Application of Parameterized Hesitant Fuzzy Soft Set Theory in Decision Making

In this paper, by combining hesitant fuzzy soft sets (HFSSs) and fuzzy parameterized, we introduce the idea of a new hybrid model, fuzzy parameterized hesitant fuzzy soft sets (FPHFSSs). The benefit of this theory is that the degree of importance of parameters is being provided to HFSSs directly from decision makers. In addition, all the information is represented in a single set in the decision making process. Then, we likewise ponder its basic operations such as AND, OR, complement, union and intersection. The basic properties such as associative, distributive and de Morgan’s law of FPHFSSs are proven. Next, in order to resolve the multi-criteria decision making problem (MCDM), we present arithmetic mean score and geometry mean score incorporated with hesitant degree of FPHFSSs in TOPSIS. This algorithm can cater some existing approach that suggested to add such elements to a shorter hesitant fuzzy element, rendering it equivalent to another hesitant fuzzy element, or to duplicate its elements to obtain two sequence of the same length. Such approaches would break the original data structure and modify the data. Finally, to demonstrate the efficacy and viability of our process, we equate our algorithm with existing methods.


Introduction
The concept of fuzzy sets presented by Zadeh [1] has conquered an enormous achievement in numerous fields. The extension of the fuzzy sets and one that integrated with other theories have been applied by some researchers, including the intuitionistic fuzzy sets [2]- [6], fuzzy multiset [7]- [9] and fuzzy soft sets [10]- [14]. Torra and Norakawa [15]- [16] introduced hesitant fuzzy sets (HFSs) in which the membership of an object to a concept is presented by a series of some different values between 0 to 1. HFSs can mirror human's hesitancy further objectively than the other usual extensions of fuzzy sets. Since its introduction, researchers have used it to answer numerous decision-making issues [17]- [38]. Aside from that, some researchers integrated HFSs with some other extensions of fuzzy sets. Lv et al. [39] studied on hesitant fuzzy information measures and their clustering application. Xu & Zhang [40] made an overview on the applications of the hesitant fuzzy sets in group decision-making. Babitha and John [41] first studied the hesitant fuzzy soft sets (HFSSs) which are the hybrid structure between HFSs and fuzzy soft set. They proposed the basic operation of HFSSs such as union, intersection, complement and proved the De Morgan's law. Consequently, Wang et al. [42] proposed HFSSs and their operations such as "AND", "OR" complement, union and intersection and their basic law properties. Beg and Rashid [43] presented the idea of an HFSSs where adaptation to manage the conditions in which experts assess an alternative giving to finite criteria in all possible values. They also proposed the distance measure between any two elements of the HFSSs. Rezaei andRezaei [44] proposed distance and similarity measures for HFSSs by using well-known Hamming, Euclidean, and Minkowski distance measures while Li et al. [45] extended the concept of HFSS to generalized HFSSs.
Among the significant milestones in the development of hesitant fuzzy soft sets and their generalizations is the introduction of the fuzzy parameterized aspect. The fuzzy parameterized aspect was firstly established by Cagman et al. [46] who proposed the fuzzy parameterized fuzzy soft (FPFS) sets and their basic operations and followed by others. [47]- [54] .
Characteristics of this work are as follows: 1. We extend the definitions of HFSS [41], [42]  FPHFSSs. We investigate certain connections between two FFPHFLTSSs and certain plain, binary-based set operations for FPHFSSs. There is also a mention of the property of operator. 2. Some of the previous methods added the maximum value, minimum value or any value to the shorter one until both have the same HFE length. These methods remove and alter data knowledge from the original data structure. In order to fill this gap, the TOPSIS algorithm is presented based on the FPHFSS's arithmetical mean and geometry mean without adding an element to HFE. This approach is simple and easy to understand.
The presentation of this article is as follows. In section 2, we call some basic concepts of hesitant fuzzy sets, fuzzy soft sets and hesitant fuzzy soft sets. In section 3, we proposed the concept of fuzzy parameterized hesitant fuzzy soft set (FPHFSSs) which is the combination of hesitant fuzzy soft set and fuzzy parameterized in which we provide the degree of importance for each alternative. We also expressed the proposed concept's basic operations namely intersection, union, and complement and then study some of their properties. In section 4, we introduce the TOPSIS based score index of FPHFSSs. Then we give numerical example of the FPHFSSs in solving decision-making problem and make a comparison analysis with other existing methods. Finally, we give the conclusion of our study and recommendation for further research.

Preliminaries
In this section, we recall some basic notions and definitions of hesitant fuzzy set and fuzzy soft set that will be used in this paper.  be an HFE. The following functions can be considered as the score index for HFEs: The arithmetic mean score index: The geometric-mean score index:

Fuzzy Parameterized Hesitant Fuzzy Soft Set
In this section, we shall define fuzzy parameterized hesitant fuzzy soft set and their operations with examples.
It should be noted that the set of all FPHFSs over U will be denoted by FPHFSS(U). Example 1. Let be a set of parameters 1 x for the price, 2 x for the location and 3 x for the size. Suppose Then the FPHFSS set is given by Definition 7. Two FPHFSSs are said to be equal, and we write  if the following conditions are satisfied:  be any three of FPHFSSs. Then the following results hold: The proof is straightforward.
where c is a fuzzy complement and c is a hesitant fuzzy soft complement. Example 2. Consider example 1. By using Definition 1, we have

Proposition 2 Let
K  be an FPHSS. Then the following results hold: The proof is straightforward.
following Definition 11 is given as Then the following results hold: 1.
The proof is straightforward.
 be any three of FPHFSSs. Then the following results hold: 1.
The proof is straightforward. Proposition 5. Let The proof is straightforward. Definition 13. The operationAND for two FPHFSSs

Theorem 4. (De Morgan Law of FPHFSSs). Let
K  and L  be two FPHFSSs over ; U Then the following results hold: U Then the following results hold:

Applications
In this part, we will explain how to utilize this algorithm in solving MCDM. We extend the theory of HFSSs by giving importance weight for each parameter in decision making process. We utilize the TOPSIS algorithm with arithmetic mean and geometric mean score to solve problem in FPHFSSs environment. It should be noted that some of the existing operations, arrangement and various measures of hesitant fuzzy sets need the hesitant fuzzy elements to have exactly the same length. In practice, however, the length of the hesitant fuzzy element may vary. The approach proposed in [56] applied those elements to the shorter hesitant fuzzy element, rendering its equivalent to another hesitant fuzzy element, or repeating its elements to obtain two sequence of the same length. These approaches would break the original data structure and alter the data details [39]. In this paper, we presented an algorithm based on the two score functions of HFE by Farhadinia [57]. Based on this argument, we materialize with our algorithm for FPHFSS given as below.
Algorithm: The TOPSIS based score index of FPHFSSs.
Step 1. Transform the FPHFSSs into fuzzy decision matrix. The FPHFSSs matrix of decision K can be represented as Step 4. Calculate distance from the PIS and NIS to score values. Step 5. Calculate the CI of every alternative Step 6. Rank the alternatives.

Numerical Examples
This section provides a numerical illustration of the viability of the proposed score index-TOPSIS approach (modified from Xu & Xia, 2011a) in the FPHFSSs decision-making problems.A comparative analysis is provided to validate its reasonableness and usefulness with other existing methods.
Energy is an important element for community social and economic growth. Considering five energy projects to be invested Pi(i=1,2,3,4,5) the decision-makers will assess five potential projects anonymously from the four following criteria, technological (C1), environmental (C2), socio-political (C3), and economic (C4). The modified part is where the degree of importance for each criteria is directly decided by the decision makers.
We present our methods to solve the MCDM problem above using the steps presented before Step 1. Construct the FPHFSSs into fuzzy decision matrix as shown in Table 1.
Step 2. Find the score index of the hesitant fuzzy element of FPHFSSs according to eq.(3). The arithmetic mean score of FPHFSSs are shown in Table 2. Step 3. Define the PIS and NIS based on arithmetic mean score as shown in Table 3 using equation (5) and (6). Step 4 and step 5. To determine the separation d + and dof each alternative Pi from the PIS and the NIS, the eq.(7) and eq.(8) shall be used. Then, calculate the relative closeness CI of each alternative using Equation (9). Step 6. Based on the value of CI the ranking is 3 5 4 1 2 P P P P P .

Comparison Analysis
In this section, we will compare the ranking results with other existing methods. Table 5 compares the ranking of alternatives given by our proposed method and other existing methods.

P P P P P
The preference weighted generalize distance with distance parameter = 6 and preference parameter = 0.1 Sun et al. [22] Similarity like positive correlation decision making factor = 0.7. Weight of the distance like positive correlation decision making factor = 0.6. This table shows some of the rankings provided by our proposed methods and another established method of selecting the energy site. The distance between the hesitant evaluation values and the ideal reference intervals is uncertain on the basis of Liu and Wang [46]. For example, the hesitant evaluations for { 0.7, 0.6 } and { 0.7, 0.5 } with the ideal reference [ 0.7,0.5 ] are the same even if the hesitant values are different. While the ranking given by Sun et al. [22] contradicts with the ranking from other methods, in which the 3 P is ranked in the final position although mostly 3 P is placed first by other methods. The ranking given by Li et al [60] is quite compatible with some other algorithms, even so their algorithms are complicated for decision-making because it is difficult to determine the value of , a  and  . Xu and Xia [55] presented the following rules for proper operation: the shorter one is expanded by inserting a minimum value, maximum value, or any value thereof until it has the same length as the longer one. The choice of this value depends mainly on the risk preferences of decision-makers. Optimists are expected to achieve desirable results and may bring maximum value and pessimists predict to contribute minimum value. Obviously, the original data structure will be broken and the data details modified [30].

Conclusions
In this paper, we consider the parameterized hesitant fuzzy soft set which includes the combination of the hesitant fuzzy set and fuzzy soft sets where an important degree is given for each element in the set of parameters. The introduction of elements of fuzzy parameters are used to avoid the degree of importance to which criteria are created from the rating of each alternative. Decision makers should decide which criteria are more important than others. Next, we study some of the algorithm's properties. The complement, union and intersection, AND and OR operation have been defined on the FPHFSSs. In addition, the algorithm given can cater for certain methods which added or repeated the hesitant fuzzy elements to the same length as this technique could destroy the original information. Finally, we provided an example which demonstrates that this theory can be used to solve MCDM problems. Comparisons are made to show feasibility and viability of our proposed method. We hope that our work could enhance the study on hesitant fuzzy soft sets which could further be applied in many other areas such as data analysis and forecasting. At the same time, it is anticipated that the fuzzy parameterized concept can extend to other generalization of fuzzy sets such as interval fuzzy set, Application of Parameterized Hesitant Fuzzy Soft Set Theory in Decision Making 251 intuinistic fuzzy set, hesitant fuzzy set and others.