Research Article | Open Access

# Extended Heronian Mean Based on Hesitant Fuzzy Linguistic Information for Multiple Attribute Group Decision-Making

**Academic Editor:**Ludovico Minati

#### Abstract

This paper investigates the hesitant fuzzy linguistic multiple attribute group decision-making (MAGDM) problem with the heterogeneous relationship among the attribute variables that cannot be solved by most existing decision-making methods. To address this problem, a new operator is introduced based on partitioning attribute variables into different sets according to their interrelationship. This operator is called the extended Heronian mean (EHM) operator. To obtain each expert’s comprehensive values of the alternatives in the hesitant fuzzy linguistic MAGDM problem, we investigate the EHM operator under a hesitant fuzzy linguistic environment and propose the hesitant fuzzy linguistic EHM operator and the hesitant fuzzy linguistic linear support degree weighted EHM operator. In addition, the axiom definition of a linguistic type similarity measure of hesitant fuzzy linguistic term sets is proposed. The weight of the experts can be determined based on this type similarity measure. Finally, a practical case is presented to demonstrate the steps of our method, and a comparison analysis illustrates its feasibility and effectiveness.

#### 1. Introduction

In such a complicated and changeable economic and social environment, it is difficult for a single expert to fully understand and master the decision-making problem. Therefore, it is necessary that experts from different disciplines take part in the decision-making process, which evolves the process into multiple attribute group decision-making (MAGDM). MAGDM has been applied in many fields, such as academic assessments of higher education institutions [1], financial risk evaluation [2, 3], investment objective selection [4], and green supply chain management [5]. The decision-maker may use qualitative values instead of quantitative values to express his/her assessment information due to characteristics of decision-making objects, information integrality cannot be guaranteed, and human knowledge is limited. For example, when evaluating the degree of economic activity in one location, people may use linguistic terms such as “low,” “medium,” and “high.”

A reasonable selection of the means of information representation is essential to solving linguistic MAGDM problems. In early research, fuzzy linguistic term sets (FLTS) were commonly used to represent decision-making information and were described and analyzed by Zadeh [6]. Based on FLTS, many other types of information representation models have been proposed in recent decades, such as the 2-tuple fuzzy linguistic representation model [7], the linguistic intuitionistic fuzzy set [8, 9], the uncertain linguistic fuzzy soft set [10], the interval neutrosophic uncertain linguistic model [11], the probabilistic linguistic term set [12], and picture 2-tuple linguistic set [13]. However, when experts vacillate among several linguistic terms, it is quite difficult for them to model such situations by the abovementioned representation models. Therefore, Rodriguez et al. [14] presented hesitant fuzzy linguistic term set (HFLTSs) which can reflect the hesitant psychology feature of the decision-maker. Hesitant fuzzy linguistic decision-making has received much attention by many scholars and has produced a large amount of research results [15–17].

Similarity measure is a very important research topic in the field of linguistic decision-making. It has been widely used in many fields, such as semantic translation, cluster analysis, and machine learning. For example, Chen et al. [18] introduced a new similarity measure formula for interval linguistic terms based on the likelihood of the comparison between two intervals. For the trapezoid fuzzy linguistic variables, Xu [19] put forward the similarity measure calculation method. Based on Xu’s work, Liao et al. [20] further proposed the axioms of similarity measures for HFLTSs and then used them to sort alternatives in multiple attribute decision-making (MADM). To enrich the hesitant fuzzy linguistic similarity measure calculation, Hesamian et al. [21] proposed Gower-Legendre and Tversky similarity measures, Liao et al. [22] introduced cosine similarity measures, Gou et al. [23] developed some cross-entropy measures, Farhadinia et al. [24] defined some entropy measures, and Song et al. [25] proposed vector similarity measures.

Information aggregation is also an important component of linguistic decision-making. An operator is a primary tool for linguistic information aggregation, which is widely accepted and used in practical decision-making. In earlier research, linguistic aggregation operators were made based on the assumption that the input arguments are independent of each other, such as the linguistic geometric averaging operator [26], the linguistic hybrid geometric operator [27], and the uncertain linguistic ordered weighted averaging operator [28]. However, in recent years, scholars have gradually realized the significance of considering the associated relationship in information aggregation [29, 30]. The Heronian mean (HM) operator and Bonferroni mean (BM) operator can capture the interrelationship of the aggregation variables through the crossover operation of aggregation variables. Wei et al. and Tian et al. investigated a weighted BM operator under uncertain linguistic fuzzy environment [31] and simplified neutrosophic linguistic environment [32], respectively. Nie et al. proposed a Pythagorean fuzzy partitioned normalized weighted Bonferroni mean [33]. Liu et al. [34] analyzed the structure principle of the BM operator and HM operator and then pointed out that the HM operator can avoid the disadvantage of redundant information and make the information fusion more efficient. Li et al. [35] developed some 2-tuple linguistic HM aggregation operators and studied their properties. Liu et al. [36] proposed some new HM operators, based on the HM operator and the geometric HM operator, to solve intuitionistic uncertain linguistic MAGDM problems. Yu et al. [37] defined some of the reducible weighted linguistic hesitant fuzzy HM operators and then discussed their special cases and desirable properties.

Analyzing the existing research literature on linguistic similarity measures and linguistic aggregation operators shows that the following problems can be found:

(1) Most of the existing studies on the similarity measures of linguistic fuzzy sets use a crisp number to measure the degree of similarity. Whether this numerical value measurement is appropriate for linguistic input arguments is worth discussing because it conflicts with the motivation for using linguistic fuzzy set in decision-making that is close to the cognitive and understanding structure of people [38]. For example, consider a medical and health institution that wants to develop an online medical diagnosis system (OMDS) to provide the results of identified illnesses in an easily understandable way. The OMDS may suggest “similar,” “moderately similar,” and “high degree of similarity” as description forms. Thus, it is necessary to carry out research on the linguistic similarity measures of HFLTS to enrich and develop the theories and means of hesitant fuzzy linguistic decision-making.

(2) Although existing HM operators consider the interrelationship of the argument variables, they are built on the assumption that a relationship exists between any two input arguments. However, in some real-world problems, this assumption is not always true. For example, consider an international company selecting a suitable department manager according to the following attributes: : Educational background; : Work experience; : Professional knowledge; : Working skill; : Age; and : Health. Based on their interrelationship, the attributes can be divided into two classes: and . It is easy to see that the elements in the same set are connected, but there is no relationship between the elements from different sets. Therefore, if we use the existing HM operators, they would generate disturbing information in the process of aggregation which would affect the reliability of decision-making.

In accordance with the above analysis, this paper begins by providing the axiom definition of linguistic similarity measures for HFLTS and then proposes a new operator under the inspiration of previous studies. This new operator is called extended Heronian mean (EHM). Next, we investigate the EHM operator in a hesitant fuzzy environment and propose some operators to infuse HFLTSs. Based on these operators and linguistic similarity measures, a new method for solving the hesitant fuzzy linguistic MAGDM problem is designed. Weights that are completely unknown or represented by a crisp numbers (linguistic terms) situation are taken into consideration in the decision-making process. This makes the new method more practical. The framework of this article is as follows:

Section 2 reviews some concepts of HFLTSs and proposes linguistic similarity measures for HFLTS and EHM operators. Section 3 introduces the HFLEHM and HFLLSDWEHM operators and investigates their properties. Section 4 designs a group decision-making method under hesitant fuzzy linguistic environments. Section 5 provides a practical example to illustrate the method and discuss the effect of parameters on decision-making results. Section 6 illustrates the feasibility and effectiveness of the proposed operators and method through comparison and analysis. Lastly, Section 7 summarizes this study.

#### 2. Preliminaries

##### 2.1. Hesitant Fuzzy Linguistic Term Set

To build a linguistic decision model, a set of evaluation scales can be built as , where denotes a linguistic measurement level of evaluation objectives, and is an even number which always takes the values of 4, 6, and 8 [26]. This requires the following properties to be satisfied: (1) and (2) . To make the linguistic decision model have an obvious logicality and system, Xu extended the above discrete evaluation scales set to a continuous set and defined the following operational laws [39]:

*Definition 1. *Let ; then, the following are given.

(1) ; (2)

(3) ; (4)

With the decision environment becoming more and more complex and uncertain, this kind of linguistic term sets is difficult to satisfy with the request of decision modeling. Because of this, Rodriguez et al. [14] proposed the HFLTS, and then Liao et al. [20] further put forward its mathematical form.

*Definition 2 (see [20]). *Let be a universe of discourse, and let be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS) on X is characterized by the following:where is a subset of , and denotes the number of linguistic terms in . For convenience, we call a hesitant fuzzy linguistic element (HFLE), denoted by .

*Definition 3 (see [20]). *Let be a linguistic term set, and let be a HFLE. is called the score of , and is called the deviation degree of . Then, the comparison law between two HFLEs ( and ) can be defined as follows:(a)If , then (b)If , then(i)if , then (ii)if , then

There are also other comparison rules of HFLEs in the literature. For example, Liu et al. proposed a comparison rule between HFLEs based on a linguistic scale function [40]. Wei et al. introduced a novel comparison rule between HFLEs, which takes into account the average linguistic term and the hesitant degree [41]. Motivated by the possibility degree of intervals, Lee et al. introduced a likelihood based comparison rule between HFLEs [42]. In consideration of both the hesitant degrees and the unbalanced linguistic terms in evaluations, Liao et al. introduced a new comparison rule between HFLEs based on the psychological characteristics of experts [43]. On the basis of pairwise comparison of each linguistic term in the two HELEs, Huang et al. offered a comparison rule between HELEs [44]. For details, please refer to the survey papers [45]. However, the comparison rule proposed by Liao et al. [20], as shown in Definition 3, is simple and convenient and is thus widely used in decision-making.

##### 2.2. A New Similarity Measure for HFLTSs

The similarity measure is an important tool for studying and applying linguistic fuzzy sets. Liao et al. proposed the numeric axiom definition of similarity measures of HFLTSs [22]. Based on this, we propose a linguistic axiom definition as follows:

*Definition 4. *Let and be two linguistic term sets, and and are two HFLTSs on ; then the linguistic similarity measure between and is defined as , which satisfies the following:

(1) ; (2) iff; (3) .

*Remark 5. * denotes a linguistic similarity measure level and and represent the maximum and minimum linguistic similarity measure levels, respectively. Moreover, must have the same number of elements as .

*Definition 6. *Let and be two linguistic term sets, and are two HFLTSs on , , and . Then is called the linguistic similarity measure between and .

*Remark 7. *If , then the linguistic similarity measure between two HFLEs, and , is defined as .

*Example 8. *Let , , and be three HFLEs on . can be built as extremely far, far, slightly far, medium, slightly close, close, and extremely . According to the definition of linguistic similarity measure of HFLTSs, we obtain the following:This means that the similarity degree between and is higher than close, and the similarity degree between and is close. Therefore, .

Utilizing the similarity measure defined by Hesamian et al. [21], we can then obtain the following: which means that .

Utilizing the similarity measure defined by Liao et al. based on the Hamming distance [20], we can then obtain the following:which means that .

From the above calculation results, we can see that the method proposed in this paper and the method given by Liao et al. have a high degree of discrimination. However, the representation of similarity degree of the two methods is quite different. One is a quantitative representation and the other is a qualitative representation. Users should reasonably choose according to the application environment.

##### 2.3. Extended Heronian Mean Operator

*Definition 9 (see [34]). *Let , , and be nonnegative real numbers. Then is called Heronian mean (HM).

The HM operator is based on the assumption that a correlation exists between any two input arguments. However, in some real-world applications the above assumption does not hold because a partial correlation may exist between input arguments. Hence, an EHM is introduced to provide more precise aggregate information.

*Definition 10. *Let , , and be nonnegative real numbers. Then, the extended Heronian mean (EHM) operator is defined as follows:where set consists of some elements of which have a correlation with , and represents the cardinality of .

*Remark 11. *If , that is to say, all input arguments are independent, then the EHM operator reduces to the following:which is called generalized arithmetic averaging (GAA) operator.

*Remark 12. *If , that is to say, all input arguments are related to each other, then EHM operator reduces to the HM operator.

Obviously, the EHM operator has the following properties:

(1) If , then

(2) If , then

*Remark 13. *From properties (1) and (2), we can get

In the following section, we further extend the EHM operator to the hesitant fuzzy linguistic environment and discuss its relevant properties.

#### 3. Hesitant Fuzzy Linguistic Extended Heronian Means

##### 3.1. HFLEHM Operator

*Definition 14. *Let be a collection of HFLEs, for any ,We call a hesitant fuzzy linguistic extended Heronian mean (HFLEHM) operator, where set consists of some elements of which have correlation with , and represents the cardinality of .

*Remark 15. *If , that is to say, all are independent, then HFLEHM operator reduces towhich is called hesitant fuzzy linguistic generalized arithmetic averaging (HFLGAA) operator.

*Remark 16. *If , that is to say, all are related to each other, then HFLEHM operator reduces to HFLHM operator.

*Remark 17. *If and , then HFLEHM operator reduces towhich is called hesitant fuzzy linguistic generalized linear decrease weighted averaging (HFLGLDWA) operator, where meet the following conditions: (1) ; (2) ; (3) .

*Remark 18. *If and , then HFLEHM operator reduces towhich is called hesitant fuzzy linguistic generalized linear increase weighted averaging (HFLGLIWA) operator, where meet the following conditions: (1) ; (2) ; (3) .

*Property 19. *Let be a collection of HFLEs; if , then

*Property 20. *Let and be hesitant fuzzy linguistic evaluation values of same attributes, and ; then .

*Property 21. *Let be a collection of HFLEs, , , , and ; then .

##### 3.2. HFLWEHM Operator

It should be noted that the HFLEHM operator considers the aggregate variables to have the same importance. In real decision-making it is difficult to satisfy this condition. To reflect the importance of the aggregate variables in the aggregation operator, two weighted forms of hesitant fuzzy linguistic EHM operators are introduced in this subsection.

*Definition 22. *Let be a collection of HFLEs, with the weight , , . For any ,which is called the hesitant fuzzy linguistic weighted extended Heronian mean (HFLEHM) operator, where and .

It is worth noting that if the importance of the aggregate variables is given by the form of HFLEs, for example, unimportant, unimportant, slightly unimportant, medium, slightly important, important, very , being a collection of HFLEs on , then , , and .

*Remark 23. *If , that is to say, all are independent, then HFLWEHM operator reduces towhich is called hesitant fuzzy linguistic generalized weighted arithmetic averaging (HFLGWAA) operator.

*Remark 24. *If , then

*Property 25. *If , then

*Property 26. *Let be a collection of HFLEs with the weight , , . If , then

The proofs of Properties 27 and 28 are similar to those of Properties 20 and 21; therefore we omit the details.

*Property 27. *Let and be two collections of HFLEs, with the same weight , , . If , , , then .

*Property 28. *Let be a collection of HFLEs with the weight , , . , , , and ; then .

Due to the complexity of the decision-making system, people may encounter a situation wherein the importance of the aggregate variables is unknown. Inspired by the work of Xu and Yager [46], we introduce a new operator to deal with this kind of problem in decision-making.

*Definition 29. *Let be a collection of HFLEs. For any ,which is called the hesitant fuzzy linguistic linear support degree weighted extended Heronian mean (HFLLSDWEHM) operator, where is the support degree of in , , , , and has the following properties. (1) , ; (2) , , if . Similarly, is the support degree of in .

*Property 30. *Let be a collection of HFLEs. If , then .

*Property 31. *Let be a collection of HFLEs, , , , and ; then

We should emphasize that, unlike the HFLWEHM operator, the HFLLSDWEHM operator does not have the monotonicity property.

#### 4. An Approach to MAGDM with Hesitant Fuzzy Linguistic Information

The critical components of solving MAGDM problems include attribute weights, expert weights, and attribute evaluation values. In this section, we utilize the proposed operators to solve MAGDM problems in which attribute weights are known or unknown, and the evaluation values are expressed by HFLTSs.

For a MAGDM problem, assume that are decision experts with the weight vectors and is a finite set of alternatives, and is a set of attributes, whose weight vector is . Set is constructed using the elements of which have a correlation with and . Suppose that the decision experts provide the hesitant fuzzy linguistic information to evaluate the characteristics of the alternatives under attribute , denoted by a HFLE ; then the hesitant fuzzy linguistic decision matrix is constructed.

An approach to multiple attribute group decision-making problems is provided in the following steps:

*Step 1 (normalize the evaluation information matrices). *For benefit attributes, higher evaluation values indicate the better alternative. However, for cost attributes, smaller evaluation values indicate the better alternative. All attributes of the alternative should have the same judgment standard in decision-making [47]. Therefore, decision matrix is transformed into the normalized decision matrix , where , for benefit criteria ; , for cost criteria , ; ; and .

*Step 2. *Calculate each expert’s comprehensive values of the alternatives.

*Case 1. *If the weighting vectors of the attributes are known, then the HFLWEHM operator is used to calculate the