Z-Score Functions of Dual Hesitant Fuzzy Set and Its Applications in Multi-Criteria Decision Making

Dual hesitant fuzzy set (DHFS) consists of two parts: membership hesitant function and non-membership hesitant function. This set supports more exemplary and flexible access to set degrees for each element in the domain and can address two types of hesitant in this situation. It can be considered a powerful tool for expressing uncertain information in the decision-making process. The function of z-score, namely z-arithmetic mean, z-geometric mean, and z-harmonic mean, has been proposed with five important bases, these bases are hesitant degree for dual hesitant fuzzy element (DHFE), DHFE deviation degree, parameter  , (the importance of the hesitant degree), parameter  , (the importance of the deviation degree) and parameter , (the importance of membership (positive view) or non-membership (negative view). A comparison of the z-score with the existing score function was made to show some of their drawbacks. Next, the z-score function is then applied to solve multi-criteria decision making (MCDM) problems. To illustrate the proposed method's effectiveness, an example of MCDM specifically in pattern recognition has been shown.


Introduction
Dual hesitant fuzzy set (DHFS) [1] is a recent generalization of fuzzy set (FS) [2] which consists of membership hesitant function and non-membership hesitant function' DHFS confronts several different potential values that indicate whether certainty or ambiguity is the epistemic degree. This includes FS, intuitionistic fuzzy set (IFS) [3], [4] and hesitant fuzzy set (HFS) [5], [6] so that ambiguous knowledge can be treated more flexibly in the decision-making process. Numerous researchers applied DHFS to solve problems in multi-criteria decision making (MCDM) [7]- [14]. Usually all forms of MCDM need to evaluate and identify the DHFE information from a single expert or community expert for better potential option. It is therefore very necessary to differentiate between any two DHFEs and rank them efficiently. One the technique for differentiate the DHFE is score functions. Chen [15] proposed a score function and accuracy function to rank DHFE based on average value of DHFE, while Ren et al. [16] suggested that the score feature relies on three aspects: (1) the hesitant degree of membership and non-membership functions; (2) the mean hesitant degree of DHFE arising from one minus mean membership and non-membership values; (3) the non-membership feature should be deemed more significant. Numerous researchers applied the score functions of HFS and DHFS in their research [17], [18], [27]- [29], [19]- [26]. While many excellent contributions to score functions of varying types were made in the sense of DHFS, certain critical issues remain to be tackled. Are all decision makers (DMs) prefer the negative view (non-membership) compared to positive view (membership)? How important is the hesitant degree in determining the score for DHFE? This research aims to examine the essence of DHFE in DHFS score functions from a DM perspective and analyze the versatility of the newly proposed score functions.
The rest of this paper is structured as follows. In the preliminaries section, we present several simple definitions of HFS, DHFS and score functions of DHFE. For the next section, we propose z-score function namely z-arithmetic mean, z-geometric mean and z-harmonic mean and make some sensitivity analysis and compare them with existing score functions. Then, we suggest weighted z-score algorithms to solve MCDM problems, in particular pattern recognition in the DHFS environment. Lastly, we conclude the paper and provide some relevant remarks.

Materials and Methods
Definition 1. [5], [6] Given a fixed set X , then a HFS denoted as K on X is a term of a function that applied to X which returns a subset of [0,1] . The HFS can be written as

Z-Score Functions of Dual Hesitant Fuzzy Set
The score function of DHFE plays an important role in determining a single value to represent the element. We propose a z-score function consisting of the z-arithmetic mean, z-geometric mean and z-harmonic mean. The proposed DHFE novel z-score function considers five key aspects: The deviation degree of membership and non-membership for DHFE is a measure of the amount of variation or distribution of a set of value. A low deviation degree indicates that its elements tend to approach the mean of the set, while a high deviation degree indicates that its elements are spread over a wider range. This degree explains the importance of the level of hesitant for membership and non-membership functions in determining the degree of mean arithmetic score, mean geometric and mean harmonic. degrees for membership and non-membership functions in determining the arithmetic-z mean score degree, z-geometric mean and z-harmonic mean.
is the degrees used by decision makers to determine the degree of a z-score. The higher the degree, the more important the deviation degree in determining the z-score for DHFE. e) Parameter  : the importance of membership or non-membership degree.
This parameter explains the importance of membership degree (positive view) and non-membership (negative view) degree in determining the arithmetic-z mean score degree, z-geometric mean and z-harmonic mean.
is used by decision makers to determine the z-score value. If 0.5

 
, it indicates that the degree of membership is as important as the degree of non-membership in determining the score for DHFE. If [0,0.5)

 
, it indicates that the non-membership degree (negative view) is more important in determining the z-score for DHFE as suggested by Ren, Xu, and Wang [16]. Whereas when (0.5,1]

 
, it shows that membership degree (positive view) is more important than non-membership degree.
Then the following is called the z-arithmetic mean score is a HFE. Then the following is called the z-geometric mean score Then the following is called the z-harmonic mean score  1 2 hh .
Example 1 [16]. Suppose The z-score value for DHFE and the ranking order for example 1 are given according to Definition 5 -Definition 7 in Table 1-Table 6. Table 1-Table 6 show the z-score ranking for ( , ) ( dd . Based on Defintion 4 [16] for Example 1, the ranking order is 5 This suggests that extra attention need to be given to the assessments and non-membership functions are considered as negative assessments. In practice, negative judgments have more important effect than good judgments. Parameters  may be considered as a priority step of DM rather than non-membership functions. The greater the parameter,  , the more important for DM to consider negative comments. Therefore, the parameter  is different because DMs differ in the decision-making process. The ranking order for the z-score methods has been reserved that 5 d

Z-Score Algorithm in MCDM
In this section, we break it down into two parts, namely the problem-solving algorithm in MCDM and the numerical example to show the effectiveness of the proposed algorithm. The algorithm of z-score function in MCDM is shown as follows; Step 1. Set up DHFS decision matrix Step 2. Calculate the weighted z-score function a) Weighted z-arithmetic mean Step 3. Rank the alternatives based on the weighted z-score function For this section, we take numerical examples from Chen et al. [15]. Here we consider the problem of pattern recognition on the classification of building materials. Given four classes of metal materials, each represented by a hesitant dual set in the feature space 1 2 3 4 , , , C C C C . We know there is an unknown type of metal material A in the set, and our goal is to explain which class belongs to A. All five possible metal materials . The DHFS decision matrix is given as in Table 7.  Table 8, Table 9 and Table 10 respectively. Table 8, Table 9 and Table 10 show the same ranking order for weighted z-arithmetic mean score, weighted z-geometric mean and weighted z-harmonic mean respectively for ( , ) (1,1)

 
. The ranking order for (0.1,0.3,0.5,0.9)   is same for all methods in z-score functions. However for 0.7

 
, weighted z-arithmetic mean and weighted z-geometric mean score rankings are l l l l l which is equivalent to the algorithmic approach for the z-score function. This shows that the proposed algorithm can solve MCDM problems.

Conclusions
In this paper, we have suggested three z-score functions for DHFE namely z-arithmetic mean, z-geometric mean and z-harmonic mean. These three score functions are based on five important foundations, namely: a) Hesitant degree for membership and non-membership b) Deviation degree for membership and non-membership: the higher the deviation degree, the less consistent the decision maker is in making an assessment of an object or criterion. We then performed a sensitive analysis to determine the z-score degree of each DHFE to verify whether the parameters have changed and it is proven that the parameters chosen play an important role in determining the ranking order for DHFE. Then a comparative analysis is performed with the existing score function and it is proven that the proposed z-score is more flexible and able to address some of the weaknesses experienced in the existing method. We applied the proposed z-arithmetic mean, z-geometric mean and z-harmonic mean to develop the proposed algorithm to solve MCDM problems, especially pattern recognition in the DHFS environment and it turns out that the proposed method can solve the problem given and the rankings order is consistent with the existing method.