Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making

Definition 2.1

Assume that Ψ is a universe set. Let μ ¯ Ω ^ : Ψ → [ 0 , 1 ] be the degree of membership and μ ̲ Ω ^ : Ψ → [ 0 , 1 ] be the degree of nonmembership of ζ ∈ Ψ to Ω ^ . Then, Ω ^ = { ⟨ ζ , μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ⟩ : ζ ∈ Ψ } is called

1. an IFS [2] if μ ¯ Ω ^ + μ ̲ Ω ^ ≤ 1 ;

2. a PFS [3] if ( μ ¯ Ω ^ ) 2 + ( μ ̲ Ω ^ ) 2 ≤ 1 ;

3. a q-ROFS [4] if ( μ ¯ Ω ^ ) q + ( μ ̲ Ω ^ ) q ≤ 1 , for q > 2 ;

4. an n,m-ROFS [6] if ( μ ¯ Ω ^ ) n + ( μ ̲ Ω ^ ) m ≤ 1 , for n , m > 1 .

Definition 2.2

[28] A CIFS Ω ^ of radius ρ ^ over a universal set Ψ is recognized as follows:

when both membership μ ¯ Ω ^ and nonmembership μ ̲ Ω ^ meet the requirement μ ¯ Ω ^ ( ζ ) + μ ̲ Ω ^ ( ζ ) ≤ 1 for all ζ ∈ Ψ , and ρ ^ ∈ [ 0 , 1 ] represents the radius of a circle with the center ( μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ) . The term π Ω ^ ( ζ ) = 1 − ( μ ¯ Ω ^ ( ζ ) + μ ̲ Ω ^ ( ζ ) ) expresses the degree of hesitation of an element ζ ∈ Ψ .

Definition 2.3

[34] A CPFS Ω ^ of radius ρ ^ over a universal set Ψ is recognized as follows:

when both membership μ ¯ Ω ^ and nonmembership μ ̲ Ω ^ meet the requirement μ ¯ Ω ^ 2 ( ζ ) + μ ̲ Ω ^ 2 ( ζ ) ≤ 1 for all ζ ∈ Ψ , and ρ ^ ∈ [ 0 , 2 ] represents the radius of a circle with the center ( μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ) . The term π Ω ^ ( ζ ) = 1 − ( μ ¯ Ω ^ 2 ( ζ ) + μ ̲ Ω ^ 2 ( ζ ) ) expresses the degree of hesitation of an element ζ ∈ Ψ .

Definition 3.1

A circular n,m-rung orthopair fuzzy set (Cn,m-ROFS) Ω ^ of radius ρ ^ over a universal set Ψ is recognized as follows:

when both membership μ ¯ Ω ^ and nonmembership μ ̲ Ω ^ meet the requirement μ ¯ Ω ^ n ( ζ ) + μ ̲ Ω ^ m ( ζ ) ≤ 1 for all ζ ∈ Ψ , and ρ ^ ∈ [ 0 , 2 ] represents the radius of a circle with the center ( μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ) . The term π Ω ^ ( ζ ) = 1 − ( μ ¯ Ω ^ n ( ζ ) + μ ̲ Ω ^ m ( ζ ) ) n + m expresses the degree of hesitation of an element ζ ∈ Ψ .

The calculation for ζ , Ω ^ = ( ζ , μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ; ρ ^ ) represents the circular n,m-rung orthopair fuzzy value (Cn,m-ROFV), which is a circle with radius ρ ^ at center ( μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ) . A Cn,m-ROFV expressed in an abstract environment is written as ( μ ¯ Ω ^ , μ ̲ Ω ^ ; ρ ^ ) instead of the ( ζ , μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ; ρ ^ ) expression linked to a different ζ . When ρ ^ is equal to zero, the Cn,m-ROFS functions transform into a regular n,m-ROFS.

Definition 3.2

Let Ω ^ = { ( μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ; ρ ^ ) ∣ ζ ∈ Ψ } , Ω ^ 1 = { ( μ ¯ Ω ^ 1 ( ζ ) , μ ̲ Ω ^ 1 ( ζ ) ; ρ ^ 1 ) ∣ ζ ∈ Ψ } and Ω ^ 2 = { ( μ ¯ Ω ^ 2 ( ζ ) , μ ̲ Ω ^ 2 ( ζ ) ; ρ ^ 2 ) ∣ ζ ∈ Ψ } be three Cn,m-ROFSs. Then, the next operations are recognized:

1. Ω ^ c = μ ̲ Ω ^ ( ζ ) m n , μ ¯ Ω ^ ( ζ ) n m ; ρ ^ ∣ ζ ∈ Ψ .

2. Ω ^ 1 ⊆ Ω ^ 2 if and only if ρ ^ 1 ≤ ρ ^ 2 , μ ¯ Ω ^ 1 ( ζ ) ≤ μ ¯ Ω ^ 2 ( ζ ) and μ ̲ Ω ^ 1 ( ζ ) ≥ μ ̲ Ω ^ 2 ( ζ ) .

3. Ω ^ 1 ≤ Ω ^ 2 if Ω ^ 1 ⊆ Ω ^ 2 .

4. Ω ^ 1 ∪ ̲ max Ω ^ 2 = { ( max { μ ¯ Ω ^ 1 ( ζ ) , μ ¯ Ω ^ 2 ( ζ ) } , min { μ ̲ Ω ^ 1 ( ζ ) , μ ̲ Ω ^ 2 ( ζ ) } ; max { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } ,

   and

   Ω ^ 1 ∪ ̲ min Ω ^ 2 = { ( max { μ ¯ Ω ^ 1 ( ζ ) , μ ¯ Ω ^ 2 ( ζ ) } , min { μ ̲ Ω ^ 1 ( ζ ) , μ ̲ Ω ^ 2 ( ζ ) } ; min { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } .

5. Ω ^ 1 ∩ ¯ max Ω ^ 2 = { ( min { μ ¯ Ω ^ 1 ( ζ ) , μ ¯ Ω ^ 2 ( ζ ) } , max { μ ̲ Ω ^ 1 ( ζ ) , μ ̲ Ω ^ 2 ( ζ ) } ; max { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } ,

   and

   Ω ^ 1 ∩ ¯ min Ω ^ 2 = { ( min { μ ¯ Ω ^ 1 ( ζ ) , μ ¯ Ω ^ 2 ( ζ ) } , max { μ ̲ Ω ^ 1 ( ζ ) , μ ̲ Ω ^ 2 ( ζ ) } ; min { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } .

Example 3.3

Let Ψ = { ζ 1 , ζ 2 } , then

and

be two C2,3-ROFSs in Ψ . Then,

and

Definition 3.4

Let Ω ^ = { ( μ ¯ Ω ^ ( ζ ) , μ ̲ Ω ^ ( ζ ) ; ρ ^ ) ∣ ζ ∈ Ψ } , Ω ^ 1 = { ( μ ¯ Ω ^ 1 ( ζ ) , μ ̲ Ω ^ 1 ( ζ ) ; ρ ^ 1 ) ∣ ζ ∈ Ψ } , and Ω ^ 2 = { ( μ ¯ Ω ^ 2 ( ζ ) , μ ̲ Ω ^ 2 ( ζ ) ; ρ ^ 2 ) ∣ ζ ∈ Ψ } be three Cn,m-ROFSs. Then, the next operations are recognized:

1. Ω ^ 1 ⊕ max Ω ^ 2 = { ( μ ¯ Ω ^ 1 n ( ζ ) + μ ¯ Ω ^ 2 n ( ζ ) − μ ¯ Ω ^ 1 n ( ζ ) μ ¯ Ω ^ 2 n ( ζ ) n , μ ̲ Ω ^ 1 ( ζ ) μ ̲ Ω ^ 2 ( ζ ) ; max { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } ,

   and

   Ω ^ 1 ⊕ min Ω ^ 2 = { ( μ ¯ Ω ^ 1 n ( ζ ) + μ ¯ Ω ^ 2 n ( ζ ) − μ ¯ Ω ^ 1 n ( ζ ) μ ¯ Ω ^ 2 n ( ζ ) n , μ ̲ Ω ^ 1 ( ζ ) μ ̲ Ω ^ 2 ( ζ ) ; min { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } .

2. Ω ^ 1 ⊗ max Ω ^ 2 = { ( μ ¯ Ω ^ 1 ( ζ ) μ ¯ Ω ^ 2 ( ζ ) , μ ̲ Ω ^ 1 m ( ζ ) + μ ̲ Ω ^ 2 m ( ζ ) − μ ̲ Ω ^ 1 m ( ζ ) μ ̲ Ω ^ 2 m ( ζ ) m ; max { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } ,

   and

   Ω ^ 1 ⊗ min Ω ^ 2 = { ( μ ¯ Ω ^ 1 ( ζ ) μ ¯ Ω ^ 2 ( ζ ) , μ ̲ Ω ^ 1 m ( ζ ) + μ ̲ Ω ^ 2 m ( ζ ) − μ ̲ Ω ^ 1 m ( ζ ) μ ̲ Ω ^ 2 m ( ζ ) ; m min { ρ ^ 1 , ρ ^ 2 } ) ∣ ζ ∈ Ψ } .

3. ε Ω ^ = { ( 1 − ( 1 − μ ¯ Ω ^ n ( ζ ) ) ε n , μ ̲ Ω ^ ε ( ζ ) ; ρ ^ ) ∣ ζ ∈ Ψ } ,

   and

   Ω ^ ε = { ( μ ¯ Ω ^ ε ( ζ ) , 1 − ( 1 − μ ̲ Ω ^ m ( ζ ) ) ε m ; ρ ^ ) ∣ ζ ∈ Ψ } ,

   for ε > 0 .

Example 3.5

Assume that both Ω ^ 1 = ( 0.89 , 0.61 ; 0.02 ) and Ω ^ 2 = ( 0.74 , 0.83 ; 0.04 ) are C4,2-ROFVs for Ψ = { ζ } . Then,

1. Ω ^ 1 ⊕ max Ω ^ 2 = ( μ ¯ Ω ^ 1 n + μ ¯ Ω ^ 2 n − μ ¯ Ω ^ 1 n μ ¯ Ω ^ 2 n n , μ ̲ Ω ^ 1 μ ̲ Ω ^ 2 ; max { ρ ^ 1 , ρ ^ 2 } ) = ( 0.8 9 4 + 0.7 4 4 − ( 0.89 ) 4 ( 0.74 ) 4 4 , ( 0.61 ) ( 0.83 ) ; max { 0.02 , 0.04 } ) ≈ ( 0.9272 , 0.5063 ; 0.04 ) .

2. Ω ^ 1 ⊕ min Ω ^ 2 = ( μ ¯ Ω ^ 1 n + μ ¯ Ω ^ 2 n − μ ¯ Ω ^ 1 n μ ¯ Ω ^ 2 n n , μ ̲ Ω ^ 1 μ ̲ Ω ^ 2 ; min { ρ ^ 1 , ρ ^ 2 } ) = ( 0.8 9 4 + 0.7 4 4 − ( 0.89 ) 4 ( 0.74 ) 4 4 , ( 0.61 ) ( 0.83 ) ; min { 0.02 , 0.04 } ) ≈ ( 0.9272 , 0.5063 ; 0.02 ) .

3. Ω ^ 1 ⊗ max Ω ^ 2 = ( μ ¯ Ω ^ 1 μ ¯ Ω ^ 2 , μ ̲ Ω ^ 1 m + μ ̲ Ω ^ 2 m − μ ̲ Ω ^ 1 m μ ̲ Ω ^ 2 m m ; max { ρ ^ 1 , ρ ^ 2 } ) = ( ( 0.89 ) ( 0.74 ) , 0.6 1 2 + 0.8 3 2 − ( 0.61 ) 2 ( 0.83 ) 2 ; max { 0.02 , 0.04 } ) ≈ ( 0.6586 , 0.8970 ; 0.04 ) .

4. Ω ^ 1 ⊗ min Ω ^ 2 = ( μ ¯ Ω ^ 1 μ ¯ Ω ^ 2 , μ ̲ Ω ^ 1 m + μ ̲ Ω ^ 2 m − μ ̲ Ω ^ 1 m μ ̲ Ω ^ 2 m m ; min { ρ ^ 1 , ρ ^ 2 } ) = ( ( 0.89 ) ( 0.74 ) , 0.6 1 2 + 0.8 3 2 − ( 0.61 ) 2 ( 0.83 ) 2 ; min { 0.02 , 0.04 } ) ≈ ( 0.6586 , 0.8970 ; 0.02 ) .

5. ε Ω ^ 1 = ( 1 − ( 1 − μ ¯ Ω ^ 1 n ) ε n , μ ̲ Ω ^ 1 ε ; ρ ^ ) = ( 1 − ( 1 − 0.8 9 4 ) 5 4 , 0.6 1 5 ; 0.02 ) ≈ ( 0.9982 , 0.0845 ; 0.02 ) , for ε = 5 .

6. Ω ^ 1 ε = ( μ ¯ Ω ^ 1 ε , 1 − ( 1 − μ ̲ Ω ^ 1 m ) ε m ; ρ ^ ) = ( 0.8 9 5 , 1 − ( 1 − 0.6 1 2 ) 5 ; 0.02 ) ≈ ( 0.5584 , 0.9499 ; 0.02 ) , for ε = 5 .

Theorem 3.6

If Ω ^ = ( μ ¯ Ω ^ , μ ̲ Ω ^ ; ρ ^ ) is a Cn,m-ROFV, then Ω ^ c is also a Cn,m-ROFV and ( Ω ^ c ) c = Ω ^ .

Proof

Since 0 ≤ μ ¯ Ω ^ n ≤ 1 , 0 ≤ μ ̲ Ω ^ m ≤ 1 and 0 ≤ μ ¯ Ω ^ n + μ ̲ Ω ^ m ≤ 1 , then

and hence,

Thus, Ω ^ c is a Cn,m-ROFS, and it is clear that ( Ω ^ c ) c = ( μ ̲ Ω ^ m n , μ ¯ Ω ^ n m ; ρ ^ ) c = ( ( μ ¯ Ω ^ n m ) m n , ( μ ̲ Ω ^ m n ) n m ; ρ ^ ) = ( μ ¯ Ω ^ , μ ̲ Ω ^ ; ρ ^ ) .□

Theorem 3.7

If Ω ^ 1 = ( μ ¯ Ω ^ 1 , μ ̲ Ω ^ 1 ; ρ ^ 1 ) and Ω ^ 2 = ( μ ¯ Ω ^ 2 , μ ̲ Ω ^ 2 ; ρ ^ 2 ) are two Cn,m-ROFVs, then Ω ^ 1 ⊕ max Ω ^ 2 , Ω ^ 1 ⊕ min Ω ^ 2 , Ω ^ 1 ⊗ max Ω ^ 2 , and Ω ^ 1 ⊗ min Ω ^ 2 are also Cn,m-ROFVs.

Proof

For Cn,m-ROFSs Ω ^ 1 = ( μ ¯ Ω ^ 1 , μ ̲ Ω ^ 1 ; ρ ^ 1 ) and Ω ^ 2 = ( μ ¯ Ω ^ 2 , μ ̲ Ω ^ 2 ; ρ ^ 2 ) it is clear that the following relationships exist,

and

Then, we have

and