Z-Score Functions of Hesitant Fuzzy Sets

The hesitant fuzzy set (HFS) concept as an extension of fuzzy set (FS) in which the membership degree of a given element, called the hesitant fuzzy element (HFE), is defined as a set of possible values. A large number of studies are concentrating on HFE and HFS measurements. It is not just because of their crucial importance in theoretical studies, but also because they are required for almost any application field. The score function of HFE is a useful method for converting data into a single value. Moreover, the scoring function provides a much easier way to determine each alternative's ranking order for multi-criteria decision-making (MCDM). This study introduces a new hesitant degree of HFE and the z-score function of HFE, which consists of z-arithmetic mean, z-geometric mean, and z-harmonic mean. The z-score function is developed with four main bases: a hesitant degree of HFE, deviation value of HFE, the importance of the hesitant degree of HFE, , and importance of the deviation value of HFE, . These three proposed scores are compared with the existing scores functions to identify the proposed z-score function's flexibility. An algorithm based on the z-score function was developed to create an algorithm solution to MCDM. Example of secondary data on supplier selection for automated companies is used to prove the algorithms’ capability in ranking order for MCDM.


Introduction
In several fields, the fuzzy set (FS) [1] theory is commonly and successfully used to model some kinds of uncertainty. Still, due to various causes of vagueness that exist in the idea, the downside of this method is that it will be much tougher to cope with imprecise and uncertain information.
Researchers recognized that FS theory has many disadvantages, but several enhancements have been done to improve the method's usage in real life applications. Some of the most known extensions of FSs include intuitionistic fuzzy sets (IFS) [2], interval-valued intuition fuzzy set (IVIFS) [3], fuzzy multiset [4] and fuzzy soft sets [5]. The hesitant fuzzy set (HFS) was initially implemented by Torra [6], [7]. They expanded FS to HFS since they discovered that an element's membership to a set is difficult to decide under a group setting due to doubts between a few different values. More and more scholars have been attracted to do research on HFS [8]- [17] and answer numerous decision-making issues in MCDM [18]- [24].
The measurements for HFEs and HFSs are the subject of an increasing number of studies. This is not only due to their basic significance in scientific research, but also because they are invaluable in nearly all fields of use [25]. Xia and Xu [26] defined a widely used score function which, by averaging the membership values in HFEs, the function can help to draw the required comparison. Liao and Xu [27] developed a two-step algorithmic technique to rank HFEs. The Score value-Variance was developed by α β considering Xia's score function and the variance value of the memberships. Zhang [28] developed a sort of score function that is a generalization of the score function of Xia by taking into account the variance of HFE memberships. A new enhanced score feature with role weighting details was proposed by Farhadinia [29]. More recently, Farhadinia [30] presented a collection of HFE score functions and defined the scheme for selecting a suitable score function from a set of score functions. A score function integrated with presented by Zhang and Xu [28] is a parameter defined by the decision makers that can be adjusted according to the reality of the situation. Even though various types of excellent score functions have been generated in the sense of HFE, certain critical problems still need to be discussed. What if the HFEs cannot be distinguished by these score functions of various lengths and deviation? What if the various HFEs have the same score? The goal of this study is to suggest a score function incorporated with HFE's hesitant degree and deviation value that is both flexible and efficient and addresses certain limitations of the current score functions.
The presentation of this article is as follows. In section 2, we recall some basic concepts of HFSs and existing score functions. In section 3, we propose the z-score function of the HFEs which consists of z-arithmetic mean, z-geometric mean and z-harmonic mean. In section 4, the weighted z-score algorithm used in MCDM is presented and examples of automotive industry supplier selection are provided to show the usability of the suggested algorithm. Finally, in section 5, we give the conclusion of this study.

Preliminaries
This section is dedicated to explaining the definitions relating to HFS, existing HFE score function, hesitant degree of HFE and deviation value of HFE. ii. If , then .
Definition 6 [30] Let be an HFE. The following functions can be considered as the score index for HFEs: 1. The arithmetic-mean score function 2. The geometric-mean score function: 3. The minimum score function:

Z-Score Functions of HFEs
The score function for the HFE plays an important role in determining the single value to represent the elements. The z-score function of the HFEs consists of z-arithmetic mean, z-geometric mean and z-harmonic mean. The suggested z-score functions take into account four fundamental aspects;

Hesitant degree of HFEs
The varying hesitant degree of HFE depends on the number of memberships for the element. The greater the membership of an element, the greater the value of the hesitant degree of HFE.

Deviation value of HFEs
This value measures the total variation or spread of a group of membership in elements. Low deviation value indicates that membership tends to approach the mean from the set, while high deviation value indicates that its membership spreads across a wider range.

Importance of the hesitant degree of HFE,
Parameter explains the importance of the hesitant degree of HFEs in determining the z-arithmetic mean, z-geometric mean, and z-harmonic mean score values. The higher the parameter, , the more important the hesitant degree of HFE in determining the z-score.

Importance of the deviation value of HFE,
This value explains the importance of the deviation value of HFE in determining the z-arithmetic mean, z-geometric mean, and z-harmonic mean score values. The parameter is the value used by the decision maker to determine the z-score-z value. The higher the value of means the more important the deviation value of HFE in determining the z-score.
Definition 10 Let is an HFE and is the length of . The hesitant degree of HFEs, is denoted by: where is determined by . If , then , and if then . The greater the value of , the more unconvinced the decision maker in assessing an object or criterion. We believe that each unit increase in membership of element (length of ) is equivalent to the increase in the hesitant degree of HFEs.  The value of the hesitant degree of HFE depends on the number of membership in HFE. The larger the number of membership means more sceptical the decision makers are about assessing an object or criterion. For example, when the length of is 1, then the hesitant degree of HFE is 0 which means that the decision makers are not hesitant in assessing an object or criterion. Based on Definition 8, decision makers tend to choose smaller numbers of membership due to significant differences in each membership addition, for example, if the number of elements is two or more, the hesitant value will be equivalent to 0.5 or greater. This will result in the purpose of HFS to be eroded. Definition 11 Let be an HFE, z-arithmetic mean score function for the HFE is given as:

Definition 12
Let be an HFE, z-geometric mean score function for the HFE is given as: If is the hesitant full element, then, , and . Proof. The proof is obvious.
Example 3.2 [28] Let and be three HFEs. The ranking of proposed z-score functions is given in Table 1 until Table  3.
The tables show that the ranking for the three HFEs are the same for each with except for z-arithmetic mean with , where the ranking is and equals to Definition 2.   Table 4.
Xu and Xia [26] approach only considers the mean value of all elements in HFE and does not distinguish between these three HFEs. This approach does not take into account the length of the HFEs. Meanwhile, the ranking based on Farhadinia's score approach [29] is inconsistent with others. Since the methods of Liao and Xu [27] are compatible with others, it takes time. The methodology is based on two principles, and if we need to address the issue in MCDM, it is very complicated to implement.
The score function of Zhang and Xu [28] is consistent with most of the existing score functions. However, the function of this score does not consider the spread of membership of HFEs. For the minimum and maximum score function, the ranking is consistent but does not consider the deviation value and length of HFEs. This score functions only consider the value of the first and last

Example 3.3. Let
, and be a three HFEs. The following comparisons are between the score functions of the z-arithmetic mean using the Zhang and Xu [28] approach. Table 5 is the score value and HFE ranking for both approaches.

Example 3.4 Let
, , , and be a five HFEs. The following comparisons are between score functions of z-arithmetic mean with Zhang and Xu [28] as per Table 6.
Based on Table 5 and  Table 6 shows a comparison between the geometric mean score function and z-geometric mean.
Based on Table 7, the geometric score function is the same for all three HFEs, although the grading arrangement for the z-geometric mean is different that is . The z-geometric mean takes into account the deviation degree and hesitant degree for these three HFEs.

Application of Z-Score Functions in MCDM
The weighted z-score algorithm in MCDM is covered in this section, and examples of automotive industry supplier selection are provided to demonstrate the usability of the proposed algorithms.

Definition 3.12
Let be an HFS. The function of the weighted z-arithmetic mean score for HFS is: Let be an HFS. The function of the weighted z-geometric mean score for HFS is: Definition 3.14 Let be an HFS. The function of the weighted z-harmonic mean score for HFS is:

Z-Score Algorithm in MCDM;
Step 1. Build a decision matrix of HFS.
Step 2. Identify benefit criteria and cost criteria. If there is a cost criterion change to complement set.
Step 3. Calculate the weighted z-score function for HFS as in Definition 3.13-Definition 3.15.
Step 4. Determine the ranking for the alternatives involved.
Step 5. Choose the best alternative.
Example 3.6. [33] To describe the proposed results, examples adapted from [34] are used. This example assumes that an automotive company seeks to choose a supplier that is best suited for one of the key elements of its manufacturing process. After the initial assessment, four suppliers are shortlisted for further assessment. For more accurate assessments on different suppliers, four of the most important criteria have been selected as follows : product quality; : relationship closeness; : delivery reputation; and : price. Clearly, , and are benefit criteria, and is cost criterion. To comprehensively evaluate various suppliers, experts who have different skills, experience and knowledge values were invited to conduct an assessment. The criteria weight vector is w = (0.2249,0.2357,0.2560,0.2833) [35]. The information provided by the experts is presented in the form of HFE matrix as provided in Table 8. Based on the proposed algorithm, Table 9 to Table 13 show z-score and ranking values for different .    Table 9 - Table 13 show the alternative ranking when parameters are evaluated differently. Decision makers with different subjective options can choose certain parameters according to their experience and attitude. This means that the proposed z-score is beneficial for a combination of subjective and objective decision-making information.
Sensitivity analysis is done by modifying parameters (level of interest for the hesitant degree and deviation degree) by increasing and decreasing the parameters. The ranking for alternatives is the same as obtained by Xu and Xia [33] and Hu et.al [35].

Conclusions
Some of the conclusions of this study are as follows: 1. The novel hesitant degree of HFE can escape the propensity of decision makers to evaluate an entity or criteria by restricting hesitancy. 2. The arithmetic mean, geometric mean and harmonic mean score functions can be integrated with the hesitant degree of HFE and deviation of HFE and are used to solve some of the challenges encountered by the current score function to make it more flexible. Decision makers set certain parameters based on their expertise or knowledge to measure the z-score functions. 3. By utilizing the recommended z-score function, the MCDM solution process can be achieved. The provided algorithm is simpler and compatible with current algorithms. 4. The proposed algorithm may prevent MCDM from applying any membership to a shorter HFE and render it equivalent to another HFE, or repeating their membership to achieve two series of the same length, thus losing the original data structure and modifying the HFE data details [36].
Future experiments might be extended to different fields for the data recruitment process and different algorithms such as Technique for Order of Preference by