Cubic soft expert sets and their application in decision making

Abstract

The Cubic soft sets are primarily concerned with generalizing the soft sets by using fuzzy sets and interval valued fuzzy sets. We introduce the concept of cubic soft expert sets (CSESs) which can be considered as a generalization of both soft expert and cubic soft expert sets. The notions of internal cubic soft expert sets (ICSESs), external cubic soft expert sets (ECSESs), P-order, P-union, P-intersection, P-AND, P-OR and R-order, R-union, R-intersection, R-AND, R-OR have been defined for cubic soft expert sets (CSESs). We also investigate structural properties of these operations on cubic soft expert sets (CSESs). It has also been proved that cubic soft expert sets (CSESs) satisfy commutative, associative, De Morgan’s, distributive, idempotent and absorption laws. In last section, we provide the application of a cubic soft expert sets (CSESs) in multi-criteria decision making problem. We present the algorithm of a cubic soft expert decision making and give the numerical application.

Keywords

Cubic soft expert sets internal cubic soft expert sets external cubic soft expert sets interval valued fuzzy sets

1 Introduction

Mathematicians develop important analytical skills and problem-solving strategies to assess a broad range of some issues in commerce, science and the arts. Mathematical models and simulations, and the interpretation of their results, are being called on increasingly in global decisions, as business, politics and management all become more quantitative in their methods. The application of mathematics is also in demand in the social sciences, particularly economics, where mathematical tools are used to formulate models of the complex interactions in an economic system. In 1965 Zadeh initiated Fuzzy sets [27]. Fuzzy sets deal with possibilistic uncertainty, connected with imprecision of states, perceptions and preferences. Zadeh extended the concept of Fuzzy set by Interval valued fuzzy set [28]. Interval-valued fuzzy sets have been used in medicine [10]. Turken [23 –25] discussed interval valued fuzzy sets in detail. Molodtsov [16] introduced the concept of soft sets that can be seen as a new mathematical theory for dealing with uncertainty. Molodtsov applied this theory to several directions [17] and [18]. Molodtsov has been given soft sets technique and its applications in [18]. M. Irfan Ali et al. [1] discussed some new operations in soft set theory. Sezgin and Atagun [22] studied soft set operations. Maji et al. [14] studied soft set theory. He also defined fuzzy soft set theory and some properties of fuzzy soft sets in [13]. Cagman studied fuzzy soft set theory and its application in [4]. Pei et al. [20] and chen et al. [5] improved Maji’s work. Li has been given an approach to fuzzy multi-attribute decision making under uncertanity in [12]. Gorzalczany discussed a method of inference in approximate reasoning based on interval valued fuzzy sets in [7]. Xu has been given methods for aggregating interval valued intuitionistic fuzzy information and their application to decision making in [26]. Soft set theory has been applied to decision making problems in [3 , 21]. In [8] Jun et al. defined cubic sets. Khan et al. [9] discussed the generalized version of Jun’s cubic sets in semigroups. In [19] Muhiuddin and Al-roqi have introduced the concept of cubic soft sets with applications in BCI/BCK-algebras. In [2] the concept of soft expert sets has been introduced.

In this paper we define cubic soft expert sets by using fuzzy set and interval valued fuzzy set as opinion of experts. Corresponding to each attribute every expert give his expertise in relevant field through fuzzy sets and interval valued fuzzy sets. We define internal, external CSESs, P-order, P-union, P-intersection, P-AND, P-OR and R-order, R-union, R-intersection, R-AND, R-OR. We also investigate properties of these operations on CSESs. CSESs satisfy commutative, associative, De Morgan’s, distributive, idempotent and absorption laws. We derive the conditions for P-OR, P-AND of two internal cubic soft expert (ICSE) sets to be internal cubic soft expert set. We also give conditions for the P-OR, R-OR and R-AND of two external cubic soft expert sets (ECSESs) to be an external cubic soft expert set. We provide conditions for the R-AND and P-AND of two cubic soft expert sets to be an internal cubic soft expert set (ICSES) and an external cubic soft expert set (ECSES). At the end an algorithm has been presented to support our structure in decision analysis.

2 Preliminaries

In this section we give some basic concepts related to soft expert set, interval valued fuzzy set and cubic set.

Definition 2.1. [2] Let U be a universe, E be a set of parameters, and X be a set of experts. Let O = {0 = disagree, 1 = agree} be a set of two valued opinion, Z = E × X × O and A ⊆ Z. A pair (F, A) is called a soft expert set over U, where F is a mapping given by F : A → P (U) where P (U) denotes the power set of U.

Definition 2.2. [8] A fuzzy set in a set U is defined to be a function λ : U → I where I = [0, 1]. Denote by I^U the collection of all fuzzy sets in a set U. For any λ, μ ∈ I^U define a relation ≤ on I^U as follows: λ ≤ μ ⇔λ (u) ≤ μ (u) ∀u ∈ U . The complement of λ, denoted by λ^c, is defined by $λ^{c} (u) = 1 - λ (u) \forall u \in U$

For a family {λ_i|i ∈ Λ} of fuzzy sets in U, we define the join (∨) and meet (∧) operations as follows: $\begin{matrix} (\underset{i \in Λ}{⋁} λ_{i}) (u) & = & \sup {λ_{i} (u) | i \in Λ}, (\underset{i \in Λ}{⋀} λ_{i}) (u) \\ = & \inf {λ_{i} (u) | i \in Λ} . \end{matrix}$ respectively, for all u ∈ U.

Definition 2.3. [7] Let U be a non-empty set. A function A : U → Int ([0, 1]) where Int ([0, 1]) stands for the set of all closed sub intervals of [0, 1], the set of all interval-valued fuzzy sets on U is denoted by [I] ^U . For every A ∈ [I] ^U and u ∈ U, A (u) = [A⁻ (u) , A⁺ (u)] is called the degree of membership of an element u to A, where A⁻ : U → I and A⁺ : U → I are fuzzy sets in U which are called lower fuzzy set and upper fuzzy set in U, respectively. For simplicity, we denote A = [A⁻, A⁺]. For every A, B ∈ [I] ^U, we define

A^c (u) = [1 − A⁺ (u) , 1 + A⁻ (u)] ,

inf {A (u) , B (u)} = [inf {A⁻ (u) , B⁻ (u)} , inf {A⁺ (u) , B⁺ (u)}] ,

sup {A (u) , B (u)} = [sup {A⁻ (u) , B⁻ (u)} , sup {A⁺ (u) , B⁺ (u)}] ,

A (u) ⪯ B (u) if and only if A⁻ (u) ≤ B⁻ (u) and A⁺ (u) ≤ B⁺ (u) ,

A (u) ≽ B (u) if and only if A⁻ (u) ≥ B⁻ (u) and A⁺ (u) ≥ B (u) ,

A ⊆ B if and only if A (u) ⪯ B (u) ,

for all u ∈ U .

Definition 2.4. [8] Let U be a non-empty set. By a cubic set in U, we mean a structure α = {< u, A (u) , λ (u) > : u ∈ U} in which A is an interval valued fuzzy sets in U (briefly, IVF set) and λ is a fuzzy set in U. A cubic set α = {< u, A (u) , λ (u) > : u ∈ U} is simply denoted by α =< A, λ >.

3 Cubic soft expert sets

In this section we define the concept of cubic soft expert set, give its types and definitions of its basic operations namely, P-order, R-order, P-containment, R-containment, P-union, P-intersection, R-union, R-intersection, complement, P AND, P-OR, R-AND and R-OR. Several laws and related results have also been investigated.

Definition 3.1. Let U be a finite universe set containing n alternatives, E; a set of criteria and X; a set of experts (or decision makers). A pair (β, E, X) is called a cubic soft expert set over U if and only if β : E × X ⟶ CP (U) is a mapping into the set of all cubic sets in U . Cubic soft expert set is denoted and defined as $\begin{matrix} (β, E, X) & = & {β (e, x) = {< u, A_{(e, x)} (u), λ_{(e, x)} (u) > \\ : u \in U, (e, x) \in E \times X} . \end{matrix}$ where A_(e,x) (u) is an interval valued fuzzy set and λ_(e,x) (u) is a fuzzy set.

Example 1. Let U = {u₁, u₂, u₃} be the set of countries, E = {e₁ =Physiological natality, e₂ = Potential mortality} be the set of factors affecting population, X = {x₁, x₂} be the set of experts. Let E × X = {(e₁, x₁) , (e₁, x₂) , (e₂, x₁) , (e₂, x₂)} . Then the cubic soft expert set (β, E, X) in U is given by $\begin{matrix} β (e_{1}, x_{1}) & = & {(u_{1}, [0.07, 0.09], 0.09), \\ (u_{2}, [0.06, 0.08], 0.02), \\ (u_{3}, [0.03, 0.06], 0.04)}, \\ β (e_{2}, x_{1}) & = & {(u_{1}, [0.03, 0.05], 0.06), \\ (u_{2}, [0.05, 0.06], 0.03), \\ (u_{3}, [0.07, 0.08], 0.05)}, \\ β (e_{1}, x_{2}) & = & {(u_{1}, [0.05, 0.08], 0.08), \\ (u_{2}, [0.06, 0.09], 0.07), \\ (u_{3}, [0.05, 0.08], 0.06)}, \\ β (e_{2}, x_{2}) & = & {(u_{1}, [0.07, 0.09], 0.02), \\ (u_{2}, [0.05, 0.08], 0.08), \\ (u_{3}, [0.04, 0.07], 0.04)} . \end{matrix}$

In this example interval valued fuzzy set indicates the experts opinion for future time period and fuzzy set indicates the experts opinion for present time period under the different circumstances related to the given problem.

Definition 3.2. A cubic soft expert set is said to be an internal cubic soft expert (ICSE) set if $A_{(e, x)}^{-} (u) \leq$ $λ_{(e, x)} (u) \leq A_{(e, x)}^{+} (u)$ for all (e, x) ∈ E × X and for all u ∈ U .

Definition 3.3. A cubic soft expert set is said to be an external cubic soft expert (ECSE) set, if $λ_{(e, x)} (u) \notin] A_{(e, x)}^{-} (u), A_{(e, x)}^{+} (u) [$ for all (e, x) ∈ E × X and for all u ∈ U .

Definition 3.4. Let (β, E, X) be a CSES over U. For any e₁, e₂ ∊ E, x₁, x₂ ∊ X if $β (e_{1}, x_{1}) = \{< u, A_{1 (e_{1}, x_{1})} (u), λ_{1 (e_{1}, x_{1})} (u) > : u ∊ U\}$ and $β (e_{2}, x_{2}) = \{< u, A_{2 (e_{2}, x_{2})} (u), λ_{2 (e_{2}, x_{2})} (u) > : u ∊ U\}$ then P-order (R-order) denoted by $β (e_{1}, x_{1}) \subseteq_{P} β (e_{2}, x_{2})$ $(β (e_{1}, x_{1}) \subseteq_{R} β (e_{2}, x_{2}))$ , are defined as below:

$A_{1 (e_{1}, x_{1})} (u) \underline{≺} A_{2 (e_{2}, x_{2})} (u) (A_{1 (e_{1}, x_{1})} (u) \underline{≺} A_{2 (e_{2}, x_{2})} (u)), \forall u ∊ U .$

$λ_{1 (e_{1}, x_{1})} (u) \leq λ_{2 (e_{2}, x_{2})} (u) (λ_{1 (e_{1}, x_{1})} (u) \geq λ_{2 (e_{2}, x_{2})} (u)), \forall u ∊ U .$

Definition 3.5. A CSES (β₁, E₁, X₁) over U is said to be P-order (R-order) contained in another CSES (β₂, E₂, X₂) over U, denoted by (β₁, E₁, X₁) ⊆ _P (β₂, E₂, X₂) , (β (e₁, x₁) ⊆ _Rβ (e₂, x₂)) , are defined as below:

E₁ ⊆ E₂, (E₁ ⊆ E₂),

X₁ ⊆ X₂, (X₁ ⊆ X₂)

β₁ (e, x) ⊆ _Pβ₂ (e, x) , (β₁ (e, x) ⊆ _Rβ₂ (e, x)) for all e∊E₁, x∊X₁ .

Definition 3.6. Two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U, are equal, denoted by (β₁, E₁, X₁) = (β₂, E₂, X₂) , if E₁ = E₂, X₁ = X₂, and β₁ (e, x) = β₂ (e, x) for all e∊E₁ = E₂, x∊X₁ = X₂ .

Corollary 3.7.For any twoCSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) overU ; If (β₁, E₁, X₁) ⊆ _P (β₂, E₂, X₂) and (β₂, E₂, X₂) ⊆ _P (β₁, E₁, X₁) , then(β₁, E₁, X₁) = (β₂, E₂, X₂). Similar result holds for R-order.

Definition 3.8. The P-union of two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U is denoted by (β₃, F, Y) = (β₁, E₁, X₁) ∪_P (β₂, E₂, X₂) where F = E₁ ∪ E₂, Y = X₁ ∪ X₂ and for all g ∈ F and z ∈ Y, it is defined as: $β_{3} (g, z) = {\begin{matrix} {< u, A_{1 (g, z)} (u), λ_{1 (g, z)} (u) >} & if (g, z) \in (E_{1} \times X_{1}) \ (E_{2} \times X_{2}) \\ {< u, A_{2 (g, z)} (u), λ_{2 (g, z)} (u) >} & if (g, z) \in (E_{2} \times X_{2}) \ (E_{1} \times X_{1}) \\ {< u, sup {A_{1 (g, z)} (u), A_{2 (g, z)} (u)} & if (g, z) \in (E_{1} \cap E_{2} \times X_{1} \cap X_{2}), \\ sup {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$ whenever β₁ (g, z) = {< u, A_1(g,z) (u) , λ_1(g,z) (u) > : u ∈ U} and β₂ (g, z) = {< u, A_2(g,z) (u) , λ_2(g,z) (u) > : u ∈ U} .

Definition 3.9. The P-intersection of two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U is denoted by (β₃, F, Y) = (β₁, E₁, X₁) ∩_P (β₂, E₂, X₂) where F =E₁ ∩ E₂, Y = X₁ ∩ X₂ and for all g ∈ F and z ∈ Y, it is defined as: $\begin{matrix} β_{3} (g, z) & = & {< u, inf {A_{1 (g, z)} (u), A_{2 (g, z)} (u)}, \\ inf {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$

Definition 3.10. The R-union of two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U is denoted by (β₃, F, Y) = (β₁, E₁, X₁) ∪_R (β₂, E₂, X₂) where F=E₁ ∪ E₂, Y = X₁ ∪ X₂ and for all g ∈ F and z ∈ Y, it is defined as: $β_{3} (g, z) = {\begin{matrix} {< u, A_{1 (g, z)} (u), λ_{1 (g, z)} (u) >} & if (g, z) \in (E_{1} \times X_{1}) \ (E_{2} \times X_{2}) \\ {< u, A_{2 (g, z)} (u), λ_{2 (g, z)} (u) >} & if (g, z) \in (E_{2} \times X_{2}) \ (E_{1} \times X_{1}) \\ {< u, sup {A_{1 (g, z)} (u), A_{2 (g, z)} (u)} & if (g, z) \in (E_{1} \cap E_{2} \times X_{1} \cap X_{2}), \\ inf {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$

Definition 3.11. The R-intersection of two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U is denoted by (β₃, F, Y) = (β₁, E₁, X₁) ∩_R (β₂, E₂, X₂) where F =E₁ ∩ E₂, Y = X₁ ∩ X₂ and for all g ∈ F and z ∈ Y, it is defined as: $\begin{matrix} β_{3} (g, z) & = & {< u, inf {A_{1 (g, z)} (u), A_{2 (g, z)} (u)}, \\ sup {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$

Definition 3.12. The complement of a CSES (β, E, X) is denoted and defined as (β, E, X) ^c = (β^c, E^c, X) where β^c: E^c × X ⟶ CP (U) is a mapping given as $\begin{array}{c} β^{c} (e^{c}, x) = {< u, A_{(e, x)}^{c} (u), λ_{(e, x)}^{c} (u) > : u ∊ U, \\ (e^{c}, x) ∊ E^{c} \times X}, \end{array}$ where $A_{(e, x)}^{c} (u) = [1 - A_{(e, x)}^{+} (u), 1 - A_{(e, x)}^{-} (u)]$ and $λ_{(e, x)}^{c} (u) = 1 - λ_{(e, x)} (u)$ whenever β (e, x) = {< u, A_(e,x) (u) , $λ_{(e, x)} (u) > : u ∊ U}$

Theorem 3.13.For anyCSESs (β₁, E₁, X₁) , (β₂, E₂, X₂) , (β₃, E₃, X₃) and (β₄, E₄, X₄) overUthe following properties hold

Idempotent (β₁, E₁, X₁) ∪ _P (β₁, E₁, X₁) = (β₁, E₁, X₁) = (β₁, E₁, X₁) ∩ _P (β₁, E₁, X₁ .

Commutative (β₁, E₁, X₁) ∪ _P (β₂, (E₂, X₂) = (β₂, E₂, X₂) ∪ _P (β₁, E₁, X₁) .

Associative ((β₁, E₁, X₁) ∪ _P (β₂, E₂, X₂)) ∪ _P (β₃, E₃, X₃) = (β₁, E₁, X₁) ∪ _P ((β₂, E₂, X₂) ∪ _P (β₃, E₃, X₃)) .

Distributive (β₁, E₁, X₁) ∪ _P ((β₂, E₂, X₂) ∩ _P (β₃, E₃, X₃)) = ((β₁, E₁, X₁) ∪ _P (β₂, E₂, X₂)) ∩ _P ((β₁, E₁, X₁) ∪_P (β₃, E₃, X₃)) , (β₁, E₁, X₁) ∩ _P ((β₂, E₂, X₂) ∪ _P (β₃, E₃, X₃)) = ((β₁, E₁, X₁) ∩ _P (β₂, E₂, X₂)) ∪ _P ((β₁, E₁, X₁) ∩ _P (β₃, E₃, X₃)) .

De Morgan’s laws ((β₁, E₁, X₁)) ∪ _P (β₂, E₂, X₂)) ^c = (β₁, E₁, X₁) ^c ∩ _P (β₂, E₂, X₂) ^c, ((β₁, E₁, X₁) ∩ _P (β₂,, E₂, X₂)) ^c = (β₁, E₁, X₁) ^c ∪ _P (β₂, E₂, X₂) ^c

Involution law ((β₁, E₁, X₁) ^c) ^c = (β₁, E₁, X₁) .

Similar results holds for R-order, R-union andR-intersection.

Proof. These properties can be verified by using Definitions 3.8, 3.9, 3.10, 3.11 and 3.12. □

Proposition 3.14. For any two CSES (β₁, E₁, X₁) and (β₂, E₂, X₂) over U the following are equivalent

(β₁, E₁, X₁) ⊆ _P (β₂, E₂, X₂) , 2) (β₁, E₁, X₁) ∩ _P (β₂, E₂, X₂) = (β₁, E₁, X₁)

(β₁, E₁, X₁) ∩ _P (β₂, E₂, X₂) = (β₁, E₁, X₁)

(β₁, E₁, X₁) ∪ _P (β₂, E₂, X₂) = (β₂, E₂, X₂)

Proof. 1) ⇒2) By Definition 3.9 we have (β₁, E₁, X₁) ∩_P (β₂, E₂, X₂) = (β₁ ∩ _Pβ₂, E₁ ∩ E₂, X₁ ∩ X₂) = (β₁ ∩ _Pβ₂, E₁, X₁) as E₁ ⊆ E₂ and X₁ ⊆ X₂ by hypothesis. Now for any (e, x) _inE1 × X₁, since β₁ (e, x) ⊆ _Pβ₂ (e, x) , Definition 3.4implies that A_1(e,x) (u) ⪯ A_2(e,x) (u) and λ_1(e,x) (u) ≤ λ_2(e,x) (u) for any u ∈ U, where β₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > u ∈ U} . By Definition 3.4, we have $A_{1 (e, x)}^{-} (u) \leq A_{2 (e, x)}^{-} (u)$ and $A_{1 (e, x)}^{+} (u) \leq A_{2 (e, x)}^{+} (u) .$ Thus inf ${A_{1 (e, x)} (u), A_{2 (e, x)} (u)} = [inf {A_{1 (e, x)}^{-} (u) \leq A_{2 (e, x)}^{-} (u)}, inf {A_{1 (e, x)}^{+} (u) \leq A_{2 (e, x)}^{+} (u)}] = [A_{1 (e, x)}^{-} (u) \leq A_{1 (e, x)}^{+} (u)]$ and inf{λ_1(e,x) (u) , λ_2(e,x) (u)} = λ_1(e,x) (u) . By using Definition 3.9 β₁ (e, x) ∩ _Pβ₂ (e, x) = {< u, inf {A_1(e,x) (u) , A_2(e,x) (u)} , inf {λ_1(e,x) (u) , λ_2(e,x) (u)} > : u ∈ U} = {< u, A_1(e,x) (u), λ_1(e,x) (u)}> : u ∈ U} = β₁ (e, x) . Hence β₁ (e, x) ∩ _Pβ₂ (e, x) = β₁ (e, x) .

2) ⇒3) By Definition 3.8 we have (β₁, E₁, X₁) ∪ _P (β₂, E₂, X₂) = (β₁ ∪ _Pβ₂, E₁ ∪ E₂, X₁ ∪ X₂) = (β₁ ∪ _Pβ₂, E₂, X₂) as E₁ ∩ E₂ = E₁ and X₁ ∩ X₂ = X₁ by hypothesis. Now for any(e, x) ∈ E₁ × X₁, since β₁ (e, x) ∩ _Pβ₂ (e, x) = β₁ (e, x) , by Definition 3.9 we have inf {A_1(e,x) (u) , A_2(e,x) (u)} = A_1(e,x) (u) and inf {λ_1(e,x) (u) , λ_2(e,x) (u)} = λ_1(e,x) (u) this implies that sup{A_1(e,x) (u) , A_2(e,x) (u)} = A_2(e,x) (u) and sup{λ_1(e,x) (u) , λ_2(e,x) (u)} = λ_2(e,x) (u) . Thus we have β₁ (e, x) ∪ _Pβ₂ (e, x) = {< u, sup {A_1(e,x) (u) , A_2(e,x) (u)} , sup {λ_1(e,x) (u) , λ_2(e,x) (u)} > : u ∈ U} = {< u, A_2(e,x) (u) , λ_2(e,x) (u)} > : u ∈ U} = β₂ (e, x) . Hence β₁ (e, x) ∪ _Pβ₂ (e, x) = β₂ (e, x) .

3) ⇒1) By hypothesis we have (β₁, E₁, X₁) ∪_P (β₂, E₂, X₂) = (β₁ ∪ _Pβ₂, E₁ ∪ E₂, X₁ ∪ X₂) = (β₁ ∪ _Pβ₂, E₂, X₂) as E₁ ∪ E₂ = E₂ and X₁ ∪ X₂ = X₂ ⇒ E₁ ⊆ E₂ and X₁ ⊆ X₂ ..Also, β₁ (e, x) ∪ _Pβ₂ (e, x) = {< u, sup {A_1(e,x) (u) , A_2(e,x) (u)} , sup {λ_1(e,x) (u) , λ_2(e,x) (u)}> : u ∈ U} = {< u, A_2(e,x) (u) , λ_2(e,x) (u)} > : u ∈ U} = β₂ (e, x)⇒ A_1(e,x) (u) ⪯ A_2(e,x) (u) and λ_1(e,x) (u) ≤ λ_2(e,x) (u) for any u ∈ U . Hence (β₁, E₁, X₁) ⊆ _P (β₂, E₂, X₂) . □

Corollary 3.15.If we takeX₁ = X₂ = Xin above proposition then the following are equivalent

(β₁, E₁, X) ⊆ _P (β₂, E₂, X)

(β₁, E₁, X) ∩ _P (β₂, E₂, X) = (β₁, E₁, X)

(β₁, E₁, X) ∪ _P (β₂, E₂, X) = (β₂, E₂, X)

(β₂, E₂, X) ^c ⊆ _P (β₁, E₁, X) ^c

Definition 3.16. Let {_ℒi} _i∈ℑ = {(β_i, E_i, X_i)} _i∈ℑ be a family of CSESs over U, where β_i (e, x) = {< u, A_i(e,x) (u) , λ_i(e,x) (u) > u ∈ U, for any e ∈ E_i, x ∈ X_i)}. Then P-union, P-intersection, R-union andR-intersection are defined as below:

$\underset{i \in ℑ}{\cup_{P}} {ℒ_{i}} = {< u, (sup_{i \in ℑ} A_{i (e, x)}) (u), (⋁_{i \in ℑ} λ_{i (e, x)}) (u) > u \in U} .$

$\underset{i \in ℑ}{\cap_{P}} {ℒ_{i}} = {< u, (inf_{i \in ℑ} A_{i (e, x)}) (u), (⋀_{i \in ℑ} λ_{i (e, x)}) (u) > u \in U} .$

$\underset{i \in ℑ}{\cup_{R}} {ℒ_{i}} = {< u, (sup_{i \in ℑ} A_{i (e, x)}) (u), (⋀_{i \in ℑ} λ_{i (e, x)}) (u) > u \in U} .$

$\underset{i \in ℑ}{\cap_{R}} {ℒ_{i}} = {< u, (inf_{i \in ℑ} A_{i (e, x)}) (u), (⋁_{i \in ℑ} λ_{i (e, x)}) (u) > u \in U} .$

Theorem 3.17.Let {_ℒi} _i∈ℑ = {(β_i, E_i, X_i)} _i∈ℑbe a family ofICSESs over U, whereβ_i (e, x) = {< u, A_i(e,x) (u) , λ_i(e,x) (u) > u ∈ U, for anye ∈ E_i, x ∈ X_i)}. Then the $\underset{i \in ℑ}{\cup_{P} {} ℒ_{i}}$ and $\underset{i \in ℑ}{\cap_{P}} {ℒ_{i}}$ are ICSESs over U .

Proof. By using Definitions 3.16 and 3.2 we can easily prove this theorem. □

Theorem 3.18.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoICSESsoverU, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈U} for any (e, x) ∈ E₁ × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂. Then the P-union of (β₁, E₁, X₁) and (β₂, E₂, X₂) is also anICSESover U.

Proof. Since (β₁, E₁, X₁) and (β₂, E₂, X₂) are ICSESs over U so $A_{1 (g, z)}^{-} (u) \leq λ_{1 (g, z)} (u) \leq A_{1 (g, z)}^{+} (u)$ for all u ∈ U and $A_{2 (g, z)}^{-} (u) \leq λ_{2 (g, z)} (u) \leq A_{2 (g, z)}^{+} (u)$ for all u ∈ U . Then we have $sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{-} (u)} \leq (λ_{1 (g, z)} \lor λ_{2 (g, z)}) (u) \leq sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{+} (u)}$ for all u ∈ U and (g, z) ∈ (E₁ ∪ E₂ × X₁ ∪ X₂) . By definition of 3.8 we have (β₃, F, Y) = (β₁, E₁, X₁) ∪_P (β₂, E₂, X₂) where F =E₁ ∪ E₂ and Y = X₁ ∪ X₂ and for any g ∈ F and z ∈ Y . $β_{3} (g, z) = {\begin{matrix} {< u, A_{1 (g, z)} (u), λ_{1 (g, z)} (u) >} & if (g, z) \in (E_{1} \times X_{1}) \ (E_{2} \times X_{2}) \\ {< u, A_{2 (g, z)} (u), λ_{2 (g, z)} (u) >} & if (g, z) \in (E_{2} \times X_{2}) \ (E_{1} \times X_{1}) \\ {< u, sup {A_{1 (g, z)} (u), A_{2 (g, z)} (u)}, & if (g, z) \in (E_{1} \cap E_{2} \times X_{1} \cap X_{2}) \\ sup {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$ if (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) , then β₃ (g, z) = {< u, sup {A_1(g,z) (u) , A_2(g,z) (u)} , (λ_1(g,z)∨ λ_2(g,z)) (u) > : u ∈ U } . Thus (β₁, E₁, X₁) ∪_P (β₂, E₂, X₂)) is an ICSE set. If (g, z) ∈ (E₁ × X₁) \ (E₂ × X₂) or if (g, z) ∈ (E₂ × X₂) \ (E₁ × X₁) , then the result is trivial. Hence (β₁, E₁, X₁) ∪_P (β₂, E₂, X₂) is an ICSES over U . □

Theorem 3.19.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoICSESs over U, where. β₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) ∈ E₁ × X₁ and β₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂ . Then the P-intersection of (β₁, E₁, X₁) and (β₂, E₂, X₂) is also anICSES.

Proof. By similar way as Theorem 3.18 we can prove this theorem. □

The following theorem gives the condition that R-union of two ICSESs is also an ICSES .

Theorem 3.20.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoICSESsoverU, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) ∈ E₁ × X₁ and β₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{-} (u)} \leq (λ_{1 (g, z)} \land λ_{2 (g, z)}) (u)$ for allu ∈ Uand (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂). Then the R-union of (β₁, E₁, X₁) and (β₂, E₂, X₂) is also anICSES .

Proof. By Definitions 3.10 and 3.2 we can easily prove this theorem. □

The following theorem gives the condition that R-intersection of two ICSESs is also an ICSES .

Theorem 3.21.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoICSESsoverU, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) ∈ E₁ × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $inf {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{+} (u)} \geq (λ_{1 (g, z)} \lor λ_{2 (g, z)}) (u)$ for allu ∈ Uand (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂). Then the R-intersection of (β₁, E₁, X₁) and (β₂, E₂, X₂) is also anICSESoverU.

Proof. By Definition 3.11, we have (β₃, E₃, X₃) = (β₁, E₁, X₁) ∩ _R (β₂, E₂, X₂) where E₃ = E₁ ∩ E₂ and X₃ = X₁ ∩ X₂, g ∈ E₃ and z ∈ X₃ . $\begin{matrix} β_{3} (g, z) & = & {< u, inf {A_{1 (g, z)} (u), A_{2 (g, z)} (u)}, \\ sup {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$

Since (β₁, E₁, X₁) and (β₂, E₂, X₂) are ICSESs over U . So we have $A_{1 (g, z)}^{-} (u) \leq λ_{1 (g, z)} (u) \leq A_{1 (g, z)}^{+} (u)$ for all u ∈ U and $A_{2 (g, z)}^{-} (u) \leq λ_{2 (g, z)} (u) \leq A_{2 (g, z)}^{+} (u)$ for all u ∈ U . Also $inf {A_{1 (g, z)}^{-} (u),$ $A_{2 (g, z)}^{-} (u)} \leq$ (λ_1(g,z) ∨ λ_2(g,z)) (u) ≤ inf ${A_{1 (g, z)}^{+} (u),$ $A_{2 (g, z)}^{+} (u)}$ for all u ∈ U and (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) .Hence (β₁, E₁, X₁) ∩ _R (β₂, E₂, X₂) is an ICSES over U . □

Theorem 3.22.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoECSESsoverU, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) ∈ E₁ × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $\begin{matrix} (λ_{1 (g, z)} \land λ_{2 (g, z)}) (u) \in \\ {\begin{matrix} inf {sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}}, \\ {inf {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, inf {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}} \end{matrix} \end{matrix}$

for all (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) andu ∈ U . Then (β₁, E₁, X₁) ∪ _R (β₂, E₂, X₂) is also anECSESoverU .

Proof. By Definition 3.10, we have (β₃, E₃, X₃) = (β₁, E₁, X₁) ∪ _R (β₂, E₂, X₂) where $β_{3} (g, z) = {\begin{matrix} {< u, A_{1 (g, z)} (u), λ_{1 (g, z)} (u) >} & if (g, z) \in (E_{1} \times X_{1}) \ (E_{2} \times X_{2}) \\ {< u, A_{2 (g, z)} (u), λ_{2 (g, z)} (u) >} & if (g, z) \in (E_{2} \times X_{2}) \ (E_{1} \times X_{1}) \\ {< u, sup {A_{1 (g, z)} (u), A_{2 (g, z)} (u)}, & if (g, z) \in (E_{1} \cap E_{2} \times X_{1} \cap X_{2}) \\ inf {λ_{1 (g, z)} (u), λ_{2 (g, z)} (u)} >} \end{matrix}$ if (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂), take $ℏ = inf {sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)},$ $sup {A_{1 (g, z)}^{-} (u),$ $A_{2 (g, z)}^{+} (u)}}$ and $R =$ sup {inf ${A_{1 (g, z)}^{+} (u),$ $A_{2 (g, z)}^{-} (u)},$ inf ${A_{1 (g, z)}^{-} (u),$ $A_{2 (g, z)}^{+} (u)}} .$ Then ℏ is one of $A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u),$ $A_{1 (g, z)}^{-} (u),$ $A_{2 (g, z)}^{+} (u)$ we only consider $ℏ = A_{2 (g, z)}^{-} (u)$ or $A_{2 (g, z)}^{+} (u)$ because remaining cases are similar to this one. If $ℏ = A_{2 (g, z)}^{-} (u)$ then $A_{1 (g, z)}^{-} (u)$ ≤ $A_{1 (g, z)}^{+} (u)$ ≤ $A_{2 (g, z)}^{-} (u)$ ≤ $A_{2 (g, z)}^{+} (u)$ and so $R = A_{1 (g, z)}^{+} (u) .$ Thus (sup {A_1(g,z), $A_{2 (g, z)}})^{-} (u) = A_{2 (g, z)}^{-} (u) = ℏ > (λ_{1 (g, z)} \land λ_{2 (g, z)}) (u) .$ Hence (λ_1(g,z) ∧ λ_2(g,z)) (u) ∉ ((sup {A_1(g,z), A_2(g,z)} ⁻ (u) , sup {A_1(g,z), A_2(g,z)} ⁺ (u)) . if ℏ = $A_{2 (g, z)}^{+} (u)$ then $A_{1 (g, z)}^{-} (u) \leq A_{2 (g, z)}^{+} (u)$ $\leq A_{1 (g, z)}^{+} (u)$ so $R = sup {A_{1 (g, z)}^{-} (u),$ $A_{2 (g, z)}^{-} (u)} .$ Assume $R = A_{1 (g, z)}^{-} (u),$ then we have $A_{2 (g, yz)}^{-} (u)$ ≤ $A_{1 (g, z)}^{-} (u)$ ≤ (λ_1(g,z)∧ λ_2(g,z)) (u) < $A_{2 (g, z)}^{+} (u) \leq$ $A_{1 (g, z)}^{+} (u) .$ So we can write $A_{2 (g, z)}^{-} (u)$ $\leq A_{1 (g, z)}^{-} (u)$ < (λ_1(g,z) ∧ λ_2(g,z)) (u) $< A_{2 (g, z)}^{+} (u)$ $\leq A_{1 (g, z)}^{+} (u)$ or $A_{2 (g, z)}^{-} (u)$ ≤ $A_{1 (g, z)}^{-} (u) = (λ_{1 (g, z)} \land λ_{2 (g, z)}) (u)$ $\leq A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u) .$

For the case $A_{2 (g, z)}^{-} (u)$ ≤ $A_{1 (g, z)}^{-} (u) <$ (λ_1(g,z) ∧ λ_2(g,z)) (u) < $A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u)$ which contradict the fact that (β₁, E₁, X₁) and (β₂, E₂, X₂) are ECSESs. For the case $A_{2 (g, z)}^{-} (u)$ < $A_{1 (g, z)}^{-} (u) =$ (λ_1(g,z) ∧ λ_2(g,z)) (u) $\leq A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u)$ we have (λ_1(g,z) ∧ λ_2(g,z)) (u) ∉ ((sup {A_1(g,z), A_2(g,z)}) ⁻ (u) , (sup {A_1(g,z), A_2(g,z)}) ⁺ (u)) because (sup {A_1(g,z), A_2(g,z)}) ⁻ (u) = $A_{1 (g, z)}^{-} (u)$ = (λ_1(g,z) ∧ λ_2(g,z)) (u) . Again assume that $R = A_{2 (g, z)}^{-} (u),$ then we have $A_{1 (g, z)}^{-} (u)$ ≤ $A_{2 (g, z)}^{-} (u)$ ≤ (λ_1(g,z) ∧ λ_2(g,z)) (u) ≤ $A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u) .$ we can write $A_{1 (g, z)}^{-} (u)$ $\leq A_{2 (g, z)}^{-} (u)$ < (λ_1(g,z) ∧ λ_2(g,z)) (u) $< A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u)$ or $A_{1 (g, z)}^{-} (u)$ ≤ $A_{2 (g, z)}^{-} (u)$ = (λ_1(g,z) ∧ λ_2(g,z)) (u) < $A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u) .$ For the case $A_{1 (g, z)}^{-} (u)$ ≤ $A_{2 (g, z)}^{-} (u) <$ (λ_1(g,z) ∧ λ_2(g,z)) (u) $< A_{2 (g, z)}^{+} (u)$ $\leq A_{1 (g, z)}^{+} (u)$ which contradict the fact that (β₁, E₁, X₁) and (β₂, E₂, X₂) are ECSESs. For the case $A_{1 (g, z)}^{-} (u)$ $\leq A_{2 (g, z)}^{-} (u)$ = (λ_1(g,z)∧ λ_2(g,z)) (u) ≤ $A_{2 (g, z)}^{+} (u)$ ≤ $A_{1 (g, z)}^{+} (u)$ we have (λ_1(g,z)∧ λ_2(g,z)) (u) ∉ (sup {A_1(g,z), A_2(g,z)}) ⁻ (u) , (sup {A_1(g,z), A_2(g,z)}) ⁺ (u) because (sup {A_1(g,z), A_2(g,z)}) ⁻ (u) = $A_{2 (g, z)}^{-} (u) =$ (λ_1(g,z) ∧ λ_2(g,z)) (u) . if (g, z) ∈ (E₁ × X₁) \ (E₂ × X₂) or (g, z) ∈ (E₂ × X₂) \ (E₁ × X₁) then the result holds trivially. Hence (β₁, E₁, X₁) ∪ _R (β₂, E₂, X₂) is an ECSES over U . □

Theorem 3.23.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoECSESsoverU, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) ∈ E₁ × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $\begin{matrix} (λ_{1 (g, z)} \lor λ_{2 (g, z)}) (u) \in \\ {\begin{matrix} inf {sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}}, \\ {inf {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, inf {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}} \end{matrix} \end{matrix}$

for all (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) andu ∈ U . Then (β₁, E₁, X₁) ∩ _R (β₂, E₂, X₂) is also anECSESoverU .

Proof. We can proof this theorem by similar way to Theorem 3.22. □

In next theorem we derive condition that P-union of two ECSESs are ECSES.

Theorem 3.24.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoECSESs over U, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) _inE1 × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $\begin{matrix} (λ_{1 (g, z)} \lor λ_{2 (g, z)}) (u) \in \\ {\begin{matrix} inf {sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}}, \\ {inf {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, inf {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}} \end{matrix} \end{matrix}$

for all (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) andu ∈ U . Then (β₁, E₁, X₁) ∪ _P (β₂, E₂, X₂) is anECSES over U .

Proof. By Definition 3.8 we can also proof above theorem. □

In next theorem we derive condition thatP-intersection of two CSESs are ECSES (ICSES).

Theorem 3.25.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoCSESs over U, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) _inE1 × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $\begin{matrix} (λ_{1 (g, z)} \land λ_{2 (g, z)}) (u) \\ = {\begin{matrix} inf {sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}}, \\ {inf {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, inf {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}} \end{matrix} \end{matrix}$

for all (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) andu ∈ U . Then (β₁, E₁, X₁) ∩ _P (β₂, E₂, X₂)) is both anECSESandICSESoverU .

Proof. We can proof this theorem by similar way to Theorem 3.22. □

Theorem 3.26.Let (β₁, E₁, X₁) and (β₂, E₂, X₂) are twoCSESs over U, whereβ₁ (e, x) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > : u ∈ U} for any (e, x) _inE1 × X₁andβ₂ (f, y) = {< u, A_2(f,y) (u) , λ_2(f,y) (u) > : u ∈ U} for any (f, y) ∈ E₂ × X₂such that $\begin{matrix} (λ_{1 (g, z)} \lor λ_{2 (g, z)}) (u) \\ = {\begin{matrix} inf {sup {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, sup {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}}, \\ {inf {A_{1 (g, z)}^{+} (u), A_{2 (g, z)}^{-} (u)}, inf {A_{1 (g, z)}^{-} (u), A_{2 (g, z)}^{+} (u)}} \end{matrix} \end{matrix}$

for all (g, z) ∈ (E₁ ∩ E₂ × X₁ ∩ X₂) andu ∈ U . Then (β₁, E₁, X₁) ∩ _R (β₂, E₂, X₂) is both anECSEset andICSESoverU .

Proof. We can proof this theorem by similar way to Theorem 3.22. □

Theorem 3.27.For any two cubic soft expert sets (β₁, E₁, X₁) and (β₂, E₂, X₂) the following absorption law hold

(β₁, E₁, X₁) ∪ _P ((β₁, E₁, X₁) ∩ _P (β₂, E₂, X₂)) = (β₁, E₁, X₁) ,

(β₁, E₁, X₁) ∩ _P ((β₁, E₁, X₁) ∪ _P (β₂, E₂, X₂)) = (β₁, E₁, X₁) ,

(β₁, E₁, X₁) ∪ _R ((β₁, E₁, X₁) ∩ _R (β₂, E₂, X₂)) = (β₁, E₁, X₁) ,

(β₁, E₁, X₁) ∩ _R ((β₁, E₁, X₁) ∪ _R (β₂, E₂, X₂)) = (β₁, E₁, X₁) .

Proof. 1) By Definitions 3.8 and 3.9 we have (β₁, E₁, X₁) ∪ _P ((β₁, E₁, X₁) ∩ _P (β₂, E₂, X₂)) = (β₃, E₁ ∪ _P (E₁ ∩ _PE₂) , X₁ ∪ _P (X₁ ∩ _PX₂)) = (β₃, E₁, X₁)

such that for any g ∈ E₁ and z ∈ X₁ we have $\begin{matrix} β_{3} (g, z) & = & β_{1} (g, z) \cup_{P} ((β_{1} (g, z) \cap_{P} \\ β_{2} (g, z)) if (g, z) \in E_{1} \times X_{1} \end{matrix}$ β₁ (g, z) ∪ _P ((β₁ (g, z) ∩ _Pβ₂ (g, z)) = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > , u ∈ U, (e, x) ∈ E₁ × X₁} ∪ _P {{< u, A_1(e,x) (u) , λ_1(e,x) (u) > , u ∈ U, (e, x) ∈ E₁ × X₁} ∩_P {< u, A_2(f,y) (u) , λ_2(f,y) (u) > , u ∈ U, (f, y) _inE2 × X₂}} = {< u, A_1(e,x) (u) , λ_1(e,x) (u) > , u ∈ U, (e, x) ∈ E₁ × X₁} ∪_P {< u, r inf {A_1(e,x) (u) , A_2(f,y) (u)} , inf {λ_1(e,x) (u) , λ_2(f,y) (u)} >} = {< u, r sup {A_1(e,x) (u) , r inf {A_1(e,x) (u) , A_2(f,y) (u)}} , sup {λ_1(e,x) (u) , inf {λ_1(e,x) (u) , λ_2(f,y) (u)}} >} ⊆ {< u, A_1(e,x) (u) , λ_1(e,x) (u) > , u ∈ U, (e, x) ∈ E₁ × X₁} = β₁ (e, x) ⊆ {< u, r inf {A_1(e,x) (u) , r sup {A_1(e,x) (u) , A_2(f,y) (u)}} , inf {λ_1(e,x) (u) , sup {λ_1(e,x) (u) , λ_2(f,y) (u)}}> = {< u, r sup {A_(e,x) (u) , r inf {A_(e,x) (u) , A_(f,y) (u)}} , sup {λ_(e,x) (u) , inf {λ_(e,x) (u) , λ_(f,y) (u)}} >} = β₁ (e, x) ∪ _P ((β₁ (e, x) ∩ _Pβ₂ (f, y)) .

In second case when (g, z) ∈ (E₁ × X₁) \ (E₂ × X₂) , using Definitions 3.8, 3.9, we have β₁ (e, x) ∪ _P ((β₁ (e, x) ∩ _Pβ₂ (f, y)) = β₁ (e, x) ∪ _Pβ₁ (e, x) = β₁ (e, x) which is our required result for both the cases. Similarly, we can prove 2) , 3) and 4). □

Definition 3.28. For two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U, P-AND is denoted and defined as $\begin{matrix} (β_{1}, E_{1}, X_{1}) ⋀_{P} (β_{2}, E_{2}, X_{2}) \\ = (β_{3}, (E_{1} \times E_{2}), (X_{1} \times X_{2})) \end{matrix}$ where β₃ ((e, f) , (x, y)) = β₁ (e, x) ∩ _Pβ₂ (f, y) for all ((e, f) , (x, y))∈ ((E₁ × E₂) × (X₁ × X₂)) .

Definition 3.29. For two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U, R-AND is denoted and defined as $\begin{matrix} (β_{1}, E_{1}, X_{1}) ⋀_{R} (β_{2}, E_{2}, X_{2}) \\ = (β_{3}, (E_{1} \times E_{2}), (X_{1} \times X_{2})) \end{matrix}$ where β₃ ((e, f) , (x, y)) = β₁ (e, x) ∩ _Rβ₂ (f, y) for all ((e, f) , (x, y)) ∈ ((E₁ × E₂) × (X₁ × X₂)) .

Definition 3.30. For two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U, P-OR is denoted and defined as $\begin{matrix} (β_{1}, E_{1}, X_{1}) ⋁_{P} (β_{2}, E_{2}, X_{2}) \\ = (β_{3}, (E_{1} \times E_{2}), (X_{1} \times X_{2})) \end{matrix}$ where β₃ ((e, f) , (x, y)) = β₁ (e, x) ∪ _Pβ₂ (f, y) for all ((e, f) , (x, y)) ∈ ((E₁ × E₂) × (X₁ × X₂)) .

Definition 3.31. For two CSESs (β₁, E₁, X₁) and (β₂, E₂, X₂) over U, R-OR is denoted and defined as $\begin{matrix} (β_{1}, E_{1}, X_{1}) ⋁_{R} (β_{2}, E_{2}, X_{2}) \\ = (β_{3}, (E_{1} \times E_{2}), (X_{1} \times X_{2})) \end{matrix}$ where β₃ ((e, f) , (x, y)) = β₁ (e, x) ∪ _Rβ₂ (f, y) for all ((e, f) , (x, y)) ∈ ((E₁ × E₂) × (X₁ × X₂)) .

Example 2. Let U = {u₁, u₂, u₃} be the initial universe, E = {e₁, e₂} be the set of attributes, X = {x₁, x₂} be the set of experts. Then the cubic set (β₁, E, X) over U is given below: $\begin{matrix} β_{1} (e_{1}, x_{1}) & = & {(u_{1}, [0.5, 0.8], 0.7), (u_{2}, [0.6, 0.7], \\ 0.8), (u_{3}, [0.4, 0.8], 0.5)}, \\ β_{1} (e_{2}, x_{2}) & = & {(u_{1}, [0.3, 0.8], 0.4), (u_{2}, [0.2, 0.9], \\ 0.7), (u_{3}, [0.3, 0.7], 0.6)} . \end{matrix}$

Let U = {u₁, u₂, u₃} be the initial universe, F = {f₁, f₂} be the set of attributes and Y = {y₁, y₂} be the set of experts. Then the cubic set (β₂, F, Y) over U is given below: $\begin{matrix} β_{2} (f_{1}, y_{1}) & = & {(u_{1}, [0.5, 0.8], 0.4), (u_{2}, [0.6, 0.9], \\ 0.9), (u_{3}, [0.4, 0.7], 0.8)}, \\ β_{2} (f_{2}, y_{2}) & = & {(u_{1}, [0.3, 0.8], 0.2), (u_{2}, [0.6, 0.9], \\ 0.4), (u_{3}, [0.2, 0.7], 0.8)} . \end{matrix}$

By using Definitions 3.28, 3.29, 3.30 and 3.31 we have $\begin{matrix} (β_{1}, E, X) ⋀_{P} (β_{2}, F, Y) \\ = (β_{3}, (E \times F), (X \times Y) \\ = β_{3} ((e_{1}, f_{1}), (x_{1}, y_{1})) \\ = {(u_{1}, [0.5, 0.8], 0.4), (u_{2}, [0.6, 0.7], 0.8), \\ (u_{3}, [0.4, 0.7], 0.5)}, \\ β_{3} ((e_{2}, f_{2}), (x_{2}, y_{2})) \\ = {(u_{1}, [0.3, 0.8], 0.2), (u_{2}, [0.2, 0.9], 0.4), \\ (u_{3}, [0.2, 0.7], 0.6)} . \\ (β_{1}, E, X) ⋀_{R} (β_{2}, F, Y) \\ = (β_{3}, (E \times F), (X \times Y) \\ = β_{3} ((e_{1}, f_{1}), (x_{1}, y_{1})) \\ = {(u_{1}, [0.5, 0.8], 0.7), (u_{2}, [0.6, 0.7], 0.9), \\ (u_{3}, [0.4, 0.7], 0.8)}, \\ β_{3} ((e_{2}, f_{2}), (x_{2}, y_{2})) \\ = {(u_{1}, [0.3, 0.8], 0.4), (u_{2}, [0.2, 0.9], 0.7), \\ (u_{3}, [0.2, 0.7], 0.8)} . \\ (β_{1}, E, X) ⋁_{P} (β_{2}, F, Y) \\ = (β_{3}, (E \times F), (X \times Y) \\ = β_{3} ((e_{1}, f_{1}), (x_{1}, y_{1})) \\ = {(u_{1}, [0.5, 0.8], 0.7), (u_{2}, [0.6, 0.9], 0.9), \\ (u_{3}, [0.4, 0.8], 0.8)}, \\ β_{3} ((e_{2}, f_{2}), (x_{2}, y_{2})) \\ = {(u_{1}, [0.3, 0.8], 0.4), (u_{2}, [0.6, 0.9], 0.7), \\ (u_{3}, [0.3, 0.7], 0.8)} . \\ (β_{1}, E, X) ⋁_{R} (β_{2}, F, Y) \\ = (β_{3}, (E \times F), (X \times Y) \\ = β_{3} ((e_{1}, f_{1}), (x_{1}, y_{1})) \\ = {(u_{1}, [0.5, 0.8], 0.4), (u_{2}, [0.6, 0.9], 0.8), \\ (u_{3}, [0.4, 0.8], 0.5)}, \\ β_{3} ((e_{2}, f_{2}), (x_{2}, y_{2})) \\ = {(u_{1}, [0.3, 0.8], 0.2), (u_{2}, [0.6, 0.9], 0.4), \\ (u_{3}, [0.3, 0.7], 0.6)} . \end{matrix}$ respectively.

Theorem 3.32.Let (β₁, E₁, X₁) be aCSES over U. If (β₁, E₁, X₁) is anICSES (ECSES) then (β₁, E₁, X₁) ^cICSES (ECSES) respectively.

Proof. By using Definitions 3.2 and 3.3 we can proof this theorem. □

Definition 3.33. Let A_{(e, x_i)}, λ_{(e,x_i)} ∈ CSES over U, 1 ≤ i ≤ n . The cubic soft expert weighted average quotient operator (CSEWAQO) is denoted and defined as $\begin{matrix} P_{w_{i}} (A_{(e, x_{i})}, λ_{(e, x_{i})}) \\ = ([\frac{\overset{n}{\underset{i = 1}{Π}} (1 + A_{(e, x_{i})}^{-} (u))^{w_{i}} - \overset{n}{\underset{i = 1}{Π}} (1 - A_{(e, x_{i})}^{-} (u))^{w_{i}}}{\overset{n}{\underset{i = 1}{Π}} (1 + A_{(e, x_{i})}^{-} (u))^{w_{i}} + \overset{n}{\underset{i = 1}{Π}} (1 - A_{(e, x_{i})}^{-} (u))^{w_{i}}}, \\ \frac{\overset{n}{\underset{i = 1}{Π}} (1 + A_{(e, x_{i})}^{+} (u))^{w_{i}} - \overset{n}{\underset{i = 1}{Π}} (1 - A_{(e, x_{i})}^{+} (u))^{w_{i}}}{\overset{n}{\underset{i = 1}{Π}} (1 + A_{(e, x_{i})}^{+} (u))^{w_{i}} + \overset{n}{\underset{i = 1}{Π}} (1 - A_{(e, x_{i})}^{+} (u))^{w_{i}}}], \\ \underset{i = 1}{\overset{n}{Π}} {λ_{(e, x_{i})} (u)}^{w_{i}}) \end{matrix}$ where w_i is the weight of experts opinion, w_i ∈ [0, 1] and $\underset{i = 1}{\overset{n}{Σ}} w_{i} = 1$ .

Definition 3.34. Let β = $< [A_{(e, x)}^{-},$ $A_{(e, x)}^{+}],$ λ_(e,x)> be a CSE value. A score functions $\overset{ˇ}{S}$ of CSES set value is defined as $\overset{ˇ}{S} (β) = \frac{A_{(e, x)}^{-} + A_{(e, x)}^{+} - λ_{(e, x)}}{3}$ where $\overset{ˇ}{S} (β) \in [- 1, 1]$

3.1 Decision making problem based on Multicriteria cubic soft expert set

Fuzzy soft set theoretic approach has been used in decision making problems by Roy et al. [21]. In this section, we give an application of CSES set theory in a decision making problem. Now we are going to present multicriteria cubic soft expert set in decision making along with weights and score function.

Step 1: Input the cubic soft expert set (β₁, E, X) .

Step 2: Utilize the opinions of experts in the form of CSESs to determine the opinions regarding given criteria. Make a separate tables for the opinion of each expert.

Step 3: Assign weight to each expert according to their expertise.

Step 4: Apply cubic soft expert weighted average operator to each above table and find the cubic soft expert weighted average corresponding to each attribute.

Step 5: Calculate the $\underset{P}{\lor}$ of ${\hat{U}}_{j} .$

Step 6: Calculate the scores of each ${\hat{U}}_{j} .$

Step 7: Generate the non-increasing order of all the alternatives according to their scores.

Example 3. Let U = {u₁ = Guinea, u₂ = Liberia, u₃ = Sierraleone, u₄ = Nigeria} be the set of countries, E = {e₁ = Diarrhea, e₂ = SevereHeadache, e₃ = Explainedbleeding, e₄ = FeverandVomiting} be the set of symptoms of Ebola patients, X = {x₁, x₂, x₃} be the set of Physciatians.

Step 1: $\begin{matrix} β_{1} (e_{1}, x_{1}) \\ = {(u_{1}, [0.4, 0.6], 0.8), (u_{2}, [0.1, 0.5], 0.3), \\ (u_{3}, [0.6, 0.7], 0.5), (u_{4}, [0.1, 0.9], 0.8)} \\ β_{1} (e_{2}, x_{1}) \\ = {(u_{1}, [0.3, 0.7], 0.4), (u_{2}, [0.7, 0.9], 0.8), \\ (u_{3}, [0.3, 0.9], 0.5), (u_{4}, [0.4, 0.6], 0.5)}, \\ β_{1} (e_{3}, x_{1}) \\ = {(u_{1}, [0.5, 0.6], 0.6), (u_{2}, [0.5, 0.7], 0.6), \\ (u_{3}, [0.2, 0.6], 0.4), (u_{4}, [0.3, 0.5], 0.4)}, \\ β_{1} (e_{4}, x_{1}) \\ = {(u_{1}, [0.3, 0.9], 0.5), (u_{2}, [0.2, 0.8], 0.6), \\ (u_{3}, [0.5, 0.7], 0.9), (u_{4}, [0.4, 0.8], 0.7)}, \\ β_{1} (e_{1}, x_{2}) \\ = {(u_{1}, [0.3, 0.6], 0.4), (u_{2}, [0.6, 0.9], 0.3), \\ (u_{5}, [0.4, 0.7], 0.3), (u_{5}, [0.4, 0.6], 0.4)}, \\ β_{1} (e_{2}, x_{2}) \\ = {(u_{1}, [0.7, 0.9], 0.2), (u_{2}, [0.5, 0.8], 0.8), \\ (u_{3}, [0.4, 0.7], 0.4), (u_{4}, [0.6, 0.9], 0.7)}, \\ β_{1} (e_{3}, x_{2}) \\ = {(u_{1}, [0.6, 0.8], 0.4), (u_{2}, [0.3, 0.7], 0.5), \\ (u_{3}, [0.5, 0.9], 0.7), (u_{4}, [0.4, 0.8], 0.8)}, \\ β_{1} (e_{4}, x_{2}) \\ = {(u_{1}, [0.5, 0.8], 0.8), (u_{2}, [0.8, 0.9], 0.4), \\ (u_{3}, [0.7, 0.9], 0.8), (u_{4}, [0.5, 0.6], 0.6)}, \\ β_{1} (e_{1}, x_{3}) \\ = {(u_{1}, [0.6, 0.8], 0.5), (u_{2}, [0.5, 0.7], 0.5), \\ (u_{3}, [0.7, 0.8], 0.6), (u_{4}, [0.6, 0.9], 0.6)}, \\ β_{1} (e_{2}, x_{3}) \\ = {(u_{1}, [0.2, 0.7], 0.5), (u_{2}, [0.3, 0.7], 0.5), \\ (u_{3}, [0.7, 0.8], 0.9), (u_{4}, [0.2, 0.5], 0.3)}, \\ β_{1} (e_{3}, x_{3}) \\ = {(u_{1}, [0.8, 0.9], 0.9), (u_{2}, [0.6, 0.7], 0.6), \\ (u_{3}, [0.4, 0.8], 0.5), (u_{4}, [0.1, 0.6], 0.4)}, \\ β_{1} (e_{4}, x_{3}) \\ = {(u_{1}, [0.1, 0.9], 0.4), (u_{2}, [0.2, 0.9], 0.8), \\ (u_{3}, [0.5, 0.9], 0.6), (u_{4}, [0.4, 0.8], 0.5)} . \end{matrix}$

Step 2: Opinion of expert x₁

	u₁	u₂	u₃	u₄
(e₁, x₁)	([0.4, 0.6], 0.8)	([0.1, 0.5], 0.3)	([0.6, 0.7], 0.5)	([0.1, 0.9], 0.8)
(e₂, x₁)	([0.3, 0.7], 0.4)	([0.7, 0.9], 0.8)	([0.3, 0.9], 0.5)	([0.4, 0.6], 0.5)
(e₃, x₁)	([0.5, 0.6], 0.6)	([0.5, 0.7], 0.6)	([0.2, 0.6], 0.4)	([0.3, 0.5], 0.4)
(e₄, x₁)	([0.3, 0.9], 0.5)	([0.5, 0.7], 0.6)	([0.5, 0.7], 0.9)	([0.4, 0.8], 0.7)

Opinion of expert x₂

	u₁	u₂	u₃	u₄
(e₁, x₂)	([0.3, 0.6], 0.4)	([0.6, 0.9], 0.3)	([0.4, 0.7], 0.3)	([0.4, 0.6], 0.4)
(e₂, x₂)	([0.7, 0.9], 0.2)	([0.5, 0.8], 0.8)	([0.4, 0.7], 0.4)	([0.6, 0.9], 0.7)
(e₃, x₂)	([0.6, 0.8], 0.4)	([0.3, 0.7], 0.5)	([0.5, 0.9], 0.7)	([0.4, 0.8], 0.8)
(e₄, x₂)	([0.5, 0.8], 0.8)	([0.8, 0.9], 0.4)	([0.7, 0.9], 0.8)	([0.5, 0.6], 0.6)

Opinion of expert x₃

	u₁	u₂	u₃	u₄
(e₁, x₃)	([0.6, 0.8], 0.5)	([0.5, 0.7], 0.5)	([0.7, 0.8], 0.6)	([0.6, 0.9], 0.6)
(e₂, x₃)	([0.2, 0.7], 0.5)	([0.3, 0.7], 0.5)	([0.7, 0.8], 0.9)	([0.2, 0.5], 0.3)
(e₃, x₃)	([0.8, 0.9], 0.9)	([0.6, 0.7], 0.6)	([0.4, 0.8], 0.5)	([0.1, 0.6], 0.4)
(e₄, x₃)	([0.1, 0.9], 0.4)	([0.2, 0.9], 0.8)	([0.5, 0.9], 0.6)	([0.4, 0.8], 0.5)

Step 3:W = (0.36, 0.21, 0.43) where weight 0.36 assign to expert x₁, weight 0.21 assign to expert x₂ and weight 0.43 assign to expert x₃ .

Step 4: The cubic soft expert weighted average of each attribute.

	u₁	u₂	u₃	u₄
e₁	([0.47, 0.70], 0.56)	([0.39, 0.70], 0.37)	([0.61, 0.74], 0.48)	([0.40, 0.86], 0.61)
e₂	([0.36, 0.75], 0.38)	([0.50, 0.81], 0.65)	([0.51, 0.83], 0.61)	([0.36, 0.66], 0.43)
e₃	([0.67, 0.80], 0.65)	([0.50, 0.70], 0.57)	([0.35, 0.77], 0.49)	([0.23, 0.62], 0.46)
e₄	([0.26, 0.88], 0.50)	([0.47, 0.85], 0.62)	([0.55, 0.85], 0.73)	([0.42, 0.76], 0.58)

Step 5: Calculate the $\underset{P}{\lor}$ of 1st, 2nd, 3rd and 4th column of above table by using Definition 3.30. So we have $\begin{matrix} {\hat{U}}_{1} & = & \underset{j = 1}{\overset{4}{\lor}} {(e_{j}, u_{1})} = ([0.67, 0.88], 0.65) \\ {\hat{U}}_{2} & = & \overset{4}{\underset{j = 1}{\lor}} {(e_{j}, u_{2})} = ([0.50, 0.85], 0.65) \\ {\hat{U}}_{3} & = & \underset{j = 1}{\overset{4}{\lor}} {(e_{j}, u_{3})} = ([0.61, 0.85], 0.73) \\ {\hat{U}}_{4} & = & \overset{4}{\underset{j = 1}{\lor}} {(e_{j}, u_{4})} = ([0.42, 0.86], 0.61) . \end{matrix}$

Step 6: Now calculate the score of above CSES elements by using Definition 3.34. $\begin{matrix} \overset{ˇ}{S} ({\hat{U}}_{1}) = 0.30 \\ \overset{ˇ}{S} ({\hat{U}}_{2}) = 0.23 \\ \overset{ˇ}{S} ({\hat{U}}_{3}) = 0.24 \\ \overset{ˇ}{S} ({\hat{U}}_{4}) = 0.22 \end{matrix}$

Step 7: Generate the non-decreasing order of the score of CSES set values.

Corresponding to $\underset{P}{\lor}$ we have the following order $u_{1} > u_{3} > u_{2} > u_{4} .$

In above example we want to check which country is more affected by ebola. Hence Guinea is more affected by ebola.

4 Conclusions

In this paper, CSES has been discussed which can be used in decision analysis. Some basic operations have been defined for CSES. Several properties have been investigated. We derive different conditions for different operations of two ICSESs (ECSESs).to be an ICSESs (ECSESs) and so many methods to solve decision making problems in various fields but this technique is more suitable because in decision analysis there are some problems in which decision makers take decision on the basis of different conditions such as climate condition, time period condition and geographical conditions. If a decision maker wants to take a decision in some problems on the basis of such conditions then this structure is very useful. At the end, an algorithm has been presented along with an illustrative example. In future we aim to study TOPSIS for group decision making with CSES also we want to define different aggregation operators similarity and distance measures and distances and similarity degrees between CSESs.

References

Ali

M.I.

, Feng

, Liu

, Min

W.K.

, and Shabir

, On some new operations in soft set theory, Comput Math Appl57 (2009), 1547–1553.

Alkhazaleh

, and Salleh

A.R.

, Soft expert sets, Adv Decis Sci (2011), 1–12(Article ID 757868).

Cagman

, and Enginoglu

, Soft set theory uni-int decision making, European J Oper Res207 (2010), 845–855.

Cagman

, Enginoglu

, and Citak

, Fuzzy soft set theory and its application, Iran J Fuzzy Syst8(3) (2011), 137–147.

Chen

, Tsang

E.C.C.

, Yeung

D.S.

, and Wang

, The parameterization reduction of soft sets and its applications, Comput Math Appl49 (2005), 757–763.

Feng

, Jun

Y.B.

, Liu

, and Li

, An adjustable approach to fuzzy soft set based decision making, J Comput Appl Math234(1) (2010), 10–20.

Gorzaldivczany

M.B.

, A method of inference in approximate reasoning based on interval valued fuzzy sets, Fuzzy Sets Syst21 (1987), 1–17.

Jun

Y.B.

, Kim

C.S.

, and Yang

K.O.

, Cubic sets, Ann Fuzzy Math Inform4 (2012), 83–98.

Khan

, Jun

Y.B.

, Gulistan

, and Yaqoob

, The generalized version of Jun.s cubic sets in semigroups, J Intell Fuzzy Syst28(2) (2015), 947–960.

10.

Kohout

L.J.

, and Bandler

, Fuzzy interval inference utilizing the checklist paradigm and BK relational products, in: Kearfort

R.B.

, et al., (Eds.)Appl Interval Comput, Kluwer, Dordrecht, 1996, pp. 291–335.

11.

Kong

, Gao

, and Wang

, A fuzzy soft set theoretic approach to decision making problems, J Comput Appl Math223(2) (2009), 540–542.

12.

D.F.

, An approach to fuzzy multiattribute decision making under uncertainty, Inform Sci169 (2005), 97–112.

13.

Maji

P.K.

, Biswas

, and Roy

A.R.

, Fuzzy soft sets, J Fuzzy Math9(3) (2001), 589–602.

14.

Maji

P.K.

, and Roy

A.R.

, Soft set theory, Comput Math Appl45 (2003), 555–562.

15.

Maji

P.K.

, Roy

A.R.

, and Biswas

, An application of soft sets in a decision making problem, Comput Math Appl44(8–9) (2002), 1077–1083.

16.

Molodtsov

D.A.

, Soft set theory-first results, Comput Math Appl37 (1999), 19–31.

17.

Molodtsov

D.A.

, The description of a dependence with the help of soft sets, J Comput Sys Sc Int40(6) (2001), 977–984.

18.

Molodtsov

D.A.

, Yu Leonov

, and Kovkov

D.V.

, Soft sets technique and its application, Nechetkie Sistemi I Myakie Vychisleniya1 (2006), 8–3.

19.

Muhiuddin

, and Al-roqi

A.M.

, Cubic soft sets with applications in BCK/BCI-algebras, Ann Fuzzy Math Inform8(2) (2014), 291–304.

20.

Pei

, and Miao

, From soft sets to information systems, in Proceedings of the IEEE International Conference on Granular Computing2 (2005), 617–621.

21.

Roy

A.R.

, and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, J Comput Appl Math203(2) (2007), 412–418.

22.

Sezgin

, and Atagun

A.O.

, On operations of soft sets, Comput Math Appl61(5) (2011), 1457–1467.

23.

Turksen

I.B.

, Interval-valued fuzzy sets based on normal forms, Fuzzy Sets Syst20 (1986), 191–210.

24.

Turksen

I.B.

, Interval-valued fuzzy sets and compensatory AND, Fuzzy Sets Syst51 (1992), 295–307.

25.

Turksen

I.B.

, Interval-valued strict preference with Zadeh triples, Fuzzy Sets Syst78 (1996), 183–195.

26.

Z.S.

, Methods for aggregating interval valued intuitionistic fuzzy information and their application to decision making, Control Decis22(2) (2007), 215–219.

27.

Zadeh

L.A.

, Fuzzy sets, Inform Control8 (1965), 338–353.

28.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning-I, Inform Sci8 (1975), 199–249.