Interval-Valued Fermatean Hesitant Fuzzy Sets and Infectious Diseases Application

The Hesitant Fuzzy Set, which is a generalization of fuzzy sets, is an important tool in dealing with the difﬁculties that arise in determining the membership of an element to a set when there is doubt between several different values in decision-making problems. In this study, Fermatean hesitant fuzzy set is given to ensure to operate the conditions in which professionals evaluate an alternative in probable membership values and non-membership values. Aggregation operators of newly deﬁned sets are deﬁned to implement to multi-attributed group decision-making problems. The main properties of the new sets were examined. A new score function and accuracy function are given to compare two interval-valued numbers. Finally, a numeric example exactly demonstrates the feasibility, practicality, and effectiveness of the offered technique.


Introduction
The reasoning and decision-making (DM ) processes of people in the face of daily events are studied by many disciplines, including psychology, philosophy, cognitive science, and artificial intelligence. These processes are generally tried to be described based on various mathematical and statistical models. In this process, the problem of decision-making arises. DM is defined as the operation of selecting one or more of the alternative forms of behavior faced by a person or an institution in order to achieve a specific goal. Research shows that while it is sufficient to make many daily decisions intuitively, this path alone is not enough for complex and vital decisions. Multi-Attribute Decision Making (M A DM ) refers to the decision-making process in discrete situations where the alternatives examined in the decision problem are finite and clearly defined. In M A DM problems, the alternatives are a predetermined number. M A DM approaches are frequently used in decision problems such as choosing among alternatives, ranking, and comparing alternatives. They are frequently preferred methods in that they allow quick decision-making without requiring heavy mathematical operations and using a package program. There is only one purpose in the M A DM method. The aim is to determine the most ideal (most benefit, least cost) alternative for the decision problem. For the example problem above, the purpose of the decision problem can be expressed as "determination of the most suitable supplier alternative".
Group decision making (GDM) is about using the unified wisdom and experience of those involved in the group to make decisions that are likely to provide affirmative benefits. One of the key advantages of GDM is its potential to involve people from different backgrounds and thought processes so that the issues facing the group can be explored from a wider range of perspectives. Individuals want to overcome the difficulties they face to reach their goals. Sometimes this task becomes so large and complex that the individual can't solve it alone. In such cases, it is a more rational approach to making decisions by using group power. Whether working around a desk or dispersed in digital environments, the synergy that emerges as a group is an important tool in improving decisions and solving problems. Thus, individuals achieve some of their needs and goals that they cannot achieve alone through groups. Thus, although group members have their own thoughts and motivations, when they want to solve a problem, the problem will no longer be the process of choosing the best option according to a single decision-maker. The resulting group decision-making process involves the conflicts of different interest groups, different goals, and objectives, different criteria, political behaviour, etc. would be expanded to take into account. At this point, the final solution is not left to the initiative of a single decision maker, that is, the responsibility of all decision makers occurs.
In general, uncertainty is the situation in which a given event may have different consequences and there is no information about the probabilities of those consequences. Therefore, uncertainty is a very important notion for the DM process. It is not easy to know the probabilities of events happening in real-life. Therefore, the DM process occurs under uncertainty. Fuzzy logic theory [36] proposes a strong logical inference structure in the face of uncertain and imprecise knowledge. Fuzzy logic theory gives computers the ability to process people's linguistic data and work using people's experiences. While gaining this ability, it uses symbolic expressions instead of numerical expressions. These symbolic expressions are called fuzzy sets(FS). It is understood that the elements of fuzzy sets are actually decision variables containing probability states. Instead of probability values of possibilities, fuzzy sets arise by assigning membership degrees to each of them objectively.
Yager [35] introduced the q-step orthopair fuzzy set. The basic rule in this set theory is that the sum of MD with ND should not be greater than 1. Based on this idea, Senepati and Yager [21] introduced the Fermatean fuzzy set(FFS) and examined its basic features. In [22], Fermatean arithmetic mean, division, and subtraction which are new transactions for FFS, are defined and some of their properties are examined. In [23], new weighted aggregated operators related to FFSs are defined. [13] have defined Fermatean fuzzy soft set(FFSS) and entropy measures. Shahzadi and Akram [24] offered a new decision support algorithm with respect to the FFSS and defined the new aggregated operators. Garg et al. [6] new FFS type aggregated operators were defined by utilizing the t-norm and t-conorm.
The FS notion was generalized to the HFS notion by Torra [27]. This new set of the FS can handle the situations that the complexity in building the MD does not get up from a margin of error or a certain probability distribution of the probable values, however, originates from hesitation among a few several values [37]. Hence the HFS can more precisely reflect the people's hesitancy in stating their preferences over objects, compared to the FS and its other generalizations. Later, HFS and IFS were combined to obtain a new HFS which is called IHFS [20]. The fundamental notion is to form the situation in which instead of a individual MD and ND, human beings hesitate among a set of MD and ND and they require to symbolize such a hesitation. In [39], the notion of a dual HFS was improved and was given some properties. As an extension of the dual IVHFS, the HIVIFS approach was given [16]. IIn [17], the notion of IHFS to GDM problems using fuzzy cross-entropy was applied. The PFHS was initially given by Khan et al [7]. PHFS compensates the case that the sum of its MDs is less than 1. The Fermatean hesitant fuzzy set has been defined by Kirisci [15].
This work is dedicated to extending FHFSs to IVFH and improving M A G DM processes to IVFHF environments by aggregation operators. Score functions and accuracy functions are defined. The basic properties are studied together with the definition of IVFH. An algorithm is given by introducing the scenario describing the idea of M A G DM in IVFHF environments. A medical application showing the feasibility and applicability of the offered technique is given.

Preliminaries
Throughout the paper, U, as the initial universe set, respectively will be denoted. Thr ID of u to F is described as θ F (u) = 3 1 − (ζ 3 F (u) + η 3 F (u)), for any FFS F and u ∈ U.
, some operations as follows [21]: The properties of complement of FFS as follows [21]: is said to be a score function.
Take the two FFSs If the following condition (A) is hold, then it is called a natural quasi-ordering concerning the FFS [21]: For the two FFSs F 1 and F 2 ; is said to be an accuracy function.
For the two FFSs For two FFS F , G , from a binary relation ≤ (SF,AF) , it may be shown as F ≤ (SF,AF) G iff the condition (B) holds:

Definition 2.3. [28] The set
is called HFS, where τ Γ (u) indicates the set of some values in unit interval, that is probable MD of u ∈ U to Γ.
From now on, HFN will be used as τ = τ Γ (u) throughout the paper.
Definition 2.4. The following operations are hold for three HFNs τ, τ 1 , τ 2 : Definition 2.5. The set , showing a probable MD and ND of u ∈ U in P Γ respectively.

New Hesitant Fuzzy Sets
In this section, IVFH will be introduced and its properties will be examined in order to get better results in preventing information loss and to increase the flexibility and applicability of decision-making techniques when dealing with qualitative information.
where ζ F (u) is the possible Fermatean membership interval and η F (u) is the possible Fermatean non-membership intervals of F . Throughout this article, ϒ will show the set of all IVFHs.
Apparently, if There is only one pair of intervals in h F (u), the IVFH converts into an IVFFS, if both ( ζ F (u), η F (u)) converts one singleton, the IVFH may be viewed as an FHFS, if η F (u)) = [0, 0], the IVFH may be viewed as an IVHFS, if ζ + F + η + F ≤ 1 the IVFH can be seen as an IVIHFS, for each u ∈ U.
and α > 0. Then, we get: Proof. It is clear that F C is an IVFE.
is called a GIFHG operator, where GIFHG : ϒ n → ϒ and´ F σ (i) is the largest ith of` F k = ( F ) nω k (k = 1, 2, · · · , n). Using the Definition 3.9, P(SC(` The sets of alternatives, attributes and l professional persons denoted by A = {A i : i = 1, 2, · · · , m}, K = {K j : j = 1, 2, · · · , n}, P = {P k : k = 1, 2, · · · , l}. i j )}, (i = 1, 2, · · · , m; j = 1, 2, · · · , n) is an IVFE served by the professional person P k , in which ( ζ (k) i j ) points out the probable membership intervals that the alternative A i satisfies the attribute K j and ( η For MAGDM, we can say that the larger the value of the attribute, the benefit attribute, and the smaller the value of the attribute, the cost attribute. Therefore, convert the cost attribute values into the benefit attribute values and normalize the IVFHM for benefit attributeC j , Based on these considerations, a new technique was constructed for M A G DM in IVFHF environments. The algorithm for this technique is as follows: Algorithm: Step 1: Step 2: For ρ = (ρ 1 , ρ 2 , · · · , ρ l ) T is the weight vector of professionals P k (k = 1, 2, · · · , l), employ the GIFHA (or GIFHG) operator to , and the certain operation is as follows: i j , · · · , F (n) Step 3: For the associated weight vector κ = (κ 1 , κ 2 , · · · , κ n ) T , utilize the GIFHA (or GIFHG) operator to collect all the preference values F i all over. The certain operation is as follows: Here, ω is the weight vector of the attributes K j . Hence,` ) can be defined as Step 4: Calculate the score values SC( F i ) and the accuracy values AF( F i ).
Step 5: Obtain the priority of the alternatives A i by ranking SC( F i ).
Example 5.1. Let's choose the A i (i = 1, 2, 3) as the set of alternatives made up of hospital management system software. Denote the set P k (k = 1, 2, 3) three physicians and the set K three criteria. The first criterion is "price", which is the cost type. The second and third criterion are "speed" and "efficiency" respectively, which are the benefit type. The weight vector of the physicians is ρ = (0. 18 (Tables 1-3), where F (k) i j is an IVFE offered by the professionals P k (Tables 4-7).
Step 1. Convert the matrix D (k) into the matrix E (k) = F (k) i j m×n (Tables 4-6).
Step Step 5: Obtain the priority of the alternatives by ranking the score functions. Then, the ranking order A 1 > A 3 > A 2 . So the optimal scheme is A 1 .