Hesitant Fuzzy Decision-Making Method Based on Correlation Coefficient under Confidence Levels with Application to Multisensor Electronic Reconnaissance

In this paper, we focus on the hesitant fuzzy decision-making method based on correlation coefficient under confidence levels. Firstly, we propose several correlation coefficients based on confidence levels, and based on the correlation coefficient between attribute-evaluated values and the ideal values, several optimal attribute weights models are constructed. Secondly, we have defined the concepts of module, weight module, projection, and weighted projection in hesitant fuzzy sets on account of the projection theory. Finally, on the strength of our research results above, we construct a hesitant fuzzy multiattribute decisionmaking method and apply it to the multiattribute decision-making problem in multisensor electronic reconnaissance. Comparative analyses are made via simulation to test the rationality and validity of the proposed method.


Introduction
In the field of multiattribute decision-making, the decisionmaking process is complex and confuses researchers. Due to different knowledge backgrounds and thinking patterns of decision makers, the degree of membership may vary for the same elements and collections. For example, some people give 0.5, other people give 0.6 or 0.7. Especially, all decision makers insist on their own opinions, and they are deadlocked. To deal with this complex situation, the concept of hesitant fuzzy sets was proposed by Torra et al. [1,2], which allowed the membership of an element in a given set to be different values. Xu and Xia [3,4] proposed multiple distance measures for hesitant fuzzy sets. Farhadinia [5] studied the relationship between entropy, similarity measure and distance measure of hesitant fuzzy sets (HFSs), and intervalvalued hesitant fuzzy sets (IVHFSs). Zhu et al. [6] and Yu et al. [7] proposed dual hesitant fuzzy sets (DHFSs) and researched the basic operation and characteristics of DHFs. e problem of fuzzy multiattribute group decision-making (MAGDM) with different priorities of attribute and expert was studied by Wei [8]. Yu [9] proposed a hesitant fuzzy multicriteria group decision-making method for performance and development review (PADR) evaluation. Farhadinia [10,11] presents a series of scoring functions for hesitant fuzzy sets (HFSs) and extended the hesitant fuzzy set (HFS) to its higher order type. Rodriguez et al. [12] present a marvellous overview on hesitant fuzzy sets.
Many classic decision-making methods have been extended to uncertain environments, and some fuzzy multiattribute decision-making methods have been obtained; attribute weights play an important role in summarizing expert assessments in the decision-making process and attracts many researchers' attention. For example, Xu [13] studied some of the group decision-making problems concerning attribute weight information. Park [14] investigated the group decision-making problems, in which all the information provided by the decision makers is presented in the form of interval-valued intuitionistic fuzzy decision matrices. And fuzzy TOPSIS method under incomplete weight information [15][16][17], fuzzy ELECTRE II methods [18], fuzzy PROMETHEE method [19], fuzzy VIKOR method [20,21], fuzzy QUALIFLEX method [22], fuzzy TWO-SIDED MATCHING model [23], etc. It can be said that the research on fuzzy decision-making methods has formed a relatively complete system, which can basically cover decision-making problems in fuzzy environments. However, among these methods, the research on the hesitant fuzzy decision-making method under confidence levels has just started, and the content is less.
In many practical decision-making problems, the evaluation experts are required to provide two types of information such as the performance of the evaluation objects and the familiarity with the evaluation fields (called confidence levels). In this paper, we propose several correlation coefficients based on confidence levels. en, we have defined the concepts of module, weight module, projection, and weighted projection in hesitant fuzzy sets under confidence levels on account of the projection theory. Finally, we construct a hesitant fuzzy multiattribute decision-making method and apply it to the multiattribute decision-making problem in multisensor electronic reconnaissance.

Preliminaries
Zadeh [24] first attempted to use membership function for uncertain information. However, in the decision-making process, because there is a lot of uncertain information, experts usually hesitate for one thing or another, and it is difficult to reach a final consensus, while hesitant fuzzy sets can effectively solve such problems.
Definition 1 (see [1,2]). Let X � x 1 , x 2 , . . . , x n be a fixed set; then, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a sunset of [0, 1].
For ease of understanding, Xia and Xu [25] represent the HFS with a mathematical symbol: where h E (x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set E. For convenience, Xia and Xu [25] call h � h E (x) a hesitant fuzzy element (HFE) and H the set of all the HFEs. In many practical decision-making problems, most existing hesitant fuzzy aggregation operators do not take into account the confidence level of aggregation parameters provided by information providers. Xia and Xu [26] propose a hesitant fuzzy aggregation operator considering absolute confidence levels.

Models and Method for Determining the Attribute Weights
In a multiattribute decision-making, the expert group is sometimes asked to provide the weights for selected attributes. At all events, the attribute weights are always incomplete because of the short of knowledge and data and the limitation of decision makers' expertise and time. erefore, how to carry out the alternatives sorting by the known information is an interesting and important question. In this section, we propose a method based on the correlation coefficients of HFEs.

Correlation Coefficients of HFEs.
We first introduce the concept of correlation coefficient, then give several correlation coefficient formulas, and discuss their properties.
Definition 4 (see [25]). For two HFEs h 1 and h 2 , the correlation coefficient of h 1 and h 2 , denoted as C(h 1 , h 2 ), should satisfy the following properties: In most cases, k h1 ≠ k h 2 , and for convenience, let k � max k h1 , k h 2 . For proper operation, when comparing them, extend the shorter one to the point where both have the same length. e best way to extend the shorter one is to add a value to it. In fact, we can extend shorter values by adding shorter ones, depending on the DM's risk appetite. e optimists expect the desired outcome and may increase the maximum, while the pessimists expect the adverse outcome and may increase the minimum. Based on Definition 4, in [4], the following correlation coefficients for HFEs are given: where h are the jth largest values in h 1 and h 2. We give another correlation coefficient for HFEs: Although all three formulas satisfy the properties of Definition 4, they each have their own characteristics. We will discuss this issue below.

Theorem 1. Let h 1 and h 2 be two HFEs; then,
is completes the proof of eorem 1. Considering the importance of the confidence levels, we propose several hesitant fuzzy correlation coefficients based on relative confidence levels and absolute confidence levels and discuss their relationship.

Definition 5.
Let h 1 and h 2 be two HFEs, k � max(k h 1 , k h 2 ) , and k h 1 and k h 2 be the length in h 1 and h 2 . en, we can construct several correlation coefficients under confidence levels for HFEs: where h

Theorem 2.
Let h 1 and h 2 be two HFEs, then is, the better the alternative A i will be. erefore, a reasonable weight vector w * � (w * 1 , w * 2 , . . . , w * n ) should be determined so that all the correlations are as bigger as possible, which means maximizing the following correlation vector [27][28][29].
under the condition w ∈ H, where H is the set of the known weight information.
To this end, we set up the following multiobjective optimization model.
Model 2 can be easily solved by the single-objective programming model; then, we can get the optimal solution w * � (w * 1 , w * 2 , . . . , w * n ), which can be used as the weight vector of attributes. If the information about attribute weights is completely unknown, the following multiobjective programming model can be established.
To solve this model, we construct the Lagrange function: where λ is the Lagrange multiplier. Differentiating equation (16) with respect to w j (j � 1, 2, . . . , n) and λ, set these partial derivatives to zero, which gives the following equation: By solving equation (17), we obtain a simple and precise formula to determine the attribute weights as shown below: 4 Mathematical Problems in Engineering which can be used as the weight vector of attributes. Apparently, w * j ≥ 0, for all j.
If C 1 (a ij , a * j ) is replaced with formula (4), then we have rough the above analysis, we know that, under the hesitant fuzzy information, both Model 2 and Model 3 can be used to determine attribute weights with incomplete weight information. Model 2 can be used when the weight information is incomplete, and the model can be constructed in various forms. Model 3 can be used when the information about attribute weights is completely unknown, which can be solved easily and obtained as a simple and accurate formula for determining the weight of an attribute [29].

Projection Method under Hesitant Fuzzy Environment
Wang [30] proposed a projection method for multiattribute group decision-making, the projection values of evaluation alternatives, and ideal alternatives are used to get the ideal sort. As an excellent measuring tool, the projection method not only reflects the closeness degree between the two objects but also reflects the intrinsic nature of the objects. Now, it is being applied to the decision-making method [31][32][33]. In this section, we focus on the projection method under hesitant fuzzy environment.
Definition 6 (see [30][31][32][33]). Suppose x � (x 1 , x 2 , . . . , x n ) and y � (y 1 , y 2 , . . . , y n ) are two vectors, then cosine of included angle between vectors x and y can be defined as follows: Evidently, cosine of included angle is in this range: 0 < cos(x, y) ≤ 1, and the larger the value of cos(x, y) is, the more the direction of vector x and y is consistent.
Definition 7 (see [30][31][32][33]). Suppose x � (x 1 , x 2 , . . . , x n ), then |x| � ����� � n i�1 x 2 i is a norm of vector x. A vector consists of two parts, the direction and the norm. e cosine of the angle cos(x, y) can only measure whether the direction is consistent, but it cannot reflect the magnitude of norm. In order to consider the norm magnitude and cosine of included angle together, the proximity of two vectors can be measured by the projection value, and the definition of projection is given as follows: Definition 8 (see [30][31][32][33]). Suppose x � (x 2 , x 2 , . . . , x n ) and y � (y 1 , y 2 , . . . y n ) are two vectors, and the projection of vector x onto vector y can be defined: Apparently, the larger the value of Prj y x is, the closer the vector x is to the vector y. For the MADM problem, if the vector y obtains a positive ideal solution and the vector x is used for each alternative, we can obtain the projection of each alternative on the positive ideal solution. e larger the projection value is, the better the alternative is. Now, we extend the projection theory to the hesitant fuzzy sets and give the concepts of the hesitant fuzzy module, weighted module, projection, and weighted projection.
Definition 9 (see [34]). Suppose H � (h 1 , h 2 , . . . , h n ) be a vector of HFSs; then, is called the module of vector H. In many cases, the weight of h j should be considered. For example, in the multiple attribute decision-making, sometimes the evaluation experts are asked to provide the weight of every attribute, so we should define the hesitant fuzzy weighted module.
Definition 11 (see [34] , e 2 , . . . , e n ) be two vectors of HFSs. |E| w is the weighted module of vector E. Let w i and t i be the weight of h i and e i , such that en, the weighted projection of vector H onto vector E can be defined:

An Approach to Multiple Attribute Decision-Making with Hesitant Fuzzy Information under Confidence Levels
Based on the above models, we develop a practical approach to solve the group decision-making problems, where the information about attribute weights is not completely known or completely unknown, and the attribute values take the form of hesitant fuzzy information. e approach includes the following steps: Step 1: we construct the decision matrix A (k) � (a ij ) (k) m×n with confidence levels, where all the arguments a ij (i � 1, 2, . . . , m; j � 1, 2, . . . , n) are HFSs, given by the decision makers, for the alternative A i ∈ A with respect to the attribute G j ∈ G, and the decision makers need to give out the confidence levels l k (0 ≤ l k ≤ 1)(k � 1, 2, . . . , p).
Step 3: we can get the ideal of attribute values A * � a * 1 , a * 2 , . . . , a * n . If the information about the attribute weights is entirely unknown, then we can get the attribute weights w � (w 1 , w 2 , . . . , w n ) T by using Model 3. If the information about the attribute weights is partly known, then we solve model 2 to obtain the attribute weights w � (w 1 , w 2 , . . . , w n ) T .
Step 4: apply formula (25) to calculate the relative weighted projection Prj A * w A i of each alternative A i : Step 5: rank the alternatives A i (i � 1, 2, . . . , m) according to the relative weighted projection Prj A * w A i . e larger the Prj A * w A i is, the better the alternative A i is.

Numerical Example
In this section, we apply the proposed hesitant fuzzy multiattribute decision-making method based on signed correlation to multisensor electronic reconnaissance decision problems. In these problems, multiattribute information about the enemy, such as pulse repetition frequency, carrier frequency, power, and pulse width, are detected using various sensors such as Radar, Electronic Support Measures (ESMs), and Infrared, and multiattribute decisions must be made regarding the kind of airborne platform the enemy is using based on these pieces of attribute information. However, in the actual measurement process under a complex electromagnetic environment, the electronic reconnaissance equipment is often interfered by air clutter and enemy signals, which causes a certain degree of information uncertainty that is in line with the characteristics of HFEs. us, it is more appropriate to describe these attribute decisions using HFE [35].
Assume that four types of airborne platform (i.e., A i (i � 1, 2, 3, 4)) are present based on the information transmitted by each electronic reconnaissance sensor to the fusion center and that each type of airborne platform has four types of attribute, i.e., pulse repetition frequency (G 1 ), carrier frequency (G 2 ), power (G 3 ), and pulse width (G 4 ). e fusion center must perform multiattribute fusion judgment on the four types of airborne platform with uncertainty based on the four attribute types. e electronic reconnaissance equipment measures the value of each attribute G j ∈ G of each scheme A i ∈ A, which forms a matrix based on hesitant fuzzy information. (H � (h ij ) m×n ).
To obtain the type of airborne platform, the specific multiattribute decision-making steps are as follows: Step 1: decision makers assess the alternatives A i (i � 1, 2, 3, 4) with respect to the attributes G j (j � 1, 2, 3, 4), and the decision makers give the confidence levels l p (0 ≤ l p ≤ 1). en, we set up the following hesitant fuzzy decision matrix A (p) � (a ij ) (p) 4×4 , andd the first components in the two types of information are the confidence levels (see Table 1).
Step 3: we can obtain the ideal of attribute values A * � 0.36, 0.39, 0.38, 0.39 { }. Because the information about the attribute weights is entirely unknown, we can get the attribute weights t � (0.24, 0.23, 0.27, 0.26) T by using Model 3.
Step 4: use formula (25) to calculate the relative weighted projection Prj A * w A i of each alternative A i : Step 5: rank the alternatives A i (i � 1, 2, 3, 4) according to the relative weighted projection Prj A * w A i : so the airborne platform type is A 1 .
If we do with the HFWA operator to aggregate the hesitant fuzzy decision matrix without considering the confidence levels, the main steps are as follows: Step 1': decision makers evaluate the alternatives A j ′ (i � 1, 2, 3, 4) with respect to the attributes G j ′ (j � 1, 2, 3, 4) without considering the confidence levels.
Step Step 4': use formula (25) to calculate the relative weighted projection Prj A ′ * w A i ′ of each alternative A i ': Step 5': rank the alternatives A i ′ (i � 1, 2, 3, 4) according to the relative weighted projection Prj A ′ * w A i ′ : so the airborne platform type is A 3 . If we do with the HFGA operator to aggregate the hesitant fuzzy decision matrix without considering the confidence levels, the main steps are as follows: Step 1": decision makers evaluate the alternatives A i ′ (i � 1, 2, 3, 4) with respect to the attributes G j ′ (j � 1, 2, 3, 4) without considering the confidence levels.
Step 4": use formula (25) to calculate the relative weighted projection Prj A ″ * w A i ″ of each alternative A i ″ :    Step 5": rank the alternatives A i ″ (i � 1, 2, 3, 4) according to the relative weighted projection Prj A ″ * w A i ″ : so the airborne platform type is A 3 . If we do not consider the factor of confidence levels, our proposed operators are simplified to the existing hesitant fuzzy aggregation operators. Nevertheless, the experts' group is completely unaware of the alternatives so that the results are different. In view of these situations, the hesitant fuzzy aggregation operators at the confidence levels proposed in this paper are useful tools. As can be seen from the above analysis, compared with the traditional hesitant fuzzy decision-making methods, the main advantages of our methods are not only that they can adapt to the hesitant fuzzy environment but also that they take into account the confidence levels between decision makers, which makes more feasibility and practicality.

Conclusion
In this paper, a new decision analysis method under confidence levels is proposed for decision-making (MADM) problems where attribute values are in the form of hesitant fuzzy numbers and unknown attribute weights. Firstly, several hesitant fuzzy correlation coefficient based on confidence levels are proposed. Secondly, some simple optimization models based on the correlation coefficients under confidence levels for HFEs are used to determine the attribute weight, and the projection method for HFEs is proposed. Finally, a numerical example is used to illustrate the practicability and effectiveness of the method. e results show that the main advantages of the decision analysis methods under confidence levels are not only that they can adapt to the hesitant fuzzy environment but also that they take into account the confidence levels between decision makers, which makes the results more feasibility and practicality. However, expert credibility needs expert's subjective judgment, and there will be some errors in decision-making problems. In the future work, we will try our best to correct expert credibility and study probabilistic hesitant fuzzy environment under confidence levels [36,37].

Data Availability
All data included in this study are available from the corresponding author upon request. Most of the data are already in the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.