An adjustable weighted soft discernibility matrix based on generalized picture fuzzy soft set and its applications in decision making

1 Introduction

The real world is full of imprecision, vagueness and uncertainty. In our daily life, we mostly deal with unclear concepts rather than exact ones. Dealing with imprecision is a big problem in many areas such as economics, medical science, social science, environmental science and engineering. In recent years, model vagueness has become interested in many authors. Many classical theories such as fuzzy set theory [1], probability theory, vague set theory [2], rough set theory [3], intuitionistic fuzzy set [4] and interval-valued fuzzy set [5] are well known and effectively model uncertainty. These approaches show their inherent difficulties as pointed out by Molodtsov [6], because of intensive quantity and type of uncertainties. In [6], Molodtsov defines the soft set which is an absolutely new logical instrument for dealing uncertainties. Soft set theory attracts many authors because it has a vast range of applications in many areas like the smoothness of functions, decision-making, probability theory, data analysis, measurement theory, forecasting and operations research [6–10].

Nowadays, many authors work on aggregation operators to solve decision-making problems using intuitionistic, hesitant, Pythagorean, picture fuzzy environments [11–15]. Agarwal et al. [16] defines the generalized intuitionistic fuzzy soft set (GIFSS) which has some problems pointed out by Feng et al. [17] and redefined GIFSS. Khalil et al. [18], define the interval-valued picture fuzzy sets and interval-valued picture fuzzy soft sets.

In [19], Coung defines the picture fuzzy set which is an extension of the fuzzy soft set and intuitionistic fuzzy set. In [20], Sing defines Correlation coefficients of PFS and their applications in clustering analysis. Zhang et. al. [21], proposed an algorithm for location selection of offshore wind power station by consensus decision framework using picture fuzzy modelling. Generalized dice similarity measures for picture fuzzy sets are defined by Wei and Gao [22]. Wei expand the TODIM model to the multi-attribute decision-making (MADM) with the picture fuzzy numbers [23]. The EDAS (evaluation based on distance from average solution) model with picture 2-tuple linguistic numbers is proposed by Zhang et al. [24]. A picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project is proposed by Wang et al. [25]. Wei [26] solves MADM problems by picture fuzzy Hamacher aggregation operators. Wei [27], defines the picture fuzzy aggregation operators method and using them to MADM for ordering enterprise resource planning (ERP) structures. Wei [28], in the light of the picture fuzzy weighted cross-entropy a basic leadership technique is researched and used it to order the choices. In [29], Garg defines aggregation operations on picture fuzzy soft set (PFSS) and used them to multi-criteria decision-making (MCDM) problems. In [30], Khan et al. defines the generalized picture fuzzy soft set (GPFSS) and applied them to decision-making problems. For study more about decision-making, we refer to [31–40].

First, Skowron and Rauszer [41], initiated the concept of discernibility matrix and extensively used in rough sets to solve attribute reduction, and the influence of it are significant and easy to understand. The soft matrix and its operations in a soft set are defined by Cagman [42]. The notion of soft discernibility matrix (SDM) is given by Feng and Zhou, which not only provide the best choice but also an order relation among all alternatives [43]. In [44], Feng et al. defines an adjustable approach for the fuzzy soft set and an adjustable approach for the intuitionistic fuzzy soft set is defined by Jiang et al. [45]. Yang et al. [46], define an adjustable soft discernibility matrix (SDM) for picture fuzzy soft sets. For study more about soft matrix, we refer to [47–49].

The purpose of this paper is to use WSDM for GPFSSs using adjustable perspective to solve decision-making problems. In literature, GPFSS set is defined and applied for decision-making problems using picture fuzzy weighted averaging operators. This technique can not only give the best alternative but also an order relation of all alternatives easily by scanning the WSDM at most one time. The idea of GPFSS is very encouraging in decision-making since it considers how to capitalize an additional picture fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists. Also, in our daily life decision-making problems, different attributes are not of equal importance. Some are more important than others, therefore, the decision maker assigns different values (weights) to different attributes and imposes different threshold functions when we need a restriction to a positive membership function and negative membership function. In this paper, we use an adjustable perspective to GPFSS and get level soft sets. Then each GPFSS can be seen as a level soft set and composed a crisp soft set, therefore, for solving decision-making problems we apply weighted soft discernibility matrix (WSDM).

The rest of this paper is organized as follows. Section 2 consists of the preliminaries which include the basic definitions related to the fuzzy sets, intuitionistic fuzzy set and SDM. Section 3 is devoted to the threshold functions and their level soft sets. In Section 4, two algorithms are proposed on the basis of WSDM to solve decision-making problem using GPFSS. Section 5 consist of a case study of scholarship for a doctoral degree. Finally, comparison and conclusion are given in Sections 6, 7.

2 Preliminaries

In this section, we present the basic definitions of fuzzy set, intuitionistic fuzzy set, soft set, picture fuzzy set, picture fuzzy soft set, generalized picture fuzzy soft set, soft discernibility matrix and weighted soft discernibility matrix.

Throughout this paper, finite set Y ˆ = { ℏ 1 , ℏ 2 , . . . , ℏ n } and P ˆ = { ε 1 , ε 2 , . . . , ε m } represents the set of n alternatives and m attributes (parameters). The abbreviations IFS, PFS, PFSS, GPFSS, SDP, SDM and WSDM represents the intuitionistic fuzzy set, picture fuzzy set, picture fuzzy soft set, generalized picture fuzzy soft set, soft discernibility parameters, soft discernibility matrix and weighted soft discernibility matrix, respectively. Moreover, the abbreviation “w.r.t." is used for “with respect to."

A fuzzy set is defined by Zadeh [1], which handles uncertainty based on the view of gradualness effectively.

Definition 1. [1] A membership function ξ B ˆ : Y ˆ → [ 0 , 1 ] defines the fuzzy set B ˆ over the Y ˆ , where ξ B ˆ ( ℏ ) particularized the membership of an element ℏ ∈ Y ˆ in fuzzy set B ˆ .

Like a membership degree on an element in a fuzzy set, human intuition suggests that there is a non-membership degree of an element in a set. In [4], an IFS defined by Atanassov to sketch the imprecision of human beings when needed the judgments over the elements.

Definition 2. [4] An IFS B ˆ over the universal set Y ˆ is defined as B ˆ = { ( ℏ , ξ B ˆ , ϑ B ˆ ) | ℏ ∈ Y ˆ } ,(1) where ξ B ˆ : Y ˆ → [ 0 , 1 ] and ϑ B ˆ : Y ˆ → [ 0 , 1 ] are the degree of positive membership and degree of negative membership, respectively. Furthermore, it is required that 0 ≤ ξ B ˆ + ϑ B ˆ ≤ 1 .

A soft set is defined by Molodtsov [6], which provides an effective framework to dealings with imprecision with the parametric point of view, i.e. each element is judged by some criteria of attributes.

Definition 3. [6] Let Y ˆ be a universal set, P ˆ a parameter space, B ˆ ⊆ P ˆ and P ( Y ˆ ) be the power set of Y ˆ . A pair ( F ˆ , B ˆ ) is called a soft set over Y ˆ , where F ˆ is a set valued mapping given by F ˆ : B ˆ → P ( Y ˆ ) .

In [19], Cuong defines the PFS, which is an extension of fuzzy set and applicable in many real life problems.

Definition 4. [19] A PFS B ˆ over the universal set Y ˆ is defined as B ˆ = { ( ℏ , ξ B ˆ , η B ˆ , ϑ B ˆ ) | ℏ ∈ Y ˆ } ,(2) where ξ B ˆ : Y ˆ → [ 0 , 1 ] , η B ˆ : Y ˆ → [ 0 , 1 ] and ϑ B ˆ : Y ˆ → [ 0 , 1 ] are the degrees of positive membership, neutral membership and negative membership, respectively. Furthermore, it is required that 0 ≤ ξ B ˆ + η B ˆ + ϑ B ˆ ≤ 1 .

In [46], Yang defines the PFSS as follows.

Definition 5. [46] Let Y ˆ be a universal set, P ˆ a parameter space, B ˆ ⊂ P ˆ and PF ( Y ˆ ) be the set of all PFSs over Y ˆ . A pair ( F ˆ , B ˆ ) is called a PFSS over Y ˆ , where F ˆ is a set valued mapping given by F ˆ : B ˆ → PF ( Y ˆ ) .

In [30], Khan et al. defines the GPFSS. The idea of GPFSS is very encouraging in decision-making since it considers how to capitalize an additional picture fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists.

Definition 6. [30] Let Y ˆ be a universal set, B ˆ ⊂ P ˆ a parametric set. By a GPFSS we mean a triple ( F ˆ , B ˆ , ρ ˆ ) , where ( F ˆ , B ˆ ) is a PFSS over Y ˆ and ρ ˆ : B ˆ → PF ( B ˆ ) is a PFS in B ˆ .

Where ( F ˆ , B ˆ ) is called basic picture fuzzy soft set (BPFSS) and ρ ˆ is called the parametric picture fuzzy set (PPFS). Definition 7. [50] Let Y ˆ be a universal set, P ˆ a parameter space, B ˆ ⊆ P ˆ and P ( Y ˆ × Y ˆ ) be the power set of Y ˆ × Y ˆ . If F ˆ : B ˆ → P ( Y ˆ × Y ˆ ) is a mapping then the soft set ( F ˆ , B ˆ ) is called a soft binary relation over Y ˆ .

Definition 8. [50] We called a soft binary relation ( F ˆ , B ˆ ) to the soft equivalence relation over Y ˆ if F ˆ ( ε ) ≠ ∅ is an equivalence relation on Y ˆ for all ε ∈ B ˆ .

Definition 9. [50] Let ( F ˆ , B ˆ ) be a soft set. The F ˆ ( ε ) , ε ∈ B ˆ divides Y ˆ into two classes and defines an equivalence relation. Then in-discernibility relation is denoted and defined as the intersection of all equivalence relation as IND ( F ˆ , B ˆ ) = ∩ ε ∈ B ˆ F ˆ ( ε ) .

In [43], Feng et al. defines the SDM for soft sets which provide not only the best alternative but also an order relation among all the alternatives.

Definition 10. [43] Let ( F ˆ , B ˆ ) be a soft set over Y ˆ . F ˆ determined the partition U | IND ( F ˆ , B ˆ ) = { N i : i ≤ | Y ˆ | } of Y ˆ . The SDM is defined as M = ( M ( N i , N j ) ) i , j ≤ | Y ˆ | , where M (N _i , N _j ) is called the SDPs among N _i and N _j and defined as M ( N i , N j ) = { E ˆ i ∪ E ˆ j : i , j ≤ | Y ˆ | } .(3) In which E ˆ i = { ε p i : F ˆ ( ℏ i , ε p ) = 1 and F ˆ ( ℏ j , ε p ) = 0 , ∀ ℏ i ∈ N i , ∀ ℏ j ∈ N j } , E ˆ j = { ε p j : F ˆ ( ℏ j , ε p ) = 1 and F ˆ ( ℏ i , ε p ) = 0 , ∀ ℏ j ∈ N j , ∀ ℏ i ∈ N i } . The symbol ε p i (or ε p j ) represents the elements in N _i (or N _j ) have the value 1 at the attribute ε _p , that is, F ˆ ( ℏ i , ε p ) = 1 , ℏ _i  ∈ N _i (or F ˆ ( ℏ j , ε p ) = 1 , ℏ _j  ∈ N _j ).

The WSDM is defined as follows.

Definition 11. [43] Let ( F ˆ , B ˆ ) be a soft set over Y ˆ . F ˆ determined the partition U | IND ( F ˆ , B ˆ ) = { N i : i ≤ | Y ˆ | } of Y ˆ . The WSDM is defined as M = ( M ( N i , N j ) ) i , j ≤ | Y ˆ | , where M (N _i , N _j ) is called the SDPs among N _i and N _j and defined as M ( N i , N j ) = { E ˆ i ∪ E ˆ j : i , j ≤ | Y ˆ | } .(4) In which E ˆ i = { ε p i * ω ˆ i : F ˆ ( ℏ i , ε p ) = 1 and F ˆ ( ℏ j , ε p ) = 0 , ∀ ℏ i ∈ N i , ∀ ℏ j ∈ N j } , E ˆ j = { ε p j * ω ˆ j : F ˆ ( ℏ j , ε p ) = 1 and F ˆ ( ℏ i , ε p ) = 0 , ∀ ℏ j ∈ N j , ∀ ℏ i ∈ N i } . The symbol ε p i * ω ˆ i (or ε p j * ω ˆ j ) represents the elements in N _i (or N _j ) have the value 1 at the parameter ε _p , that is, F ˆ ( ℏ i , ε p ) = 1 , ℏ _i  ∈ N _i (or F ˆ ( ℏ j , ε p ) = 1 , ℏ _j  ∈ N _j ).

In [43], Feng et al. also give some properties of SDM and WSDM.

Proposition 1. [43] Let ( F ˆ , B ˆ ) be a soft set over Y ˆ , where Y ˆ = { ℏ 1 , ℏ 2 , . . . , ℏ n } and φ (N _i , N _j ) = |M (N _i , N _j ) |. Then the SDM has the following characteristics:

M (N _i , N _i ) =∅ (∀i ≤ n);

Decision-making relies on the evaluation of all available alternatives with respect to certain criteria. Some of these problems are essentially humanistic and thus subjective in nature. In general, there does not exist a unique/uniform principle for making the optimal decision. To overcome all the above difficulties, we should use an adjustable framework to solve fuzzy soft set based decision-making problems. A proposal of flexible feature was initiated by the authors in [44], using the following novel concept called level soft sets. Level soft sets can be viewed as “soft generalizations” of classical level (cut) sets in fuzzy set theory. For real-life applications, this threshold is chosen by a decision maker; it represents the personal requirement on the level of membership degrees. It is natural to say that an alternative satisfies a criterion if it meets the desirable level required by a decision maker.

Definition 12. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. For a quadruple (k, l, m, α), where (k, l, m) , α ∈ [0, 1] ³, the (k, l, m, α)-level soft set of Γ ˆ is a crisp soft set L ( Γ ˆ ; ( k , l , m , α ) ) = ( F ˆ ( k , l , m ) , B ˆ , ρ ˆ α ) defined by

L ( Γ ˆ ; k , l , m ) = ( F ˆ ( k , l , m ) , B ˆ ) is a (k, l, m) level-soft set of a picture fuzzy soft set defined by F ˆ ( k , l , m ) ( ε ) = L ( F ˆ ( ε ) ; k , l , m ) = { ℏ ∈ Y ˆ | ξ F ˆ ( ε ) ( ℏ ) ≥ k , η F ˆ ( ε ) ( ℏ ) ≤ l and ϑ F ˆ ( ε ) ( ℏ ) ≤ m } , ∀ ε ∈ B ˆ .

Example 1. Department appointment to senior positions in an institute and an applicant may be judged by criteria such as “creativity" and “managerial skills" and so forth. Suppose that there are five candidates Y ˆ = { ℏ 1 , ℏ 2 , ℏ 3 , ℏ 4 , ℏ 5 , ℏ 6 } for senior positions and B ˆ = { ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 } is a criteria for evaluating candidates, where each ε _i stands for “creativity", “managerial skills", “intuition", “research productivity", “ability to work under pressure" and “knowledge", respectively. The tabular representation of GPFSS ( F ˆ , B ˆ , ρ ˆ ) is shown in Table 1. Now we take k = 0.5, l = 0.2, m = 0.3 and α = (0.5, 0.2, 0.2), then we have the following: L ( F ˆ ( ε 1 ) ; 0.5 , 0.2 , 0.3 ) = { ℏ 1 , ℏ 6 } , L ( F ˆ ( ε 2 ) ; 0.5 , 0.2 , 0.3 ) = { ℏ 1 , ℏ 4 , ℏ 5 , ℏ 6 } , L ( F ˆ ( ε 3 ) ; 0.5 , 0.2 , 0.3 ) = { ℏ 3 , ℏ 5 } , L ( F ˆ ( ε 4 ) ; 0.5 , 0.2 , 0.3 ) = { ℏ 1 , ℏ 3 } , L ( F ˆ ( ε 5 ) ; 0.5 , 0.2 , 0.3 ) = { ℏ 2 , ℏ 3 , ℏ 4 , ℏ 6 } , L ( F ˆ ( ε 6 ) ; 0.5 , 0.2 , 0.3 ) = { ℏ 1 , ℏ 4 , ℏ 5 , ℏ 6 } , and L ( ρ ˆ ; 0.5 , 0.2 , 0.2 ) = { ε 2 , ε 4 } .(5) Hence the (0.5, 0.2, 0.3, (0.5, 0.2, 0.2))-level soft set of Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) is a soft set L = ( Γ ˆ ; 0.5 , 0.2 , 0.3 , ( 0.5 , 0.2 , 0.2 ) ) = ( F ˆ ( 0.5 , 0.2 , 0.3 ) , B ˆ , ρ ˆ ( 0.5 , 0.2 , 0.2 ) ) , where the set valued mappings F ˆ ( 0.5 , 0.2 , 0.3 ) : B ˆ → P ( Y ˆ ) and ρ ˆ ( 0.5 , 0.2 , 0.2 ) : B ˆ → P ( B ˆ ) are defined as F ˆ ( 0.5 , 0.2 , 0.3 ) ( ε ) = L ( F ˆ ( ε ) ; 0.5 , 0.2 , 0.3 ) for all ε ∈ B ˆ and ρ ˆ ( 0.5 , 0.2 , 0.2 ) = L ( ρ ˆ ; ( 0.5 , 0.2 , 0.2 ) ) , where P ( Y ˆ ) and P ( B ˆ ) are power sets of Y ˆ and B ˆ , respectively. The tabular representation of (0.5, 0.2, 0.3, (0.5, 0.2, 0.2))-level soft set is given in Table 2.

Now we show some properties of (k, l, m, α)-level soft sets.

Theorem 1. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. Let L ( Γ ˆ ; k 1 , l 1 , m 1 , α 1 ) and L ( Γ ˆ ; k 2 , l 2 , m 2 , α 2 ) are (k₁, l₁, m₁, α₁) and (k₂, l₂, m₂, α₂)-level soft set of Γ ˆ , respectively, where k₁, l₁, m₁, k₂, l₂, m₂ ∈ [0, 1] and α₁ = (u₁, v₁, w₁) , α₂ = (u₂, v₂, w₂) ∈ [0, 1] ³.

If k₁ ≥ k₂, l₂ ≥ l₁, m₂ ≥ m₁, u₁ ≥ u₂, v₂ ≥ v₁ and w₂ ≥ w₁ then we have L ( Γ ˆ ; k 1 , l 1 , m 1 , α 1 ) ⊆ L ( Γ ˆ ; k 2 , l 2 , m 2 , α 2 ) .

Proof. To complete the proof, we need to show

L ( Γ ˆ ; k 1 , l 1 , m 1 ) ⊆ L ( Γ ˆ ; k 2 , l 2 , m 2 ) ,

Since k₁ ≥ k₂, l₂ ≥ l₁ and m₂ ≥ m₁ then for all ε ∈ B ˆ we have the following:

{ ℏ ∈ Y ˆ | ξ F ˆ ( ε ) ( ℏ ) ≥ k 1 , η F ˆ ( ε ) ( ℏ ) ≤ l 1 , ϑ F ˆ ( ε ) ( ℏ ) ≤ m 1 } ⊆ { ℏ ∈ Y ˆ | ξ F ˆ ( ε ) ( ℏ ) ≥ k 2 , η F ˆ ( ε ) ( ℏ ) ≤ l 2 , ϑ F ˆ ( ε ) ( ℏ ) ≤ m 2 }

F ˆ ( k 1 , l 1 , m 1 ) ( ε ) ⊆ F ˆ ( k 2 , l 2 , m 2 ) ( ε ) L ( Γ ˆ ; k 1 , l 1 , m 1 ) ⊆ L ( Γ ˆ ; k 2 , l 2 , m 2 ) . Let α₁ = (u₁, v₁, w₁) and α₂ = (u₂, v₂, w₂). Since u₁ ≥ u₂, v₂ ≥ v₁ and w₂ ≥ w₁. We have { ε ∈ B ˆ | ξ ρ ˆ ( ε ) ≥ u 1 , η ρ ˆ ( ε ) ≤ v 1 and ϑ ρ ˆ ( ε ) ≤ w 1 } ⊆ { ε ∈ B ˆ | ξ ρ ˆ ( ε ) ≥ u 2 , η ρ ˆ ( ε ) ≤ v 2 and ϑ ρ ˆ ( ε ) ≤ w 2 } L ( ρ ˆ ; α 1 ) ⊆ L ( ρ ˆ ; α 2 ) . Hence we have L ( Γ ˆ ; k 1 , l 1 , l 1 , α 1 ) ⊆ L ( Γ ˆ ; k 2 , l 2 , m 2 , α 2 ) .(6) □

Definition 13. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. Then the threshold function ψ : B ˆ → [ 0 , 1 ] 3 , i.e., ψ (ε) = (k (ε) , l (ε) , m (ε)), ψ ρ ˆ = ( u ψ , v ψ , w ψ ) and k (ε) , l (ε) , m (ε) , u _ψ , v _ψ , w _ψ  ∈ [0, 1] for all ε ∈ B ˆ . The level soft set of Γ ˆ with respect to ψ is a crisp soft set L ( Γ ˆ ; ψ ) = ( F ˆ ψ , B ˆ , ρ ˆ ψ ) defined by L ( Γ ˆ ; ψ ( ε ) ) = ( F ˆ ψ ( ε ) , B ˆ ) , where F ˆ ψ ( ε ) = L ( F ˆ ( ε ) ; ψ ( ε ) ) = { ℏ ∈ Y ˆ | ξ F ˆ ( ε ) ( ℏ ) ≥ k ( ε ) , η F ˆ ( ε ) ( ℏ ) ≤ l ( ε ) and ϑ F ˆ ( ε ) ( ℏ ) ≤ m ( ε ) | for all ε ∈ B ˆ } and ρ ˆ ψ = L ( ρ ˆ ; ψ ) = { ε ∈ B ˆ | ξ ρ ˆ ( ε ) ≥ u ψ , η ρ ˆ ( ε ) ≤ v ψ and ϑ ρ ˆ ( ε ) ≤ w ψ | for all ε ∈ B ˆ } .

Definition 14. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. Then mid threshold function mid Γ ˆ : B ˆ → [ 0 , 1 ] 3 , i.e., mid Γ ˆ ( ε ) = ( k mid Γ ˆ ( ε ) , l mid Γ ˆ ( ε ) , m mid Γ ˆ ( ε ) ) and mid Γ ˆ ( ρ ˆ ) = ( u mid Γ ˆ , v mid Γ ˆ , w mid Γ ˆ ) for all ε ∈ B ˆ , where k mid Γ ˆ ( ε ) = 1 | Y ˆ | ∑ ℏ ∈ Y ˆ ξ F ˆ ( ε ) ( ℏ ) , l mid Γ ˆ ( ε ) = 1 | Y ˆ | ∑ ℏ ∈ Y ˆ η F ˆ ( ε ) ( ℏ ) , m mid Γ ˆ ( ε ) = 1 | Y ˆ | ∑ ℏ ∈ Y ˆ ϑ F ˆ ( ε ) ( ℏ ) , u mid Γ ˆ = 1 | B ˆ | ∑ ε ∈ B ˆ ξ ρ ˆ ( ε ) , v mid Γ ˆ = 1 | B ˆ | ∑ ε ∈ B ˆ η ρ ˆ ( ε ) and w mid Γ ˆ = 1 | B ˆ | ∑ ε ∈ B ˆ ϑ ρ ˆ ( ε ) . L ( Γ ˆ ; mid ) is a notation used to represents the corresponding level soft set of Γ ˆ and called the mid-level soft set of Γ ˆ .

Definition 15. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. Then top-bottom-bottom threshold function tbb Γ ˆ : B ˆ → [ 0 , 1 ] 3 , i.e., tbb Γ ˆ ( ε ) = ( k tbb Γ ˆ ( ε ) , l tbb Γ ˆ ( ε ) , m tbb Γ ˆ ( ε ) ) and tbb Γ ˆ ( ρ ˆ ) = ( u tbb Γ ˆ , v tbb Γ ˆ , w tbb Γ ˆ ) for all ε ∈ B ˆ , where k tbb Γ ˆ ( ε ) = max ℏ ∈ Y ˆ ξ F ˆ ( ε ) ( ℏ ) , l tbb Γ ˆ ( ε ) = min ℏ ∈ Y ˆ η F ˆ ( ε ) ( ℏ ) , m tbb Γ ˆ ( ε ) = min ℏ ∈ Y ˆ ϑ F ˆ ( ε ) ( ℏ ) , u tbb Γ ˆ = max ε ∈ B ˆ ξ ρ ˆ ( ε ) , v tbb Γ ˆ = min ε ∈ B ˆ η ρ ˆ ( ε ) , w tbb Γ ˆ = min ε ∈ B ˆ ϑ ρ ˆ ( ε ) . L ( Γ ˆ ; tbb ) is a notation used to represents the corresponding level soft set of Γ ˆ and called the top-bottom-bottom-level soft set of Γ ˆ .

Definition 16. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. Then top-top-top threshold function ttt Γ ˆ : B ˆ → [ 0 , 1 ] 3 , i.e., ttt Γ ˆ ( ε ) = ( k ttt Γ ˆ ( ε ) , l ttt Γ ˆ ( ε ) , m ttt Γ ˆ ( ε ) ) and ttt Γ ˆ ( ρ ˆ ) = ( u ttt Γ ˆ , v ttt Γ ˆ , w ttt Γ ˆ ) for all ε ∈ B ˆ , where k ttt Γ ˆ ( ε ) = max ℏ ∈ Y ˆ ξ F ˆ ( ε ) ( ℏ ) , l ttt Γ ˆ ( ε ) = max ℏ ∈ Y ˆ η F ˆ ( ε ) ( ℏ ) , m ttt Γ ˆ ( ε ) = max ℏ ∈ Y ˆ ϑ F ˆ ( ε ) ( ℏ ) , u ttt Γ ˆ = max ε ∈ B ˆ ξ ρ ˆ ( ε ) , v ttt Γ ˆ = max ε ∈ B ˆ η ρ ˆ ( ε ) , w ttt Γ ˆ = max ε ∈ B ˆ ϑ ρ ˆ ( ε ) . L ( Γ ˆ ; ttt ) is a notation used to represents the corresponding level soft set of Γ ˆ and called the top-top-top-level soft set of Γ ˆ .

Definition 17. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ be a set of parameters. Then bottom-bottom-bottom threshold function bbb Γ ˆ : B ˆ → [ 0 , 1 ] 3 , i.e., bbb Γ ˆ ( ε ) = ( k bbb Γ ˆ ( ε ) , l bbbb Γ ˆ ( ε ) , m bbbb Γ ˆ ( ε ) ) and bbb Γ ˆ ( ρ ˆ ) = ( u bbb Γ ˆ , v bbb Γ ˆ , w bbb Γ ˆ ) for all ε ∈ B ˆ , where k bbb Γ ˆ ( ε ) = min ℏ ∈ Y ˆ ξ F ˆ ( ε ) ( ℏ ) , l bbb Γ ˆ ( ε ) = min ℏ ∈ Y ˆ η F ˆ ( ε ) ( ℏ ) , m bbb Γ ˆ ( ε ) = min ℏ ∈ Y ˆ ϑ F ˆ ( ε ) ( ℏ ) , u bbb Γ ˆ = min ε ∈ B ˆ ξ ρ ˆ ( ε ) , v bbb Γ ˆ = min ε ∈ B ˆ η ρ ˆ ( ε ) , w bbb Γ ˆ = min ε ∈ B ˆ ϑ ρ ˆ ( ε ) . L ( Γ ˆ ; bbb ) is a notation used to represents the corresponding level soft set of Γ ˆ and called the bottom-bottom-bottom-level soft set of Γ ˆ .

Definition 18. Let Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) be a GPFSS over Y ˆ and B ˆ ⊆ P ˆ a set of parameters. Then med threshold function med Γ ˆ : B ˆ → [ 0 , 1 ] 3 , i.e., med Γ ˆ ( ε ) = ( k med Γ ˆ ( ε ) , l med Γ ˆ ( ε ) , m med Γ ˆ ( ε ) ) and med Γ ˆ ( ρ ˆ ) = ( u med Γ ˆ , v med Γ ˆ , w med Γ ˆ ) , for all ε ∈ B ˆ , where for all ε ∈ B ˆ . If we align ascendingly (or descendingly) the value of membership, neutral and non-membership functions of all elements of BPFSS, then k med Γ ˆ ( ε ) , l med Γ ˆ ( ε ) and m med Γ ˆ ( ε ) represents the medians, where k med Γ ˆ ( ε ) = { ξ F ˆ ( ε ) ( ℏ ( | Y ˆ | + 1 2 ) ) , if | Y ˆ | is odd , ( ξ F ˆ ( ε ) ( ℏ ( | Y ˆ | 2 ) ) + ξ F ˆ ( ε ) ( ℏ ( | Y ˆ | 2 + 1 ) ) ) / 2 , if | Y ˆ | is even , l med Γ ˆ ( ε ) = { η F ˆ ( ε ) ( ℏ ( | Y ˆ | + 1 2 ) ) , if | Y ˆ | is odd , ( η F ˆ ( ε ) ( ℏ ( | Y ˆ | 2 ) ) + η F ˆ ( ε ) ( ℏ ( | Y ˆ | 2 + 1 ) ) ) / 2 , if | Y ˆ | is even , m med Γ ˆ ( ε ) = { ϑ F ˆ ( ε ) ( ℏ ( | Y ˆ | + 1 2 ) ) , if | Y ˆ | is odd , ( ϑ F ˆ ( ε ) ( ℏ ( | Y ˆ | 2 ) ) + ϑ F ˆ ( ε ) ( ℏ ( | Y ˆ | 2 + 1 ) ) ) / 2 , if | Y ˆ | is even . If we align ascendingly (or descendingly) the value of membership, neutral and non-membership functions of all elements of PPFS, then u med Γ ˆ , v med Γ ˆ and w med Γ ˆ represents the medians, where u med Γ ˆ ( ε ) = { ξ ρ ˆ ( ε ) ( ε ( | B ˆ | + 1 2 ) ) , if | B ˆ | is odd , ( ξ ρ ˆ ( ε ) ( ε ( | B ˆ | 2 ) ) + ξ ρ ˆ ( ε ) ( ε ( | B ˆ | 2 + 1 ) ) ) / 2 , if | B ˆ | is even , v med Γ ˆ ( ε ) = { η ρ ˆ ( ε ) ( ε ( | B ˆ | + 1 2 ) ) , if | B ˆ | is odd , ( η ρ ˆ ( ε ) ( ε ( | B ˆ | 2 ) ) + η ρ ˆ ( ε ) ( ε ( | B ˆ | 2 + 1 ) ) ) / 2 , if | B ˆ | is even , w med Γ ˆ ( ε ) = { ϑ ρ ˆ ( ε ) ( ε ( | B ˆ | + 1 2 ) ) , if | B ˆ | is odd , ( ϑ ρ ˆ ( ε ) ( ε ( | B ˆ | 2 ) ) + ϑ ρ ˆ ( ε ) ( ε ( | B ˆ | 2 + 1 ) ) ) / 2 , if | B ˆ | is even . L ( Γ ˆ ; med ) is a notation used to represents the corresponding level soft set of Γ ˆ and called the med-level soft set of Γ ˆ .

Example 2. Let us consider the GPFSS Γ ˆ = ( F ˆ , B ˆ , ρ ˆ ) in Example 1. The threshold functions mentioned above and their level soft sets with tabular representation (Tables 3-7) are given as follows.

mid Γ ˆ = { ( 0.38 , 0.25 , 0.25 ) / ε 1 , ( 0.45 , 0.18 , 0.3 ) / ε 2 , ( 0.4 , 0.13 , 0.4 ) / ε 3 , ( 0.38 , 0.13 , 0.32 ) / ε 4 , ( 0.5 , 0.17 , 0.23 ) / ε 5 , ( 0.48 , 0.22 , 0.25 ) / ε 6 } , mid Γ ˆ ( ρ ˆ ) = ( 0.52 , 0.18 , 0.28 ) .