A Heterogeneous Multiattribute Group Decision-Making Method Based on Intuitionistic Triangular Fuzzy Information

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Introduction
Group decision making (GDM) has been known as a popular method for finding the best alternative from a set of alternatives through aggregating decision information given in a group of experts, in which the evaluation of alternatives may involve multiple attributes including objective and subjective information [1][2][3].Due to the limited cognition and preference of decision maker, it is hard for different attributes to use the same information format to express the evaluation.For instance, in an online seller evaluation, the service attitude of the seller is suited to be described by triangular fuzzy numbers (TFNs) since the service of the seller is generally stable, but sometimes it is excellent and sometimes bad.It is convenient to describe the shipping speed of seller with interval numbers (INs) since it is not fixed but fluctuates in a certain range.These types of GDM problems with multiple conflicting attributes whose values are given by decision makers (DMs) may be represented in the form of multiple formats, such as real numbers (RNs), INs, TFNs, trapezoidal fuzzy numbers (TrFNs), and linguistic values (LVs), called heterogeneous multiattribute group decision making (HMAGDM) problems [4].
In this recent research, HMAGDM methods have been successfully applied to various fields, such as supply chain coordination [5], business processes [6], and software quality evaluation [7][8][9].The key to tackling such problems is how to fuse various types of attribute values [10].So far, many useful and valuable methods have been developed to study the fusion process of heterogeneous information, which can be roughly classified into three main categories [4,10].(1) The indirect approaches [11][12][13][14], in which the heterogeneous decision information given by DMs is converted into uniformed information by transformation methods.Wang and Cai [13] developed a generic distance-based VIKOR which can use aggregation function to convert heterogeneous 2 Complexity information into a uniform nonfuzzy degree and applied it to deal with emergency supplier selection.Using transformation function, Zhang et al. [14] transformed the multigranular linguistic decision matrices (LDMs) into uniform LDMs.Then, a new optimization consensus model was constructed for 2-Rank multigranular linguistic MAGDM problems.(2) The optimization-based approaches [5,[15][16][17], in which the heterogeneous information is integrated by constructing different multiple objective optimization models.Dong et al. [15] proposed a new complex and dynamic HMAGDM method to deal with the differences between individual sets of attributes and heterogeneous information.Zhang et al. [16] developed a HMAGDM method with aspirations information by combining the prospect theory and a biobjective intuitionistic fuzzy programming model.Yu et al. [17] incorporated risk attitude and preference deviation of experts into the mathematical programming models to solve the HMAGDM problems with RNs, INs, Ifs, LVs, and TFNs.(3) The direct approaches [10,[18][19][20][21].In the direct approach, the collective decision information is obtained by aggregating standardized individual decision information.Then the heterogeneous information is transformed into some comparable preference information.Yue and Jia [10] introduced a projection measure to aggregate decision information including IFNs and IVIFs.Yue [18] proposed a direct projection-based group decisionmaking methodology with RNs and INs.To overcome the irrationality of the classical projection formulae in RN and IN vector settings, Yue [19] presented a normalized projection measure and applied it to solve HMAGDM problems with RNs and INs.In order to integrate heterogeneously interrelated attributes in the HMAGDM problem, Das et al. [20] develop an Atanassov's intuitionistic fuzzy extended Bonferroni mean based on a strict t-conorm.Li et al. [21] proposed a new HMAGDM method using weighted power average operator to integrate the heterogeneous decision data.
These achievements have provided the foundation of the HMAGDM problems.It is noticed that methods [1, [5][6][7][8][15][16][17][18][19][20][21] are on the basis of the hypothesis that the ratings provided by DMs are completely affirmative, neglecting the judgment subjectivity; thus, the impreciseness and uncertainty of original decision information cannot be captured.Methods [10][11][12][13][14] turned the heterogeneous information into a unified form of linguistic terms, which are subjective and cannot measure quantitatively and intuitively the uncertainty of attribute values.Intuitionistic fuzzy (IF) sets (IFSs) [22,23] and interval-valued intuitionistic fuzzy sets (IVIFSs) [24] can be viewed as an effective tool to describe the uncertainty and ambiguity, which has led to the wide applications of IFSs and IVIFSs [25][26][27].To fill these research gaps, many practical studies have been proposed to aggregate decision data into IFSs [22][23][24], which can be divided into two categories: (1) the methods for aggregating RNs into intuitionistic fuzzy number (IFN) based on Golden Section idea [28,29], Minimax Criterion [30], and statistical theory [31]; (2) the methods for aggregating RNs or INs into interval-valued intuitionistic fuzzy number (IVIFN) [1] based on Minimax Criterion [32], linear transformation [33], and mean and standard deviation [34].However, these methods [28][29][30][31][32][33][34] cannot be suitable for HMAGDM problems.More recently, Xu et al. [35] presented a general method to aggregate decision information into IFN and applied it to select cloud computing service providers wherein the assessments take the form of RNs, INs, TFNs, TrFNs, and LVs.Combined with the relative closeness in technique for order preference by similarity to ideal solution (TOPSIS) and statistical theory [36], Wan et al. [37] developed a new general method to aggregate the attribute value vector into IVIFNs and used it for HMAGDM problem with RNs, INs, TFNs, and TrFNs.
The aforementioned methods [35,37] have made deep discussions to HMAGDM problems based on aggregating decision data into IF information, but these aggregation techniques [28][29][30][31][32][33][34][35]37] still suffer from some deficiencies.(1) They cannot deal with more complicated attribute values represented by triangular intuitionistic fuzzy number (TIFNs) and trapezoidal intuitionistic fuzzy numbers (TrIFNs).( 2) They ignore the influence of different experts in aggregation process which may lead to unreasonable results.(3) The membership degree and nonmembership degree of integrated value in [28][29][30][31][32][33][34][35]37] cannot reflect the distribution characteristics of the data like normal distribution.In many real decision situations, the evaluations of decision maker are based on a number of historical feedbacks on the corresponding attribute.Studies showed that the distribution of historical feedbacks is generally close to a normal distribution when the number of feedbacks is larger.It may be effective to use TFNs to model the integrated value instead of crisp value and interval-value since TFNs contain more information and are more consistent with normal distribution characteristics.Thus, when an assessment vector is aggregated into an IFN and IVIFN, the loss of information is likely to occur.Intuitionistic triangular fuzzy numbers (ITFNs) introduced by Liu and Yuan [38], as an extension of IFSs, can express more information from different dimension decision information [39] than IFNs and IVIFNs since its prominent characteristic is that the corresponding membership degree and nonmembership degree are described by TFNs [40].Thus, ITFNs can not only depict the fuzzy concept of "good" or "excellent," but also outstand the satisfaction and dissatisfaction information with the maximum probability and also recoup the deficiency due to the loss of the center of gravity in IVIFNs [41][42][43].For instance, in a trustworthy seller selection example, the service attitude may be expressed by an ITFN ((0.4,0.6,0.7),(0.1,0.2,0.3)), which contains two aspects of implication in the historical ratings of a seller: one is that users' satisfactory degree is between 0.4 and 0.7; the most possible satisfactory degree is 0.6; the other is that users' dissatisfactory degree is between 0.1 and 0.3; the most possible dissatisfactory degree is 0.2.Some theories and GDM methods based on ITFNs have been developed.Wang [40] defined score function and accuracy function to compare the ITFNs and developed several ITFN geometric aggregation operators.Wei [41] proposed the ITFN weighted averaging operator and ITFN ordered weighted averaging operator and applied them to solve GDM problems.To consider the interaction among attributes, Gao et al. [42] presented some ITFN aggregation operators with interaction.Yu and Xu [43] investigated a series of intuitionistic multiplicative triangular fuzzy aggregation operators.Although these studies [38][39][40][41][42][43] focused on different aspects of ITFNs, it can only aggregate ITFNs.Therefore, to push ahead with the application of the above aggregations, it is necessary to aggregate multiple types of decision information into ITFNs, which is very interesting yet relatively sophisticated to dispose of.
To do that, this paper aims to propose a novel HMAGDM method based on ITFNs.The primary contributions of this paper can be illuminated briefly as follows.
(i) A new elicitation of the support, opposite, and uncertain information based on distance is introduced, which can accommodate more complicated attribute values including TIFNs and TrIFNs since it just needs to calculate the distance from decision data to the maximum and minimum grade.
(ii) A new construction approach of ITFN is presented by group consistency which not only takes into account expert's weight but can overcome the shortcoming of the hypothesis of the normal distribution.
(iii) It can not only effectively avoid the loss of original information, but also reflect the distribution characteristics of the original decision data.
(2) A new similarity measure of ITFNs is developed and applied to construct a multiple objective linear programming to determine attribute weights in ITFN environment with incomplete information.The determination method of attribute weights can effectively avoid the subjectivity brought by the given attribute weights in advance.
(3) Based on the aforesaid provision, a new method to deal with HMAGDM problems with RNs, INs, TFNs, TrFNs, and TIFNs is proposed.The comprehensive evaluation value of the alternative is an ITFN, which preserves more useful information.
The remainder of this paper is set out as follows.Section 2 briefly introduces related basic concepts.Section 3 presents an approach to aggregating heterogeneous decision data into ITFNs.Section 4 builds a multiple objective linear programming model to determine attribute weights and propose a HMAGDM method.Section 5 provides a numerical example to illustrate the feasibility and reasonableness of the proposed method.Section 6 makes our conclusions.

Preliminary
In this section, some basic concepts of ITFN and distance measures are briefly described below.

. . Intuitionistic Triangular Fuzzy Number
Definition (see [38]).A triangular fuzzy number (TFN) A is a special fuzzy set on a real number set R; its membership function is defined by where 0 ≤   ≤   ≤  ℎ ≤ 1,   and  ℎ present the lower limit and upper limit of A, respectively, and   is the mode, which can be denoted as a triplet (  ,   ,  ℎ ).
Proof.It is easy to see that the proposed similarity measure ( α1 , α2 ) meets the third property of Theorem 6.We only need to prove (i), (ii), and (iv).
For (i).By (2), we have It is easy to see that Thus we get And then the inequality 0 ≤ (α 1 , α2 ) ≤ 1 is established.
For (ii).When ( α1 , α2 ) = 1, if and only if Apparently, it is easy to derive Thus we get And then α1 = α2 .

A New Method for Heterogeneous MAGDM Problems
In this section, the presentation of heterogeneous MAGDM problems is given first.Then, an approach to aggregating heterogeneous information into ITFNs is developed.
Since there are multiple formats of rating values, the attribute set A = { 1 ,  2 , . . .,   } is divided into four subsets sets for subsets Â ( = 1, 2, 3, 4, 5) by Hence, a group decision matrix of alternative   can be expressed as To reduce information loss and simplify the focused problems, the group decision matrices   = (   ) × ( = 1, 2, . . ., ) can be integrated into a collective ITFN decision matrix.The key to addressing this issue lies in an effective approach for constructing ITFNs based on the experts' assessment expressed in different types of data.
. .An Approach to Aggregating Heterogeneous Information into ITFNs.To facilitate the calculation, denote the jth column vector in the matrix   as which is the normalized assessment vector of alternative   on attribute   given by all DMs   ( = 1, 2, . . ., ).Let  max  and  min  be the largest grade and smallest grade employed in the rating system.For example, if the assessments in   are TFNs, then  max The implementation of the aggregation approach involves a four-stage framework (see Figure 1): (1) Elicit Rsd, Rdd, and Rud.In this process, we use the TOPSIS method to obtain the rating satisfactory degree (Rsd) and rating dissatisfactory degree (Rdd) of    and construct the support set   and opposition set   of   .The rating uncertain degree (Rud) of    and the corresponding uncertain set   are derived by geometry entropy.(2) Calculate mode.Combining the group consistency and mean method, the modes of the above sets   ,   , and   are computed in this stage.(3) Construct Qst and Qdt.According to the Min-Max method, the quasisatisfactory triangular (Qst) and quasi-dissatisfactory triangular (Qdt) of   can be built.( 4) Induce an ITFN.The ITFN of   can be obtained through a linear transformation in this process.
. . .Elicit the Rsd, Rdd, and Rud.Consider that (1) the relative closeness [45] from    to  max  implies the satisfaction of DM; (2) the relative closeness from    to  min  implies the dissatisfaction of DM; and (3) according to the ratiobased measure of fuzziness [46,47], the ratio of distances from    to  min  and from    to  max  can also express the fuzziness degree of    .Thus, combining the relative closeness of TOPSIS [36] and geometry entropy method [46], the Rsd, Rdd, and Rud of    can be elicited as follows.
Definition .Let   be a benefit attribute vector, and let    be an arbitrary element in   .The Rsd, Rdd, and Rud of    are defined as

𝑗
of the attribute   .However, it is hard to determine the middle grade for some sets of TIFNs [43] and TrIFNs [49,50].Hence, the proposed extraction method is more effective and simple.

Complexity
Remark .When   is a cost attribute vector, the Rud of    can be derived by (21) [49,50], the more consistent it is with the rest of   , the greater the importance of the Rsd    given by DM   .That is to say, the weighted average of the collection   can be regarded as its mode.Here, we utilize the distance between    and    to define the consistency degree of   on support set   to the rest of experts, which can be obtained by where (⋅) is the distance between    and other Rsds in   .Clearly, 0 ≤   ≤ 1.
Assume that the weight vector  = ( 1 ,  2 , . . .,   ) of attributes is fully known; by (4), we can easily obtain the comprehensive rating   = ((   ,    ,  ℎ  ), (   ,    ,  ℎ  )) of the alternative   .When  is incompletely known, we may utilize the following programming model to establish the attribute weights.

Complexity
For a multiple objective programming, there are several solution methods.Here, we apply the Max-Min method.Let  = min{ +  − −  }.By using the Max-Min method, (36) can be solved by the following single objective linear programming model: By plugging (34) and ( 35) in (37), it is easily seen that we can derive the attribute weights since the optimal solution of ( 37) is a Pareto optimal solution of (36).
Remark .To obtain the weights of attributes in the intuitionistic triangular fuzzy environment, the current methodology first combines a multiple objective mathematical programming model with TOPSIS idea based on the above collective decision matrix.Then, this model can be solved by Max-Min method, which is relatively simple.However, Li and Chen [49] determined the attribute weights by expected weight value that involved options of decision makers.Shan and Xu [42] gave the attribute weights in advance.Therefore, our method could be more reasonable and objective.
Thus, the ranking order of the alternative   can be conducted by the following relative closeness coefficient (RCC): where 0 ≤   ≤ 1, ( ∈ ).It is obvious that the larger   , the better the alternative   .
. .Procedure of the Proposed Aggregation Method for Heterogeneous MAGDM.On the basis of the above analysis, a new approach to heterogeneous MAGDM problems involves the following primary steps.
Step . .Calculate the mode of the support set   and opposition set   of   by ( 25)- (27).
Step .Determine the attribute weights by constructing a multiple objective programming model.The detailed steps are as follows.
Step . .Compute the similarity degrees  +  and  −  from the elements at the kth row of the collective decision matrix  = (  ) × to PIS  + and NIS  − by ( 34) and (35), respectively.
Step . .Construct a multiple objective programming model based on (36).
Step . .Convert the above model into a single objective programming model by (37).
Step . .Obtain the optimal weights of attributes by solving the linear programming model.
Step .Calculate the similarity degrees  +  and  −  of alternative   ( ∈ ) by the obtained attribute weights and (34) and (35).
Step .Rank the alternatives according to the RCC and select the best one.
The decision procedure of the proposed algorithm may be depicted in Figure 3.

A Trustworthy Seller Selection Problem and Comparison Analyses
To demonstrate the efficacy of the proposed HMAGDM method, this section gives a trustworthy seller selection example and conducts comparison analyses with the ones of the existing methods [34,35,37].
. .A Trustworthy Seller Selection Example and Its Solution Procedure.Online service trading usually takes place between parties who are autonomous, in an environment where the buyer often has not enough information about the seller and goods.Many scholars think that trust is a prerequisite for successful trading.Therefore, it is very important that buyers can identify the most trustworthy seller.Suppose that a consumer desires to select a trustworthy seller.After preliminary screening, four candidate sellers  1 ,  2 ,  3 , and  4 remain to be further evaluated.Based on detailed seller ratings, the decision-making committee assesses the four candidate sellers according to the five trust factors, including product quality (A ), service attitude (A ), website usability (A ), response time (A ), and shipping speed (A ).Product quality (A ), service attitude (A ), and website usability (A ) given by DMs with a one-mark system are all benefit attributes.It is better to use TFNs to assess product quality (A ).For service attitude (A ), the experts like to provide the lower and upper limits and the most possible intervals; thus the assessments of A can be represented by TrFNs.The website usability (A ) is expressed by TIFNs, while response time (A ) and shipping speed (A ) given by DMs with a ten-mark system both are cost attributes.The assessments of the sellers on A can be represented by RNs.Due to the uncertainty of shipping speed, INs are suitably utilized to represent the assessments of shipping speed (A ).The assessments of four sellers on five attributes given by five experts are listed in Table 1.The attributes' importance is incomplete and experts give incomplete information on the attributes' importance as follows: Obviously, the decision problem mentioned above is a heterogeneous MAGDM problem involving five different formats of data: RNs, INs, TFNs, TrFNs, and TIFNs.
To address this problem, we apply the proposed decision method to the selection of the trustworthy sellers below.Step .The group decision matrices are obtained as in Table 1.
Step .Due to the ratings of A , A , A given by DMs based on the one-mark system and the ratings of A , A , with the ten-mark system, we have Obtain the aggregated ITFNs corresponding to the attribute vectors.
Step . .By ( 25)-( 27), we can obtain the modes of   and   , which are listed in Table 3.
Step . .Using ( 28) and ( 29), the Qst   and Qdt   of   can be constructed, and the results are presented in Table 3.
Step . .Based on (30), the aggregated ITFNs of   are also shown in Table 3.
Step .Determine the attribute weights.
Complexity 13 Step . .By using (36), a multiple objective programming model is expressed as follows: former, the integrated information of experts on the same attribute is a TIFN, whereas that in the latter is an IVIFN.Hence, the current methodology can express vagueness information of reality more accurately and abundantly.(ii) The former takes into account the weights of experts by their consistency degree, whereas there is no consideration in the latter.So, the current methodology is more reasonable.(iii) The same as the above, the latter is only suitable for the HMAGDM problem with RNs, INs, TFNs, and TrFNs.Thus, the latter cannot solve the abovementioned example.
. .Comparison Analyses with Existing HMAGDM Methods.In this section, we compare the proposed method with other two methods for HMAGDM problems; one is the complex and dynamic MAGDM method developed by Dong et al. [15] that is an optimization-based approach, and the other is the GDM method based on integrating heterogeneous information introduced by Li [21] that is a direct approach.For simplicity, the comparative analysis methods are denoted as CD-GDM and IGI-GDM.The highlighted features of the proposed methods can be summarized as follows.
(1) During the initial phase, CD-GDM and IGI-GDM need to standardize the decision data, whereas there is no need for standardization in the proposed method which is relatively simple.
(2) The GDM matrix of the proposed method is an ITFN decision-making matrix containing ITFNs only which is easy to handle, whereas that of IGI-GDM remains a heterogeneous decision-making matrix which is difficult to deal with.
(3) In CD-GDM and IGI-GDM, the integrated information of experts is expressed by original decision data types and real number type which contains less information, while the integrated information of experts in the proposed method is represented by ITFNs which can express intuitively and describe satisfaction, dissatisfaction, and distribution of experts.
(4) The weights of attributes in IGI-GDM are given by decision makers in advance which are subjective; CD-GDM determine the weights of attributes by nonlinear programming model, whereas they construct a multiple objective linear programming model to establish attributes' weights.Thus the proposed method is more objective and effective.

Conclusions
In this paper, we put forward a new aggregation approach to solve such heterogeneous MAGDM problems in which the weights of the attributes are incompletely known.The key features of the proposed method are listed as follows: (1) a new similarity measure of ITFNs is proposed; (2) a new general approach to aggregating decision information into ITFNs is proposed.It can not only accommodate more complicated data of types, including INs, TFNs, TrFNs, and TIFNs, but also take importance of experts into account; (3) a new multiple objective mathematical programming model is developed for determining the attribute weights objectively under intuitionistic triangular fuzzy environment; (4) a new method is presented to solve heterogeneous MAGDM problems, which considers fully the indeterminacy of the DMs in the assessment; thus the final decision results derived by the proposed method are more reasonable.Additionally, the proposed method can be also appropriate for the complex multiattribute large-group decision-making problems [51].Future research will extend the developed method to heterogeneous MAGDM with complete unknown weight information under complex fuzzy environment.Meanwhile, as the scale of group increases and the decision makers have different backgrounds and levels of knowledge, it is difficult to achieve consensus among decision makers [52,53].Therefore, it will be very interesting in future studies to discuss the consensus reaching mechanism in the large-scale HMAGDM.

Figure 2 :
Figure 2: Distances ratio-based rating uncertain degree for    .

)
Similarly, we have the mode (  ) of opposition set   .Note that the membership degree and nonmembership degree of a TIFN are TFNs rather than real numbers.Moreover,    and    are real numbers which are difficult to express the imprecise and vague experts' subjective judgment.By doing this, the TFNs of    and    are commonly used to represent Qsd and Qdd of   since TFN is characterized by a membership function.Thus, it is necessary to construct the Qst and Qdt of   .As per the definition of TFN, the corresponding TFNs of   and   can be constructed as follows.Definition .For the attribute vector   , the Qst   and Qdt   of alternative   on attribute   are defined as   = (min {  } ,  (  ) , max {  }) ,   = (min {  } ,  (  ) , max {  }) ,  } and max{  } are the minimum value and maximum value of the support set   and min{  } and max{  } are the minimum value and maximum value of opposition set   .For the convenience of discussion, the pair (  ,   ) is called a quasi-ITFN.  = (1/5)(min{  }+(  )+min{  }+ (  ) + mean(  )) + max{  } + max{  }. is an ITFN.Namely, all the attribute values in the vector   can be aggregated into an ITFN   .
. ..Construct Qst and Qdt.Remark .To calculate the mode of triangular fuzzy numbers   and   , this paper employs the weighted averaging value that considers the distribution of ratings, whereas some works used the mean value method.The essential difference is that the current method takes the consistency of the group into account, while the mean value method is based on statistical assumptions.. . .Inducing an ITFN.Finally, an ITFN is induced from the Qst and Qdt of alternative   on the attribute   by the following normalized method.Let .1;  2 −  1 ≤ 0.05;  3 −  1 ≤ 0.05;  5 −  1 ≤ 0.05;  1 +  3 +  5 ≤ 0.6;  4 ≤ 0.2;  1 +  2 +  3 +  4 +  5 = 1.

Table 1 :
The decision matrix of four alternatives.

Table 2 :
Rsd, Rdd and Rud of each attribute value.