In 2023, Alkhazaleh
Many real-world problems are characterized by uncertainty, which complicates traditional decision-making methods in fields such as economics, engineering, and medicine. To address these challenges, Zadeh
Further advancing these concepts, Molodtsov
Alkhazaleh and Salleh
Definition 1 (
Definition 2 (
Definition 3 (
Definition 4 (
μi: V × Q → I , i = 1,2, … , k.
The function (μ (v, q), μ (v, q), … , μ (v, q)) is called the membership function of multi Q-fuzzy set A ̃ : and
1 2 k Q
μ (v, q) + μ (v, q) + ⋯ + μ (v, q) ≤ 1, k is called the dimension of A ̃ . The set of all multi Q-fuzzy sets of
1 2 k Q
dimension k in V and Q is denoted by M QF(V).
Definition 5 (
Definition 6 (
uj F(s)Λ = { : uj ∈ U, s ∈ S}.
TU(u )
For all s ∈ S and for all ak ∈ C, we have
We will now propose the definition of Effective Q-Fuzzy Soft Expert Set (EQFSES) and propose some of its properties. Throughout the discussion, V is the initial universe, E is the set of parameters, Q be a set of supplies, Λ is the set of effective parameters, X is the set of experts, and O = {agree = 1, disagree = 0} a set of suggestions. Let A ⊆ Z where Z = E × X × O and S ⊆ Z.
Definition 7. (FQ, A) is a QFSES over V, where FQ is the mapping FQ: A → QFSES such that QFSES is the set of all QFSES over V.
m×k m×k
FQ (S) = ⋀ FQ(et, pj, 1) , ⋀ FQ(et, pj, 0) , t = 1,2, … , m, j = 1,2, … , k.
Λ Λi Λi i=1 i=1
uj FQ(et. pj. 1) = { : uj ∈ U, s ∈ S }.
Λi v
U(uj)
uj FQ(et. pj. 0) = { : uj ∈ U, s ∈ S}.
Λi v
U(uj)
For all s ∈ S and for all ak ∈ C, we have
for all (ak) ∈ Λ, and |Λ| is a cardinality Λ.
Example 1. Suppose a customer who wants to build a new house wants to get feedback from several experts. Let V = {v1, v2} be the set of houses, Q = {k1, k2} be the set of construction companies, E = {e1, e2} be the set of decision parameters, and the set of effective parameters is represented by Λ = {l1, l2}. Let X = {p1, p2} be the set of experts. Assume that
l1 l2 Λ (v1 . k1, p1) = { . }.
0,5 0,4 l1 l2 Λ (v1 . k2, p1) = { . }.
0,3 0,8 l1 l2 Λ (v1 . k1, p2) = { . }.
0,7 0,3 l1 l2 Λ (v1 . k2. p2) = { . }.
0,2 0,6 l1 l2 Λ (v2 . k1. p1) = { . }.
0,1 0,4 l1 l2 Λ (v2 . k1. p2) = { . }.
0,9 0,4 l1 l2 Λ (v2 . k2. p1) = { . }.
0,1 0,2 l1 l2 Λ (v2 . k2. p2) = { . }.
0,9 0,7 Let F be the QFSES defined as follows:
FQ(e1, p1, 1) = {( , , , )}.
FQ(e1, p2, 1) = {( , , , )}.
FQ(e2, p1, 1) = {( , , , )}.
FQ(e2, p2, 1) = {( , , , )}.
FQ(e1, p1, 0) = {( , , , )}.
FQ(e1, p2, 0) = {( , , , )}.
FQ(e2, p1, 0) = {( , , , )}.
FQ(e2, p2, 0) = {( , , , )}.
Then, by applying Definition 6, we get
(v1, k1) 0.5 + 0.4 0.6 + [(1 − 0.6) ] (v1, k2) 0.5 + 0.4 0.7 + [(1 − 0.7) ] (v1, k1) (v1, k2) (v2, k1) (v2, k2) ( ) 2 FQ e1, p1, 1 1 = = , , , .
(v , k ) 0.8 0.8 0.7 0.6 0.5 + 0.4 0.4 + [(1 − 0.4) ] (v2, k2) 0.5 + 0.4 { 0.2 + [(1 − 0.2) ] } Similarly, when the calculations are continued, the EQFSES is found as follows:
0.6, 0.7 0.6 0.5 0.7 0.8 0.6 0.4,
0.3 0.8 0.8 0,7 0.6 0.8 0.9 0,9
Definition 8. For two EQFSES (FQ, A) i and (GQ, B) i over V, (FQ, A) i is called an EQFSE subset of
Λ Λ Λ
(GQ, B) i if
I. B ⊆ A.
II. GQ(ε) i is an effective Q-fuzzy soft expert subset FQ(ε) i for all ε ∈ B.
Example 2. Consider Example 1, where A = {(e1, p1, 1), (e2, p1, 1)}.
and B = {(e1, p1, 1)}.
B is an effective Q-fuzzy soft expert subset of A, hence B ⊆ A. Define (GQ, B) 3 and (FQ, A) 3 as follows:
(F , A) = {(e , p , 1) 3 = , , , .
Q 3 1 1 Λ
(e , p , 1) 3 = , , , }.
2 1 Λ
(G , B) = {(e , p , 1) 3 = , , , }.
Q 3 1 1 Λ
Therefore (G , B) ⊆ ̃ (F , A) .
Q 3 Q 3
Definition 9. Two EQFSES (FQ, A) i and (GQ, B) i over V are equal if (FQ, A) i is an EQFSES subset of
Λ Λ Λ
(GQ, B) i and (GQ, B) i is an EQFSES subset of (FQ, A) i for all i.
Λ Λ Λ
Definition 10. Agree-EQFSES (FQ, A) over V is an EQFSES subset of (FQ, A) defined as:
(FQ, A) = {FQ (∝)Λ:∝∈ E × X × {1}}.
Example 3. Using our previous Example 1, the agree-EQFSES (FQ, Z) over V is:
1 (v1, k1) (v1, k2) (v2, k1) (v2, k2) (F , Z) = {(e , p , 1) 1 = , , , .
(e , p , 1) 1 = , , , .
1 2 Λ
(e , p , 1) 1 = , , , .
2 1 Λ
(e , p , 1) 1 = , , , .
2 2 Λ
(e , p , 1) 2 = , , , .
1 1 Λ
(e , p , 1) 2 = , , , .
1 2 Λ
(e , p , 1) 2 = , , , .
2 1 Λ
(e , p , 1) 2 = , , , .
2 2 Λ
(e , p , 1) 3 = , , , .
1 1 Λ
(e , p , 1) 3 = , , , .
1 2 Λ 0.7 0.8 0.6 0.4,
F (e , p , 1) 8 = , , , }.
Definition 11. A disagree-EQFSES (FQ, Z) over V is an EQFSES subset of (FQ, Z) defined as:
(FQ, Z) = {FQ (∝)Λ: ∝∈ E × X × {0}}.
Example 4. Using our previous Example 1, the disagree-EQFSES (FQ, Z) over V is:
0 (v1, k1) (v1, k2) (v2, k1) (v2, k2) (F , Z) = {F (e , p , 0) 1 = , , , .
Q Q 1 1 Λ
0.6, 0.7 0.6 0.5 0.3 0.8 0.8 0,7
0.6 0.8 0.9 0,9
F (e , p , 0) 8 = , , , }.
Definition 12. The complement of an EQFSES (FQ, A) is
(FQ, A) = (FQ , ¬A) .
Such that FQ : ¬A → EQFSE(V) a mapping
FQ (∝)Λ = {1 − μ F (∝) }.
For each ∝∈ E. It is clear that it is ((FQ, A) ) = (FQ, A) .
Example 5. Using our previous Example 1, the complement of the EQFSES FQ denoted by FQ is given as follows:
c (v1, k1) (v1, k2) (v2, k1) (v2, k2) (F , Z) = {¬(e , p , 1) 1 = , , , .
¬(e , p , 1) 1 = , , , .
2 1 Λ
And so on.
Definition 13. The union of two EQFSES (FQ, A) and (GQ, B) over V, denoted by
(FQ, A) (GQ, B) ,
is the EQFSES (HQ, C) such that C = A ∪ B as follows:
μF (e)(m), if e ∈ A − B, μG (e) (m), if e ∈ B − A, μH (e) = { Q max (μF (e)(m), μG (e)(m)) , if e ∈ A ∩ B.
Example 6. Suppose that (FQ, A) and (GQ, B) are two EQFSES over V, such that
(FQ, A) = {[(e1, q, 0) ( , , , )].
[(e2, q, 0), ( , , , )]}.
(GQ, B) = {[(e1, q, 0) ( , , , )].
[(e2, q, 1), ( , , , )]}.
Then (FQ, A) (GQ, B) = (HQ, C) where
Λ Λ Λ
(HQ, C) = {[(e1, q, 0), ( , , , )].
[(e2, q, 0), ( , , , )].
[(e2, q, 1), ( , , , )]}.
Proposition 1. If (FQ, A) , (GQ, B) and (HQ, C) are three EQFSES over V, then
Λ Λ Λ
I. ((FQ, A) (GQ, B) ) (HQ, C) = (FQ, A) ((GQ, B) (HQ, C) ).
Λ Λ Λ Λ Λ Λ
II. (F , A) (F , A) ⊆ ̃ (F , A) .
Q Q Q Λ Λ Λ
Definition 14. Suppose (FQ, A) and (GQ, B) are two EQFSES over the common universe V. The
intersection of (FQ, A) and (GQ, B) is (FQ, A) (GQ, B) = (KQ, C) as follows:
Λ Λ Λ Λ Λ
μF (e)(m), if e ∈ A − B, μG (e) (m), if e ∈ B − A, μK (e) = { Q min (μF (e)(m), μG (e)(m)) , if e ∈ A ∩ B.
Example 7. Suppose that (FQ, A) and (GQ, B) are two EQFSES over V, such that
(v1, k1) (v2, k1) (FQ, A) = {[(e1, p, 1), ( , )].
Λ 0.3 0.9 (v1, k1) (v2, k1) [(e2, q, 1), ( , )].
0.8 0.3 (v1, k1) (v2, k1) [(e2, q, 0), ( , )]}.
0.7 0.8 (v1, k1) (v2, k1) (GQ, B) = {[(e1, p, 1), ( , )]}.
Λ 0.4 0.7
Then (FQ, A) (GQ, B) = (KQ, C) where
Λ Λ Λ
(v1, k1) (v2, k1) (KQ, C) = {[(e1, p, 1), ( , )]}.
Λ 0.3 0.7
Proposition 2. If (FQ, A) , (GQ, B) and (KQ, C) are three EQFSES over V, the following properties hold
Λ Λ Λ
true.
I. ((FQ, A) (GQ, B) ) (KQ, C) = (FQ, A) ((GQ, B) (KQ, C) ).
Λ Λ Λ Λ Λ Λ
II. (FQ, A) (FQ, A) ⊆ (FQ, A) .
Λ Λ Λ Proposition 3. If (FQ, A) , (GQ, B) and (KQ, C) are three EQFSES over V, then
Λ Λ Λ
I. ((FQ, A) (GQ, B) ) (KQ, C) = ((FQ, A) (KQ, C) ) ((GQ, B) (KQ, C) ) .
Λ Λ Λ Λ Λ Λ Λ
II. ((FQ, A) (GQ, B) ) (KQ, C) = ((FQ, A) (KQ, C) ) ((GQ, B) (KQ, C) ) .
Λ Λ Λ Λ Λ Λ Λ
Definition 15. If (FQ, A) and (GQ, B) are two EQFSES over V, then (FQ, A) and (GQ, B) is
Λ Λ Λ Λ
(FQ, A) ∧ (GQ, B) = (HQ, A × B) .
Λ Λ Λ
Such that HQ(α, β) = FQ(α) ∩ GQ(β) and memberships of (HQ, A × B) is as follows:
μH (α,β)(m) = min(μF (α)(m), μG (β)(m)),
Q Q Q
where for all α ∈ A,for all β ∈ B.
Example 8. Suppose that (FQ, A) and (GQ, B) are two EQFSES over V, such that
(FQ, A) = {[(e1, p, 1), ( , , , )]}.
Λ 0.6 0.4 0.4, 0.8
(GQ, B) = {[(e1, p, 1), ( , , , )].
Λ 0.5 0.9, 0.3 0.1
[(e2, q, 0), ( , , , )]}.
Then (FQ, A) ∧ (GQ, B) = (HQ, A × B) where
Λ Λ Λ
(HQ, A × B) = {[(e1, p, 1), (e1, p, 1), ( , , , )].
[(e1, p, 1), (e2, q, 0), ( , , , )]}.
Definition 16. If (FQ, A) and (GQ, B) are two EQFSES over V, then (FQ, A) OR (GQ, B) is
Λ Λ Λ Λ
(FQ, A) ∨ (GQ, B) = (KQ, A × B) .
Λ Λ Λ
Such that KQ(α, β) = FQ(α) ∪ GQ(β) and the memberships of truth, indeterminacy, and falsity of (KQ, A × B) are as follows:
μK (α,β)(m) = max (μF (α)(m), μG (β)(m)),
Q Q Q
where for all α ∈ A, for all β ∈ B.
Example 9. Suppose that (FQ, A) and (GQ, B) are two EQNSES over V, such that
(FQ, A) = {[(e1, p, 1), ( , , , )]}.
(GQ, B) = {[(e1, p, 1), ( , , , )].
[(e2, q, 0), ( , , , )]}.
Then (FQ, A) ∨ (GQ, B) = (KQ, A × B) where
Λ Λ Λ (KQ, A × B) = {[(e1, p, 1), (e1, p, 1) ( , , , )].
[(e1, p, 1), (e2, q, 0), ( , , , )]}.
Proposition 4. If (FQ, A) and (GQ, B) are EQNSES over V, then
c c c
I. ((FQ, A) ∧ (GQ, B) ) = (FQ, A) ∨ (GQ, B) .
Λ Λ Λ Λ c c c
II. ((FQ, A) ∨ (GQ, B) ) = (FQ, A) ∧ (GQ, B) .
Λ Λ Λ Λ
We introduced the concept of EQFSES theory as a new mathematical tool to deal with uncertainty. Furthermore, we presented some of its properties and defined its basic operations as complement, union, intersection, AND, and OR. As a future direction, researchers can develop this concept into an effective neutrosophic vague soft expert set.
Conceptualization, Z. B. and V. U. Methodology, Z. B. writing-reviewing and editing, Z. B. and V. U. All authors have read and agreed to the published version of the manuscript.
The author has given consent for the publication of this manuscript.
This work is purely theoretical and does not involve studies on humans or animals.
The authors declare that no external funding or support was received for the research.
All data supporting the reported findings in this research paper are provided within the manuscript.
The authors declare no conflict of interest.