Linguistic Weighted Aggregation under Confidence Levels

1College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China 2College of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, China 3Center for Applied Statistics, Renmin University of China, Beijing 100872, China 4College of Computer and Information, Zhejiang Wanli University, Ningbo 310015, China 5Research Institute of Economic and Social Development, Zhejiang University of Finance and Economics, Hangzhou 310018, China


Introduction
Information aggregation is a technique that analyzes the information in order to provide a final result.A variety of aggregation operators have been developed in the past few decades.One of the most common aggregation operators is the weighted average (WA) operator [1].Another interesting one is the ordered weighted averaging (OWA) operator, originally introduced by Yager [2].Its main advantage is that it provides a parameterized family of aggregation operators between the minimum and the maximum [3].Since their introduction, many new extensions of the WA and OWA operator have been proposed, such as the weighted OWA (WOWA) operator [4], the OWA weighted averaging (OWAWA) operator [5], the induced ordered weighted averaging (IOWA) operator [6], the generalized OWA (GOWA) operator [7], the induced generalized OWA (IGOWA) operator [8], the power ordered weighted average (POWA) operator [9], and the continuous generalized ordered weighted averaging (CGOWA) operator [10].
Due to time pressure and decisions maker's limited expertise related with the problem domain, decision information about alternatives is often uncertain or fuzzy.As a result, it is more suitable to provide preferences by using linguistic variables rather than numerical ones.The use of the linguistic variable provides a direct way to assess vague environments where the information is provided by using expressions such as low, high, good, or bad [11,12].In order to aggregate the linguistic information, many linguistic aggregation operators have been developed, such as the linguistic weighted ordered weighted averaging (LOWA) operator [13,14], linguistic ordered weighted geometric averaging (LOWGA) operator [15], the induced linguistic generalized ordered weighted averaging (ILGOWA) operator [16], the linguistic generalized ordered weighted averaging (LGOWA) operator [17], and the linguistic generalized power ordered weighted average (LGPOWA) operator [18].Recently, some new aggregation operators are developed to aggregate linguistic information.For example, Zhou and Chen [19] developed the induced linguistic continuous ordered weighted geometric (ILCOWG) operator, which is very suitable for group decision-making (GDM) problems taking the form of uncertain multiplicative linguistic preference relations.Wei et al. [20] developed a new aggregation operator called the belief structure generalized linguistic hybrid averaging (BS-GLHA) operator.Wang et al. [21] developed some linguistic cloud aggregation operators including the cloud weighted arithmetic averaging (CWAA) operator, cloud ordered weighted arithmetic averaging (COWA) operator, and cloud hybrid arithmetic (CHA) operator.Based on probabilistic information and induced aggregation operators, Merigó et al. [22] developed the induced linguistic probabilistic ordered weighted average (ILPOWA), which uses probabilities and OWA operators in the same formulation considering the degree of importance that each concept has in the formulation.
Most of the existing linguistic aggregation operators do not consider the confidence level of the aggregated arguments provided by the decision makers.However, in many real decision-making problems, such as the blind peer review of doctoral dissertation in China, the evaluation experts are requested to provide two types of information such as the performance of the evaluation objects and the familiarity with the evaluation areas (called confidence levels) [23,24].To overcome this issue, Yu [24] and Xia et al. [23] developed some induced aggregation operators under confidence levels, which can take into account the confidence levels of the aggregated arguments, while after reviewing the existing literature, it seems that there is no investigation on linguistic information aggregation under belief levels, which is an interesting and important issue.In this paper, we focus on the linguistic information aggregation issue in the situation where the confidences levels of the aggregated arguments are asked to be considered, and we develop a series of linguistic aggregation operators considering the confidence levels of the aggregated arguments, such as the confidence linguistic weighted averaging (CLWA) operator, the confidence linguistic ordered weighted averaging (CLOWA) operator, and the confidence generalized linguistic ordered weighted averaging (CGLOWA) operator.We also apply the developed operators to decision-making with linguistic information.Finally, an illustrative example has been given to show the developed method.
This paper is organized as follows.First, we briefly review some basic concepts such as the linguistic approach to be used throughout the paper, the LWA, and the LOWA operator.Second, we present the CLWA, CLOWA, and CGLOWA operator.Third, we discuss the applicability of the CGLOWA operator with a multicriteria decision-making example and we end the paper summarizing the main conclusions.

Preliminaries
This section briefly reviews the linguistic approach, the linguistic weighted average, and the linguistic OWA operator.

Linguistic Approach.
The linguistic approach is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables.For computational convenience, let  = {  |  = 1, 2, . . ., } be a finite and totally ordered discrete term set, where   represents a possible value for a linguistic variable.For example, a set of nine terms  could be given as follows: Note that EL means extremely low, VL very low, L low, LM low-medium, M medium, MH medium-high, H high, VH very high, and EH extremely high.Usually, in these cases, it is required that in the linguistic term set there exist the following: (i) a negation operator: Neg(  ) =   such that  = +1−; (ii) the set ordered   ≤   if and only if  ≤ ; In order to preserve all the given information, Xu [25] extended the discrete term set  to a continuous term set Ŝ = {  |  ∈ [1, ]}, where if   ∈ , then we call   the original term, and otherwise, we call   the virtual term.In general, the decision maker uses the original linguistic terms to evaluate alternatives, and the virtual linguistic terms can only appear in the actual calculation [26].
Consider any two linguistic terms   ,   ∈ Ŝ and  > 0; we define some operational laws as follows: Note that this model is very useful for computing with words because it is very easy to use and it follows a similar methodology as the numerical information.

2.2.
The Linguistic Weighted Average.The linguistic weighted average (LWA) [14] is a linguistic aggregation operator that uses the weighted average under uncertain environments assessed with linguistic information.It is defined as follows.
Definition 2. A LOWA operator of dimension  is a mapping LOWA: Ŝ → Ŝ, which has an associated weighting vector  with   ∈ [0, 1] and where    is the th largest of the    .
The LOWA operator has been also extended under different frameworks including the use of Dempster-Shafer theory [20,39], induced aggregation operators [16,26,40], generalized aggregation operators [17,41], distance measures [42], and power aggregation operators [18].However, most of the existing linguistic aggregation operators do not consider the confidence level of the aggregated arguments provided by the information providers.Therefore, in the next section, we will develop some new linguistic aggregation operators, which can take into account the confidence levels of the aggregated arguments.

Linguistic Information Aggregation Operators under Confidence Levels
In some real decision-making problems, the evaluation experts are requested to provide two types of information such as the performance of the evaluation objects and the familiarity with the evaluation areas (called confidence levels).In this Section, we investigate the linguistic information aggregation methods under confidence levels and proposed a series of new linguistic aggregation operators.

Confidence Linguistic Weighted Averaging (CLWA) Operator.
In the following, we propose the confidence linguistic weighted averaging (CLWA) operator and study the desirable properties of the proposed operator.The definition of the CLWA operator is given as follows.
Definition 3. A CLWA operator of dimension  is a mapping CLWA:   × Ŝ → Ŝ, which has an associated weighting vector  with   ∈ [0, 1] and where   is the belief levels of linguistic variable    , 0 ≤   ≤ 1.
In the following example, we present a simple numerical example showing how to use the CLWA operator in an aggregation process.
The CLOWA operator is a mean or averaging operator.This is a reflection of the fact that the operator is monotonic, idempotent, bounded, and commutative.These properties are proven in the following theorems.

Confidence Generalized Linguistic Ordered Weighted
Averaging (CGLOWA) Operator.The confidence generalized linguistic ordered weighted averaging (CGLOWA) operator is an extension of the OWA operator that uses the main characteristics of both the CLOWA and the GOWA operator.Then, we can obtain a generalization that includes the CLOWA operator and many other situations, such as confidence linguistic ordered weighted geometric (CLOWG) operator and confidence linguistic ordered weighted harmonic averaging (CLOWHA) operator.It can be defined as follows.
Note that the different aggregated results can be obtained if the parameter  takes different values.The selection of the particular parameter  depends on the particular interest of the decision maker in the specific problem considered.
Similar to the CLOWA operator, the CGLOWA operator is also monotonic, idempotent, bounded, and commutative.If we analyze different values of the parameter , we obtain a group of particular cases.For example, we have the following.

An Application of the Proposed Operator to Multicriteria Decision-Making
The CGLOWA (or CLOWA) operator is applicable in a wide range of situations, such as decision-making, statistics, engineering, and economics.In summary, all of the studies that use the OWA operator can be revised and extended by using this new approach.
In the following, we are going to develop a decisionmaking method about the use of the CGLOWA in a multicriteria decision-making problem.For a multicriteria decisionmaking problem, let  = { 1 ,  2 , . . .,   } be a discrete set of alternatives and let  = { 1 ,  2 , . . .,   } be the set of criteria.The main steps of the decision-making methods are as follows.
Step 1.The decision makers provide their evaluations and belief levels about the alternative   under the attribute   , forming the decision matrix  = (  ) × , where   ∈  is a preference value, which takes the form of linguistic variable.
Example 13.Let us consider a blind peer review of doctoral dissertation evaluation problem in China's university (adapted from [24]).In many Chinese universities, the doctoral dissertation will be reviewed by several experts anonymously.And they will review dissertation according to five criteria, including topic selection and literature review ( 1 ), innovation ( 2 ), theory basis and special knowledge ( 3 ), capacity of scientific research ( 4 ), and theses writing ( 5 ).Unlike many existing evaluation methods, the experts not only required to provide the evaluation results of the doctoral dissertation but also asked to give the degrees to which they are familiar with the research topics (called belief levels).Suppose there are five declarations that need to be reviewed by expert; the degree of familiarity of the five declarations provided by expert is   = (0.8, 0.7, 0.9, 0.8, 0.6).Suppose the experts give the decision matrix under a linguistic framework of nine linguistic terms in the set as explained in Section 2.1, shown in Table 1.
Suppose that the weighting vector associating with the CGLOWA operator is  = (0.11, 0.24, 0.30, 0.24, 0.11), which is derived by using the normal distribution based method [1].With this information, it is possible to aggregate the available information in order to take a decision.By using some key particular cases of the CGLOWA operators to aggregate the linguistic variables for five declarations, the aggregated results and the ranking of the declarations are shown in Table 2.
As we can see, depending on the aggregation operator used, the results and decisions may be different.Therefore, the decision about which declarations to select may be also different.Note that, in this example, the optimal choice is the same for all aggregation operators.
If we do not consider the confidence levels factor, in other words, all the criteria of the evaluated objects are treated with sure familiar by the decision maker, then our proposed CGLOWA operator is reduced to the existing GLOWA operator [17].The aggregated results of this example by the types of the GLOWA operators, such as linguistic OWA (LOWA) operator, the linguistic OWG (LOWG) operator, the linguistic OWHA (LOWHA) operator, and the linguistic ordered weighted quadratic averaging (LOWQA) operator, are listed in Table 3.
It can be easily seen that the ranking of the five doctoral dissertations obtained by the proposed CLGOWA operator is similar to the result by the LGOWA operator.However, in real-life decision process, the decision maker(s) may not be familiar with the doctoral dissertation absolutely.To deal with such situations, the proposed CLGOWA operator is useful tool.From the above analysis, we can see that the main advantage of using the CGLOWA operator is that it can consider the belief levels of the decision maker(s).Another main advantage is that it includes a wide range of particular cases such as the CLOWA, the CLOWQA, and the CLOWG operator.Due to the fact that each particular family of CGLOWA operator may give different results, the decision maker(s) will select for his decision the one that is closest to his interests.

Concluding Remarks
In this paper, we have developed some new linguistic information aggregation operators by introducing the belief levels.The confidence linguistic weighted averaging (CLWA) operator, the confidence linguistic ordered weighted averaging (CLOWA) operator, and the confidence generalized linguistic ordered weighted averaging (CGLOWA) operator have been introduced.We have studied various properties of the developed operators.We have also presented an application of the CGLOWA operator to a multicriteria decision-making problem concerning the dissertation evaluation problem.We have seen that the CGLOWA is very useful because it represents very well the uncertain information by using linguistic labels, as well as considering the belief levels of the decision makers.Moreover, it includes a wide range of particular cases such as the CLOWA, the CLOWQA, and the CLOWG operator.Due to the fact that each particular family of CGLOWA operator may give different results, the decision maker gets a more complete view of the decision problem and is able to select the alternative that is closest to his interests.
It is worth pointing out that we can extend the confidence aggregation, using a similar method, to deal with the other fuzzy situations where the arguments are expressed in interval values or triangular fuzzy values.In future research, we expect to develop further extensions by adding new characteristics in the problem such as the use of inducing variables or probabilistic aggregations.

Table 1 :
Linguistic fuzzy decision making metric.