Spherical fuzzy soft sets and its applications in decision-making problems

Abstract

In this paper, the definition of spherical fuzzy soft sets and some of its properties are introduced. Spherical fuzzy soft sets are a generalization of soft sets. Notably, we showed DeMorgan’s laws that are valid in spherical fuzzy soft set theory. Also, we propose an algorithm to solve the decision-making problem based on adjustable soft discernibility matrix. It gives an order relation between all the objects of our universe. Finally, an illustrative example is discussed to prove that they can be effectively used to solve problems with uncertainties.

Keywords

Soft sets spherical fuzzy sets spherical fuzzy soft sets soft discernibility matrix decision-making problem

1 Introduction

Decision-making problems are a huge part of human society and applied widely to practical fields like economics, management, engineering. However, with the development of science and technology, the uncertainty also plays a dominant factor during the decision-making (DM) analysis. Further, the role of the decision-makers during the process is so challenging to collect precise data. Most of the information collected from the various resources are either uncertain or imprecise and hence leads to inaccurate results. Thus, the task of the DM process is to find the finest objects among the available by utilizing this imprecise or uncertain information. Traditionally, the information extracted from the various resources is crisp, deterministic and precise, which is less adaptable to apply in real-life problems due to the uncertainties arises everywhere. To deal with such uncertainties, many theories such as fuzzy set [1], rough set [2], vague set [3], etc., came into existence. But these theories have their limitations and inherent difficulties. For example, fuzzy sets developed by zadeh [1] was initially used for many applications. But the non-membership value of an element in the fuzzy sets could not be derived from its membership value. So to overcome some drawbacks of fuzzy sets, Atanassov developed intuitionistic fuzzy sets (IFSs) [4] by introducing two indices, membership degree (P (x)) and non-membership degree (N (x)) such that 0 ≤ P (x) + N (x) ≤1. Later, the interval-valued intuitionistic fuzzy sets were developed by Atanassov and Gargov [5]. Interval-valued intuitionistic fuzzy sets are the extension of fuzzy sets and intuitionistic fuzzy sets. But in some real-life cases, the decision-makers deal with the situations in which the sum of membership grade and non-membership grade exceed 1. To overcome the situation Yager developed the Pythagorean fuzzy sets [6] with the condition 0 ≤ P² (x) + N² (x) ≤1. It is a generalization of IFSs. But these theories cannot show the neutral state (which neither favor nor disfavor). Based on these circumstances, to overcome the situation the idea of picture fuzzy set was developed by Cuong and Kreinovich [7] by utilized three indices, positive membership degree (P (x)), neutral membership degree (I (x)) and negative membership degree (N (x)), such that 0 ≤ P (x) + I (x) + N (x) ≤1. Surely it is more useful and it has more applications than intuitionistic fuzzy sets and Pythagorean fuzzy sets. The application of these sets for solving the decision-making problems is widely applied in the different fields [8 –11].

However, from these theories and their applications, we have observed that in real-life decision-making problems, there are several kinds of problems which can’t be handled by these IFSs, Pythagorean fuzzy sets or picture fuzzy sets. That is, the case that the sum of the positive membership degree, neutral membership degree, and negative membership degree exceeds 1. So to overcome this situation, Ashraf et al. [12] introduced the idea of spherical fuzzy sets with the condition 0 ≤ P² (x) + I² (x) + N² (x) ≤1. The notion of spherical sets gives us a new way of representation to some of the problems which look quite difficult to explain using existing other extensions of fuzzy set theory, such as problems involving human opinions, involving different responses of the type yes, abstain, no and refusal. At the same time, another version of the spherical fuzzy set was introduced by Kutlu Gündoğdu and Kahraman [13]. Instead of neutral membership degree, they used hesitancy degree (π (x)) as one of the three indices. The other two indices are positive membership degree (P (x)) and negative membership degree (N (x)). Also, all the above three indices satisfies 0 ≤ P² (x) + π² (x) + N² (x) ≤1. Later, an extension of WASPAS with spherical fuzzy sets, VIKOR method using spherical fuzzy sets and correlation coefficients were presented by the researchers in [14 –17].

Apart from these sets, Moltodtsov’s introduced another theory to deal with uncertainty and named as soft set theory [18] to deal with the parameterizations factor during the analysis. By embedding the ideas of fuzzy sets with the soft set, Maji et al. [19] introduced the concept of fuzzy soft sets (FSSs). Later, a concept of intuitionistic FSS (IFSS) is introduced by Maji et al. [20]. Yang et al. [21] combined the concept of a soft set with an interval-valued fuzzy set. Since its appearance, several researchers have presented different kinds of methods and algorithms for solving the decision-making problems under the soft set environment. For example, Feng and Zhou [22] presented the soft discernibility matrix. Çağman and Enginoğlu [23] presented the concept of the soft matrix. Arora and Garg [24] presented an algorithm for solving the decision-making problems based on the aggregation operators under the IFSSs. Apart from them, for some other approaches under the soft set environment, we may refer to read the papers [25 –31]. However, some extension models of soft sets are rapidly developed such as soft rough sets [32], generalized FSS [33], group generalized FSS [34], Pythagorean fuzzy soft sets [35], picture FSS [30], interval-valued picture FSS [36].

In an imprecise real world, the decision-making problem has found preeminent importance in recent years. The decision-making problem introduced as an application of soft sets by Maji et al. [37] with the help of rough mathematics by Pawlak [2]. They used an almost analogous representation of the soft sets in the form of a binary information table. Later, Roy and Maji [25] presented some results on an application of FSSs in decision-making problem. Kong et al. [38] presented the modified form of the Roy and Maji [25] algorithm. Later, an adjustable approach to IFSSs based on decision-making is presented by Jiang et al. [39]. Peng and Garg [40] presented an algorithm for interval-valued FSSs in emerging decision-making with some new information measures. Yang et al. [30] presented a decision-making approach based on adjustable soft discernibility matrix on picture FSSs. Subsequently, Kamacı et al. [41] proposed two new methods called a soft max-row decision-making method and a multi-soft distributive max-min decision-making methods. Recently, Kamacı et al. [42] presented a novel decision-making algorithm using matrix representation of the inverse soft set. Further, Kamacı [43] presented a bijective soft decision system based on the bijective soft matrix theory.

The objective of this paper is to combine spherical fuzzy sets and soft sets, from which a new soft set model named Spherical fuzzy soft sets are proposed. This new model is more realistic, practical and accurate in some cases than the existing other soft set models. Through this paper, the aim is to solve the decision-making problem using adjustable soft discernibility matrix. The presentation of the rest of this paper is structured as follows. In Section 2, the definitions of soft sets, discernibility matrix, soft discernibility matrix, fuzzy soft sets, Pythagorean fuzzy soft sets, and spherical fuzzy sets are recalled. In Section 3, the definition and some operations of spherical fuzzy soft sets are included. In Section 4, a decision-making problem is analyzed as an application of spherical fuzzy soft sets and an adjustable algorithm is discussed. Finally, the conclusion is drawn inSection 5.

2 Preliminaries

As preliminary, we recall the basic concept of soft sets [18], soft discernibility matrices [30], fuzzy soft sets [19], Pythagorean fuzzy soft sets [35] and Spherical fuzzy sets [12].

Let U be an initial universe set of objects and E the set of parameters in relation to objects in U. Parameters are often attributes, characteristics or properties of objects. Let P (U) denote the power set of U and A ⊆ E.

Definition 2.1. [18] A pair 〈F, A〉 is called a soft set over U, where F is a mapping given by F : A → P (U). In other words, a soft set over U is a parametrized family of subsets of the universe U. For e ∈ A, F (e) may be considered as the set of e-approximate elements of the soft set 〈F, A〉.

For two soft sets 〈F, A〉 and 〈G, B〉 over U, 〈F, A〉 is called a soft subset of 〈G, B〉 if,

A ⊆ B and

∀e ∈ A, F (e) ⊆ G (e).

Definition 2.2. [44] If F : A → P (U × U) be a mapping from a subset of parameter A to the power set of U × U, then the soft set 〈F, A〉 over U × U is called a soft binary relation over U.

Definition 2.3. [45] A soft binary relation 〈F, A〉 over a set U is called a soft equivalence relation over U, if F (e) ≠ φ is an equivalence relation on U for all e ∈ A.

Definition 2.4. [45] Suppose 〈F, A〉 is a soft equivalence relation over U, for each equivalence relation F (e), the notion of equivalence class over it is defined as [x] _F(e) = {y : (x, y) ∈ F (e) , y ∈ U}.

We also observe that there is an indiscernibility relation induced by the soft set 〈F, A〉 itself. This indiscernibility relation is obtained by intersection of all equivalence relations defined by parameters. Let us say IND〈F, A〉 = ⋂ _{e_i∈A}F (e_i).

Suppose 〈F, A〉 is a soft set over U, where U ={h₁, h₂, …, h_m}. The partition of U determined by the indiscernibility relation IND〈F, A〉 can be denoted by U|IND〈F, A〉, and U|IND〈F, A〉 ={C₁, C₂, …, C_i} (i ≤ m), where C_i = {[h_j] _IND〈F,A〉 : h_j ∈ U}.

Example 2.1. Let U = {h₁, h₂, h₃, h₄, h₅} be a set of houses under consideration, where E = {e₁, e₂, e₃, e₄, e₅} be a set of parameters and A = {e₁, e₂, e₃} be a subset of parameters for selection of the house. Let e₁ stands for beautiful houses, e₂ stands for low cost houses and e₃ stands for eco-friendly houses. Let 〈F, A〉 be the soft set to categorize the houses with respect to parameters given by set A, such that F (e₁) = {h₁, h₂, h₃, h₄}, F (e₂) = {h₂, h₃}, F (e₃) = {h₂, h₃, h₅}. The tabular representation of the soft set 〈F, A〉 is given in Table 1.

Table 1
Tabular form of soft set 〈F, A〉

U e ₁ e ₂ e ₃

h ₁ 1 0 0

h ₂ 1 1 1

h ₃ 1 1 1

h ₄ 1 0 0

h ₅ 0 0 1

U	e ₁	e ₂	e ₃
h ₁	1	0	0
h ₂	1	1	1
h ₃	1	1	1
h ₄	1	0	0
h ₅	0	0	1

From this table, it is obvious that each of the parameter e_i ; i = 1, 2, 3 induces an equivalence relation on U. So we have a soft equivalence relation say 〈F, A〉 over U. Hence we get the following equivalence classes for each of the equivalence relations as

for F (e₁), the equivalence classes are {h₁, h₂, h₃, h₄}, {h₅},

for F (e₂), the equivalence classes are {h₁, h₄, h₅}, {h₂, h₃},

for F (e₃), the equivalence classes are {h₁, h₄}, {h₂, h₃, h₅}.

The indiscernibility relation IND〈F, A〉 = ⋂ _{e_i∈A}F (e_i) and hence obtained as

\begin{matrix} IND 〈 F, A 〉 \\ = {\begin{matrix} (h_{1}, h_{1}), (h_{2}, h_{2}), (h_{3}, h_{3}), (h_{4}, h_{4}), (h_{5}, h_{5}), \\ (h_{1}, h_{4}), (h_{4}, h_{1}), (h_{2}, h_{3}), (h_{3}, h_{2}) \end{matrix}} \end{matrix}

Therefore, the partition of U obtained by indiscernibility relation IND〈F, A〉 is U|IND〈F, A〉 = {h₁, h₄}, {h₂, h₃}, {h₅}. So the indiscernibility relation over 〈F, A〉 partition the set U into three classes, that is,

\begin{matrix} C_{1} = {h_{1}, h_{4}}; C_{2} = {h_{2}, h_{3}}; C_{3} = {h_{5}} . \end{matrix}

Definition 2.5. [22] Let 〈F, A〉 be a soft set over U. Partition U|IND〈F, A〉 = {C_i : i ≤ |U|} is determined by F. $D = (D (C_{i}, C_{j}))_{i, j \leq | U |}$ is called partition class-based discernibility matrix over 〈F, A〉, where D (C_i, C_j) = {e_l ∈ A : F (h_i, e_l) ≠ F (h_j, e_l) , ∀ h_i ∈ C_i, ∀ h_j ∈ C_j} is called the set of discernibility parameter between C_i and C_j, and F (h_i, e_l) is the value of objects in C_i associated with parameter e_l.

Example 2.2. Consider Example 2.1. The discernibility matrix is represented in a tabular form as shown in Table 2.

Table 2

Discernibility matrix of Table 1

	C ₁	C ₂	C ₃
C ₁	φ
C ₂	{e₂, e₃}	φ
C ₃	{e₁, e₃}	{e₁, e₂}	φ

Definition 2.6. Let 〈F, A〉 be a soft set over U. Partition U|IND〈F, A〉 = {C_i : i ≤ |U|} is determined by F. The soft discernibility matrix is defined as $D = (D (C_{i}, C_{j}))_{i, j \leq | U |}$ , where D (C_i, C_j) = {Eⁱ ∪ E^j : i, j ≤ |U|} is called the set of soft discernibility parameters between C_i and C_j in which $E^{i} = {e_{l}^{i} : F (h_{i}, e_{l}) = 1$ and F (h_j, e_l) =0, ∀ h_i ∈ C_i, ∀ h_j ∈ C_j, e_l ∈ A}, and $E^{j} = {e_{l}^{j} : F (h_{j}, e_{l}) = 1$ and F (h_i, e_l) =0, ∀ h_i ∈ C_i, ∀ h_j ∈ C_j, e_l ∈ A}, the symbol $e_{l}^{i}$ (or $e_{l}^{j}$ ) represents the objects in C_i (or C_j) have the value 1 at parameter e_l, that is, F (h_i, e_l) =1, h_i ∈ C_i (or F (h_j, e_l) =1, h_j ∈ C_j).

Example 2.3. Let us reconsider Example 2.1. The soft discernibility matrix can be represented in Table 3.

Table 3

The soft discernibility matrix of Table 1

	C ₁	C ₂	C ₃
C ₁	φ
C ₂	${e_{2}^{2}, e_{3}^{2}}$	φ
C ₃	${e_{1}^{1}, e_{3}^{3}}$	${e_{1}^{2}, e_{2}^{2}}$	φ

By introducing the concept of fuzzy sets into soft set theory, Maji et al. [19] proposed the notions of FSSs and by introducing the concept of Pythagorean fuzzy sets into soft set theory, Peng et al. [35] proposed the notions of Pythagorean fuzzy soft sets as follows.

Definition 2.7. Let $F (U)$ be the set of all fuzzy subsets of U. Let E be a set of parameters and A ⊆ E. A pair 〈F, A〉 is called a fuzzy soft set over U, where F is a mapping given by $F : A \to F (U)$ .

For ∀e ∈ A, F (e) is a fuzzy subset of U and it is called fuzzy value set of parameter E. We can see that every soft set may be considered as a fuzzy soft set. If ∀e ∈ A, F (e) is a crisp subset of U, then 〈F, A〉 is degenerated to be a standard soft set.

Definition 2.8. [35] Let U be the universe set and E be a set of parameters. $PF (U)$ denotes the set of all Pythagorean fuzzy sets of U. Let A ⊆ E. A pair 〈F, A〉 is called a Pythagorean fuzzy soft set over U, where F is a mapping is given by $F : A \to PF (U)$ .

For any parameter e ∈ A, F (e) is a Pythagorean fuzzy subsets of U and it is called Pythagorean fuzzy value set of parameter e.

A generalization of fuzzy sets and Pythagorean fuzzy sets are the following notion of spherical fuzzy sets.

Definition 2.9. [12] A Spherical fuzzy set (SFS), $S$ on a universe U is an object of the form, $S$ = ${(x, μ_{S} (x), η_{S} (x), ϑ_{S} (x)) | x \in U}$ where, $μ_{S} (x) \in [0, 1]$ is called the “degree of positive membership of x in U", $η_{S} (x) \in [0, 1]$ is called the “degree of neutral membership of x in U" and $ϑ_{S} (x) \in [0, 1]$ is called the “degree of negative membership of x in U" and where $μ_{S}$ , $η_{S}$ and $ϑ_{S}$ satisfy the following condition, $\forall x \in U, μ_{S}^{2} (x) + η_{S}^{2} (x) + ϑ_{S}^{2} (x) \leq 1$ .

Then for x ∈ U, $π_{S} (x) = \sqrt{1 - μ_{S}^{2} (x) - η_{S}^{2} (x) - ϑ_{S}^{2} (x)}$ is called the degree of refusal-membership of x in U. For simplicity, we call $S (μ_{S} (x), η_{S} (x), ϑ_{S} (x))$ a spherical fuzzy number (SFN) denoted by e = $S (μ_{e}, η_{e}, ϑ_{e})$ where μ_e, η_e, ϑ_e ∈ [0, 1], $π_{e} = \sqrt{1 - (μ_{e})^{2} - (η_{e})^{2} - (ϑ_{e})^{2}}$ , and (μ_e) ²+ (η_e) ² +(ϑ_e) ² ≤ 1.

Definition 2.10. [12] For every two SFSs A and B over a universe U, the union, intersection and complement are defined as follows:

A ⊆ B if ∀x ∈ U, μ_A (x) ≤ μ_B (x), η_A (x) ≤ η_B (x) and ϑ_A (x) ≥ ϑ_B (x).

A = B if and only if A ⊆ B and B ⊆ A.

A ∪ B = {(x, max {μ_A (x) , μ_B (x)} , min {η_A (x) , η_B (x)}, min {ϑ_A (x) , ϑ_B (x)}) |x ∈ U}.

A ∩ B = {(x, min {μ_A (x) , μ_B (x)} , min {η_A (x) , η_B (x)}, max {ϑ_A (x) , ϑ_B (x)}) |x ∈ U}.

Co(A) = A^c = {(x, ϑ_A (x) , η_A (x) , μ_A (x)) |x ∈ U}.

Let SFS (U) denote the set of all spherical fuzzy sets of U.

Proposition 2.1. Let A, B, C ∈ SFS (U). Then

If A ⊆ B and B ⊆ C, then A ⊆ C.

(A^c) ^c = A.

operations ∩ and ∪ are commutative, associative and distributive.

operations ∩ and ∪ satisfy DeMorgan’s laws.

Proof 2.1. Follows directly from the definitions of SFSs.

3 Spherical fuzzy soft set

In this section, we present a novel concept of spherical FSS (SFSS) and investigated their several properties over the universal set U.

Definition 3.1. Let U be an initial universal set, E be a set of parameters and A ⊆ E. A pair 〈F, A〉 is called a spherical fuzzy soft set (SFSS) over U, where F is a mapping given by F : A → SFS (U).

Here, for any parameter e ∈ E, F (e) can be written as a spherical fuzzy set such that $F (e) = {(x, μ_{F (e)} (x), η_{F (e)} (x), ϑ_{F (e)} (x)) | x \in U}$ where μ_F(e) (x) is the degree of positive membership, η_F(e) (x) is the degree of neutral membership and ϑ_F(e) (x) is the degree of negative membership function respectively with the condition, $μ_{F (e)}^{2} (x) + η_{F (e)}^{2} (x) + ϑ_{F (e)}^{2} (x) \leq 1$ . If for any parameter e ∈ A and for any x ∈ U, η_F(e) (x) =0, then F (e) will become a Pythagorean fuzzy set and 〈F, A〉 will become a Pythagorean fuzzy soft set if it is true for all e ∈ A. We denote the set of all Spherical fuzzy soft sets over U by SFSS (U).

Example 3.1. Consider a Spherical fuzzy soft set 〈F, A〉 where U is the set of five houses under consideration, which is denoted by U = {h₁, h₂, h₃, h₄, h₅} and A is the parameter set, where A = {e₁, e₂, e₃, e₄}, e₁ = beautiful, e₂ = in the green surroundings, e₃ = cheap, e₄ = wooden. The Spherical fuzzy soft set 〈F, A〉 describes the “attractiveness of the houses”. Suppose that $\begin{matrix} F (e_{1}) = {\begin{matrix} (h_{1}, 0.8, 0.1, 0.4), (h_{2}, 0.5, 0.2, 0.6), (h_{3}, 0.7, 0.15, 0.3), \\ (h_{4}, 0.6, 0.25, 0.31), (h_{5}, 0.55, 0.12, 0.5) \end{matrix}} \\ F (e_{2}) = {\begin{matrix} (h_{1}, 0.9, 0.1, 0.3), (h_{2}, 0.7, 0.2, 0.45), (h_{3}, 0.55, 0.03, 0.12), \\ (h_{4}, 0.65, 0.15, 0.56), (h_{5}, 0.0.8, 0.17, 0.3) \end{matrix}} \\ F (e_{3}) = {\begin{matrix} (h_{1}, 0.52, 0.05, 0.45), (h_{2}, 0.7, 0.12, 0.31), (h_{3}, 0.45, 0.15, 0.5), \\ (h_{4}, 0.6, 0.2, 0.4), (h_{5}, 0.75, 0.05, 0.25) \end{matrix}} \\ F (e_{4}) = {\begin{matrix} (h_{1}, 0.3, 0.2, 0.7), (h_{2}, 0.45, 0.1, 0.55), (h_{3}, 0.56, 0.15, 0.49), \\ (h_{4}, 0.25, 0.05, 0.3), (h_{5}, 0.51, 0.1, 0.25) \end{matrix}} \end{matrix}$ The Spherical fuzzy soft set 〈F, A〉 is a parameterized family {F (e_i) : i = 1, 2, 3, 4} of SFSs over U. The matrix form of the SFSS 〈F, A〉, is given as follows: $\begin{matrix} 〈 F, A 〉 = \\ e_{1} & e_{2} \\ h_{1} & (0.80, 0.10, 0.40) & (0.90, 0.10, 0.30) \\ h_{2} & (0.50, 0.20, 0.60) & (0.70, 0.20, 0.45) \\ h_{3} & (0.70, 0.15, 0.30) & (0.55, 0.03, 0.12) \\ h_{4} & (0.60, 0.25, 0.31) & (0.65, 0.15, 0.56) \\ h_{5} & (0.55, 0.12, 0.50) & (0.80, 0.17, 0.30) \\ e_{3} & e_{4} \\ h_{1} & (0.52, 0.05, 0.45) & (0.30, 0.20, 0.70) \\ h_{2} & (0.40, 0.12, 0.31) & (0.45, 0.10, 0.55) \\ h_{3} & (0.45, 0.15, 0.50) & (0.56, 0.15, 0.49) \\ h_{4} & (0.60, 0.20, 0.40) & (0.25, 0.05, 0.80) \\ h_{5} & (0.75, 0.05, 0.25) & (0.51, 0.10, 0.25) \end{matrix}$

Definition 3.2. Let 〈F, A〉 and 〈G, B〉 be two SFSSs over U. Then 〈F, A〉 is called a spherical fuzzy soft subset of 〈G, B〉, if

A ⊆ B, and

∀e ∈ A, F (e) ⊆ G (e).

Definition 3.3. Two SFSSs 〈F, A〉 and 〈G, B〉 over U are called to be spherical fuzzy soft equal, if and only if 〈F, A〉 is a spherical fuzzy soft subset of 〈G, B〉 and 〈G, B〉 is a spherical fuzzy soft subset of 〈F, A〉.

Definition 3.4. Let 〈F, A〉 be a SFSS over U. The complement of 〈F, A〉, denoted by 〈F, A〉^c, is defined by 〈F, A〉^c = 〈F^c, A〉, where F^c : A → SFS (U) is a mapping given by F^c (e) = (F (e)) ^c for every e ∈ A.

Since (F^c (e)) ^c is equal to the F (e), we get (〈F, A〉^c) ^c = 〈F, A〉.

Definition 3.5. Let 〈F, A〉 and 〈G, B〉 be two SFSSs over U, then 〈F, A〉 AND 〈G, B〉 is a spherical fuzzy soft set denoted by 〈F, A〉 ∧ 〈G, B〉 is defined by 〈F, A〉 ∧ 〈G, B〉= 〈H, A × B〉 where H (α, β) = F (α) ∩ G (β) ∀α, β ∈ A × B. That is, H (α, β) (x) = (x, min {μ_F(α) (x), μ_G(β) (x)}, min {η_F(α) (x), η_G(β) (x)}, max {ϑ_F(α) (x), ϑ_G(β) (x)}), ∀α, β ∈ A × B and ∀x ∈ U.

Definition 3.6. Let 〈F, A〉 and 〈G, B〉 be two SFSSs over U, then 〈F, A〉 OR 〈G, B〉 is a spherical fuzzy soft set denoted by 〈F, A〉 ∨ 〈G, B〉 is defined by 〈F, A〉 ∨ 〈G, B〉 = 〈H, A × B〉 where H (α, β) = F (α) ∪ G (β) ∀α, β ∈ A × B. That is, H (α, β) (x) = (x, max {μ_F(α) (x), μ_G(β) (x)}, min {η_F(α) (x), η_G(β) (x)}, min {ϑ_F(α) (x), ϑ_G(β) (x)}), ∀α, β ∈ A × B and ∀x ∈ U.

Theorem 3.1. Let 〈F, A〉 and 〈G, B〉 be two SFSSs over U. Then

(〈F, A〉 ∧ 〈G, B〉) ^c = 〈F, A〉^c ∨ 〈G, B〉^c

(〈F, A〉 ∨ 〈G, B〉) ^c = 〈F, A〉^c ∧ 〈G, B〉^c

Proof 3.1. We shall prove the part (i), while other can be obtained similarly.

Suppose that 〈F, A〉 ∧ 〈G, B〉 = 〈H, A × B〉, where H (α, β) = F (α) ∩ G (β) ∀ (α, β) ∈ A × B. That is, H (α, β) = (x, min {μ_F(α) (x), μ_G(β) (x)}, min {η_F(α) (x), η_G(β) (x)}, max {ϑ_F(α) (x), ϑ_G(β) (x)}), ∀ (α, β) ∈ A × B and ∀x ∈ U.

Now (〈F, A〉 ∧ 〈G, B〉) ^c = 〈H, A × B〉^c = 〈H^c, A × B〉. That is, ∀ (α, β) ∈ A × B and ∀x ∈ U, we have

$\begin{matrix} H^{c} (α, β) & = & (x, max {ϑ_{F (α)} (x), ϑ_{G (β)} (x)}, \\ min {η_{F (α)} (x), η_{G (β)} (x)}, \\ min {μ_{F (α)} (x), μ_{G (β)} (x)}) \end{matrix}$ (1) Next consider, 〈F, A〉^c ∨ 〈G, B〉^c = 〈F^c, A〉 ∨ 〈G^c, B〉 = 〈I, A × B〉 where I (α, β) = F^c (α) ∪ G^c (β) ∀ (α, β) ∈ A × B. That is, for x ∈ U, we get

$\begin{matrix} I (α, β) & = & (x, max {μ_{F^{c} (α)} (x), μ_{G^{c} (β)} (x)}, \\ min {η_{F^{c} (α)} (x), η_{G^{c} (β)} (x)}, \\ min {ϑ_{F^{c} (α)} (x), ϑ_{G^{c} (β)} (x)}) \\ = & (x, max {ϑ_{F (α)} (x), ϑ_{G (β)} (x)}, \\ min {η_{F (α)} (x), η_{G (β)} (x)}, \\ min {μ_{F (α)} (x), μ_{G (β)} (x)}) \end{matrix}$ (2) From (1) and (2) we get (〈F, A〉 ∧ 〈G, B〉) ^c = 〈F, A〉^c ∨ 〈G, B〉^c.

Definition 3.7. The union of two SFSSs 〈F, A〉 and 〈G, B〉 over a common universe U is a SFSS 〈H, C〉 where C = A ∪ B and ∀e ∈ C. $\begin{matrix} H (e) = {\begin{matrix} F (e), & if e \in A - B \\ G (e), & if e \in B - A \\ F (e) \cup G (e), & if e \in A \cap B \end{matrix} \end{matrix}$ That is, ∀e ∈ A ∩ B, we have F (e) ∪ G (e) = {x, max {μ_F(e) (x) , μ_G(e) (x)}, min {η_F(e) (x), η_G(e) (x)}, min {ϑ_F(e) (x), ϑ_G(e) (x)} |x ∈ U}. This relation is denoted by 〈F, A〉 ∪ 〈G, B〉 = 〈H, C〉.

Theorem 3.2. The union 〈H, C〉 of two SFSSs 〈F, A〉 and 〈G, B〉 is a SFSS.

Proof 3.2. From Definition 3.7, we have ∀e ∈ C if e ∈ A - B or e ∈ B - A then H (e) = F (e) or H (e) = G (e). So in either case, we get H (e) is a spherical fuzzy soft set.

If e ∈ A ∩ B, for a fixed x ∈ U, without loss of generality, assume μ_F(e) (x) ≤ μ_G(e) (x), then we have, $\begin{matrix} μ_{H (e)}^{2} (x) + η_{H (e)}^{2} (x) + ϑ_{H (e)}^{2} (x) \\ = & {μ_{F (e)}^{2} (x) \lor μ_{G (e)}^{2} (x)} + {η_{F (e)}^{2} (x) \land η_{G (e)}^{2} (x)} \\ + {ϑ_{F (e)}^{2} (x) \land ϑ_{G (e)}^{2} (x)} \\ = & {μ_{F (e)}^{2} (x)} + {η_{F (e)}^{2} (x) \land η_{G (e)}^{2} (x)} \\ + {ϑ_{F (e)}^{2} (x) \land ϑ_{G (e)}^{2} (x)} \\ \leq & μ_{G (e)}^{2} (x) + η_{G (e)}^{2} (x) + ϑ_{G (e)}^{2} (x) \\ \leq & 1 \end{matrix}$ Therefore 〈H, C〉 is a SFSS.

Definition 3.8. The intersection of two spherical fuzzy soft sets 〈F, A〉 and 〈G, B〉 over a common universe U is a spherical fuzzy soft set 〈H, C〉 where C = A ∪ B and ∀e ∈ C $\begin{matrix} H (e) = {\begin{matrix} F (e), & if e \in A - B \\ G (e), & if e \in B - A \\ F (e) \cap G (e), & if e \in A \cap B \end{matrix} \end{matrix}$ That is ∀e ∈ A ∩ B, we have F (e) ∩ G (e) = {x, min {μ_F(e) (x), μ_G(e) (x)}, min {η_F(e) (x), η_G(e) (x)}, max {ϑ_F(e) (x), ϑ_G(e) (x)} |x ∈ U}. This relation is denoted by 〈F, A〉 ∩ 〈G, B〉 = 〈H, C〉.

Theorem 3.3. The intersection 〈H, C〉 of two SFSSs 〈F, A〉 and 〈G, B〉 is a SFSS.

Proof 3.3. From Definition 3.8, we have ∀e ∈ C if e ∈ A - B or e ∈ B - A then H (e) = F (e) or H (e) = G (e). So in either case, we get H (e) is a SFSS.

If e ∈ A ∩ B, for a fixed x ∈ U, without loss of generality, assume ϑ_F(e) (x) ≤ ϑ_G(e) (x), then we have, $\begin{matrix} μ_{H (e)}^{2} (x) + η_{H (e)}^{2} (x) + ϑ_{H (e)}^{2} (x) \\ = & {μ_{F (e)}^{2} (x) \land μ_{G (e)}^{2} (x)} + {η_{F (e)}^{2} (x) \land η_{G (e)}^{2} (x)} \\ + {ϑ_{F (e)}^{2} (x) \lor ϑ_{G (e)}^{2} (x)} \\ = & {μ_{F (e)}^{2} (x) \lor μ_{G (e)}^{2} (x)} + {η_{F (e)}^{2} (x) \land η_{G (e)}^{2} (x)} \\ + {ϑ_{G (e)}^{2} (x)} \\ \leq & μ_{G (e)}^{2} (x) + η_{G (e)}^{2} (x) + ϑ_{G (e)}^{2} (x) \\ \leq & 1 \end{matrix}$ Therefore 〈H, C〉 is a SFSS.

Theorem 3.4. Let 〈F, A〉, 〈G, B〉 and 〈H, C〉 be SFSSs over U. Then

〈F, A〉 ∪ 〈F, A〉 = 〈F, A〉.

〈F, A〉 ∩ 〈F, A〉 = 〈F, A〉.

〈F, A〉 ∪ 〈G, B〉 = 〈G, B〉 ∪ 〈F, A〉.

〈F, A〉 ∩ 〈G, B〉 = 〈G, B〉 ∩ 〈F, A〉.

(〈F, A〉 ∪ 〈G, B〉) ∪ 〈H, C〉 = 〈F, A〉 ∪ (〈G, B〉∪〈H, C〉).

(〈F, A〉 ∩ 〈G, B〉) ∩ 〈H, C〉 = 〈F, A〉 ∩ (〈G, B〉∩〈H, C〉).

Proof 3.4. Follows from the definitions of union and intersection and Proposition 2.1.

Theorem 3.5. Let 〈F, A〉, 〈G, B〉 and 〈H, C〉 be spherical fuzzy soft sets over U. Then

〈F, A〉 ∪ (〈G, B〉 ∩ 〈H, C〉) = (〈F, A〉 ∪ 〈G, B〉)∩ (〈F, A〉 ∪ 〈H, C〉).

〈F, A〉 ∩ (〈G, B〉 ∪ 〈H, C〉) = (〈F, A〉 ∩ 〈G, B〉)∪ (〈F, A〉 ∩ 〈H, C〉).

Proof 3.5. It is obtained directly obtained from the fact that the union and intersection of SFSS are distributive in Proposition 2.1.

Theorem 3.6. Let 〈F, A〉 and 〈G, B〉 be two SFSSs over U, then we have the following properties

(〈F, A〉 ∩ 〈G, B〉) ^c = 〈F, A〉^c ∪ 〈G, B〉^c.

(〈F, A〉 ∪ 〈G, B〉) ^c = 〈F, A〉^c ∩ 〈G, B〉^c

Proof 3.6. We shall prove only the part (i), while other can be obtained similarly.

We have 〈F, A〉 ∩ 〈G, B〉 =〈H, C〉 where C = A ∪ B and ∀e ∈ C

H (e) = ${\begin{matrix} F (e), & if e \in A - B \\ G (e), & if e \in B - A \\ F (e) \cap G (e), & if e \in A \cap B \end{matrix}$

That is, ∀e ∈ A ∩ B, we have F (e) ∩ G (e) = {x, min {μ_F(e) (x), μ_G(e) (x)}, min {η_F(e) (x), η_G(e) (x)}, max {ϑ_F(e) (x), ϑ_G(e) (x) |x ∈ U}. So that (〈F, A〉 ∩ 〈G, B〉) ^c = 〈H, C〉^c = 〈H^c, C〉 and H^c (e) = (H (e)) ^c. Therefore,

(H (e)) ^c = ${\begin{matrix} (F (e))^{c}, & if e \in A - B \\ (G (e))^{c}, & if e \in B - A \\ (F (e) \cap G (e))^{c}, & if e \in A \cap B \end{matrix}$ That is, ∀e ∈ A ∩ B, we get $\begin{matrix} (F (e) \cap G (e))^{c} \\ = & {x, min {μ_{F (e)} (x), μ_{G (e)} (x)}, min {η_{F (e)} (x), \\ η_{G (e)} (x)}, max {ϑ_{F (e)} (x), ϑ_{G (e)} (x) | x \in U}^{c} \\ = & {x, max {ϑ_{F^{c} (e)} (x), ϑ_{G^{c} (e)} (x)}, min {η_{F^{c} (e)} (x), \\ η_{G^{c} (e)} (x)}, min {μ_{F^{c} (e)} (x), μ_{G^{c} (e)} (x) | x \in U} \end{matrix}$ Now 〈F, A〉^c = 〈F^c, A〉 and 〈G, B〉^c = 〈G^c, B〉. So that 〈F, A〉^c ∪ 〈G, B〉^c = 〈F^c, A〉 ∪ 〈G^c, B〉 = 〈H^c, C〉 where C = A ∪ B and

H^c (e) = ${\begin{matrix} (F^{c} (e)), & if e \in A - B \\ (G^{c} (e)), & if e \in B - A \\ (F^{c} (e) \cup G^{c} (e)), & if e \in A \cap B \end{matrix}$

That is, ∀e ∈ A ∩ B, (F^c (e) ∪ G^c (e)) = {x, max{ϑ_F^c(e) (x), ϑ_G^c(e) (x)}, min {η_F^c(e) (x), η_G^c(e) (x)}, min {μ_F^c(e) (x), μ_G^c(e) (x) |x ∈ U}. Therefore, we get (〈F, A〉 ∩ 〈G, B〉) ^c = 〈F, A〉^c ∪ 〈G, B〉^c

Example 3.2. Let U = {x₁, x₂, x₃, x₄} and E = {e₁, e₂, e₃, e₄, e₅}, Suppose that 〈F, A〉, 〈G, B〉 and 〈I, D〉 are three SFSSs over U given by A = {e₁, e₂}, B = {e₁, e₂, e₄}, and D = {e₁, e₃, e₄} defined as

$\begin{matrix} 〈 F, A 〉 = & e_{1} & e_{2} \\ x_{1} & (0.3, 0.2, 0.9) & (0.1, 0.2, 0.8) \\ x_{2} & (0.6, 0.1, 0.5) & (0.2, 0.3, 0.6) \\ x_{3} & (0.8, 0.2, 0.3) & (0.2, 0.0, 0.9) \\ x_{4} & (0.2, 0.2, 0.7) & (0.5, 0.1, 0.8) \end{matrix}$

$\begin{matrix} 〈 G, B 〉 = & e_{1} & e_{2} & e_{4} \\ x_{1} & (0.4, 0.2, 0.7) & (0.4, 0.2, 0.7) & (0.3, 0.2, 0.7) \\ x_{2} & (0.7, 0.2, 0.4) & (0.8, 0.3, 0.5) & (0.6, 0.1, 0.5) \\ x_{3} & (0.9, 0.3, 0.2) & (0.7, 0.1, 0.6) & (0.4, 0.3, 0.6) \\ x_{4} & (0.3, 0.3, 0.6) & (0.6, 0.2, 0.7) & (0.5, 0.2, 0.4) \end{matrix}$

and

$\begin{matrix} 〈 I, D 〉 = & e_{1} & e_{3} & e_{4} \\ x_{1} & (0.3, 0.2, 0.6) & (0.6, 0.2, 0.4) & (0.9, 0.1, 0.3) \\ x_{2} & (0.7, 0.2, 0.4) & (0.1, 0.3, 0.7) & (0.5, 0.4, 0.2) \\ x_{3} & (0.8, 0.1, 0.3) & (0.2, 0.2, 0.8) & (0.3, 0.3, 0.7) \\ x_{4} & (0.4, 0.2, 0.6) & (0.7, 0.2, 0.4) & (0.1, 0.1, 0.9) \end{matrix}$

Then 〈F, A〉 is a spherical fuzzy soft subset of 〈G, B〉. The complement of 〈F, A〉 is given by,

$\begin{matrix} 〈 F, A 〉^{c} = 〈 F^{c}, A 〉 \\ = & e_{1} & e_{2} \\ x_{1} & (0.9, 0.2, 0.3) & (0.8, 0.2, 0.1) \\ x_{2} & (0.5, 0.1, 0.6) & (0.6, 0.3, 0.2) \\ x_{3} & (0.3, 0.2, 0.8) & (0.9, 0.0, 0.2) \\ x_{4} & (0.7, 0.2, 0.2) & (0.8, 0.1, 0.5) \end{matrix}$

The AND and OR operations are given by,

$\begin{matrix} 〈 F, A 〉 \land 〈 G, B 〉 = 〈 H, A \times B 〉 \\ = & (e_{1}, e_{1}) & (e_{1}, e_{2}) & (e_{1}, e_{4}) \\ x_{1} & (0.3, 0.2, 0.9) & (0.3, 0.2, 0.9) & (0.3, 0.2, 0.9) \\ x_{2} & (0.6, 0.1, 0.5) & (0.6, 0.1, 0.5) & (0.6, 0.1, 0.5) \\ x_{3} & (0.8, 0.2, 0.3) & (0.7, 0.1, 0.6) & (0.4, 0.2, 0.6) \\ x_{4} & (0.2, 0.2, 0.7) & (0.2, 0.2, 0.7) & (0.2, 0.2, 0.7) \\ (e_{2}, e_{1}) & (e_{2}, e_{2}) & (e_{2}, e_{4}) \\ x_{1} & (0.1, 0.2, 0.8) & (0.1, 0.2, 0.8) & (0.1, 0.2, 0.8) \\ x_{2} & (0.2, 0.2, 0.6) & (0.2, 0.3, 0.6) & (0.2, 0.1, 0.6) \\ x_{3} & (0.2, 0.0, 0.9) & (0.2, 0.0, 0.9) & (0.2, 0.0, 0.9) \\ x_{4} & (0.3, 0.1, 0.8) & (0.5, 0.1, 0.8) & (0.5, 0.1, 0.8) \end{matrix}$

and

$\begin{matrix} 〈 F, A 〉 \lor 〈 G, B 〉 = 〈 I, A \times B 〉 \\ = & (e_{1}, e_{1}) & (e_{1}, e_{2}) & (e_{1}, e_{4}) \\ x_{1} & (0.4, 0.2, 0.7) & (0.4, 0.2, 0.7) & (0.3, 0.2, 0.7) \\ x_{2} & (0.7, 0.1, 0.4) & (0.8, 0.1, 0.5) & (0.6, 0.1, 0.5) \\ x_{3} & (0.9, 0.2, 0.2) & (0.8, 0.1, 0.3) & (0.8, 0.2, 0.3) \\ x_{4} & (0.70.2, 0.6) & (0.6, 0.2, 0.7) & (0.5, 0.2, 0.4) \\ (e_{2}, e_{1}) & (e_{2}, e_{2}) & (e_{2}, e_{4}) \\ x_{1} & (0.4, 0.2, 0.7) & (0.4, 0.2, 0.7) & (0.3, 0.2, 0.7) \\ x_{2} & (0.7, 0.2, 0.4) & (0.8, 0.3, 0.5) & (0.6, 0.1, 0.5) \\ x_{3} & (0.9, 0.0, 0.2) & (0.7, 0.0, 0.6) & (0.4, 0.0, 0.6) \\ x_{4} & (0.5, 0.1, 0.6) & (0.6, 0.1, 0.7) & (0.5, 0.1, 0.4) \end{matrix}$

The union and intersection of SFSSs 〈F, A〉 and 〈I, D〉 are given by 〈F, A〉 ∪ 〈I, D〉 = 〈H, C〉 where C = A ∪ D = {e₁, e₂, e₃, e₄} and defined as

$\begin{matrix} 〈 F, A 〉 \cup 〈 I, D 〉 = 〈 H, C 〉 \\ = & e_{1} & e_{2} & e_{3} & e_{4} \\ x_{1} & (0.3, 0.2, 0.6) & (0.1, 0.2.0.8) & (0.6, 0.2, 0.4) & (0.9, 0.1, 0.3) \\ x_{2} & (0.7, 0.1, 0.4) & (0.2, 0.3, 0.6) & (0.1, 0.3, 0.7) & (0.5, 0.4, 0.2) \\ x_{3} & (0.8, 0.1, 0.3) & (0.2, 0.0, 0.9) & (0.2, 0.2, 0.8) & (0.3, 0.3, 0.7) \\ x_{4} & (0.4, 0.2, 0.6) & (0.5, 0.1, 0.8) & (0.7, 0.2, 0.4) & (0.1, 0.1, 0.9) \end{matrix}$ and 〈F, A〉 ∩ 〈I, D〉 = 〈H, C〉, where C = A ∪ D = {e₁, e₂, e₃, e₄} given by

$\begin{matrix} 〈 F, A 〉 \cap 〈 I, D 〉 = 〈 H, C 〉 \\ = & e_{1} & e_{2} & e_{3} & e_{4} \\ x_{1} & (0.3, 0.2, 0.9) & (0.1, 0.2.0.8) & (0.6, 0.2, 0.4) & (0.9, 0.1, 0.3) \\ x_{2} & (0.6, 0.1, 0.5) & (0.2, 0.3, 0.6) & (0.1, 0.3, 0.7) & (0.5, 0.4, 0.2) \\ x_{3} & (0.8, 0.1, 0.3) & (0.2, 0.0, 0.9) & (0.2, 0.2, 0.8) & (0.3, 0.3, 0.7) \\ x_{4} & (0.2, 0.2, 0.7) & (0.5, 0.1, 0.8) & (0.7, 0.2, 0.4) & (0.1, 0.1, 0.9) \end{matrix}$

4 Application of the spherical fuzzy soft setmodel based on adjustable softdiscernibility matrix

Combining the algorithm based on soft discernibility matrix [30] and decision-making method used in [22], we present an algorithm for spherical fuzzy soft sets. By using the new algorithm proposed, we are able to find not only the optimal object but we get an order relation between the all object in our universe.

First we present some basic concepts.

Definition 4.1. Let ϖ = 〈F, A〉 be a SFSS over U. For a triple (p, q, r) ∈ [0, 1] ³, the (p, q, r)-level soft set of ϖ is a crisp soft set L (ϖ ; (p, q, r)) = 〈F_(p,q,r), A〉 is defined by

F_(p,q,r) (e) = L (F (e) ; (p, q, r)) = {x ∈ U|μ_F(e) (x) ≥p, η_F(e) (x) ≤ q and ϑ_F(e) ≤ r}, ∀e ∈ A.

Example 4.1. Consider Example 3.1. Now we take (p, q, r) = (0.5, 0.2, 0.4), we get L (F (e₁) ; (0.5, 0.2, 0.4)) = {h₁, h₃} L (F (e₂) ; (0.5, 0.2, 0.4)) = {h₁, h₂, h₅} L (F (e₃) ; (0.5, 0.2, 0.4)) = {h₂, h₄, h₅} L (F (e₄) ; (0.5, 0.2, 0.4)) = {h₅} Here, the (0.5, 0.2, 0.4)-level soft set of ϖ = 〈F, A〉 is a crisp soft set L (F (ϖ ; (0.5, 0.2, 0.4)).

Now we show some properties of the (p, q, r)-level soft sets.

Theorem 4.1. Let ϖ = 〈F, A〉 be a spherical fuzzy soft set over U, where A ⊆ E and E is a set of parameters. L (ϖ ; (p₁, q₁, r₁)) and L (ϖ ; (p₂, q₂, r₂)) are (p₁, q₁, r₁)-level soft set and (p₂, q₂, r₂)-level soft set of ϖ, respectively, where p₁, q₁, r₁, p₂, q₂, r₂ ∈ [0, 1]. If p₂ ≤ p₁, q₂ ≥ q₁ and r₂ ≥ r₁, then we have L (ϖ ; (p₁, q₁, r₁)) ⊆ L (ϖ ; (p₂, q₂, r₂)).

Proof 4.1. Let L (ϖ ; (p₁, q₁, r₁)) = 〈F₍p₁, q₁, r₁) , A〉 where, F₍p₁, q₁, r₁) (e) = L (F (e) ; (p₁, q₁, r₁)) = {x∈U|μ_F(e) (x) ≥ p₁, η_F(e) (x) ≤ q₁ and ϑ_F(e) ≤ r₁}, for all e ∈ A. And let L (ϖ ; (p₂, q₂, r₂)) = 〈F₍p₂, q₂, r₂) , A〉 where, F₍p₂, q₂, r₂) (e) = L (F (e) ; (p₂, q₂, r₂)) = {x ∈ U|μ_F(e) (x) ≥ p₂, η_F(e) (x) ≤ q₂ and ϑ_F(e) ≤ r₂}, for all e ∈ A. Obviously, A ⊆ A. To prove the theorem it is enough to show that ∀e ∈ A, F₍p₁, q₁, r₁) (e) ⊆ F₍p₂, q₂, r₂) (e). Since p₂ ≤ p₁, q₂ ≥ q₁ and r₂ ≥ r₁, {x ∈ U|μ_F(e) (x) ≥ p₁, η_F(e) (x) ≤ q₁ and ϑ_F(e) ≤ r₁} ⊆ {x ∈ U|μ_F(e)(x) ≥ p₂, η_F(e) (x) ≤ q₂ and ϑ_F(e) ≤ r₂}. Thus we have, F₍p₁, q₁, r₁) (e) ⊆ F₍p₂, q₂, r₂) (e), ∀e ∈ A. Therefore, L (ϖ ; (p₁, q₁, r₁)) ⊆ L (ϖ ; (p₂, q₂, r₂)).

Theorem 4.2. Let ϖ = 〈F, A〉 and ς = 〈G, A〉 be two spherical fuzzy sets over U, where A ⊆ E and E is a set of parameters. L (ϖ ; (p, q, r)) and L (ς ; (p, q, r)) are (p, q, r)-level soft sets of ϖ and ς respectively, where p, q, r ∈ [0, 1]. If ϖ ⊆ ς, then L (ϖ ; (p, q, r)) ⊆ L (ς ; (p, q, r)).

Proof 4.2. Let L (ϖ ; (p, q, r)) = 〈F₍p, q, r) , A〉 where, F₍p, q, r) (e) = L (F (e) ; (p, q, r)) = {x ∈ U|μ_F(e) (x) ≥ p, η_F(e) (x) ≤ q and ϑ_F(e) ≤ r}, for all e ∈ A. And let L (ς ; (p, q, r)) = 〈G₍p, q, r) , A〉 where, G₍p, q, r) (e) = L (G (e) ; (p, q, r)) = {x ∈ U|μ_G(e) (x) ≥ p, η_G(e) (x) ≤ q and ϑ_G(e) ≤ r}, for all e ∈ A. Obviously, A ⊆ A. To prove the theorem it is enough to show that ∀e ∈ A, F₍p, q, r)(e) ⊆ G₍p, q, r) (e).

Since ϖ ⊆ ς, we have the following μ_F(e) (x)≤μ_G(e) (x), η_F(e) (x) ≥ η_G(e) (x) and ϑ_F(e) (x) ≥ ϑ_G(e) (x) for all x ∈ U and e ∈ A. Assume, x∈F₍p, q, r) (e). Then, we have, μ_F(e) (x) ≥ p, η_F(e) (x)≤q and ϑ_F(e) (x) ≤ r. Further, μ_F(e) (x) ≤ μ_G(e) (x), η_F(e) (x) ≥ η_G(e) (x) and ϑ_F(e) (x) ≥ ϑ_G(e) (x), we get μ_G(e) (x) ≥ p, η_G(e) (x) ≤ q and ϑ_G(e) (x) ≤ r. Hence x ∈ {x ∈ U|μ_G(e) (x) ≥ p, η_G(e) (x) ≤ q and ϑ_G(e) ≤ r}. Therefore, F₍p, q, r) (e) ⊆ G₍p, q, r) (e), ∀e ∈ A. Consequently, L (ϖ ; (p, q, r)) ⊆ L (ς ; (p, q, r)).

While making decisions sometimes, we need to impose different thresholds on different parameters. To overcome this situation, we use a function to replace a constant value triple as the thresholds on positive membership value, neutral membership value and negative membership value respectively.

Definition 4.2. Let ϖ = 〈F, A〉 be a SFSS over U. Let λ : A → [0, 1] ³ be a function. That is, forall e ∈ A, λ (e) = (p (e) , q (e) , r (e)), where p (e) , q (e) , r (e)∈[0, 1]. The level soft of ϖ with respect to λ is a crisp soft set L (ϖ ; λ) = 〈F_λ, A〉 defined by F_λ (e) = L (F (e) ; λ (e)) = {x∈ U|μ_F(e) (x) ≥ p (e) , η_F(e) (x) ≤q (e) and ϑ_F(e) ≤ r (e)}, ∀e ∈ A.

Remark 4.1. Here λ : A → [0, 1] ³ is the function, where p (e) is the least threshold with respect to the parameter e on the degree of positive membership, q (e) is the greatest threshold with respect to the parameter e on the degree of neutral membership and r (e) is the greatest threshold with respect to the parameter e on the degree of negative membership.

Let ϖ = 〈F, A〉 be a SFSS over U. The four threshold functions that all are already familiar with us are shown below,

The Mid-level threshold function (mid_ϖ): The mid-threshold function of ϖ = 〈F, A〉, mid_ϖ defined from A to [0, 1] ³ is given by mid_ϖ (e) = (p_{mid_ϖ} (e) , q_{mid_ϖ} (e) , r_{mid_ϖ} (e)), ∀e ∈ A, where p_{mid_ϖ} (e) = $\frac{1}{| U |} \sum_{x \in U} μ_{F (e)} (x)$ , q_{mid_ϖ} (e) = $\frac{1}{| U |} \sum_{x \in U} η_{F (e)} (x)$ , r_{mid_ϖ} (e) = $\frac{1}{| U |} \sum_{x \in U} ϑ_{F (e)} (x)$ . The level soft set with respect to mid_ϖ, L (ϖ ; mid_ϖ) is called the mid-level soft set of ϖ.

The Top-bottom-bottom-level threshold function (tbb_ϖ): The top-bottom-bottom threshold function of ϖ = 〈F, A〉, tbb_ϖ defined from A to [0, 1] ³ is given by tbb_ϖ (e) = (p_{tbb_ϖ} (e) , q_{tbb_ϖ} (e) , r_{tbb_ϖ} (e)), ∀e ∈ A, where p_{tbb_ϖ} (e) = $\max_{x \in U}$ μ_F(e) (x), q_{tbb_ϖ} (e) = $\min_{x \in U}$ η_F(e) (x), r_{tbb_ϖ} (e) = $\min_{x \in U}$ ϑ_F(e) (x). The level soft set with respect to tbb_ϖ, L (ϖ ; tbb_ϖ) is called the top-bottom-bottom-level soft set of ϖ.

The Bottom-bottom-bottom-level threshold function (bbb_ϖ): The bottom-bottom-bottom threshold function of ϖ = 〈F, A〉, bbb_ϖ defined from A to [0, 1] ³ is given by bbb_ϖ (e) = (p_{bbb_ϖ} (e), q_{bbb_ϖ} (e), r_{bbb_ϖ} (e)), ∀e ∈ A, where p_{bbb_ϖ} (e) = $\min_{x \in U}$ μ_F(e) (x), q_{bbb_ϖ} (e) = $\min_{x \in U}$ η_F(e) (x), r_{bbb_ϖ} (e) = $\min_{x \in U}$ ϑ_F(e) (x). The level soft set with respect to bbb_ϖ, L (ϖ ; bbb_ϖ) is called the bottom-bottom-bottom-level soft set of ϖ.

The Med-level threshold function (med_ϖ): The med-threshold function of ϖ = 〈F, A〉, med_ϖ defined from A to [0, 1] ³ is given by med_ϖ (e) = (p_{med_ϖ} (e) , q_{med_ϖ} (e) , r_{med_ϖ} (e)), ∀e ∈ A, where ∀e ∈ A, p_{med_ϖ} (e) , q_{med_ϖ} (e) , r_{med_ϖ} (e) are the medians by ranking the degree of positive, neutral and negative membership respectively of all alternatives according to order from large to small (or small to large), that is,

$\begin{matrix} p_{{med}_{ϖ}} (e) \\ = & {\begin{matrix} μ_{F (e)} (x_{(\frac{| U | + 1}{2})}), & if | U | is odd, \\ μ_{F (e)} (x_{(\frac{| U | + 1}{2})}) + \frac{μ_{F (e)} (x_{(\frac{| U |}{2} + 1)})}{2}, & if | U | is even . \end{matrix} \end{matrix}$

$\begin{matrix} q_{{med}_{ϖ}} (e) \\ = & {\begin{matrix} η_{F (e)} (x_{(\frac{| U | + 1}{2})}), & if | U | is odd, \\ η_{F (e)} (x_{(\frac{| U | + 1}{2})}) + \frac{η_{F (e)} (x_{(\frac{| U |}{2} + 1)})}{2}, & if | U | is even \end{matrix} \end{matrix}$

and

$\begin{matrix} r_{{med}_{ϖ}} (e) \\ = & {\begin{matrix} ϑ_{F (e)} (x_{(\frac{| U | + 1}{2})}), & if | U | is odd, \\ ϑ_{F (e)} (x_{(\frac{| U | + 1}{2})}) + \frac{ϑ_{F (e)} (x_{(\frac{| U |}{2} + 1)})}{2}, & if | U | is even . \end{matrix} \end{matrix}$

The level soft set with respect to med_ϖ, L (ϖ ; med_ϖ) is called the med-level soft set of ϖ.

Example 4.2. Let us reconsider the Example 3.1. From it, the above mentioned thresholds and the level soft sets are given below,

${mid}_{ϖ} = {\begin{matrix} 〈 e_{1}, (0.63, 0.17, 0.42) 〉, 〈 e_{2}, (0.72, 0.13, 0.35) 〉, \\ 〈 e_{3}, (0.60, 0.11, 0.38) 〉, 〈 e_{4}, (0.41, 0.12, 0.56) 〉 \end{matrix}}$

${tbb}_{ϖ} = {\begin{matrix} 〈 e_{1}, (0.8, 0.1, 0.3) 〉, 〈 e_{2}, (0.9, 0.03, 0.12) 〉, \\ 〈 e_{3}, (0.75, 0.05, 0.25) 〉, 〈 e_{4}, (0.56, 0.05, 0.25) 〉 \end{matrix}}$

${bbb}_{ϖ} = {\begin{matrix} 〈 e_{1}, (0.5, 0.1, 0.3) 〉, 〈 e_{2}, (0.55, 0.03, 0.12) 〉, \\ 〈 e_{3}, (0.52, 0.05, 0.25) 〉, 〈 e_{4}, (0.25, 0.05, 0.25) 〉 \end{matrix}}$

${med}_{ϖ} = {\begin{matrix} 〈 e_{1}, (0.6, 0.15, 0.4) 〉, 〈 e_{2}, (0.7, 0.15, 0.3) 〉, \\ 〈 e_{3}, (0.6, 0.12, 0.4) 〉, 〈 e_{4}, (0.45, 0.1, 0.55) 〉 \end{matrix}}$

From the discussions made above, when the level soft set has been introduced using threshold functions, an order relation of the objects can be easily obtained from the soft discernibility matrix as pointed out in Feng and Zhou [22]. Now we are presenting an effective algorithm based on adjustable soft discernibility matrix as follows.

Table 4
Rating values of SFSS ϖ = 〈F, A〉

e ₁ e ₂ e ₃ e ₄ e ₅

p ₁ (0.8,0.1,0.4) (0.5,0.2,0.6) (0.7,0.15,0.3) (0.6,0.25,0.31) (0.55,0.12,0.5)

p ₂ (0.9,0.1,0.3) (0.7,0.2,0.45) (0.55,0.03,0.12) (0.65,0.15,0.56) (0.8,0.17,0.3)

p ₃ (0.52,0.02,0.7) (0.7,0.12,0.31) (0.45,0.15,0.5) (0.6,0.2,0.4) (0.75,0.05,0.25)

p ₄ (0.7,0.05,0.4) (0.45,0.1,0.55) (0.56,0.15,0.49) (0.25,0.05,0.8) (0.51,0.1,0.25)

p ₅ (0.7,0.1,0.4) (0.3,0.2,0.7) (0.47,0.25,0.56) (0.31,0.15,0.6) (0.8,0.2,0.4)

p ₆ (0.55,0.25,0.65) (0.81,0.15,0.3) (0.1,0.2,0.8) (0.6,0.05,0.3) (0.65,0.1,0.5)

p ₇ (0.92,0.1,0.3) (0.2,0.25,0.6) (0.35,0.2,0.7) (0.8,0.12,0.4) (0.75,0.01,0.3)

	e ₁	e ₂	e ₃	e ₄	e ₅
p ₁	(0.8,0.1,0.4)	(0.5,0.2,0.6)	(0.7,0.15,0.3)	(0.6,0.25,0.31)	(0.55,0.12,0.5)
p ₂	(0.9,0.1,0.3)	(0.7,0.2,0.45)	(0.55,0.03,0.12)	(0.65,0.15,0.56)	(0.8,0.17,0.3)
p ₃	(0.52,0.02,0.7)	(0.7,0.12,0.31)	(0.45,0.15,0.5)	(0.6,0.2,0.4)	(0.75,0.05,0.25)
p ₄	(0.7,0.05,0.4)	(0.45,0.1,0.55)	(0.56,0.15,0.49)	(0.25,0.05,0.8)	(0.51,0.1,0.25)
p ₅	(0.7,0.1,0.4)	(0.3,0.2,0.7)	(0.47,0.25,0.56)	(0.31,0.15,0.6)	(0.8,0.2,0.4)
p ₆	(0.55,0.25,0.65)	(0.81,0.15,0.3)	(0.1,0.2,0.8)	(0.6,0.05,0.3)	(0.65,0.1,0.5)
p ₇	(0.92,0.1,0.3)	(0.2,0.25,0.6)	(0.35,0.2,0.7)	(0.8,0.12,0.4)	(0.75,0.01,0.3)

Algorithm 1: Decision making based on adjustable soft discernibility matrix.

Input the spherical fuzzy soft set ϖ = 〈F, A〉 over U, where U = {x₁, x₂, …, x_m}.

Input any one of the threshold function λ : A → [0, 1] ³ (mid-threshold function; or top-bottom-bottom-threshold function; or bottom-bottom-bottom-threshold function; or med-threshold function) or input a threshold triple (p, q, r) ∈ [0, 1] ³ for decision-making.

Compute the corresponding level soft set L (ϖ, λ) of the SFSS ϖ = 〈F, A〉.

Present the level soft set L (ϖ, λ) in tabular form.

Compute the partition of U and the discernibility matrix.

Compute the soft discernibility matrix $D = D (C_{i}, C_{j})_{i, j \leq m}$

Select the items D₁ and D₂ from the soft discernibility matrix, where D₁ = {D (C_i, C_j) : |D (C_i, C_j) |=2n, n ∈ N} D₂ = {D (C_i, C_j) : |D (C_i, C_j) |=2n + 1, n ∈ N}

For every element of D₁, make the comparison of |Eⁱ| with |E^j|. If |Eⁱ| = |E^j| then the object(s) x_i ∈ C_i and x_j ∈ C_j are kept in the same decision class. Otherwise there must exist an order relation between the objects in C_i and the objects in C_j. That is, either x_i is superior to x_j or x_j is superior to x_i.

If we get an order relation including all the objects in U, then turn to step 11; otherwise, go to the next step.

Combine with the result of step 9, find the corresponding element in D₂ to compare the order relation.

Output the order relation among all the objects by combining step 9 and step 10.

Choose the optimal object(s) from the order relation by selecting the object(s) in the first place of the order relation which is arranged from large to small. If there exist more than one optimal object take any one of them as the optimal solution.

The following example is utilized to exemplify the basic idea of Algorithm given above.

Example 4.3. A group of associates wants to start a paper producing factory in some place. They are in search of different plots to start up their factory. Let us consider a spherical fuzzy soft set ϖ = 〈F, A〉 which describes “the suitability of different plots" where they can build their factory. Suppose there are seven plots under consideration, U = {p₁, p₂, p₃, p₄, p₅, p₆, p₇}. The associates must take a decision according to criteria set A = {e₁, e₂, e₃, e₄, e₅} where e₁ stands for availability of water, e₂ stands for availability of raw materials, e₃ stands for waste management, e₄ stands for transportation facility and e₅ stands for low population density near the plot. The associates evaluates the plots p_i, (i = 1, 2, 3, 4, 5, 6, 7) with respect to the criteria e_j, (j = 1, 2, 3, 4, 5) and construct a SFSS ϖ = 〈F, A〉 given as Table 11.

Suppose that the group of associates would like to select the optimal plot to start factory. There are different rules in decision-making problem. If in this case associates deals with med-level decision rule. Clearly, the med threshold of ϖ = 〈F, A〉 is

$\begin{matrix} {med}_{ϖ} = {\begin{matrix} 〈 e_{1}, (0.7, 0.1, 0.4) 〉, 〈 e_{2}, (0.5, 0.2, 0.55) 〉, \\ 〈 e_{3}, (0.47, 0.15, 0.5) 〉, 〈 e_{4}, (0.6, 0.15, 0.4) 〉, \\ 〈 e_{5}, (0.75, 0.1, 0.3) 〉 \end{matrix}} \end{matrix}$

Using this method we shall obtain the med-level soft set L (ϖ ; med_ϖ) of ϖ with tabular representation as in Table 5.

Table 5

Med-level soft set L (ϖ ; med_ϖ) of ϖ

	e ₁	e ₂	e ₃	e ₄	e ₅
p ₁	1	0	1	0	0
p ₂	1	1	1	0	1
p ₃	0	1	0	0	1
p ₄	1	0	1	0	0
p ₅	1	0	0	0	0
p ₆	0	1	0	1	0
p ₇	1	0	0	1	1

Now it is easy to see from Table 5 that each of the parameter e_j ; j = 1, 2, 3, 4, 5 induces an equivalence relation on U. That is, p_l related to p_m if and only if F (p_l, e_j) = F (p_m, e_j) for j = 1, 2, 3, 4, 5 and ∀p_l ∈ C_l, ∀ p_m ∈ C_m, where F (p_l, e_j) is the value of objects in C_l associated with parameter e_j. Hence we get the following equivalence classes (ECs) for each of the equivalence relation as following,

For F (e₁), ECs are {p₁, p₂, p₄, p₅, p₇} , {p₃, p₆}.

For F (e₂), ECs are {p₂, p₃, p₆} , {p₁, p₄, p₅, p₇}.

For F (e₃), ECs are {p₁, p₂, p₄} , {p₃, p₅, p₆, p₇}.

For F (e₄), ECs are {p₁, p₂, p₃, p₄, p₅} , {p₆, p₇}.

For F (e₅), ECs are {p₁, p₄, p₅, p₆} , {p₂, p₃, p₇}.

We also observe that there is an indiscernibility relation defined by the soft set 〈F, A〉 itself. Which can be obtained as

$\begin{matrix} IND 〈 F, A 〉 \\ = {\begin{matrix} (p_{1}, p_{1}), (p_{1}, p_{4}), (p_{4}, p_{1}), (p_{2}, p_{2}), (p_{3}, p_{3}), \\ (p_{4}, p_{4}), (p_{5}, p_{5}), (p_{6}, p_{6}), (p_{7}, p_{7}) \end{matrix}} \end{matrix}$

Therefore the partition of U is

$\begin{matrix} U | IND 〈 F, A 〉 = {{p_{1}, p_{4}}, {p_{2}}, {p_{3}}, {p_{5}}, {p_{6}}, {p_{7}}} \end{matrix}$

From this, we can obtain the partition of U is C₁ = {p₁, p₄}, C₂ = {p₂}, C₃ = {p₃}, C₄ = {p₅}, C₅ = {p₆}, C₆ = {p₇}

Now the discernibility matrix is constructed in Table 6.

Table 6

Discernibility matrix of the considered example

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
C ₁	φ
C ₂	{e₂, e₅}	φ
C ₃	{e₁, e₂, e₃, e₅}	{e₁, e₃}	φ
C ₄	{e₃}	{e₂, e₃, e₅}	{e₁, e₂, e₅}	φ
C ₅	{e₁, e₂, e₃, e₄}	{e₁, e₃, e₄, e₅}	{e₄, e₅}	{e₁, e₂, e₄}	φ
C ₆	{e₃, e₄, e₅}	{e₂, e₃, e₄}	{e₁, e₂, e₄}	{e₄, e₅}	{e₁, e₂, e₅}	φ

Further, the soft discernibility matrix is constructed in Table 7.

Table 7

Soft discernibility matrix of the med-level soft set L〈ϖ, med_ϖ〉

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
C ₁	φ
C ₂	${e_{2}^{2}, e_{5}^{2}}$	φ
C ₃	${e_{1}^{1}, e_{2}^{3}, e_{3}^{1}, e_{5}^{3}}$	${e_{1}^{2}, e_{3}^{2}}$	φ
C ₄	${e_{3}^{1}}$	${e_{2}^{2}, e_{3}^{2}, e_{5}^{2}}$	${e_{1}^{4}, e_{2}^{3}, e_{5}^{3}}$	φ
C ₅	${e_{1}^{1}, e_{2}^{5}, e_{3}^{1}, e_{4}^{5}}$	${e_{1}^{2}, e_{3}^{2}, e_{4}^{5}, e_{5}^{2}}$	${e_{4}^{5}, e_{5}^{3}}$	${e_{1}^{4}, e_{2}^{5}, e_{4}^{5}}$	φ
C ₆	${e_{3}^{1}, e_{4}^{6}, e_{5}^{6}}$	${e_{2}^{2}, e_{3}^{2}, e_{4}^{6}}$	${e_{1}^{6}, e_{2}^{3}, e_{4}^{6}}$	${e_{4}^{6}, e_{5}^{6}}$	${e_{1}^{6}, e_{2}^{5}, e_{5}^{6}}$	φ

From Table 7, we have,

$\begin{matrix} D_{1} & = & {\begin{matrix} D (C_{2}, C_{1}), D (C_{3}, C_{1}), D (C_{3}, C_{2}), D (C_{5}, C_{1}), \\ D (C_{5}, C_{2}), D (C_{5}, C_{3}), D (C_{6}, C_{4}) \end{matrix}} \\ D_{2} & = & {\begin{matrix} D (C_{4}, C_{1}), D (C_{4}, C_{2}), D (C_{4}, C_{3}), D (C_{5}, C_{4}), \\ D (C_{6}, C_{1}), D (C_{6}, C_{2}), D (C_{6}, C_{3}), D (C_{6}, C_{5}) \end{matrix}} \end{matrix}$

In D₁, we get |E²| > |E¹| in D (C₂, C₁). So p₂ > {p₁, p₄}. Since |E³| = |E¹| in D (C₃, C₁), p₁, p₄ and p₃ are in the same decision class. That is, p₂ > {p₁, p₃, p₄}. Also |E¹| = |E⁵| in D (C₅, C₁). So objects in C₁ and objects in C₅ are in the same decision class. Thus we get p₂ > {p₁, p₃, p₄, p₆}. In D (C₆, C₄), |E⁶| > |E⁴|. So p₇ > p₅. Overall we get p₂ > {p₁, p₃, p₄, p₆} and p₇ > p₅.

Next consider D₂, We note that |E²| > |E⁶| in D (C₆, C₂), we have p₂ > p₇. Also in D (C₄, C₃), |E³| > |E⁴|. So p₃ is superior to p₅ and |E⁶| > |E³| in D (C₆, C₃), we get p₇ > p₃. That is, p₂ > p₇ > {p₁, p₃, p₄, p₆} > p₅. So we obtained an order relation and the optimal decision is to select the plot p₂. Therefore the group of associates should select the plot p₂ as the suitable plot for starting their paper producing factory.

5 Conclusions

In this paper, we introduced the definition of spherical fuzzy soft sets and discussed various operations on them. We also proved some theorems based on proposed definitions including DeMorgan’s law. Spherical fuzzy soft sets are a generalization of the concept of fuzzy soft sets. Also, this spherical fuzzy soft set model is more sensible and more accurate than existing other soft set models. Then as an application, a decision-making problem based on adjustable soft discernibility matrix on spherical fuzzy soft set is proposed.

Using this new extension of soft sets, expressions for distance measure, similarity measure, and entropy measure can be obtained. Also, the algebraic and topological structures can be studied as future work.

Footnotes

Acknowledgments

The authors are thankful to the editor and anonymous reviewers for their constructive comments and suggestions that helped us in improving the paper significantly. First author gratefully acknowledge the financial assistance provided by University Grants Commission (UGC) India, throughout the preparation of the manuscript.

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Spherical fuzzy soft sets and its applications in decision-making problems

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Tabular form of soft set 〈F, A〉 U e 1 e 2 e 3 h 1 1 0 0 h 2 1 1 1 h 3 1 1 1 h 4 1 0 0 h 5 0 0 1

Footnotes

Acknowledgments

References

Table 1
Tabular form of soft set 〈F, A〉

U e ₁ e ₂ e ₃

h ₁ 1 0 0

h ₂ 1 1 1

h ₃ 1 1 1

h ₄ 1 0 0

h ₅ 0 0 1