 3 5 , 0 s 1 5 , 0 x s 2 5 , 0 x s 1 5 , 0 s 1 5 , 0 x s 2 5 , 0 P 1 2 = s 2 5 , 0 s 2 5 , 0 s 3 5 , - 0 . 5 s 3 5 , - 0 . 4 s 2 5 , 0 s 2 5 , 0 x s 3 5 , 0 s 2 5 , - 0 . 5 x s 2 5 , 0 x s 1 5 , 0 . 4 s 1 5 , 0 x s 2 5 , 0  experte2 P 2 = s 3 7 , 0 s 2 7 , 0 s 4 7 , 0 s 3 7 , 0 s 4 7 , 0 s 3 7 , 0 s 4 7 , 0 s 4 7 , 0 s 2 7 , 0 s 2 7 , 0 s 3 7 , 0 s 2 7 , 0 s 3 7 , 0 s 2 7 , 0 s 4 7 , 0 s 3 7 , 0 P 2 2 = s 3 7 , 0 s 2 7 , 0 s 4 7 , 0 s 4 7 , - 0 . 4 s 4 7 , 0 s 3 7 , 0 s 4 7 , 0 s 4 7 , 0 s 2 7 , 0 s 2 7 , 0 s 3 7 , 0 s 3 7 , 0 s 2 7 , 0 . 4 s 2 7 , 0 s 3 7 , 0 s 3 7 , 0  experte3 P 3 = s 4 9 , 0 s 3 9 , 0 x s 5 9 , 0 s 5 9 , 0 s 4 9 , 0 s 5 9 , 0 x x s 3 9 , 0 s 4 9 , 0 s 6 9 , 0 s 3 9 , 0 x s 2 9 , 0 s 4 9 , 0 P 3 1 = s 4 9 , 0 s 3 9 , 0 x s 5 9 , 0 s 5 9 , 0 s 4 9 , 0 s 5 9 , 0 x x s 3 9 , 0 s 4 9 , 0 s 5 9 , 0 s 3 9 , 0 x s 3 9 , 0 s 4 9 , 0Step 4.
Rank alternativesAi(i=1,2,3,4) based on the final priority weight vectorAccording to the final priority weight vectorw∗=0.2714,0.3235,0.2246,0.1805T, the ranking of alternatives is A2≻A1≻A3≻A4; i.e., repairing damaged roads and transporting relief supplies by car are the best choice.


## 5. Discussion

As mentioned in introduction, there are two related studies [4, 8] to deal with IMGFLPRs in group decision-making. In this section, we carry out a comparison of our suggested method with previous related research [4, 8]. First, the dynamic regulation for experts’ weights in iterative process according to adjusted information quality is intuitively shown in Figure 2. By means of Figure 2, weight of experts is changed according to their information quality. Second, the differences between the method proposed in this paper and the above-mentioned study [4] are summarized in Table 2. From Table 2, it can be seen that the using above two strategies gets the same optimal rescue plan A2. While the sorting results for A3 and A4using above two strategies are different, the reason for different ranking for A3 and A4 may be that the method [4] has not considered the consensus reaching in GDM.Table 2
Contrast between our method and existing related study.
Methods Handling method for IMGPLPR Dynamic adjustment of experts’ weights Consensus Ranking The proposed method Optimization model Yes Yes A 2 ≻ A 1 ≻ A 3 ≻ A 4 Xu et al [4] Filling method No No A 2 ≻ A 1 ≻ A 4 ≻ A 3Figure 2
Adjust dynamically the weight of the experts in iterative process.In short, the main contributions of the paper can be summed up as follows:(i)
A consensus reaching algorithm is constructed based on direct consensus framework for IMGFLPRs in GDM, while existing related studies [4, 8] does not take into account this situation.(ii)
Previous related research [4, 8] was needed to fill the incomplete elements and revert to uniform granularity, while our proposed optimization model to deal with IMGFLPR needed not to carry out the two steps above and then obtained directly the priority weight vector of alternatives, which avoids information loss and reduces the computation complexity.(iii)
A dynamic adjustment procedure of experts’ weight is proposed according to the quality of adjusted information.


## 6. Conclusions

In this paper, we proposed a novel GDM method with incomplete multi-granular 2-tuple FLPRs based on mathematical programming. A consensus reaching algorithm based on the direct approach is constructed, which makes decision results more believable. And most of all, the uniform language granularity and estimating unknown elements are avoided by our proposed mathematical programming to reduce computation greatly. Then, a step-by-step GDM procedure based on the group consensus reaching algorithm has been concluded. Following the procedure, an emergency decision problem is worked out by means of the proposed model.In future, it is also interesting to analyze GDM problems with incomplete 2-tuple FLPRs based on multi-granular unbalanced linguistic terms [38]. Besides, the proposed model also can be applied to different problems such as Traditional Chinese Medicine decision support [39–41] and Decision Support System [42].



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*Source: 8197414-2018-08-16.xml*