#### Abstract

To address complexity information fusion problems involving fuzzy and grey uncertainty information, we develop prioritized averaging aggregation operator and Bonferroni mean aggregation operator with grey linguistic 2-tuple variables and apply them to design a new decision-making scheme. First, the grey linguistic 2-tuple prioritized averaging (GLTPA) operator is developed to characterize the prioritization relationship among experts and employed to fuse experts’ information into an overall opinion. Second, we establish dual generalized grey linguistic 2-tuple weighted Bonferroni mean (DGGLTWBM) operator to capture the interrelationship among any attribute subsets, which can be reduced to some conventional operators by adjusting parameter vector. On that basis, a flexible group decision-making approach with fuzzy and grey information is designed and applied to an evaluation problem, in which grey relationship analysis (GRA) method and a linear programming model are combined to extract attribute weights from partially known attribute information. Furthermore, an illustrative example is employed to illustrate the practicality and flexibility of the designed method by conducting the related comparative studies.

#### 1. Introduction

Multiple-attribute decision-making (MADM) methods support us to identify the solution from some possible alternatives, which are widely used in our daily life such as human resource selection [1], investment strategy analysis [2, 3], task scheduling and sort management [4], supplier identifying [5], query processing [6], and location planning voting system [7].

With the increasing uncertainty and complexity in managerial decision practices, it is usually difficult to express their preferences in the form of an accurate value. The reason for this problem is mainly derived from the cognitive uncertainty (fuzzy uncertainty) and insufficient information (grey uncertainty). Fortunately, fuzzy set theory and grey system theory have been proposed to help us characterize these uncertainties, respectively. Fuzzy set theory focuses on the problems with cognitive uncertainty (the fuzzy boundary between two concepts, such as the phrases “young men” and “old men”) [8], which describes uncertainty with the aid of membership grades. The sources of assessment information are associated with the past collected records, while the clarity of information is largely influenced by the completeness of information sources. For example, an expert will evaluate sales performances of five clothing factories based on their data records over the last five years. However, some of the factories can only provide statistical data over the past four years or three years. Due to lack of record of information, the final evaluation result comes with some level of insufficient information. The grey theory is designed for quantifying the systems involving insufficient information [9]. The grey degree can be incorporated into fuzzy decision-making procedures for qualifying the insufficiency, incompletion, or unlikelihood of information that relates to the characteristics of the cognitive activities.

Multiattribute decision-making activities involving fuzzy uncertainty and insufficiency information have two characteristics, which are called the grey fuzzy MADM problems. Chen proposed grey fuzzy sets (GFSs) [10] which employ membership grades to express individual’s uncertainty opinions and measure information content with the aid of grey grade. In recent decades, classical fuzzy sets [11] have been extended to the forms of fuzzy numbers [12], linguistic variables [13–16], multiple membership grades [17], and their hybrid forms [18, 19] to capture human cognitive information in handling real application problems. Advances of fuzzy sets come with a variety of grey fuzzy decision approaches. Linguistic term sets and GFSs have also been integrated to make full use of their advantages for managing complex uncertainty activities. Tian et al. introduced grey linguistic sets (GLSs) [20], denoted as , where represents the linguistic term with membership degree , can take some linguistic terms “low” or “excellent” etc., and the grey grade of information is . GLSs are new development forms of fuzzy sets and linguistic term sets. Recently, different extensions of GLSs have been proposed, and pertinent aggregation operators of GLSs have been established to address information fusion in managerial decision activities. Mi et al. proposed a new method for evaluating online tour review based on grey linguistic 2-tuple terms [21]. Rao et al. put forward a new representation strategy of uncertainty by combining 2-tuple terms and GLSs [22], and they established a decision-making technique by introducing an aggregation operator. Moreover, interval grey linguistic variables have been widely applied in dealing with managerial decision practices, and their aggregation operators are also proposed [23–25]. There are other hybrid methods involving information measures to address some complex managerial decision problems from different viewpoints. 2-dimensional (uncertain) linguistic variables [26, 27] express fuzzy evaluation information by adding linguistic terms in terms of the subjective reliability. The methodology of -numbers provides a strategy for describing the fuzziness with the probability reliability [28–30].

To better approximate the mechanism of group decision-making, there is an increasing special requirement on aggregation operators and attribute weights. Aggregation operators can fuse the experts’ evaluation information to the group’s opinion or integrate the performances of alternatives with respect to different attributes to an overall score. To address different issues of cognitive processes, lots of aggregation operators were developed from different viewpoints [31–33]. Prioritized averaging (PA) operator can characterize a prioritization relationship between individuals or attributes [32]. Different priority levels widely exist in daily routine [34]. Most of the existed works focused on the prioritization relationship among attributes, while that of individuals is actually encountered in real decision-making [35]. For example, in a company, the chairman has a higher priority than the general manager, and the general manager has a higher priority than the deputy. We develop PA (prioritized averaging) operator for grey linguistic 2-tuple terms to serve as a way of reflecting the priority relationship that existed in decision-makers. In reality, it is deserved to point that attributes are not independent of each other in some decision-making contexts, so we need to consider the interrelationship among any attribute subsets to make the aggregated results more realistic. Inspired by dual generalized weighted Bonferroni mean (DGWBM) operator [36], we develop a dual generalized grey linguistic 2-tuple weighted Bonferroni mean (DGGLTWBM) operator. In addition, attribute weight is a key influence factor for results of aggregation information. However, the determination of weight is associated with various factors. Within the decision context involving fuzzy uncertainty and grey uncertainty, it is impossible to assign accurate weight information directly. Based on the developed aggregation operators, we design a new technique for solving group decision problems with grey linguistic 2-tuple variables, in which grey relationship analysis (GRA) method and linear programming approach are combined to optimize attribute weights from a priori weight information. The paper is structured as follows. In Section 2, some pertinent concepts of the grey linguistic 2-tuple set, PA operator, and DGWBM operator are recalled, respectively. In Section 3, two generalized aggregation operators are proposed to fuse grey linguistic 2-tuple information, and their properties are also studied. In Section 4, a method is designed to solve group management decision problems with grey linguistic 2-tuple set. In Section 5, the developed method is exemplified by analyzing the ranking results with the change of the relevant factor and comparison with different method and different types of information fusion. Section 6 offers some conclusions.

#### 2. Related Work

In this section, some relevant definitions and operators are recalled, which serve as the foundation for establishing aggregation operators and decision-making model.

##### 2.1. The Grey Linguistic 2-Tuple Representation Approach

Due to the complexity of the external decision-making environment, it is necessary to portray the increasingly complex activities with a certain insufficiency degree of characterizing information [22]. Grey grade can better perform this function, which is a measurement to qualify the insufficiency, incompletion, or unlikelihood of information. In general, information content can be deemed as five levels including very sufficient, sufficient, common, incomplete, and very incomplete.

*Definition 1 [10]. *Let be a fuzzy subset in the space . If the membership degree of belonging to is grey in the unit interval and its grey grade is , then is a GFS in the space .

As a matter of fact, each GFS consists of set pair , where denotes the fuzzy part of and represents the grey part of .

A successful attempt was made to combine linguistic terms and GFS for improving the ability to characterize uncertainty information [20, 37]. Especially, linguistic 2-tuple variables are taken as the fuzzy part of GFS to create the concept of grey linguistic 2-tuple variables [21, 22].

*Definition 2 [22]. *Let be the collection of grey grades, be a linguistic 2-tuple variable, and , then is called a grey linguistic 2-tuple variable, where represents the greyness grade for characterizing the incompleteness level of evaluation information and is a linguistic term.

Especially, degenerates into a linguistic 2-tuple variable when . GFS is considered as the generalization of fuzzy sets and grey sets [20]. The linguistic term set should be comprehensible and predefined, by which every expert can further qualify possible alternatives with respect to multiple attributes. Usually, takes an even number in ; satisfies the following characteristics: ; there exists a negation operator: ; there exist maximum operator and minimum operator: and for .

*Definition 3 *[22]. Let and be two grey linguistic 2-tuple variables; the order relations are defined as follows:
Some fundamental operations of grey linguistic 2-tuple variables are defined as follows.

*Definition 4 [21, 22]. *Let and be two grey linguistic 2-tuple variables, the operational rules are defined as follows:
(1)(2)where “” represents the minimizing operation, with , , , and round is the map that assigns to the closest positive integer . On the contrary, is defined as .

*Definition 5 [22]. *Let and be two grey linguistic 2-tuple variables; the distance function between and is given as follows:

##### 2.2. The Prioritized Averaging Operator

In the real decision-making situation, different experts or attributes have different priority levels. In order to capture this phenomenon, a prioritized averaging (PA) operator is coined by Yager [32].

*Definition 6 [32]. *Let be a set of indexes, where (“” denotes “prior to”). The value represents the performance of the alternative with respect to and satisfies . Then
where , and

##### 2.3. The Dual Generalized Weighted Bonferroni Mean Operator

Bonferroni [38] initially put forward the Bonferroni mean (BM) operator, which provided an access to consider the interrelationships between any two attribute subsets. To further address the interrelationships among all elements involved in the aggregation process, Zhang et al. [36] created an operator called dual generalized weighted Bonferroni mean (DGWBM) operator. It not only characterizes the interrelationships among any attribute subsets but also takes the weight of the aggregated argument into account. The DGWBM operator is shown below.

*Definition 7 [36]. *Let be the parameter vector with and be a collection of nonnegative integers with weights such that and . Then

#### 3. New Aggregation Operators for Grey Linguistic 2-Tuple Variables

Aggregation is a crucial step in dealing with decision-making problems, by which the individual opinions can be integrated into a collective decision value [39]. Two new operators are developed to aggregate grey linguistic 2-tuple variables, and their related properties are investigated in this study.

##### 3.1. Grey Linguistic 2-Tuple Prioritized Averaging Aggregation Operator

In this part, we first generalize the PA operator to grey linguistic 2-tuple situation and establish grey linguistic 2-tuple prioritized averaging (GLTPA) operator.

*Definition 8. *Let be a collection of grey linguistic 2-tuple variables and be an element of , and GLTPA operator is defined as follows:
where is aggregation-associated weight with , , and .

Theorem 1. *Let be a set of grey linguistic 2-tuple variables, then the following properties of the GLTPA operator hold:
*(1)*Idempotency: let be equal to for , then
*(2)*Boundedness: let be a set of grey linguistic 2-tuple variables
where and , , , , and .*

##### 3.2. Dual Generalized Grey Linguistic 2-Tuple Weighted Bonferroni Mean Aggregation Operator

In this subsection, we develop a dual generalized grey linguistic 2-tuple weighted Bonferroni mean (DGGLTWBM) operator by incorporating the DGWBM operator (Definition 7) into a grey linguistic 2-tuple situation.

*Definition 9. *Let be a parameter vector with , be a collection of grey linguistic 2-tuple variables, and be an element of with weight , then DGGLTWBM operator is defined as follows:

Theorem 2. *Let be a parameter vector with , be a collection of grey linguistic 2-tuple variables, and be an element of with weight , then the following properties of the DGGLTWBM operator are obtained:
*(1)*Monotonicity: let and be two grey linguistic 2-tuple variables; if and for all , then
*

*Proof 1. *Due to and for all , we can get and *.*

According to Definition 3, we can obtain
where (2)Boundedness: let be a set of grey linguistic 2-tuple variables with weight , then
where and , , , , and .

*Proof 2. *Because of , , , and for all , we can obtain the following result:
where

The DGGLTWBM operator can be reduced to some classical aggregation operators by adjusting the parameter vector.

*Remark 1. *When , the DGGLTWBM operator is reduced to a generalized grey linguistic 2-tuple weighted averaging (GGLTWA) operator.

*Remark 2. *When , the DGGLTWBM operator is reduced to a grey linguistic 2-tuple weighted Bonferroni mean (GLTWBM) operator.

*Remark 3. *When , the DGGLTWBM operator is reduced to a generalized grey linguistic 2-tuple weighted Bonferroni mean (GGLTWBM) operator.
where , , and .

#### 4. A Method to Solve Group Decision-Making Problem with Grey Linguistic 2-Tuple Variables

In this section, we employ the developed operators to solve managerial decision problems involving grey linguistic 2-tuple information. Suppose a decision-making context that there includes alternatives , attributes with partial a priori weight information, and experts . Assume that represents the -th decision information matrix, where is the performance evaluated by for alternative with respect to the attribute .

Assigning weight is an essential step in managerial decision procedures, which can be determined by the objective and subjective viewpoints. The objective weights can be elicited by using entropy method, rough sets, maximum deviation square method, and granular structures. The subjective weights are usually determined by expert’s judgments. It is difficult to assign attribute weights in the form of crisp numbers in complex decision situation. In contrast, it is more reasonable to provide some partial information of attribute weights. In the sequel, we choose the grey relational analysis (GRA) ([40]) technique to extract the optimal attribute weight information under grey linguistic 2-tuple environment and partial weight information.

The Steps of Solving Group Decision Problems Based on GLTPA and DGGLTWBM Operators are listed as follows.

*Step 1. *Obtain the grey linguistic 2-tuple decision matrices.

*Step 2. *Utilize (17) to calculate the value of .

*Step 3. *Employ the GLTPA operator in (18) to fuse all individuals’ values, which are shown as follows:

*Step 4. *Utilize the GRA method to calculate the attribute weights.
(a)Determine the PIS and NIS with grey linguistic 2-tuple variables, respectively
where and .(b)Calculate the grey relational coefficient of each alternative, respectively
where , , and the identification coefficient . The distance measure is defined in Definition 5.(c)Utilize linear programming model to elicit the optimal attribute weights under grey linguistic 2-tuple variables
where represents a set of the attribute weight information. According to the abovementioned model, we can receive the optimal attribute weights .

*Step 5. *Utilize the DGGLTWBM operator to aggregate all the performances of each alternative and get the overall terms.

*Step 6. *The overall preference values of the alternatives are ranked in a descending order with the aid of Definition 3, and the best alternative can be identified.

#### 5. Exemplification and Comparison Analysis of the Designed Method

In this section, we employ an example [22] to illustrate the applicability of the designed collaborative decision-making scheme. Suppose that a company wants to procure barrels of Brent oil, and six suppliers serve as alternatives involving competition. The purchasing company invites three experts with prioritization relationship to evaluate the possible alternatives with respect to six attributes [41, 42], i.e., price, delivery time, quality, supplier reputation, carbon emission, and level of environmental management. The constraint conditions of attribute weight information are assigned in advance as below:

Experts take elements of the linguistic set to evaluate the corresponding six attributes of six alternative suppliers, respectively.

##### 5.1. The Detailed Decision-Making Steps

*Step 1. *Obtain the decision information matrices , which are described in Table 1, Table 2 and Table 3, respectively.

*Step 2. *From (17), are calculated as follows.

*Step 3. *Group aggregative values are obtained by means of the GLTPA operator, which are listed in Table 4.

*Step 4. *(a)Determine the PIS and NIS with grey linguistic 2-tuple variable, respectively
(b)Calculate all grey relationship coefficient matrices
(c)Utilize an optimization model to elicit the exact attribute weights of the grey linguistic 2-tuple variables
By solving the above model, we can get , , , , , and .

*Step 5. *Utilize the DGGLTWBM operator to obtain the overall performances of each alternative .

*Step 6. *According to Definition 3, we can rank the overall preference values and we can get the best alternative .

Firstly, in order to analyze the influence of on the ranking results, we take some values of the parameter vector of the DGGLTWBM operator to obtain the corresponding results. From Table 5, we can observe that the final rankings of alternatives are different but the best one keeps the same, which can reflect the conclusions in Remarks 1–3. Therefore, the DGGLTWBM operator with parameter vector can provide decision-makers more flexible choices by changing parameter vector .

What is more, we set the prioritization relationship of experts to and other conditions remain unchanged; the ranking result of the alternatives is obtained, which keeps the same as before. When taking some values of the parameter vector of the DGGLTWBM operator, we obtain the corresponding results, which are shown in Table 6. As can be seen from Table 6, there is minor difference between the last two alternatives in the first two lines compared with Table 5.

Attribute weights play the role of controlling the support degrees of attributes for aggregation result. To some extent, increasing or decreasing the attribute weight information range has an important influence on the ranking result. For example, when we reset the attribute weight information , the ranking result becomes . There exists a major difference between the ranking results under different attribute weight information ranges.

##### 5.2. Comparison of the Proposed Method with Other Method

In order to better illustrate the practicality and effectiveness of the proposed method, we use the GLTWA operator conceived in [22] to cope with the same problem, in which the attribute weights and expert weights are given by accurate numbers in advance and the ranking result is . In our method, the experts’ weights are figured out by prioritization relationship. And the attribute weights, which are partially known, are calculated by the linear programming model. It is worth mentioning that we consider the interrelationship among any attribute subsets by using the DGGLTWBM operator.

In addition, we can get some ranking results to consider different aggregation requirements by changing experts’ prioritization relationship or adjusting the parameter vector values of the DGGLTWBM operator. When we consider experts’ prioritization relationship and choose or we consider experts’ prioritization relationship and choose or or , the results of two types of methods are completely identical, . These discussions elucidate the practicality and robustness of the designed decision-making method.

##### 5.3. Comparison Analysis under Different Types of Information Fusion

Aggregation method can be taken different ways in managerial decision process. Some literatures further studied these aggregation approaches, including classifying and comparing them [43, 44]. In some decision-making models, the maximum and minimum values of experts’ evaluation information are required. In different aggregation scenarios, these ideal points have a different form such as a point, a vector, or a matrix [45, 46]. In this paper, we explore different sequences of aggregation sources under different forms, which are designed as follows:

The first aggregation scenario has already been explored in this paper. In what follows, we focus on the second aggregation scenario with comparison analysis on the same example [22].

*Step 1. *Establish the decision matrices .

*Step 2. *(a)Determine the PIS and NIS with grey linguistic 2-tuple variables of decision-makers. and represent the PIS and NIS of the -th decision-makers, respectively
(b)Calculate grey relationship coefficient matrices of all experts