#### Abstract

This paper investigates an approach to multiple attribute group decision-making (MAGDM) problems, in which the individual assessments are in the form of triangle interval type-2 fuzzy numbers (TIT2FNs). Firstly, some Frank operation laws of triangle interval type-2 fuzzy set (TIT2FS) are defined. Secondly, some Frank aggregation operators such as the triangle interval type-2 fuzzy Frank weighted averaging (TIT2FFWA) operator and the triangle interval type-2 fuzzy Frank weighted geometric (TIT2FFWG) operator are developed for aggregation TIT2FNs. Furthermore, some desirable properties of the two aggregation operators are analyzed in detail. Finally, an approach based on TIT2FFWA (or TIT2FFWG) operator to solve MAGDM is developed. An illustrative example about supplier selection is provided to illustrate the developed procedures. The results demonstrate the practicality and effectiveness of our new method.

#### 1. Introduction

Fuzzy multiple attribute group decision making (FMAGDM) is a hot area of research in fuzzy decision analysis theory. It involves the process to select the best alternative(s) from the set of feasible alternatives with respect to multiple attributes, in which the decision information usually with fuzziness or uncertainty is provided by a group of decision makes under fuzzy environment. In recent years, many techniques have been developed to solve FMAGDM problems [1–8]. Li and Yang [1] developed a linear programming technique for multidimensional analysis of preferences in multiple attribute group decision making under fuzzy environments. Wang and Lee [2] generalized TOPSIS method for FMAGDM based on high and low operators which satisfy the partial ordering relation on fuzzy numbers. Xu and Chen [3] established an interactive model for FMAGDM in which the information about attribute weights is partly known. Lin and Wu [4] presented a fuzzy DEMATEL method for group decision making to gather group ideas and analyzed the cause-effect relationship of complex problems in fuzzy environment. Guha and Chakraborty [5] developed a FMAGDM technique considering the degrees of confidence of experts' opinions and aggregated the decision information by using similarity. Hatami-Marbini and Tavana [6] proposed an alternative fuzzy outranking method by extending the ELECTRE I method to take into account the uncertain, imprecise, and linguistic assessments provided by a group of decision makers. Mousavi et al. [7] utilized Monte Carlo simulation technique to derive a stochastic approach for handling FMAGDM and applied it in risk selection problem in a highway project. Zhao et al. [8] utilized the fuzzy prioritized operators to develop some models for triangular FMAGDM in which both the attributes and decision makes are in different priority level.

Type-2 fuzzy set (T2FS) was first proposed by Zadeh [9], which can be viewed as an effective extension of tradition type-1 fuzzy set (T1FS). The membership function of T2FS, which is characterized by primary function, secondary function and a footprint of uncertainty (FOU). It is the new third dimension of T2FS and the footprint of uncertainty can reflect more additional degrees of freedom that make it more capable of handling imprecision and imperfect information in real-world application [10–12]. Hence, T2FS are receiving more and more attentions from researchers and have been successfully developed both in theoretical and in practical aspects [13–21]. However, because the computational complexity of general T2FS is too high, it is usually difficult to use in real practice situations [22]. To overcome the limitation of T2FS, interval type-2 fuzzy set (IT2FS), which is also called interval-valued fuzzy set (IVFS) and contains membership values that are crisp interval from zero to one, is proposed [23] and has been the most useful tool for modeling uncertainty and complexity in many domains [24–30]. In particular, many approaches have been presented in the field of multiple attribute group decision making (MAGDM). Chen and Lee [31] presented a new method based on arithmetic operations for handing multiple attribute hierarchical group decision-making problems. Chen and Lee [32] further developed an interval type-2 TOPSIS method to solve MAGDM and gave some practical examples to illustrate the procedure of the proposed method. Chen et al. [33] presented a ranking formula for IT2FS information and established a framework based on ranking method for handling MAGDM with interval type-2 fuzzy information. Hu et al. [34] presented an approach based on possibility degree for interval type-2 fuzzy multiple criteria group decision making and established a maximizing deviation optimal model for determining the attribute weights, in which the weight information is partly known. Wang et al. [35] established some weight optimal models to determine the weights of attributes and utilized the interval type-2 weighted averaging (IT2WA) operator for handling MAGDM under interval type-2 fuzzy environment. Chen [36] presented a signed-distance-based method for determining the objective importance of criteria and handling fuzzy multiple criteria group decision-making problems in a flexible and intelligent way. Chen et al. [37] investigated extended QUALIFLEX method for group decision making, in which the attribute values take the form of IT2FS, and further applied it in medical decision making. Baležentis and Zeng [38] further extended MULTIMOORA method with IT2FS and applied it in the field of human resource management and performance management. Chiao [39] introduced the concepts of triangular interval type-2 fuzzy set and developed the analytic hierarchy process (AHP) method based on logarithmic regression function for handling MAGDM problem. Ngan [40] established the probabilistic linguistic framework with IT2FS MAGDM problems, and some arithmetic operations, such as union and intersection between the interval type-2 fuzzy linguistic numbers, were also defined. Chen [41] developed a new linear assignment method based on signed distances for handling MAGDM problems with IT2FS information and applied it in the selection of landfill site. Z. M. Zhang and S. H. Zhang [42] proposed the concept of trapezoidal interval type-2 fuzzy soft set and further developed an MAGDM approach under interval type-2 fuzzy environment.

For MAGDM problems, an essential important step is how to aggregate the decision-making information in different formats with aggregation operators. As an important tool for information fusion, aggregation operators are widely used in various fuzzy set (FS), such as T1FS, intuitionistic fuzzy set (IFS), and hesitant fuzzy set (HFS). However, up till now, the researches on type-2 fuzzy aggregation operators are very rare. Wu and Mendel [43] presented the concept of linguistic weighted average (LWA), in which the attributes and the weights both are in the form of IT2FS, the algorithm was provided and verified the LWA operator can be decomposed into finding two fuzzy weighted average (FWA). Liu et al. [44] presented an analytical solution method for the FWA; the method had some good mathematical analysis properties, which can be used for interval type-2 aggregation process. Zhou et al. [45, 46] proposed the type-2 OWA operator to aggregate the linguistic opinions or preferences in human decision-making modeling by T2FS, and a direct method to aggregate IT2FS by type-2 OWA operator was provided. Liu et al. [47–49] developed some aggregation operators, such as weighted aggregation operators, geometric aggregation operators, and hybrid harmonic averaging operators to handle MAGDM problems with interval-valued trapezoidal fuzzy numbers.

All the above interval type-2 fuzzy aggregation operators are based on alpha-cut decomposition theorem [9]; the main difference between the existing interval type-2 fuzzy aggregation operators and other FS aggregation operators is that the interval type-2 fuzzy aggregation operators are derived by fuzzy extension principle while the other fuzzy aggregation operators are obtained by triangle norms [50, 51]. As the fundamental of information fusion, triangle norms play an important role and have been successfully used to derive various fuzzy aggregation operators [52–54]. However, until now, there is no research about interval type-2 aggregation operator based on triangle norm operations. In order to fill this gap, we will utilize the Frank triangle norms to develop some desirable interval type-2 fuzzy aggregation operators in this paper.

Frank triangle norms [55], which is the only one type triangle norm satisfying the compatibility. Since the Frank triangle norms involve the parameter, this can provide more flexibility and robustness in the process of information fusion and make it more adequate to model practical decision-making problems than other triangle norms [56]. Calvo et al. [57] study the functional equations of Frank and Alsina for two classes of commutative, associative, and increasing binary operators. Yager [58] introduced the extensions operations to the lattice of closed interval-valued fuzzy set based on Frank -norms and gave necessary and sufficient conditions such that these operations can form a complete algebraic structure. Casanovas and Torrens [59] investigated the additive generating function of Frank -norms and established a framework with Frank -norms in approximate reasoning. Sarkoci [60] made a comparison between the Frank -norms and the Hamacher -norms and proved that two different -norms form the same family. However, the previous research works almost focus on the basic mathematical construct and properties of Frank -norms; the researches of its applications are very rare, especially in aggregation and decision making. To date, we have not seen any researches on aggregation operators based on Frank -norms for decision-making problems. Therefore, it is meaningful to study aggregation operators based on Frank triangle norms operations and their application in MAGDM under triangle interval type-2 fuzzy environment.

The remainder of this paper is shown as follows. In Section 2, we briefly introduce some basic concepts of T2FS, triangle interval type-2 fuzzy set, and Frank triangle norms. In Section 3, we establish the Frank operational laws of triangle interval type-2 fuzzy set. In Section 4, we present triangle interval type-2 fuzzy Frank weighted averaging (TIT2FFWA) operator and triangle interval type-2 fuzzy Frank geometric average (TIT2FFWG) operator and also discuss some desirable properties in detail. In Section 5, we develop an approach based on the proposed Frank aggregation operators to MAGDM under the triangle interval type-2 fuzzy environment. A practical example is provided to illustrate the procedure of our method in Section 6. Finally, we end the paper with some conclusions and point out future researches in Section 7.

#### 2. Preliminaries

In this section, we will briefly review some basic concepts of type-2 fuzzy set and Frank triangular norms.

##### 2.1. Type-2 Fuzzy Set

*Definition 1 (see [23]). *A type-2 fuzzy set in the universe of discourse can be represented by a type-2 membership function as follows:
where denotes an interval in . Moreover, the type-2 fuzzy set can also be expressed as the following form:
where is the primary membership at , and indicates the second membership at . For discreet situations, is replaced by .

*Definition 2 (see [23]). *Let be a type-2 fuzzy set in the universe of discourse represented by a type-2 membership function . If all , then is called an interval type-2 fuzzy set (IT2FS). An interval type-2 fuzzy set can be regarded as a special case of the type-2 fuzzy set, which is defined as

It is obvious that the interval type-2 fuzzy set defined on is completely determined by the primary membership which is called the footprint of uncertainty (FOU), and the FOU can be expressed as follows:

##### 2.2. Triangular Interval Type-2 Fuzzy Set

Because the operations on interval type-2 fuzzy sets are very complex according to the decomposition theorem [23], the interval type-2 fuzzy sets are usually taken in some simplified formations in applications. Here, we follow the results of Chiao [39], which adopted triangular interval type-2 fuzzy set for solving MAGDM problems.

In [39], Chiao proposed the concept of triangular interval type-2 fuzzy set, in which the upper membership function (UMF) and the lower membership function (LMF) are represented by triangular fuzzy number.

Let be a triangular interval type-2 fuzzy set defined on , which is shown in Figure 1, where are the reference points of the triangular interval type-2 fuzzy set, satisfying . The upper and lower membership functions of are defined as

The footprint of uncertainty (FOU) of the membership function of is shown as the shaded region in Figure 1. Obviously, if , , then for all , and the triangular interval type-2 fuzzy set reduces to the triangular type-1 fuzzy set.

If is a set in which the elements are all real numbers, then a triangular interval type-2 fuzzy set in is called a triangular interval type-2 fuzzy number (TIT2FN).

Chen et al. [33] proposed the ranking formula for interval type-2 fuzzy number as follows.

*Definition 3 (see [33]). *Let be a triangular interval type-2 fuzzy number in Figure 1; the ranking value of the triangular interval type-2 fuzzy number is defined as follows:
The larger the ranking value , the greater the TIT2FN .

##### 2.3. Frank Triangular Norms

Frank operations include the Frank product and Frank sum, which are examples of triangle norms and triangle conorms, respectively. Frank product is a -norm and Frank sum is a -conorm, and the mathematical forms are defined as follows [55]:

It is pointed out that the Frank product and Frank sum have the following property [61]:

Based on the limit theory, we can easily prove the interesting results as follows [61].(1)If , then , , and the Frank product and Frank sum are reduced to the probability product and probability sum.(2)If , then , , and the Frank product and Frank sum are reduced to the Lukasiewicz product and Lukasiewicz sum.

#### 3. Frank Operations of Triangular Interval Type-2 Fuzzy Numbers

Based on Frank -corm and Frank -conorm previously mentioned, we will establish the basic operation laws of triangular interval type-2 fuzzy numbers in this section.

*Definition 4. *Let , and be three TIT2FNs, and ; then we define their operation laws as follows.

(1) Addition operation:

(2) Multiplication operation:

(3) Multiplication by an ordinary number:

(4) Power operation:

Theorem 5. *Let , , and be three TIT2FNs, and , , ; then one has*(1)*;*(2)*;*(3)*;*(4)*.**It is easy to prove the formulas in Theorem 5; thus, they are omitted in here.*

#### 4. Frank Aggregation Operators for Triangular Interval Type-2 Fuzzy Numbers

In this section, we will develop the Frank operator under triangular interval type-2 fuzzy environment and propose the triangular interval type-2 fuzzy Frank weighted averaging (TIT2FFWA) operator and the triangular interval type-2 fuzzy Frank weighted geometric (TIT2FFWG) operator based on Frank triangular norms, respectively.

##### 4.1. Triangular Interval Type-2 Fuzzy Frank Weighted Averaging Operator

*Definition 6. *Let be a collection of TIT2FNs, and let TIT2FFWA: ; if
then the function TIT2FFWA is called a triangular interval type-2 fuzzy Frank weighted averaging (TIT2FFWA) operator, where is the weight vector of , and . In particular, if , then the TIT2FFWA operator is reduced to a triangular interval type-2 fuzzy Frank arithmetic averaging (TIT2FFAA) operator of dimension , which is defined as follows:

Based on the Frank operation laws from Theorem 5, we can derive the following Theorem 7, which shows that the FWA of TIT2 fuzzy sets is also a TIT2 fuzzy set.

Theorem 7. *Let be a collection of TIT2FNs; then their aggregated value by TIT2FFWA operator is still a TIT2FN, and
**
where is the weight vector of , , and .*

*Proof. *Here, we prove Theorem 7 by using the mathematical induction method as follows.

(1) For , based on the Frank operational laws of TIT2FNs, we have
In accordance with Definition 2, we obtainIf (15) holds for , that is,
then if , based on the operational laws of the TIT2FNs, we can deduce that
That is, (15) holds for . Therefore, (15) holds for all , which completes the proof of Theorem 7.

In what follows, we can easily prove that the TIT2FFWA operator has the idempotency, boundary, and other properties that ordinary aggregation operators usually have.

Theorem 8 (idempotency). *Let , be a collection of TIT2FNs, where . If all are equal, that is, , for all , then
*

*Proof. *By using Theorem 7, we have
Thus, the proof is completed.

Theorem 9 (boundary). *Let be a collection of TIT2FNs, where , and let
**
Then
*

*Proof. *Since
for all , then it follows that
and then
Then, it follows that
Similarly, we have

Let ; then by using the ranking value formula of TIT2FN, we have
Therefore,
which completes the proof of Theorem 9.

Theorem 10. *Let be a collection of TIT2FNs, where . If is also a TIT2FN on , then
*

*Proof. *Since
according to Theorem 7, we have

On the other hand, based on Theorem 7 and the operational laws of TIT2FNs, we haveTherefore, we have
which completes the proof of Theorem 10.

Theorem 11. *Let be a collection of TIT2FNs, where . If , then
*

*Proof. *Based on the Frank operation laws of TIT2FNs, we haveAlso sincetherefore, we have
Thus, the proof is completed.

Theorem 12. *Let be a collection of TIT2FNs. If , *