On Generalised Interval-Valued Fuzzy Soft Sets

Definition 2.1 (see [8].)A pair (F, A) is called a soft set over U where F is a mapping given by F : A → P(U).

Definition 2.2 (see [29].)Let 𝒫(U) denote the set of all fuzzy subsets of U. Then a pair $$ is called a fuzzy soft set over U, where $$ is a mapping from A to 𝒫(U).

Definition 2.3 (see [32].)Let U be an initial universal set, E a set of parameters, and the pair (U, E) a soft universe. Let F : E → 𝒫(U) and μ be a fuzzy subset of E; that is, μ : E → [0,1]. Let F_μ : E → 𝒫(U)×[0,1] be a function defined as follows: F_μ(e) = (F(e), μ(e)), where F(e) ∈ 𝒫(U). Then F_μ is called a generalised fuzzy soft set over (U, E).

Definition 2.4 (see [7].)An interval-valued fuzzy set X on a universe U is a mapping X : U → Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1].

Definition 2.5 (see [7].)Let X and Y be two interval-valued fuzzy sets on universe U. Then the union, intersection, and complement of vague sets are defined as follows:

Definition 2.6 (see [31].)Let U be an initial universe, let E be a set of parameters, and let A⊆E. ℱ(U) denotes the set of all interval-valued fuzzy sets of U. A pair (F, A) is an interval-valued fuzzy soft set over U, where F is a mapping given by F : A → ℱ(U).

Definition 2.7 (see [37].)A t-norm is an increasing, associative, and commutative mapping T : [0,1]×[0,1]→[0,1] that satisfies the boundary condition: T(a, 1) = a for all a ∈ [0,1].

Definition 2.8 (see [37].)A t-conorm is an increasing, associative, and commutative mapping S : [0,1]×[0,1]→[0,1] that satisfies the boundary condition: S(a, 0) = a for all a ∈ [0,1].

Definition 3.1. Let U be an initial universe and E a set of parameters, A⊆E, $$, and let α be a fuzzy sets of A, that is, α : A → [0,1]. Define a function $$ as $$, where $$ is an interval value is called the degree of membership an element h to $$, and α(e) is called the degree of possibility of such belongness. Then $$ is called type 1 generalised interval-valued fuzzy soft set over the soft universe (U, E).

Definition 3.2. Let U be an initial universe and E a set of parameters, A⊆E, $$, and let α be an interval-valued fuzzy sets of A; that is, α : A → Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1]. Define a function $$ as $$, where $$ and α(e) = [α⁻(e), α⁺(e)] are interval values. Then $$ is called type 2 generalised interval-valued fuzzy soft set over the soft universe (U, E).

Example 3.3. Let U = {h₁, h₂, h₃} be a set of mobile telephones and A = {e₁, e₂, e₃} ∈ E a set of parameters. The e_i  (i = 1,2, 3) stand for the parameters “expensive”, “beautiful”, and “multifunctional”, respectively. Let $$ be a function given as follows:

Definition 3.4. Let $$ and $$ be GIVFS sets over (U, E). Then $$ is called a GIVFS subset of $$ if

• (1)

  A⊆B;

• (2)

  $$ is an interval-valued fuzzy subset of $$ for any e ∈ A; that is, $$ and $$ for any h ∈ U and e ∈ A;

• (3)

  α is an interval-valued fuzzy subset of β; that is, α⁻(e) ≤ β⁻(e) and α⁺(e) ≤ β⁺(e) for any e ∈ A.

In this case, the above relationship is denoted by $$. And $$ is said to be a GIVFS superset of $$.

Definition 3.5. Let $$ and $$ be GIVFS sets over (U, E). Then $$ and $$ are said to be GIVFS equal if and only if $$ and $$.

Definition 3.6. The relative complement of a GIVFS set $$ is denoted by $$ and is defined by $$, where $$ is a mapping given by $$ and α^r : A → Int([0,1]) is a mapping given by α^r(e) for all h ∈ U, e ∈ A, where $$, α^r(e) = [α^r−(e), α^r+(e)] = [1 − α⁺(e), 1 − α⁻(e)].

Example 3.7. We consider the GIVFS set $$ given in Example 3.3 and define a GIVFS set $$ as follows:

Definition 3.8. Let $$. A GIVFS set $$ over (U, E) is said to be relative absolute GIVFS set denoted by $$, if $$ and $$ for all h ∈ U and e ∈ A.

Definition 3.9. Let $$. A GIVFS set $$ over (U, E) is said to be relative null GIVFS set, denoted by $$, if $$ and $$ for all h ∈ U and e ∈ A.

Definition 3.10. The union of two GIVFS sets $$ and $$ over (U, E) denoted by $$ is a GIVFS set $$ and defined as $$ such that, for all h ∈ U and e ∈ A ∪ B,

Definition 3.11. The intersection of two GIVFS sets $$ and $$ over (U, E) denoted by $$ is a GIVFS set $$ and defined as $$ such that, for all h ∈ U and e ∈ A∩B, $$, where $$, $$ and γ(e) = T(α(e), β(e)) = [T(α⁻(e), β⁻(e)), T(α⁺(e), β⁺(e))].

Example 3.12. We consider the GIVFS sets $$ and $$ given in Examples 3.3 and 3.7, respectively, and consider S(x, y) = max {x, y} and T(x, y) = min {x, y}. Then

Proposition 3.13. Let $$ be a GIVFS set over (U, E). Then the following holds

• (1)

  $$,

• (2)

  $$,

• (3)

  $$,

• (4)

  $$.

Proof. It is easily obtained from Definitions 3.8–3.11.

Theorem 3.14. Let $$, $$, and $$ be GIVFS sets over (U, E). Then the following holds

• (1)

  $$,

• (2)

  $$,

• (3)

  $$,

• (4)

  $$.

Proof. It is easily obtained from Definitions 3.10 and 3.11.

Definition 3.15. The restricted union of two GIVFS sets $$ and $$ over (U, E) denoted by $$ is a GIVFS set $$ and defined as $$ such that, for all h ∈ U and e ∈ A∩B, $$, where $$, $$ and γ(e) = S(α(e), β(e)) = [S(α⁻(e), β⁻(e)), S(α⁺(e), β⁺(e))].

Definition 3.16. The extended intersection of two GVS sets $$ and $$ over (U, E), denoted by $$, is a GVS set $$ which is defined as, for all h ∈ U, e ∈ A  ∪  B,

Theorem 3.17. Let $$, $$, and $$ be three GIVFS sets over (U, E). Then the following holds:

• (1)

  $$,

• (2)

  $$,

• (3)

  $$,

• (4)

  $$.

Proof. It is easily obtained from Definitions 3.15 and 3.16.

Theorem 3.18. Let $$ and $$ be two GIVFS sets over (U, E). Then the following holds:

• (1)

  $$,

• (2)

  $$.

Proof. (1) Suppose that $$, then C = A∩B, and, for  all  e ∈ C,   h ∈ U,

(2) The proof is similar to that of (1).

Definition 3.19. The “AND” of two GIVFS sets $$ and $$ over (U, E), denoted by $$, is defined as $$ such that for all h ∈ U and (a, b) ∈ A × B, $$, where $$, $$ and γ(a, b) = T(α(a), β(b)) = [T(α⁻(a), β⁻(b)), T(α⁺(a), β⁺(b))].

Definition 3.20. The “OR” of two GIVFS sets $$ and $$ over (U, E), denoted by $$, is defined as $$ such that for all h ∈ U and (a, b) ∈ A × B, $$, where $$, $$ and γ(a, b) = S(α(a), β(b)) = [S(α⁻(a), β⁻(b)), S(α⁺(a), β⁺(b))].

Theorem 3.21. Let $$, $$, and $$ be three GIVFS sets over (U, E). Then the following holds

• (1)

  $$,

• (2)

  $$.

Proof. It is easily obtained from Definitions 3.19 and 3.20.

Theorem 3.22. Let $$ and $$ be two GIVFS sets over (U, E). Then the following holds

• (1)

  $$,

• (2)

  $$.

Proof. (1) Suppose that $$, then C = A × B, and, for  all  (a, b) ∈ C,   h ∈ U,

(2) The proof is similar to that of (1).

Proposition 4.1. Let $$ be a GIVFS sets over (U, E). Then the following holds

• (1)

  $$,

• (2)

  $$.

Example 4.2. We consider the GIVFS set $$ given in Example 3.3. We have that the following

• (1)

  If S(a, b) = a + b − a · b, then $$, $$, $$, and $$; that is, $$.

• (2)

  If S(a, b) = min (1, a + b), then $$, $$, $$, and $$; that is, $$.

• (3)

  if T(a, b) = a · b, then $$, $$, $$ and $$, that is, $$;

• (4)

  If T(a, b) = max (0, a + b − 1), then $$, $$, $$ and $$; that is, $$.

For convenience, let $$ denote the set of all GIVFS sets over (U, E); that is, $$.

Theorem 4.3. Let A, B⊆E, $$, and $$ be GIVFS sets over (U, E). Then the following hold:

• (1)

  $$,

• (2)

  $$,

• (3)

  $$,

• (4)

  $$.

Proof. (1) and (2) are trivial to prove. We prove only (3) since (4) can be proved similarly.

Suppose that the parameter sets of two GIVFS sets $$ and $$ are denoted by M and N, respectively. Let $$ and $$. Then M = A ∪ B, N = (A ∪ B)∩A = A. And, for each e ∈ A and h ∈ U,

• (i)

  if e ∉ B, then $$, and η(e) = T(α(e), α(e)) = min (α(e), α(e)) = α(e),

• (ii)

  if e ∈ B, then $$, $$, and η(e) = T(S(α(e), β(e)), α(e)) = min (max (α(e), β(e)), α(e)) = α(e).

Theorem 4.4. Let A, B, C⊆E, $$, $$, and $$ be GIVFS sets over (U, E). Then the following hold:

• (1)

  $$,

• (2)

  $$.

Proof. (1) Suppose that the parameter sets of two GIVFS sets $$ and $$ are denoted by M and N, respectively. Let $$ and $$. Then M = A∩(B ∪ C) = (A∩B)∪(A∩C) = N. And, for each e ∈ M, h ∈ U, it follows that e ∈ A and e ∈ B ∪ C,

• (i)

  if e ∈ A,   e ∉ B,   e ∈ C, then $$,  $$, and δ(e) = T(α(e), γ(e)) = min (α(e), γ(e)) = η(e),

• (ii)

  if e ∈ A,   e ∈ B,   e ∉ C, then $$,   $$, and δ(e) = T(α(e), β(e)) = min (α(e), β(e)) = η(e),

• (iii)

  if e ∈ A,   e ∈ B,   e ∈ C, then $$, $$,   $$, and δ(e) = T(α(e), S(β(e), γ(e))) = min (α(e), max (β(e), γ(e))) = max (min (α(e), β(e)), min (α(e), γ(e))) = S(T(α(e), β(e)), T(α(e), γ(e))) = η(e).

(2) The proof is similar to that of (1).

Theorem 4.5. (1)  $$ is a distributive lattice.

(2) Let ≤₁ be the order relation in $$ and $$. One has $$ if and only if $$ and α(e) ≤ β(e) for all e ∈ A and h ∈ U.

Proof. (1) The proof is straightforward from Theorems 3.14, 4.3, and 4.4.

(2) Suppose that $$. Then $$. So by Definition 3.10, we have A ∪ B = B, $$, and max (α(e), β(e)) = β(e) for all e ∈ A and h ∈ U. It follows that A⊆B, $$ and α(e) ≤ β(e) for all e ∈ A and h ∈ U. Conversely, suppose that A⊆B, $$ and α(e) ≤ β(e) for all e ∈ A and h ∈ U. We can easily verify that $$. Thus $$.

Theorem 4.6. Let $$ and $$ be GIVFS sets over (U, E). Then the following hold:

• (1)

  $$,

• (2)

  $$,

• (3)

  $$,

• (4)

  $$.

Theorem 4.7. Let $$, $$ and $$ be GIVFS sets over (U,E). Then the following hold:

• (1)

  $$,

• (2)

  $$.

Theorem 4.8. (1)$$ is a distributive lattice.

(2) Let ≤₂ be the order relation in $$ and $$. $$ if and only if $$ and α(e) ≤ β(e) for all e ∈ B.

Example 4.9. Let U = {h₁, h₂, h₃} be the universe, E = {e₁, e₂, e₃} the set of parameters, A = {e₁, e₂}, B = {e₂, e₃}. The GIVFS sets $$ and $$ over (U, E) are given as

Suppose that $$. Then C = A∩B = {e₂} ≠ A. So $$, that is, $$.

Again, suppose that the parameters set of a GIVFS set $$ is denoted by D, and $$. Then D = A ∪ B = {e₁, e₂, e₃} ≠ A, Therefore, $$, that is, $$.

Definition 5.1. Let $$ be a GIVFS set, h_i, h_j ∈ U,   e_k ∈ A. One says membership value of h_j lowerly exceeds or equals to the membership value of h_i with respect to the parameter e_k if $$. The corresponding characteristic function is defined as follows:

Definition 5.2. Let $$ be a GIVFS set, h_i, h_j ∈ U, e_k ∈ A. One says membership value of h_j upperly exceeds or equals to the membership value of h_i with respect to the parameter e_k if $$. The corresponding characteristic function is defined as follows:

Remark 5.3. Let $$ be a GIVFS set, h_i, h_j ∈ U,   and  e_k ∈ A. For convenience, we denote the vectors $$ and (α⁻(e_k), α⁺(e_k)) as $$ and $$, respectively.

Definition 5.4. Let $$ be a GIVFS set. The generalised comparison table about $$ is a square table in which the number of rows and number of columns are equal. Both rows and columns are labeled by the object names of the universe such as h₁, h₂, …, h_n, and the entries are C_ij, given as follows:

Remark 5.5. The generalised comparison table is different from the comparison table in [30]. First, the comparison in the generalised comparison table is between two interval values, instead of two single values. Second, the entries C_ij of the generalised comparison table are numbers of real interval [0,1] in general, instead of single values 0 and 1. Hence, the generalised comparison table is an extension of the comparison table in [30]. If each interval degenerates to a point and α(e) = 1 for each e ∈ A, then the generalised comparison table will be degenerate to the comparison table in [30].

Algorithm 5.6. (1) Input the objects set U and the parameter set A⊆E.

(2) Consider the GIVFS set $$ in tabular form.

(3) By calculating the entries C_ij, construct generalised comparison table.

(4) Compute the score of each h_i using row sum and the column sum.

(5) The optimal decision is to select h_k if the score of h_k is maximum.

(6) If k has more than one value then any one of h_k may be chosen.

Example 5.7. Let us consider a GIVFS set which describes the capability of the candidates who are wanted to fill a position for a company. Suppose that there are six candidates in the universe U = {h₁, h₂, h₃, h₄, h₅, h₆} under consideration, and E = {e₁, e₂, e₃, e₄, e₅, e₆} is the set of decision parameters, where e_i  (i = 1,2, 3,4, 5,6) stands for the parameters “experience”, “computer knowledge”, “young age”, “higher education”, “good health”, and “over-married”, respectively.