On Full Fuzzy Parameterized Soft Set

Some Years back, researchers made attempts in trying to fuzzify set of parameters of soft set. In this paper, we consider a review of “fuzzy parameterized soft set", and improve it by introducing a way to fully fuzzify the set of parameters of soft set. We deﬁne the concept “full fuzzy parametrized soft set" and study some of its basic operations. We also illustrate the concept with example


Introduction
Many fields like Economics, Engineering and Environmental Sciences deal with uncertainty that may not be successfully modeled classically.Therefore, over the years, many non-classical set theories have been developed for modeling imprecision and uncertainty.F -Set theory proved to be more accurate due to its unique way of describing each element of a set by its membership degree, [1].
But it has a difficulty of inadequacy in parametrization tool associated with the theory, which was pointed out and handled by introducing S-set theory, [2].S-set served as a mathematical tool in dealing with uncertainties in a parametric manner.The author also outlined several areas of application of the theory such as: Riemann integration, game theory, operations research, probability theory, etc.
With the rapid increase in works relating to S-set theory, Maji et al [3] defined some operations on S-set and used the theory to solve some decision making problems ( [4]).In [5], Chen et al presented a new definition of S-set as an improvement of [4].Also, scholars like Cagman & Enginoglu [6], introduced and investigated soft matrix theory and applied it to a decision making problem.In sequel, many researchers in various directions also contributed, like Ali et al [7], defined some new operations such as: restricted union, extended intersection, restricted intersection, etc.Furthermore, Researchers in a bid to solve problems which are more complicated have introduced Fset into the study of S-set theory, leading to the notion of F S-set.In this direction, Maji et al in [8] are the first contributors to defined F S-set.Following this definition, many researchers have come up with interesting applications of the theory.In [9], Roy & Maji investigated some application of F S-set.Yang et al [10,11] made some improvement on this concept.In [13], Cagman et al defined fuzzy soft set theory and its related properties, and fuzzy soft aggregation operator that allows for more efficient way in dealing with decision making problems.Again, in [14,15], Cagman et al introduced the concepts of F P S-set and F P F S-set together along with their associated properties.In a related development, Alkhazaleh et al [16] introduced the concept of fuzzy parameterized interval-valued F S-set and gave its application in decision making.In [17], the authors introduced the concept of multi Q-fuzzy parameterized soft set.
In addition, many authors consider the application parts of S-set and F S-set theories in different areas of decision making.In [18], Rodriguez et al proposed the comparison score based approach to solving F S-set based decision making problems.Also, in [19], Nasef et al present another application of S-set in a decision making problem for real estate marketing with the help of rough mathematics, and provide an algorithm to select the optimal choice of an object.Again in [20], the authors applied the notion of F S-set in Sanchez's method [21,22] of decision making.Furthermore, in [23], the authors developed a way to solve intertemporal choice problems like: savings, investments, spendings, etc., for F S-sets.Also, in [24], the authors combined hesitant F -set and multi fuzzy soft set to develop hesitant multi fuzzy soft set and presented an algorithm and a novel approach to hesitant multi fuzzy soft set based decision making problems.
In recent developments, the idea of the concept of F P S-set theory has been a great influence to numerous researchers towards solving more realistic decision making problems.For instance; the work of Rodzi and Ahmad [25] on F P HF LT S-set in multi-criteria decision making, which came up by studying the work on hesitant fuzzy linguistic term soft set [26] in a fuzzy parameterized environment.The authors also described some related concepts and consider the fundamental operations of F P HF LT S-set, and they were able to develop three different algorithm for solving group decision making.Other contributions include: [27][28][29][30][31][32][33].
Up to the present era, all these works, starting from the example given in [2], are choice based on some certain factors; the influence of choice are more to be considered as fuzzy than crisp.This is because, for instance; the word "expensive" is not well defined in a classical sense, and of course, cannot be precisely measured, given that the price of a particular house cannot be said to be just "expensive" or "not expensive".Also, the other parameters are considered not to be well defined as cannot be precisely measured.Just like in the case of F P S-set, [14], with reference to the example given in [2], the parameters were just given fuzzy values.For example; a certain parameter "expensive" is given a fuzzy value, say 0.6, then the fuzzy value of the other parameters with respect to the subset are automatically always 0, i.e to say; f (0.6/e 1 ) = {h 1 }, Thus, we consider this as a type of generalization of soft set.This is why, in this present work, we consider a more encompassing way in fuzzifying the set of parameters.In this case, the description of each element of the power set of the universe set (P (U )) is entirely based on the whole set of parameters, i.e., instead of this function f µ(e 1 )/e 1 = {h 1 } as presented in [14], we use f µ(e 1 )/e 1 , µ(e 2 )/e 2 , ..., µ(e n )/e n = {h 1 }, which FULLY describe the set {h 1 } in terms of all the parameters.One reason for our generalization is that in real life, decision making is most time based on several factors (parameters) and not just one.Clearly, the degree to which each parameter contributes to the final decision varies.Thus, our work is a kind of generalization of the work presented in [14], since in our work, the function f 0.6/e 1 , 0/e 2 , 0/e 3 , 0/e 4 = {h 1 } means that the subset {h 1 } is considered expensive (e 1 ) to the degree 0.6 and to the degree 0 for the other parameters, and is considered to be equivalent to the function f (0.6/e 1 ) = {h 1 } as presented in [14].Therefore, the decision on the selection of any element in P (U ) is based on the contribution of each of the parameters.This motivated us to introduce the new concept "F F P S-set".
The remaining part of this paper is organized as follows: Section 2 contains some basic relevant definitions.Section 3 introduces the new concept (F F P S-set).Section 4 studies some basic operations, namely: A − empty and A − U niversal F F P S-set, F F P S-subset, Complement, Intersection, Set difference and union of F F P S-set.While section 5 draws conclusion and suggestion for further research.

Preliminaries
In this section, we provide some basic definitions following from: [1], [2], [8], [14].[1] [2] (S-set): Let U be a universe set, and E be the set of parameters.A pair (F, E) is called a S-set over U if and only if F is a mapping from E into the set of all subsets of the universe set U , i.e., F : E → P (U ), where P (U ) is the power set of U .In other words, S-set over U is a parameterized family of subsets of U .Every set F (e), for every e ∈ E, from this family may be considered as the set of e-elements of the S-set (F, E) or considered as the set of e-approximate elements of the soft set.Accordingly, we can view a soft set (F, E) as a collection of approximations: (F, E) = {F (e) : e ∈ E}.
[8] (F S-set): Let I U be the set of all fuzzy sets of U .Then a pair (f, A) is called a F S-set over U , where A is a subset of the set of parameters E, and f is a mapping from A into I U .That is, f : A → I U , and for each a ∈ A, f (a) = f a : U → I, is a F -set on U .
[14] (F P S-set): Let U be an initial universe, E be the set of parameters and A be the fuzzy set over E. A F P S-set A on the universe U is defined by the set of ordered pairs Where P (U ) is the power set of U and the function Characterized by the membership function The value µ A (x) of an element x of the parameters represents its degree of importance.And it is solely based on the desirability of the decision maker.
Hence, this means that the approximate function is defined from fuzzy subset of E to the crisp subset of the Universe set U .
Note that from now on, the sets of all F P S-sets over U will be denoted by F P S(U ).
[14] (A-empty F P S-set): If A = ∅, then A is called an empty F P S-set, denoted by ∅ .
[14] (A-universal F P S-set): Let A ∈ F P S(U ).If A is a crisp subsets of E and γ A (x) = U for all x ∈ A, then A is called A-universal F P S-set, denoted by Ã.
If A ∈ E, then the A-universal F P S-set is called universal F P S-set, denoted by Ẽ [14] (Union of F P S-set): Let A , B ∈ F P S(U ).Then, union A and B , denoted by A ∪ B , is defined by for all x ∈ E.
[14] (Intersection of F P S-set): Let A , B ∈ F P S(U ).Then, intersection of A and B , denoted by A ∩ B , is an F P S-sets defined by the approximate and membership functions for all x ∈ E.

The Concept of Full Fuzzy Parameterized Soft Set
In this section, we introduce the notion of F F P S-set.
Let E be the set of all parameters and let U be an initial universe with P (U ) the power set of U .Here, we fuzzify E to Ẽ.In this case, Ẽ is considered to be the set of all possible fuzzy set over E. Therefore, a fuzzy set ŷ over E is as: characterized by the membership function This is so, since every fuzzy set is completely and uniquely defined.Hence, Ẽ is said to contain the elements y i , for i = 1, 2, 3, . . .Therefore, each ŷ ∈ Ẽ is considered as an indicator of the degree to which the parameters in E are considered intra-dependently.Meanwhile, µ ŷ (x) is the degree to which the parameter x ∈ E is considered.
[14] Let Ã ⊂ Ẽ.An F F P S-set F Ã on the universe U is given as: Where f Ã : Ẽ → P (U ) represents the approximation function of F Ã such that f Ã(ŷ) = ∅ whenever µ ŷ (x) = 0 ∀x ∈ E and ŷ ∈ Ã.Therefore, throughout this work, the set of all F F P S-set will be denoted by F F P SS(U, Ẽ).
As an illustration, we use the following example, presented in [2] for more detail discussion.Given the following initial universe, U = the set of houses under consideration for sales, E be the set of parameters.Suppose: U = {h 1 , h 2 , h 3 , h 4 , h 5 , h 6 } and E = {x 1 , x 2 , x 3 , x 4 } Where we have six houses in the defined universe, and x i ∈ E for i = 1, 2, 3, 4, stands for the parameters: x 1 = expensive, x 2 = beautiful, x 3 = wooden, x 4 = in green surrounding.

Assume:
International Journal of Mathematical Sciences and Optimization: Theory and Applications Vol. 7, No. 2, pp.76 -87 https://doi.org/10.52968/28302774 Suppose that: ) is a subset of U whose elements match the fuzzy set ŷi over E. Therefore, f Ã(ŷ 2 ) represents houses which are considered expensive to the degree 0.3 beautiful to the degree 0.5 wooden to the degree 0.1 in green surrounding 0.2 The functional value of f Ã(ŷ 2 ) is {h 2 , h 4 }.Hence, this means that ŷ2 indicates the degree to which the parameters in E intra-dependently give a description of the houses h 2 and h 4 (i.e., describes the degree of attractiveness of the houses h 2 and h 4 ).In this case, both houses are considered expensive to the degree 0.3, considered beautiful to the degree 0.5, considered wooden to the degree 0.1 and considered in a green surrounding to the degree 0.2.
Thus, considering Ẽ as a set of customers for the purchase of the houses, then the customer ŷ2 values the houses h 2 and h 4 as above.Considering the example above, it is clear in general, that crisp parameterized things contains high degree of uncertainty.Therefore, dealing with this, we will need to fuzzify the set of parameters in use.Now, considering example 2.1 in [3], parameter e 1 , is the predicate "expensive house" having the approximate value set {house h 2 ; house h 4 }.In this example, the question of uncertainty arises.Therefore, the question is "does it mean houses h 2 and h 4 can only be expensive or not?".Assuming this is so, then another question arises "what condition(s) or situation(s) necessitated this decision?".Before looking into the questions above, we first consider the work as presented by Cagman et al in [14], the authors considered dealing with the notion of "not absoluteness" of valuation (i.e., not using the valuation 0 or 1 only) by partially fuzzifying the set of parameters E. i.e, attaching an independent degree to each of parameters in E. We consider this attempt by the authors in [14] not sufficient to answer the questions.This is because all they did was to individually attach numeric values within the unit interval to the elements in E. This still have the same kind of valuation as in the classical soft set, in which case, parameters are still mapped individually.Therefore in this research, we attempt to proffer a solution to the questions above.Using example 3, we illustrate our proposed solution to the above questions.
Thus, by example 3, our predicate part for each approximation is fuzzy.In our case, unlike [14], all parameters in E are attached with choice numeric values and all together assigned against possible approximate value set.Therefore we have that parameters intra-dependently relate, i.e, putting into consideration the impact parameters have on one another.
With this, we consider that to a good extent, imprecision and uncertainty in crisp valuation are dealt with.
In this case, to find the ŷ − approximate of h 4 , we consider the arithmetic mean of ŷ2 and ŷ3 componentwise.
So h 4 approximation is: In a tabular form, we can represent the F F P S-set as follows: Table 1: Interpretation (for house h 1 ): h 1 is expensive to the degree 0.15, beautiful to the degree 0.2, wooden to the degree 0.3 and in green surrounding to the degree 0.35.
Clearly, this tabular representation of FFPS-Set is a form of generalization of table 1( Tabular representation of a soft set ) given by Maji et al in [3].
In which case: Interpretation (for house,h 1 ): h 1 is not expensive, not wooden, but beautiful, cheap and in green surrounding.Thus it is clear that the question of "to what extent...?" is not put into consideration.Therefore for example the phrase "not expensive" is absolute!Also, in the same way we represent in tabular form, the F P S-set presented by Cagman et al in [14]: Interpretation (for object u 2 ): Not x 1 , it is x 2 to the degree 0.8, it is x 3 to the degree 0.3, it is x 4 to the degree 0.5 and not x 5 .Viewing table 3 above, we still have the presence of crispness in the representation even though with the presence of other degree such as 0.2.But is not always the case that other parameters do not have some degree of influence on the value set.Thus, we consider the work by Cagman et al in [14] as a partial fuzzy parameterized soft set (P F P Sset).This gives rise to our name; F F P S-set.So, our work seek for full graded membership of set of parameters. 4 Defining Operations on F F P -soft set [14] Let F Ã ∈ F F P SS(U, Ẽ), then; Let Ã, B ⊆ Ẽ and F Ã, F B ∈ F F P SS(F, Ẽ), then F Ã is Full Fuzzy Parameterized Soft Subset (F F P S-subset) of F B , denoted by F Ã ⊆ F B if the following condition holds: ∀ y i ∈ Ã ∃ z j ∈ B such that y i ≤ z j and f Ã(y i ) ⊆ f B (z j ).

Table 3 :
F P S-set Table U/E x 1 x 2