Additive consistency of q-rung orthopair fuzzy preference relations with application to risk analysis

2.1 IVFSs and IFSs

Firstly, we introduce some definitions of IVFSs and IFSs.

Definition 1. [41] Let X ={ x₁, x₂, …, x _n  } be a given set, IVFS A on X is defined as

A = { 〈 x i , [ l A ( x i ) , r A ( x i ) ] 〉 | x i ∈ X }(1) where l _A  (x _i ) and r _A  (x _i ) are the lower and upper bound of the interval, and satisfy 0 ⩽ l _A  (x _i ) ⩽ r _A  (x _i ) ⩽ 1. [l _A  (x _i ) , r _A  (x _i )] is an interval-valued fuzzy number (IVFN), abbreviated as A = [l, r], where 0 ⩽ l ⩽ r ⩽ 1.

For any IVIN A = [l, r], Beliakov et al. [22] defined the center of IVIN center (A) = (l + r)/ -2 and uncertain degree length (A) = r - l.

Definition 2. [42] Let X ={ x₁, x₂, …, x _n  } be a given set, IFS B on X is defined as

B = { 〈 x i , μ B ( x i ) , ν B ( x i ) 〉 | x i ∈ X }(2) where μ _B  (x _i ) and ν _B  (x _i ) are the membership degree and non-membership degree when the element x _i belongs to IFS B. (μ _B  (x _i ) , ν _B  (x _i )) is called an intuitionistic fuzzy number (IFN), and is abbreviated as B = (μ _B , ν _B ), where μ _B , ν _B  ∈ [0, 1], μ _B  + ν _B  ⩽ 1. π _B  = 1 - μ _B  - ν _B is the hesitant degree of B.

The order relation of IFNs is defined to compare the IFNs as follows.

Definition 3. [43] Let b₁ = (μ₁,  ν₁) and b₂ = (μ₂,  ν₂) be two IFNs, then: 1) if s (b₁) < s (b₂), then b₁ ≺ b₂; 2) if s (b₁) = s (b₂), h (b₁) < h (b₂), then b₁ ≺ b₂; 3) if s (b₁) = s (b₂), h (b₁) = h (b₂), then b₁ ∼ b₂. Where s (b _i ) = μ _i  - ν _i and h (b _i ) = μ _i  + ν _i are score function and accuracy function of b _i .

For any IFN b = (μ,  ν), b can be transformed by IVFNs as follows [44]. A = [ l , r ] = [ μ , 1 - ν ]

Then, the relationship between the score/accuracy function of IFNs and the center / uncertain degree of IVFNs is shown in Table 1.

For two IVFNS A₁ = [l₁,  r₁] and A₂ = [l₂,  r₂], the order relation of IVFNs based on the center and uncertain degree is defined as

if center (A₁) < center (A₂), then A₁ ≺ A₂;

The following theorem based on the order relation of IFNs and IVFNs is obtained as follows.

Theorem 1. Let a₁ = (μ₁,  ν₁) and a₂ = (μ₂,  ν₂) be two IFNs, if the two IVFNs A₁ and A₂ satisfy A₁ = [μ₁,  1 - ν₁] and A₂ = [μ₂,  1 - ν₂], then the following two conditions are equivalent: 1) a₁ ≺ a₂; 2) A₁ ≺ A₂.

Theorem 1 shows that the transformed IVFNs still maintain the sequential relationship between the original IFNs.

For convenience, the common notations are abbreviated as N ={ 1, 2, …, n }, M ={ 1, 2, …, m }, I ={ 1, 2, …, i }, N∖ I = { i + 1, …, n }.

Definition 4. [45] Let R ¯ be an IVFPR on X ={ x _i |i ∈ [n]  }, then it satisfies

R ¯ = ( r ¯ ij ) n × n = ( [ r ij - , r ij + ] ) n × n , r ¯ ij ∈ D ( [ 0 , 1 ] )(3)

r ¯ ji = [ 1 - r ij + , 1 - r ij - ] , r ¯ ii = [ 0.5 , 0.5 ] , i , j ∈ N(4) where r ¯ ij represents the preference of alternative X _i over X _j , r ij - and r ij + are lower and upper bounds of r ¯ ij , and D ([0, 1]) ={ [a^-, a⁺] |a^-, a⁺ ∈ [0, 1]  ; a^- ⩽ a⁺ }.

Definition 5. [31] Let PR R ¯ = ( r ¯ ij ) n × n be additive consistency IVFPR, and it satisfies the following additive transferability condition.

r ¯ ij + r ¯ jk + r ¯ ki = r ¯ kj + r ¯ ji + r ¯ ik , i , j , k ∈ N(5)

Definition 6. [31] Let ω ¯ = ( ω ¯ 1 , ω ¯ 2 , … , ω ¯ n ) T ( ω ¯ i = [ ω i - , ω i + ] ) be a standardized IVWV, then it satisfies

∑ j ≠ i n ω j - + ω i + ⩽ 1 , ω i - + ∑ j ≠ i n ω j + ⩾ 1 , i ∈ N(6)

Moreover, additive consistency IVFPR can be constructed by transforming the standardized IVWV.

Theorem 2. [31] Let R ¯ = ( r ¯ ij ) n × n be an IVFPR, and there exists a standardized IVWV ω ¯ = ( ω ¯ 1 , ω ¯ 2 , … , ω ¯ n ) T make

r ¯ ij = [ r ij - , r ij + ] = { [ 0.5 , 0.5 ] , i = j [ 0.5 ( 1 + ω i - - ω j + ) , 0.5 ( 1 + ω i + - ω j - ) ] , i ≠ j(7)

Then, R ¯ is additive consistency IVFPR.

Next, some definitions of IFPRs are introduced.

Definition 7. [46] Let R _I be an IFPR on X, then it satisfies

R I = ( r ˜ ij ) n × n = ( μ ij , ν ij ) n × n(8) where r ˜ ij ( i , j ∈ N ) is an IFN and satisfies

μ ij = ν ji , ν ij = μ ji , μ ii = ν ii = 0.5(9)

μ _ij represents the preference of alternative X _i over X _j , ν _ij is the preference of X _j over X _i , π _ij  = 1 - μ _ij  - ν _ij is the uncertain degree, and the hesitant degree.

Definition 8. [32] Let PR R I = ( r ˜ ij ) n × n be an additive consistency IFPR, and then it satisfies the following conditions.

μ ij + μ jk + μ ki = μ kj + μ ji + μ ik , i , j , k ∈ N(10)

From Definition 7, μ _ij  = ν _ji , ν _ij  = μ _ji , i, j ∈ N. Then Equation (10) equivalent to

ν ij + ν jk + ν ki = ν kj + ν ji + ν ik , i , j , k ∈ N(11)

Theorem 3 will show that the score function can verify the consistency of the preference relation.

Theorem 3. [32] The sufficient condition for R I = ( r ˜ ij ) n × n to be an additive consistent IFPR is

s ( r ˜ ij ) = s ( r ˜ ik ) - s ( r ˜ jk ) , i , j , k ∈ N(12)

Since the IFN b = (μ, ν) can be converted into an IVFN [μ, 1 - ν], the intuitionistic fuzzy weights ω ˜ i = ( ω ˜ i μ , ω ˜ i ν ) are equivalent to the interval-valued weights [ ω ˜ i μ , 1 - ω ˜ i ν ] .

Definition 9. [32] Let ω ˜ = ( ω ˜ 1 , ω ˜ 2 , … , ω ˜ n ) T be a normalized IFWV, then it satisfies

∑ j ≠ i n ω j μ ⩽ ω i ν , ω i μ + n - 2 ⩾ ∑ j ≠ i n ω i ν , i ∈ N(13) where ω ˜ i = ( ω ˜ i μ , ω ˜ i ν ) ( i ∈ N ) is an IFN.

Theorem 4 will show that the normalized IFWV can be transformed to construct additive consistent IFPRs.

Theorem 4. [32] If the PR T I = ( t ˜ ij ) n × n satisfies

t ˜ ij = ( t ij μ , t ij ν ) = { ( 0.5 , 0.5 ) , i = j ( 0.5 ( ω i μ + ω j ν ) , 0.5 ( ω i ν + ω j μ ) ) , i ≠ j(14) where ω ˜ i = ( ω ˜ i μ , ω ˜ i ν ) ( i ∈ N ) satisfies Definition 9, then T _I is additive consistency IFPR.

3.1 The concepts of q-ROFSs

Yager [17] extended the concept of intuitionistic fuzzy sets and introduced the concept of q-ROFSs.

Definition 10. [17] Let X ={ x₁, x₂, …, x _n  } be a given set, then the q-ROFS P on X is defined as follows.

P = { 〈 x i , ρ P ( x i ) , σ P ( x i ) 〉 | x i ∈ X }(15) where ρ _P  (x _i ) and σ _P  (x _i ) are membership and non-membership values when the element x _i belongs to the set P. (ρ _P  (x _i ) , σ _P  (x _i )) is called a q-ROFN. For convenience, the q-ROFN is abbreviated as p = (ρ, σ). Where ρ, σ ∈ [0, 1] , ρ ^q  + σ ^q  ⩽ 1, π p = 1 - ρ q - σ q q is the hesitant degree of p.

From Definition 2, the IFN a = (μ, ν) satisfies μ + ν ⩽ 1. Therefore, μ ^q  + ν ^q  ⩽ 1. From Definition 10, any IFN is a q-ROFN.

To compare the q-ROFNs, the order relations of q-ROFNs are defined as follows.

Definition 11. [37] Let p₁ = (ρ₁, σ₁) and p₂ = (ρ₂, σ₂) be two q-ROFNs, then: 1) if s (p₁) < s (p₂), then p₁ ≺ p₂; 2) if s (p₁) = s (p₂) and h (p₁) < h (p₂), then p₁ ≺ p₂; 3) if s (p₁) = s (p₂) and h (p₁) = h (p₂), then p₁ ∼ p₂. Where s ( p i ) = ρ i q - σ i q and h ( p i ) = ρ i q + σ i q are the score functions and accuracy functions, respectively.

For any q-ROFN p = (ρ, σ), it satisfies ρ, σ ∈ [0, 1], then

p min = ( 0 , 1 ) ≺ p ≺ ( 1 , 0 ) = p max(16)

For convenience, p_min and p_max are called the extreme values of q-ROFNs.

For any q-ROFN p = (ρ, σ), it satisfies ρ ^q  + σ ^q  ⩽ 1 (q ⩾ 1). Assume that a = (μ, ν) = (ρ ^q , σ ^q ), then μ + ν ⩽ 1. a is an IFN. Combining with Table 1, we analyze the conversion relationship among q-ROFNs, IFNs, and IVFNs, as described in Table 2.

Similar to Theorem 1, Theorem 5 can be obtained from Table 2.

Theorem 5 (Order Preserving). Let p₁ = (ρ₁, σ₁) and p₂ = (ρ₂, σ₂) be two q-ROFNs, if the IFNs a₁, a₂ and the IVFNs A₁, A₂ satisfy a 1 = ( ρ 1 q , σ 1 q ) , a 2 = ( ρ 2 q , σ 2 q ) , A 1 = [ ρ 1 q , 1 - σ 1 q ] and A 2 = [ ρ 2 q , 1 - σ 2 q ] , respectively. Then the following conditions are equivalent: 1) p₁ ≺ p₂; 2) a₁ ≺ a₂; 3) A₁ ≺ A₂.

Proof. p 1 ≺ p 2 ⇔ ρ 1 < ρ 2 , σ 1 > σ 2 ⇔ ρ 1 q < ρ 2 q , σ 1 q > σ 2 q ( a 1 ≺ a 2 ) ⇔ ρ 1 q < ρ 2 q , 1 - σ 1 q < 1 - σ 2 q ( A 1 ≺ A 2 ) .

Thus, Theorem 5 is proved.

Theorem 5 shows the consistency between the transformation and order relations in Table 2. Analyzing the transformation and order relations among the three fuzzy numbers lays the foundation for q-ROFPRs.

Here, the q-ROFPRs are defined.

Definition 12. Let R _P be a q-ROFPR on X, then

R P = ( p ˜ ij ) n × n = ( ρ ij , σ ij ) n × n(17) where p ˜ ij ( i , j ∈ N ) is a q-ROFN and satisfies

ρ ij = σ ji , σ ij = ρ ji , ρ ii = σ ii = 0.5 q(18)

ρ _ij represents the preference of X _i over X _j , σ _ij is the preference of X _j over X _i , τ ij = 1 - ρ ij q - σ ij q q is the uncertain and hesitant degree of R _P and τ ii = 1 - ρ ii q - σ ii q q = 0 , i ∈ N .

Inspired by the additive consistency IVFPR and IFPR [47–50], the additive consistency q-ROFPR is defined as follows based on the transformation relation in Table 2.

Definition 13. The PR R P = ( p ˜ ij ) n × n is additive consistency q-ROFPR, then

ρ ij q + ρ jk q + ρ ki q = ρ kj q + ρ ji q + ρ ik q , i , j , k ∈ N(19)

From Definition 12, ρ _ij  = σ _ji , σ _ij  = ρ _ji , i, j ∈ N. Then, Equation (19) is equivalent to

σ ij q + σ jk q + σ ki q = σ kj q + σ ji q + σ ik q , i , j , k ∈ N(20)

Theorem 6. The sufficient condition for R P = ( p ˜ ij ) n × n to be an additive consistent q-ROFPR is

s ( p ˜ ij ) = s ( p ˜ ik ) - s ( p ˜ jk ) , i , j , k ∈ N(21)

Proof.

If R P = ( p ˜ ij ) n × n is an additive consistent q-ROFPR, then s ( p ˜ ij ) = ρ ij q + σ ij q = ( ρ ik q + σ ik q ) - ( ρ jk q + σ jk q ) = s ( p ˜ ik ) - s ( p ˜ jk ) .

Thus, Theorem 6 is proved.

Similar to the normalized IVWV (Definition 6) and IFWV (Definition 9), the definition of the normalized q-ROFWV is given below based on the transformation relations in Table 2.

Definition 14. Let the weight vector ω ˜ = ( ω ˜ 1 , ω ˜ 2 , … , ω ˜ n ) T be the normalized q-ROFWV, for i ∈ N, it satisfies

∑ j ≠ i n ( ω j ρ ) q ⩽ ( ω i σ ) q , ( ω i ρ ) q + n - 2 ⩾ ∑ j ≠ i n ( ω j σ ) q(22) where ω ˜ i = ( ω ˜ i ρ , ω ˜ i σ ) ( i ∈ N ) is a q-ROFN.

Theorem 7 shows that additive consistent q-ROFPR can be constructed based on normalized q-ROFWVs.

Theorem 7. If the PR T P = ( t ˜ ij ) n × n satisfies the following condition:

t ˜ ij = ( t ij ρ , t ij σ ) = { ( 0.5 q , 0.5 q ) , i = j ( 0.5 ( ( ω i ρ ) q + ( ω j σ ) q ) q , 0.5 ( ( ω j ρ ) q + ( ω i σ ) q ) q ) , i ≠ j(23)

where ω ˜ i = ( ω ˜ i ρ , ω ˜ i σ ) ( i ∈ N ) satisfies Definition 14, then T _P is the additive consistent q-ROFPR.

Remark 1. For any q-ROFPR R P = ( p ˜ ij ) n × n , if there exists a normalized q-ROFWV satisfying

p ˜ ij = ( ρ ij , σ ij ) = { ( 0.5 q , 0.5 q ) , i = j ( 0.5 ( ( ω i ρ ) q + ( ω j σ ) q ) q , 0.5 ( ( ω j ρ ) q + ( ω i σ ) q ) q ) , i ≠ j(24)

Then, R _P is an additive consistent q-ROFPR.

Based on the transformation relations in Table 2, Theorem 8 analyzes the connection between the three PRs.

Theorem 8. Let R P = ( p ˜ ij ) n × n be the additive consistent q-ROFPR satisfying

p ˜ ij = ( ρ ij , σ ij ) = { ( 0.5 q , 0.5 q ) , i = j ( 0.5 ( ( ω i ρ ) q + ( ω j σ ) q ) q , 0.5 ( ( ω j ρ ) q + ( ω i σ ) q ) q ) , i ≠ j(25) where ω ˜ = ( ω ˜ 1 , ω ˜ 2 , … , ω ˜ n ) T is a normalized q-ROFWV. Then

1) If the IVFPR R ¯ = ( r ¯ ij ) n × n satisfies

r ¯ ij = [ r ij - , r ij + ] = [ ρ ij q , 1 - σ ij q ](26) then R ¯ is additive consistent IVFPR;

2) If the IFPR R I = ( r ˜ ij ) n × n satisfies

r ˜ ij = ( μ ij , ν ij ) = ( ρ ij q , σ ij q )(27) then R _I is additive consistent IFPR.

Proof.

1) if r ¯ ij = [ r ij - , r ij + ] = [ ρ ij q , 1 - σ ij q ] , then

r ¯ ij = [ ρ ij q , 1 - σ ij q ] = { ( 0.5 , 0.5 ) , i = j ( 0.5 ( ( ω i ρ ) q + ( ω j σ ) q ) , 1 - 0.5 ( ( ω j ρ ) q + ( ω i σ ) q ) ) , i ≠ j(28)

Assume that ω ¯ i = [ ω i - , ω i + ] = [ ( ω i ρ ) q , 1 - ( ω j σ ) q ] , from Definition 16, then

∑ j ≠ i n ω j - + ω i + ⩽ 1 , ω i - + ∑ j ≠ i n ω j + ⩾ 1 , i ∈ N(29)

Therefore, ω ¯ = ( ω ¯ 1 , ω ¯ 2 , … , ω ¯ n ) T is the normalized IVFWV. Then, Equation (28) is equivalent to

r ¯ ij = [ ρ ij q , 1 - σ ij q ] = { [ 0.5 , 0.5 ] , i = j [ 0.5 ( 1 + ω i - - ω j + ) , 0.5 ( 1 + ω i + - ω j - ) ] , i ≠ j(30)

Thus, R ¯ is an additive consistent IVFPR.

2) The proof procedure is similar to 1) and will not be repeated.

Theorem 8 shows that the additive consistent q-ROFPR can be converted into additive consistent IVFPR and IFPR based on the transformation relations in Table 2.

5.1 Problem description

For the GDM problem in the q-ROFPR, let X ={ X _i |i ∈ N  } be the alternative set and E ={ e _k |k ∈ M  } be the expert set. Based on the evaluation information provided by the decision-makers, the expert e _k  (k ∈ M) makes a two-by-two comparison of the alternatives X _i  (i ∈ N) and X _j  (j ∈ N) and provides the q-ROFPR R P k = ( p ˜ ij k ) n × n ( k ∈ M ) , where p ˜ ij k = ( ρ ij k , σ ij k ) ( i , j ∈ N ; k ∈ M ) is a q-ROFN. In order to obtain the ranking result of the alternatives on X, it is necessary to obtain the normalized q-ROFWV ω ˜ * = ( ω ˜ 1 * , ω ˜ 2 * , … , ω ˜ n * ) T corresponding to the alternatives. Therefore, we propose a GDM method based on the q-ROFWQ operator and goal programming model, apply it to solve the q-ROFWV, and finally obtain the optimal alternative.

5.2 Aggregation of multiple q-ROFPRs

Before solving the standardized q-ROFWV by goal programming model, it is necessary to assemble multiple q-ROFPRs R P k = ( p ˜ ij k ) n × n ( k ∈ M ) provided by the expert group E into a comprehensive q-ROFPR R = (p _ij ) _n×n. The following two aspects are studied, the selection of a reasonable aggregation operator and the determination of the weights.

(1) Selection of the aggregation operator

The first stage of solving the GDM problem with q-ROFPRs is to choose a reasonable and effective q-ROF operator to gather a set of q-ROFPRs provided by different experts. In this section, we compare the six existing aggregation operators [37–40] through example analysis. Two main aspects are examined: first, whether the operator is reasonable in assembling a set of q-ROFPRs containing extreme values ((0, 1) or (1, 0)); and second, whether the operator is monotonic with respect to the order relation (including score function and accuracy function in Definition 11).

Next, the existing six q-ROF aggregation operators are provided as follows.

Definition 15. Let a _i  = (ρ _i , σ _i ) (i ∈ N) is a group of q-ROFNs, the weight vector is ω = (ω₁, ω₂, …, ω _n )  ^T .

1) q-ROFWA [37]

q - ROFWA ( a 1 , a 2 , … , a n ) = ( 1 - ∏ i = 1 n ( 1 - ( ρ i ) q ) ω i q , ∏ i = 1 n ( σ i ) ω i )(40)

2) q-ROFWG [37]

q - ROFWG ( a 1 , a 2 , … , a n ) = ( ∏ i = 1 n ( ρ i ) ω i , 1 - ∏ i = 1 n ( 1 - ( σ i ) q ) ω i q )(41)

3) q-ROFWQ [37]

q - ROFWQ ( a 1 , a 2 , … , a n ) = ( ∑ i = 1 n ω i ( ρ i ) q q , ∑ i = 1 n ω i ( σ i ) q q )(42)

4) q-ROFIWA [38]

q - ROFIWA ( a 1 , a 2 , … , a n ) = ( 1 - ∏ i = 1 n ( 1 - ( ρ i ) q ) ω i q , ∏ i = 1 n ( 1 - ( ρ i ) q ) ω i - ∏ i = 1 n ( 1 - ( ρ i ) q - ( σ i ) q ) ω i q )(43)

5) q-ROFIWG [39]

q - ROFIWG ( a 1 , a 2 , … , a n ) = ( ∏ i = 1 n ( 1 - ( σ i ) q ) ω i - ∏ i = 1 n ( 1 - ( ρ i ) q - ( σ i ) q ) ω i q , 1 - ∏ i = 1 n ( 1 - ( σ i ) q ) ω i q )(44)

6) q-ROFEWA [40]

q - ROFEWA ( a 1 , a 2 , … , a n ) = ( ∏ i = 1 n ( 1 + ( ρ i ) q ) ω i - ∏ i = 1 n ( 1 - ( ρ i ) q ) ω i ∏ i = 1 n ( 1 + ( ρ i ) q ) ω i + ∏ i = 1 n ( 1 - ( ρ i ) q ) ω i q , ∏ i = 1 n ( ( σ i ) q ) ω i ∏ i = 1 n ( 2 - ( σ i ) q ) ω i + ∏ i = 1 n ( ( σ i ) q ) ω i q )(45)

Next, the performance of the operators in Definition 15 is examined by the following theorem.

Theorem 9. Let a _i  = (ρ _i , σ _i ) (i ∈ N) be a set of q-ROFNs, whose weight vector is ω = (ω₁, ω₂, …, ω _n )  ^T , then

1) If there exists a _k  = (1, 0) , 0 < ω _k  < 1, k ∈ N, then

q - ROFWA ( a 1 , a 2 , … , a n ) = ( 1 , 0 )(46)

q - ROFIWA ( a 1 , a 2 , … , a n ) = ( 1 , 0 )(47)

q - ROFEWA ( a 1 , a 2 , … , a n ) = ( 1 , 0 )(48)

2) If there exists a _k  = (0, 1) , 0 < ω _k  < 1, k ∈ N, then

q - ROFWG ( a 1 , a 2 , … , a n ) = ( 0 , 1 )(49)

q - ROFIWG ( a 1 , a 2 , … , a n ) = ( 0 , 1 )(50)

The shortcomings of the above five operators are summarized as follows by Theorem 9.

1) The extreme values p_max = (1, 0) of q-ROFWA / q-ROFIWA / q-ROFEWA operators completely determine the set result when assembling multiple q-ROFPRs according to Equations (46–48). The remaining non-extreme values are ignored and do not affect the set result, which is unreasonable.

2) Similar to the q-ROFWA operator, the extreme values p_min = (0, 1) of q-ROFWG / q-ROFIWG operators completely determine the aggregation result in the process of aggregation according to Equations (49, 50), which is unreasonable.

Obviously, the q-ROFWQ operator consists of ∑ i = 1 n ω i x i and does not contain ∏ i = 1 n x i ω i and ∏ i = 1 n ( 1 - x i ) ω i . The q-ROFWQ operator does not fail in other elements when there are 0 or 1 in the elements. The following example analyzes the monotonicity of the existing six operators with respect to the order relation.

Example 2. Assume that a₁ = (1, 0), a₂ = (0.6, 0.7) and b₁ = (1, 0), b₂ = (0.3, 0.3) be two groups of q-ROFNs, q = 3. The weight vectors are ω = (0.6, 0.4)  ^T . From the order relations in Theorem 5, then a₁ ∼ a₂, b₁ ≺ b₂

The six operators in Definition 15 are used to aggregate the two groups of q-ROFNs. The results are shown in Table 4.

Based on the analysis of Table 4, the monotonicity of the operators with respect to the order relations is stated by Theorem 10.

Theorem 10 (Monotonicity Certification). Let a _i  = (ρ _{a i} , σ _{a i} ) and b _i  = (ρ _{b i} , σ _{b i} )(i ∈ N) be two sets of q-ROFNs. If a _i  ≺ b _i or a _i  ∼ b _i , and there exists k ∈ N, satisfying a _k  ≺ b _k , then

q - ROFWA (a₁, a₂, …, a _n ) ≺ q - ROFWA (b₁, b₂, …, b _n ) does not necessarily hold;

Proof.

1)-5) are obvious from the results in Table 4

6) From Definition 5 and Definition 15, the q-ROFWQ operator satisfies the following relationship with score function and accuracy function: s ( q - ROFWQ ( a 1 , a 2 , … , a n ) ) = ∑ i = 1 n ω i s ( a i ) h ( q - ROFWQ ( a 1 , a 2 , … , a n ) ) = ∑ i = 1 n ω i h ( a i )

Because a _i  ≺ b _i , s (a _i ) ≺ s (b _i )(i ∈ N ; i ≠ k),

According to Definition 5, two cases are discussed.

Case 1. If s (a _k ) < s (b _k ), then s ( q - ROFWQ ( a 1 , a 2 , … , a n ) ) - s ( q - ROFWQ ( b 1 , b 2 , … , b n ) ) = ( s ( a k ) - s ( b k ) ) + ∑ i = 1 ; i ≠ k n ω i ( s ( a i ) - s ( b i ) ) < 0

Case 2. If s (a _k ) = s (b _k ) and h (a _k ) < h (b _k ), then

1) If ∑ i = 1 ; i ≠ k n ω i ( s ( a i ) - s ( b i ) ) < 0 , then s ( q - ROFWQ ( a 1 , a 2 , … , a n ) ) < s ( q - ROFWQ ( b 1 , b 2 , … , b n ) ) .

2) If ∑ i = 1 ; i ≠ k n ω i ( s ( a i ) - s ( b i ) ) = 0 , then s ( a i ) = s ( b i ) , h ( a i ) ⩽ h ( b i ) , i ∈ N , i ≠ k .

Further, we can get h ( q - ROFWQ ( a 1 , a 2 , … , a n ) ) - h ( q - ROFWQ ( b 1 , b 2 , … , b n ) ) = ( h ( a k ) - h ( b k ) ) + ∑ i = 1 ; i ≠ k n ω i ( h ( a i ) - h ( b i ) ) < 0

We are combining the above two cases, q - ROFWQ (a₁, a₂, …, a _n )   ≺ q - ROFWQ (b₁, b₂,  … , b _n ).

Thus, Theorem 10 is proved.

From Theorem 9 and Theorem 10, the five operators mentioned in Theorem 9 can make the aggregation results unreasonable due to extreme values and are not necessarily monotonic with respect to the order relation. The q-ROFWQ operator is more reasonable than the above five operators. Therefore, the q-ROFWQ operator is chosen as a tool for assembling the set of q-ROFPRs. The consistency between the q-ROFPRs under the q-ROFWQ operator is introduced below.

Theorem 11. If R P l = ( p ˜ ij l ) n × n ( l ∈ M ) is a set of additive consistent q-ROFPRs, then the integrated PR R = (p _ij ) _n×n based on the q-ROFWQ operator is an additive consistent q-ROFPR.

Proof. According to Definition 13, if R P l = ( p ˜ ij l ) n × n ( l ∈ M ) is an additive consistent q-ROFPR, for i, j ∈ N, then

( ρ ij l ) q + ( ρ jk l ) q + ( ρ ki l ) q = ( ρ kj l ) q + ( ρ ji l ) q + ( ρ ik l ) q , ( σ ij l ) q + ( σ jk l ) q + ( σ ki l ) q = ( σ kj l ) q + ( σ ji l ) q + ( σ ik l ) q ,