A Grey Target Group Decision Method with Dual Hesitant Fuzzy Information considering Decision-Maker’s Loss Aversion

Chongqing Engineering Technology Research Center for Information Management in Development, Chongqing Technology and Business University, Chongqing, China Postdoctoral Research Station of Management Science and Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 211106, China Research Center for Economy of Upper Researches of the Yangtze River, Chongqing Technology and Business University, Chongqing, China


Introduction
Multiattribute decision-making (MADM) is an important issue of modern decision-making science. Recently, MADM has become a hotspot in the decision-making area, and a lot of academic achievements have been made in the related domain [1][2][3]. Multiattribute group decisionmaking (MAGDM) can use the wisdom of the group to promote scientific decision-making. Pattern matching optimization is an effective method, which is applied to many areas in MAGDM, such as tracking and detection [4,5], computer engineering [6], physical sciences [7], health-related issues [8], natural sciences, and industrial academic areas [9]. In addition, other methods, such as TOPSIS [10,11], probabilistic linguistic term sets [12], data-capturing devices [13][14][15][16][17], the state-of-the-art features [18][19][20][21][22], the hidden Markov model (HMM), modified HMM, embedded HMM, Gaussian mixture modal (GMM), and the support vector machine (SVM) are also widely used in the field of MAGDM [23][24][25][26]. Grey target decision is one of the main methods using grey system theory to solve the problem of uncertain MADM and MAGDM under incomplete information condition. e main idea of the grey target decision is to achieve a uniform dimension of the Euclidean space through measuring and transforming index sets in the absence of the standard mode, i.e., the grey target. All the alternatives are distributed on the grey target. A target-eye of the grey target is found as the standard mode. en, all the decision points in the grey target are compared with the target-eye to obtain relative target-eye distances, and the dominant position of each decision point is sorted by the target-eye distances.
In recent years, many scholars have carried out theoretical research on the classic grey target decision methods from different perspectives. Gupta et al. [27] presented an extended TOPSIS method under the interval-valued intuitionistic fuzzy environment. Luo and Wang [28] studied the multiattribute grey target decision method for the attribute value within the three-parameter interval grey number. Wang et al. [29] studied the multiattribute grey target decision method based on soft set theory. Qian et al. [30] built a grey target decision model based on interval grey number panel data for multiindex dynamic problems. Ye [31] proposed an extended TOPSIS method with interval-valued intuitionistic fuzzy numbers for virtual enterprise partner selection. Fu et al. [32] presented a kind of multiattribute grey target decision model based on the positive-negative target center aimed at the complexity and uncertainty of the actual decision environment. Zeng et al. [33] proposed a grey target decision-making model based on the cobweb area and gave an application of choosing the software development pattern.
Fuzzy sets (FSs) are also an effective method to deal with the uncertain problem of decision-making. FSs are used to deal with vagueness and fuzziness in real decisionmaking problems, and several extensions have been developed. Rodriguez et al. [34] studied the hesitant fuzzy linguistic term set on MADM. Traditional hesitant fuzzy decision-making methods just provide the membership degrees and without considering the importance of the nonmembership degrees. In fact, the nonmembership plays the same important role in describing the vague decision-making information, which indicates the possible degrees of one element does not belong to a fixed set. To assess the attribute values more precisely, Zhu et al. [35] developed the dual hesitant fuzzy set (DHFS), taking into account much more information given by decisionmakers, in which the membership degree and the nonmembership degree are in the form of sets of values in [0, 1]. DHFS can avoid information distortion and losing effectively in describing the vague decision-making information. en, the aggregation operators for aggregating dual hesitant fuzzy elements (DHFEs) have attracted the attention from researchers [36][37][38][39][40]. Regarding the MADM problems with dual hesitant fuzzy information, many methods have been proposed. Ren and Wei [41] developed a prioritized multiattribute decision-making method to solve dual hesitant fuzzy decision problems. en, they [42] further proposed a dual hesitant fuzzy VIKOR method for multicriteria group decision-making based on the fuzzy measure and the new comparison method. Wei et al. [43] investigated the MADM problem based on the geometric aggregation operators with interval-valued dual hesitant fuzzy linguistic information. Lu and Wei [44] investigated the MADM problem based on the arithmetic and geometric aggregation operators with dual hesitant fuzzy uncertain linguistic information.
Based on the above analysis, we can conclude that the current studies on the DHFS are mainly based on some aggregation operators which are limited to tackle the complex MAGDM problem under the dual hesitant fuzzy environment. erefore, other classical uncertain decision methods such as the grey target decision method should be further studied for real decision-making problems under dual hesitant fuzzy information. erefore, the grey target decision method based on positive and negative clouts proposed is extended to solve the MAGDM problems with dual hesitant fuzzy information. It is the major motivation of our study. To the best of our knowledge, the existing grey target decision-making method has not yet been accommodated to deal with the dual hesitant fuzzy information provided by decisionmakers. In addition, the current research on the grey target decision method has also not considered the decision-maker's loss aversion. With the development of behavioral decision theory, the MAGDM method has been widely used on the basis of prospect theory.
erefore, the proposed model considers decisionmaker's loss aversion and variable weights of attributes based on the prospect theory. e contributions of this paper can be summarized as follows. (i) We proposed a novel grey target group decision method with positive and negative clouts under the dual hesitant fuzzy environment. (ii) We introduce the prospect theory to the grey target group decision. e remainder of this paper is structured as follows. Section 2 presents the preliminaries of DHFS and DHFEs. Section 3 is the process of establishing the mathematical models. A numerical example is presented in Section 4 followed by conclusions in Section 5.

Preliminaries
In this section, some essential concepts are reviewed as follows [36]. Definition 1. Let X be a given set, DHFS; D is a function mapping to X: (1) In expression (1), h(x) and g(x) are two sets of numbers within the interval [0, 1]. h(x) and g(x), respectively, denote the possible membership degrees and nonmembership degrees under the following conditions: where r ∈ h(x), η ∈ g(x), r + ∈ h + (x) � r∈h(x) max r { }, and η + ∈ g + (x) � η∈g(x) max η for all x ∈ X. e pair d(x) � 〈h(x), g(x)〉 is called the DHFE, which can be denoted simply by d � 〈h, g〉.
Obviously, if there is only one element in both h(x) and g(x), the DHFE reduces to an intuitionistic fuzzy number.
Definition 2 (see [34]). Let d 1 � 〈h 1 , g 1 〉 and d 2 � 〈h 2 , g 2 〉, respectively, denote two DHFEs; then, the operational laws can be defined as Definition 3 (see [43]). Let d i � 〈h i , g i 〉 (i � 1, 2, · · ·, n) denote a collection of DHFEs; then, the score function S(d i ) and the accuracy function P(d i ) of d i (i � 1, 2, · · ·, n) can be defined as where #h and #g are the number of values in h(di) and g(di), respectively.

Definition 4
(see [46]). Let d 1 � < h 1 , g 1 > and d 2 � < h 2 , g 2 > be two DHFEs. e normalized Hamming distance is defined as where #h i and #g i (i � 1, 2) are the number of values in h i and g i , respectively.

A Grey Target Decision Method for MAGDM Problems with Dual Hesitant Fuzzy Information and Decision-Maker's Loss Aversion
In this section, a grey target decision method based on positive and negative clouts is developed to solve the MAGDM problem under the dual hesitant fuzzy environment and uncertain weight information. Let A � A 1 , A 2 , · · · , A m be a finite set of m alternatives and C � C 1 , C 2 , · · · , C n be the set of n attributes. Let DM � DM 1 , DM 2 , · · · , DM k be the set of decision-makers whose ij > denotes the DHFEs for the evaluation of the attribute C j (j � 1, 2, · · · , n) with respect to the alternative . en, to determine the most desirable alternatives, a novel method is proposed based on the grey target decision method with positive and negative clouts.
Step 1. Aggregate all individual dual hesitant fuzzy decision matrices Df � (d f ij ) m×n (i � 1, 2, · · · , m, j � 1, 2, · · · , n, f � 1, 2, · · · , k) into a collective dual hesitant fuzzy decision matrix by the following equation [24]: Step 2. Identifying positive and negative clouts, and defining the dual hesitant fuzzy ideal optimization scheme as the positive clout r + and the ideal inferior scheme as the negative clout r − . Positive and negative clouts in the collective dual hesitant fuzzy decision matrix D can be computed according to equations (8) and (9), respectively: For the benefit attributes, r + ij � max 1≤i≤m d ij and r − ij � min 1≤i≤m d ij , the attribute values d ij can be compared by the score function S(d ij ) and the accuracy function P(d ij ) according to eorem 1.

Scientific Programming
Step 3. Calculating positive and negative target-eye distances based on r + and r − , and the positive and negative target-eye distances of each alternative are calculated using the following equations: where ω 1 , ω 2 , · · · , ω n denote the attribute weight vector.
denote the normalized Hamming distances between the DHFEs d ij to r + ij and r − ij , respectively.
Step 4. Calculating comprehensive target-eye distance: the evaluation vector of each alternative is always between the positive and negative clouts. Luo and Wang [28] proved that the optimal solution can be obtained based on r io * which is on the line between positive and negative clouts. Obviously, the larger r io * , the better the corresponding scheme. e projection r io * is the comprehensive target-eye distance, and r io * can be calculated by the following equation: where ζ 0 is the distance between positive clout ζ + ij and negative clout ζ − ij , ζ 0 � d(ζ + ij , ζ − ij ), which can be calculated according to equation (6).
In traditional grey target decision methods, r io * can be directly calculated when the attribute weights are known. en, the alternatives can be sorted by r io * . In the case of unknown attribute weight, the method that how to establish the model to solve the optimal initial weight vector is to be given as follows.
Step 5. Optimization of initial attribute weights ω 0 : if the weight sequence ω � (ω 1 , ω 2 , · · · , ω n ) is unknown, the sequence is the grey connotative sequence, and the grey entropy can be defined as According to the maximum entropy principle, ω s (1, 2, · · · , n) should be adjusted to minimize the sequence uncertainty of ω 0 � (ω 1 , ω 2 , · · · , ω n ) and maximize H ⊗ (ω). At the same time, considering the proximity between the effect measure of each scheme and the positive clout, the distance between the positive and negative clouts is minimized, and the optimization model is proposed as follows: Let f 1 (ω) � m i�1 n j�1 r io * and f 2 (ω) � n s�1 ω s ln ω s ; f max 1 (ω) and f min 1 (ω) are the maximum and minimum values of f 1 (ω), respectively; f max 2 (ω) and f min 2 (ω) are the maximum and minimum values of f 2 (ω), respectively. e two targets are dimensionless. en, the above multiobjective optimization problems can be transformed into the following single-objective optimization problem: In view of the fair competition of optimizing objective functions, usually set λ � 0.5 [28]. e optimal initial weight vector can be obtained by using available mathematical programing software. e optimal solution ω * is the initial weight ω 0 .
Step 6. Calculating the variable weights: firstly, the relative closeness of alternatives should be calculated by the following equation: en, let z j be the relative closeness of a positive ideal solution and S(z j ) be the state variable weight vector.
In equation (17), B(z j ) is an equilibrium function. e variable weights can be calculated by the following equation:

Scientific Programming
Step 7. Calculating the loss values of attribute weights: taking the initial weight as the reference point, the weight loss matrix of the variable weight vector relative to the initial weight is established as F: Step 8. Establishing the foreground theory matrix: considering the decision-maker's different risk attitudes towards the gain and loss of weight, the prospect theory matrix is established as Step 9. Ranking the alternatives according to comprehensive target-eye distances, R * io is calculated based on the variable weight vector and the prospect theory matrix. e most desirable alternative which has a higher value of R * io should be selected: In summary, the decision process can be described as Figure 1.

A Numerical Example.
In this section, a numerical example is given to present the application of the MAGDM method and to demonstrate its feasibility and effectiveness in a realistic scenario. An investment firm would like to invest four possible alternatives within the volume of total investment. e problem could be resolved through selecting the best options. ree decision-makers DM 1 , DM 2 , DM 3 (the weight vector is λ � (0.25, 0.40, 0.35) T ) are invited to evaluate the alternatives based on the following three attributes: C j (j � 1, 2, 3), where C 1 denotes the market share analysis, C 2 denotes the market growth analysis, and C 3 denotes the benefit analysis. In this case, all attributes are positive. e attribute weight vector is unknown. e four possible alternatives A i (i � 1, 2, 3, 4) are evaluated by using the DHFEs under the above three attributes; then, the dual hesitant fuzzy decision matrices Df � (d   (7): (2) Identifying the dual hesitant fuzzy positive clout r + and negative clout r − of alternatives in the collective dual hesitant fuzzy decision matrix D by equations (8) and (9), respectively:  (9) Equations (10) and (11) Equations (12) and (13) Equation (14) Solving model (16) with the fmincon function of MATLAB Equations (17) to (19) Equation (20) Establishing the foreground theory matrix V Equation (21) Ranking the alternatives according to R + i0 Equation (22) Figure 1: Decision flow of the proposed method.    (3) Calculating the positive and negative target-eye distances of each alternative from r + and r − by equations (10) and (11), respectively: (4) Calculate comprehensive target-eye distance r * io by equation (12): where ζ 0 can be calculated by equation (6), With reference to [46,47], as well as taking p as the reference point, the S-shaped utility curve combined with the preference of decision-makers is constructed as follows: Chinese people's decision-making preference is mostly an S-shaped utility curve [44] if p � 0.5, which is positioned as the following: In equation (25), i � 1, 2, · · · , 3 and j � 1, 2, · · · , 4. en, the variable weights W can be calculated by equation (18) (7) Calculating the weight loss matrix F by equation (19): Scientific Programming (9) Ranking the alternatives according to R * io . R * io is calculated by equation (21), and e alternatives can be ranked according to R * io ; thus, A 3 ≻A 1 ≻A 2 ≻A 4 . erefore, the most desirable alternative is A 3 .

e Sensitivity Analysis.
In order to illustrate the influence of reference points on decision results, the sensitivity analysis results are shown in Table 4 by changing the values of p and repeating Steps (6) to (9). As can be seen from Table 4, in all the utility functions, the optimal alternative is the same, but the ranking of evaluation results at different reference points is different. erefore, this method fully reflects the flexibility of the process and the limited rationality of the decision-maker.

Comparison and Analysis with Previous Methods.
Compared with the grey target decision method proposed in [27] and the grey relational analysis (GRA) method based on DHFS proposed in [43], the comparison results are shown in Table 5. At the same time, an ablation study is given to analyze the impact of decision-maker's loss aversion on results. We calculate the results of the grey target decision method with positive and negative clouts without considering the loss aversion factor of decision-makers. e tackle method is described as follows. Calculating Step 1 to Step 5 in Section 3, and then substituting the weight calculated in Step 5 into Step 4 to get r * io ; the alternatives can be sorted by r * io . e results show that, without considering the decisionmaker's loss aversion, the decision results obtained by the three methods are the same. e reliability of the grey target decision method with positive and negative clouts and the DHFS proposed in this article is proved. After considering the decision-maker's loss aversion, the sorting results changed. en, the decision results reflect the limited rationality and preference of decision-makers.

Conclusions
In this paper, a novel MAGDM problem based on the grey target decision method with positive and negative clouts under the dual hesitant fuzzy environment is proposed. An optimization model is established, which is employed to determine the initial weight vector of attributes. e optimization model is constructed with considering the proximity between the effect measures of alternatives, the positive ideal value, and the uncertainty of the attribute weight. e alternatives are sorted according to the comprehensive target-eye distance. Finally, a numerical example is presented to illustrate the application of the proposed method. e method provides a scientific and practical decision support to solve the MAGDM problem with double hesitation fuzzy numbers and decision-maker's loss aversion.
is study is limited in that it cannot solve some new DHFSs, such as the probabilistic dual hesitant fuzzy set (PDHFS). Further research can introduce multiple new DHFSs to study the grey target group decision method.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.