#### Abstract

How to aggregate decision information in heterogeneous multiattribute group decision making (HMAGDM) is vital. The aim of this paper is to develop an approach to aggregating decision data into intuitionistic triangular fuzzy numbers (ITFNs) for heterogeneous MAGDM problems with real numbers (RNs), interval numbers (INs), triangular fuzzy numbers (TFNs), trapezoidal fuzzy numbers (TrFNs), and triangular intuitionistic fuzzy number (TIFNs). Using the relative closeness of technique for order preference by similarity to ideal solution (TOPSIS) and geometry entropy method, we first present a general approach to aggregating heterogeneous information into ITFNs, which takes the group consistency of experts into account. Based on the collective intuitionistic triangular fuzzy decision matrix and extended TOPSIS, a multiple objective mathematical program is constructed to determine the optimal attribute weights. Subsequently, a new method to solve HMAGDM problems is presented based on the aforementioned discussion. A trustworthy service selection example is provided to verify the practicality and effectiveness of the proposed method.

#### 1. 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 [1–3]. 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 [7–9]. 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 [11–14], 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 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, 15–17], 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, 18–21]. 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 decision-making 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, 5–8, 15–21] 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 [10–14] 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 [25–27]. To fill these research gaps, many practical studies have been proposed to aggregate decision data into IFSs [22–24], 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 [28–34] 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 [28–35, 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 [28–35, 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 [41–43]. 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 [38–43] 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.

(1) We first present an aggregation technology to aggregate heterogeneous information into TIFNs. Compared with existing methods [28–35, 37], the proposed aggregate technology has the following advantages. For more details, refer to Section 5.2.(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.

#### 2. Preliminary

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

##### 2.1. Intuitionistic Triangular Fuzzy Number

*Definition 1 (see [38]). *A triangular fuzzy number (TFN)* A* is a special fuzzy set on a real number set R; its membership function is defined bywhere , and present the lower limit and upper limit of* A*, respectively, and is the mode, which can be denoted as a triplet .

*Definition 2 (see [38]). *Let* X* be a fixed set; and are TFNs defined on the unit interval ; then an intuitionistic triangular fuzzy set over* X* is defined as where the parameters and indicate, respectively, the membership degree and nonmembership degree of the element* x* in , with the conditions .

For convenience, we call an intuitionistic triangular fuzzy number (ITFN), whereIt is clear that the largest and smallest ITFN are and , respectively.

*Definition 3 (see [38]). *Let and be two ITFNs; then the containment isSome arithmetic operations between ITFNs and are shown as below [40]:(1), ,(2), , , .

*Definition 4 (see [41]). *For a set of ITFNs that have associated an importance weight vector with and . We callan intuitionistic triangular fuzzy weighted average operator (*ITFWA*).

*Definition 5. *Let and be two ITFNs. A similarity measure between the ITFNs and is defined as follows:

Theorem 6. *The similarity measure satisfies the following properties:*(i)*.*(ii)* if and only if .*(iii)*.*(iv)*If is a ITFN and , then and .*

*Proof. *It is easy to see that the proposed similarity measure meets the third property of Theorem 6. We only need to prove (i), (ii), and (iv).*For (i)*. By (2), we haveIt is easy to see thatThus we getAnd then the inequality is established.*For (ii)*. When , if and only ifApparently, it is easy to deriveThus we get *, **, **, **, **, *. And then .*For (iv)*. Sincewe getBased on the above inequalities, it is easy to deriveThus, it holds thatThus, . In the same way, it is proved that .

##### 2.2. Distance Measures

Hamming distance is easily processed and commonly used in the process of heterogeneous information processing. For INs and (or TFNs and , TrFNs and , and TIFNs and , the distance measures can be defined as follows [35, 44]:

#### 3. 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.

##### 3.1. Heterogeneous MAGDM Problems

For the sake of convenience, some symbols are introduced to characterize the heterogeneous MAGDM problems as follows:

(1) The group of DMs .

(2) The set of attributes . Denote the attribute weight vector by , where represents the weight of such that and .

(3) The set of alternatives .

Since there are multiple formats of rating values, the attribute set is divided into four subsets , , , , and , where , ), and , is an empty set. The rating values in the subsets are in the form of RNs, INs, TFNs, TrFNs, and TIFNs, respectively. Denote the subscript sets for subsets by , , , , and , respectively.

(4) The group decision matrix.

Suppose that the rating of alternative with respect to the attribute given by DM is denoted by . If , then is a RN. If , then is an IN. If , then is a TFN. If , then is a TrFN. If , then . Namely, can be unified as follows:

Hence, a group decision matrix of alternative can be expressed as

To reduce information loss and simplify the focused problems, the group decision matrices 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.

##### 3.2. An Approach to Aggregating Heterogeneous Information into ITFNs

To facilitate the calculation, denote the* j*th column vector in the matrix aswhich is the normalized assessment vector of alternative on attribute given by all DMs . Let and be the largest grade and smallest grade employed in the rating system. For example, if the assessments in are TFNs, then and ; if the assessments in are INs, then and . To integrate the decision matrices into a collective ITFN decision matrix, all the elements in vector need to be aggregated into an ITFN. 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 quasi-satisfactory 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.

###### 3.2.1. Elicit the Rsd, Rdd, and Rud

Consider that (1) the relative closeness [45] from to implies the satisfaction of DM; (2) the relative closeness from to implies the dissatisfaction of DM; and (3) according to the ratio-based measure of fuzziness [46, 47], the ratio of distances from to and from to 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 7. *Let be a benefit attribute vector, and let be an arbitrary element in . The Rsd, Rdd, and Rud of are defined asrespectively, where , , and denote the Rsd, Rdd, and Rud given by on attribute in the alternative , is the distance between and the largest grade of the attribute , and is the distance between and the smallest grade of the attribute .

*For Example **1*. Consider that is a TFN in the ten-mark system; then , . According to (19)-(21), the Rsd, Rdd, and Rud of are calculated, respectively, as , , and .

Theorem 8. *Rud of in has the following properties.**(EP1) If or , then , which means that and are not fuzzy since and are crisp sets.**(EP2) If (namely, is the middle point), then , which means that is the fuzziest element.**(EP3) , if is less fuzzy than , i.e.,**It is easy to prove that Rud meets the properties (EP1)-(EP3), which are consistent with the axioms (1)-(3) of fuzzy entropy based on distance in [48]. From a geometric standpoint, and in the rating system are nonfuzzy which can correspond to the position M and position N (Figure 2). When a fuzzy number is moved from position M (or N) towards middle position O, the distance from to M is close to N. Meanwhile, become more and more vague; i.e., is getting bigger and bigger. Particularly, when is in position P, the distance from to M is equal to N. So, the fuzzy number is the fuzziest, i.e., . According to the analysis above, it is reasonable that Rud measures the uncertainty of the original assessment. It is worth mentioning that (21) is suitable for different forms of decision data such as INs, TFNs, TIFNs, and TrIFNs.*

*Remark 9. *From (19)-(21), the extraction method in this paper relies on just the largest grade and the smallest grade of the attribute , while the methods [35, 37] rely on and as well as the middle grade 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.

*Remark 10. *When is a cost attribute vector, the Rud of can be derived by (21). The Rsd and Rdd of can be rewritten as

*Remark 11. *For the benefit attribute vector , , , and of each element are composed of support set , opposition set , and uncertain set of which can be defined as and and , respectively.

###### 3.2.2. Calculate the Mode

For the support set , the mode of the TFN is located in the center around which the Rsd gather. Inspired by the literature [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 bywhere is the distance between and other Rsds in . Clearly, .

Generally, an expert’s Rsd is more important if he/she is more similar to the group’s Rsd. In other words, the larger the value of is, the more important is. Thus, the weight of on can be obtained by

Then the mode of for support set is derived as

Similarly, we have the mode of opposition set .

###### 3.2.3. Construct Qst and Qdt

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 12. *For the attribute vector , the Qst and Qdt of alternative on attribute are defined aswhere the and are the minimum value and maximum value of the support set and and are the minimum value and maximum value of opposition set . For the convenience of discussion, the pair is called a quasi-ITFN.

*Remark 13. *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.

###### 3.2.4. Inducing an ITFN

Finally, an ITFN is induced from the Qst and Qdt of alternative on the attribute by the following normalized method. Let be the induced ITFN by the attribute vector , and is the uncertain degree of , To satisfy the conditions in (3) and consider the influence of uncertain degree, the values of , , , , , and can be computed as follows:respectively, where .

Apparently, , , , , , and satisfy (3). Thus, is an ITFN. Namely, all the attribute values in the vector can be aggregated into an ITFN .

*For Example **2*. Consider that , , , , is a TFN vector in the ten-mark system; then , . According to (19)-(21), the support set, opposition set, and uncertain set of are calculated, respectively, as , , and . Subsequently, the mode of the sets and is calculated from (25)-(27) to be and . It follows from (28) and (29) that and . Finally, by using (29), the induced ITFN associated with is derived to be .

#### 4. A Novel Approach for Heterogeneous MAGDM Problems

According to the proposed aggregation method, the collective decision matrix is aggregated as follows:where