In this section, we provide fundamental information on PUIL q-ROFSs along with their corresponding operations and distance measures.

Definition 4.1

For a given a first hierarchy LTS \(S =\|t = -\tau , -\tau + 1,..., 0,..., \tau -1,\tau \}\) and second hierarchy LTS \(O=\|k=-\varsigma ,-\varsigma +1,...,0,...,\varsigma -1,\varsigma \}\), with \(Y= \,y_,...,y_\}\) being a finite universe of discourse, a PUIL q-ROFS A on Y is mathematically defined as follows:

$$\begin A=\},\overline}],p_),([\underline},\overline}],q_)\rangle \}, \end$$

(4.1)

where \(([\underline}, \overline}], p_)\) and \(([\underline}, \overline}], q_)\) represent the membership degree (MD) and non-membership degree (NMD) of the element y belonging to A, respectively; \(p_\) and \(q_\) are the probabilities associated with the intervals \([\underline}, \overline}]\) and \([\underline}, \overline}]\), respectively.

This structure is subject to the following constraints:

1.

The membership and non-membership degrees satisfy \(0 \le \underline}, \overline} \le 1\) and \(0 \le \underline}, \overline} \le 1\).

2.

The following condition holds for all \(q \ge 1\):

$$\begin 0 \le (u(\overline}))^ p_ + (u(\overline}))^ q_ \le 1. \end$$

In this definition, \(\underline}, \overline}, \underline}, \overline}\) are all double hierarchy unbalanced linguistic terms, where \(\underline}\) and \(\underline}\) denote the lower bounds of \(\mu _\) and \(\nu _\), respectively, while \(\overline}\) and \(\overline}\) represent the corresponding upper bounds. The parameter q is a positive integer, and the function u retrieves the semantic value of the double hierarchy unbalanced linguistic terms within the set \(U_}\) as defined in Definition 3.5.

Additionally, the interval hesitation degree of A is defined below:

$$\begin \pi _=[\underline},\overline}], \end$$

where \(\underline}=\root q \of }))^p_-(u(\overline}))^q_}\). \(\overline}=\root q \of }))^p_-(u(\underline}))^q_}\) is defined as the hesitance degree (HD) of element y in set A. \(\underline}, \overline},\underline},\overline}\in U_}\) and satisfy \(u(\underline})\le u(\overline})\), \(u(\underline})\le u(\overline})\). \(\underline}\), \(\overline}\), \(\underline}\), \(\overline}\) are represented in the form of UDHLTSs.

This section formally introduces the comparative framework for PUIL q-ROFNs. The cornerstone of this framework is a pair of metrics known as the score and accuracy functions, defined as follows.

Definition 4.2

Consider a PUIL q-ROFN \(A=},\overline}],p_),([\underline},\overline}],q_)\rangle }\). Its score function, denoted by S(A), and its accuracy function, denoted by H(A), are given by:

$$\begin S(A) = \left( \frac}) \right) ^ + \left( u(\overline}) \right) ^ } \right) p_ - \left( \frac}) \right) ^ + \left( u(\overline}) \right) ^ } \right) q_, \end$$

(4.2)

$$\begin H(A) = \left( \frac}) \right) ^ + \left( u(\overline}) \right) ^ } \right) p_ + \left( \frac}) \right) ^ + \left( u(\overline}) \right) ^ } \right) q_. \end$$

(4.3)

The score function S(A) serves as the primary indicator for ranking, whereas the accuracy function H(A) acts as a tie-breaker when scores are equal, thereby ensuring a comprehensive and unambiguous comparison methodology.

The comparison of two PUIL q-ROFNs is conducted through a two-step procedure based on their score and accuracy values, formalized in the following definition.

Definition 4.3

For any two PUIL q-ROFNs \(A_1\) and \(A_2\), their order relation is defined as follows.

Step 1: Compare the score values \(S(A_1)\) and \(S(A_2)\).

If \(S(A_1)> S(A_2)\), then \(A_1 \succ A_2\).

If \(S(A_1) = S(A_2)\), then proceed to Step 2.

Step 2: Compare the accuracy values \(H(A_1)\) and \(H(A_2)\).

If \(H(A_1)> H(A_2)\), then \(A_1 \succ A_2\).

If \(H(A_1) = H(A_2)\), then \(A_1 = A_2\).

Proposition 4.4

The score function S(A) for any PUIL q-ROFN A satisfies the following fundamental properties:

Boundedness: \(-1 \le S(A) \le 1\).

Maximal Score: \(S(A) = 1\) if and only if \(A = \langle ([1, 1], 1), ([0, 0], 0) \rangle\).

Minimal Score: \(S(A) = -1\) if and only if \(A = \langle ([0, 0], 0), ([1, 1], 1) \rangle\).

Proof

It is trivial to verify these properties by directly substituting them into the definition of the score function. \(\square\)

Having established the comparison mechanism, we now proceed to define the fundamental algebraic operations for PUIL q-ROFNs.

Definition 4.5

For two PUIL q-ROFSs \(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}\), \(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}\), the following operations are defined:

1. Addition (\(\oplus\)):

$$\begin A_ \oplus A_ & =\left\langle \left( \left( \left[ \root q \of }})^+(\underline}})^-(\underline}})^(\underline}})^}, \root q \of }})^+(\overline}})^-(\overline}})^(\overline}})^}\right] , \right. \right. \right. \\ & \qquad \left. \left. \left. \left( p_}+p_}-p_}p_}\right) \right) ,\left( [\underline}}\underline}},\overline}}\overline}}],q_}q_}\right) \right) \right\rangle \end} \end$$

(4.4)

2.Multiplication (\(\otimes\)):

$$\begin A_1 \otimes A_2 = \biggl \langle & \biggl ( \left[ \underline}\,\underline},\ \overline}\,\overline} \right] ,\ p_p_ \biggr ), \\ & \biggl ( \left[ \root q \of })^q + (\underline})^q - (\underline})^q(\underline})^q},\ \root q \of })^q + (\overline})^q - (\overline})^q(\overline})^q} \right] , \\ & \quad q_ + q_ - q_q_ \biggr ) \biggr \rangle \end} \end$$

(4.5)

3. Scalar Multiplication:

$$\begin \lambda A_=\left\langle \left( \left[ \root q \of }})^\right) ^},\root q \of }})^\right) ^}\right] ,1-(1-p_)^\right) ,\left( \left[ (\underline}})^,(\overline}})^\right] ,(q_)^\right) \right\rangle \end$$

(4.6)

4. Power Operation:

$$\begin A_^=\left\langle \left( \left[ (\underline}})^,(\overline}})^\right] ,(p_)^\right) , \left( \left[ \root q \of }})^\right) ^},\root q \of }})^\right) ^}\right] ,1-(1-q_)^\right) \right\rangle \end$$

(4.7)

To determine the optimal choice from a set of m available options, denoted as \(a =\, a_, a_,..., a_\}\), we evaluate them based on n criteria, denoted as \(c =\, c_, c_,..., c_\}\). A group of decision experts, \(e =\, e_, e_,..., e_\}\), is tasked with evaluating these alternatives. The weights assigned to the experts are denoted by \(w =(w_, w_, w_,..., w_)^\). Using the methodologies we have established and the findings from [47], we can derive the following equation:

Theorem 4.6

Let \(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}(t=1,2,...,l)\), then the generalized orthopair fuzzy weighted averaging (GOFWA) operator is defined as follows:

$$\begin \begin \textrm(A_, A_, \ldots , A_)&= \left\langle \left( \left[ \left( 1 - \prod _^(1 - (\underline})^q)^ \right) ^, \left( 1 - \prod _^(1 - (\overline})^q)^\right) ^ \right] , 1 - \prod _^(1 - p_)^ \right) , \right. \\&\quad \left. \left( \left[ \prod _^ (\underline})^, \prod _^ (\overline})^ \right] , \prod _^ \left( q_\right) ^ \right) \right\rangle . \end \end$$

(4.8)

The proof of Theorem 4.6 and the proofs of the properties of the GOFWA operator are given in the appendix.

Furthermore, the distance measure plays a fundamental role in the PUIL q-ROFSs-VIKOR approach, providing a crucial tool to quantify the disparity between the two assessments.

A relationship is established between a PUIL q-ROFS and an interval vector, enabling the calculation of distances between PUIL q-ROFSs through interval numbers. This process involves transforming PUIL q-ROFSs into a series of interval vectors and subsequently computing the distances between these interval vectors based on interval numbers.

Definition 4.7

Let \(A_\) and \(A_\) be two Probabilistic Unbalanced Interval Linguistic q-Rung Orthopair Fuzzy Sets (PUIL q-ROFSs) defined as:

\(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}\),

\(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}\).

The distance between \(A_\) and \(A_\) is given by:

$$\begin D(A_,A_)=\sqrt(D_^+D_^+D_^+D_D_+D_D_)} \end$$

(4.9)

where \(D_\) represents the distance between the degrees of interval membership, \(D_\) denotes the distance between the degrees of interval non-membership, \(D_\) indicates the distance between the degrees of interval hesitancy.

Proof

The distance between PUIL q-ROFSs \(A_1\) and \(A_2\) is calculated through the following systematic procedure:

Step 1: Vector Representation Transformation

Each PUIL q-ROFS is mapped to a three-dimensional interval vector, where the components encode the interval-valued membership, non-membership, and hesitancy degrees, respectively. The vector representations are formally defined as:

$$\begin \vec _1 = \begin \left[ \underline}, \overline}\right] \\ \left[ \underline}, \overline}\right] \\ \left[ \underline}, \overline}\right] \end, \quad \vec _2 = \begin \left[ \underline}, \overline}\right] \\ \left[ \underline}, \overline}\right] \\ \left[ \underline}, \overline}\right] \end. \end$$

(4.10)

This vector-space representation enables systematic component-wise distance analysis.

Step 2: Component Distance Computation

Following the interval number distance framework in [48], we define three signed distance components:

\(D_\): Signed distance between interval membership degrees

\(D_\): Signed distance between interval non-membership degrees

\(D_\): Signed distance between interval hesitancy degrees

The squared magnitude for each distance component is computed as:

$$\begin \begin D_^(A_,A_) =&\left[ \left( \frac}})^ + (\overline}})^} \right) p_} - \left( \frac}})^ + (\overline}})^} \right) p_} \right] ^ \\&+ \frac \left[ \left( \frac}})^ - (\underline}})^} \right) p_} - \left( \frac}})^ - (\underline}})^} \right) p_} \right] ^, \end \end$$

(4.11)

$$\begin \begin D_^(A_,A_) =&\left[ \left( \frac}})^ + (\overline}})^} \right) p_} - \left( \frac}})^ + (\overline}})^} \right) p_} \right] ^ \\&+ \frac \left[ \left( \frac}})^ - (\underline}})^} \right) p_} - \left( \frac}})^ - (\underline}})^} \right) p_} \right] ^, \end \end$$

(4.12)

$$\begin \begin D_^(A_,A_) =&\left[ \frac}})^ + (\overline}})^} - \frac}})^ + (\overline}})^} \right] ^ \\&+ \frac \left[ \frac}})^ - (\underline}})^} - \frac}})^ - (\underline}})^} \right] ^. \end \end$$

(4.13)

The sign of each distance component is determined by the relative magnitude of the weighted central values:

$$\begin \begin D_&= \sqrt^} & \text \left( (\underline}})^ + (\overline}})^\right) p_} \ge \left( (\underline}})^ + (\overline}})^\right) p_} \\ -\sqrt^} & \text \end\right. } \\ D_&= \sqrt^} & \text \left( (\underline}})^ + (\overline}})^\right) p_} \ge \left( (\underline}})^ + (\overline}})^\right) p_} \\ -\sqrt^} & \text \end\right. } \\ D_&= \sqrt^} & \text (\underline}})^ + (\overline}})^ \ge (\underline}})^ + (\overline}})^ \\ -\sqrt^} & \text \end\right. } \end \end$$

(4.14)

Step 3: Final Distance Aggregation

The overall distance between \(A_1\) and \(A_2\) is obtained by synthesizing the signed component distances (4.9).

This formulation ensures a comprehensive distance measure that accounts for interactions among all uncertainty components. \(\square\)

The distance measure between interval vectors consists of the distance between each pair of corresponding elements in two vectors. The elements of interval vectors are interval numbers, hence the distance between elements \(D_\), \(D_\), \(D_\) can be obtained by means of distance of interval numbers.

The distance between interval vectors is computed by evaluating the distances between corresponding elements in each pair of vectors. Since these elements are interval numbers, distances \(D_\), \(D_\) and \(D_\) are derived using the established interval number distance metrics.

Proposition 4.8

Let \(A_\) and \(A_\) be two PUIL q-ROFSs defined as:

\(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}\),

\(A_=\}},\overline}}],p_}),([\underline}},\overline}}],q_})\rangle \}\). the following properties are fulfilled:

The distance measure \(D(A_,A_)\) defined in (4.9) satisfies the following properties:

1.

Non-negativity: \(D(A_,A_)\ge 0\).

2.

Symmetry: \(D(A_,A_)=D(A_,A_)\).

3.

Identity of Indiscernibles: \(D(A_,A_)=0\) if and only if \(A_=A_\).

Proof

Proof of Property (1): Non-negativity

Recall the distance formula:

$$\begin D(A_, A_) = \sqrt\left( D_^ + D_^ + D_^ + D_D_ + D_D_\right) }. \end$$

We demonstrate that the radicand is non-negative through algebraic manipulation:

$$\begin \begin&\frac\left( D_^ + D_^ + D_^ + D_D_ + D_D_\right) \\&= 2\left( \fracD_ + \fracD_\right) ^ + 2\left( \fracD_ + \fracD_\right) ^. \end \end$$

Since the right-hand side of equation (2) represents a sum of squared terms with positive coefficients, we have:

$$\begin 2\left( \fracD_ + \fracD_\right) ^ + 2\left( \fracD_ + \fracD_\right) ^ \ge 0. \end$$

Therefore, the radicand in equation (1) is non-negative, which implies \(D(A_, A_) \ge 0\).

Proof of Property (2): Symmetry

The component distances exhibit anti-symmetric behavior:

$$\begin \begin D_(A_, A_)&= -D_(A_, A_), \\ D_(A_, A_)&= -D_(A_, A_), \\ D_(A_, A_)&= -D_(A_, A_). \end \end$$

Examining the distance formula components:

$$\begin \begin&D_^(A_, A_) = D_^(A_, A_), \\&D_^(A_, A_) = D_^(A_, A_), \\&D_^(A_, A_) = D_^(A_, A_) ,\\&D_(A_, A_)D_(A_, A_) = D_(A_, A_)D_(A_, A_), \\&D_(A_, A_)D_(A_, A_) = D_(A_, A_)D_(A_, A_). \end \end$$

All components in the distance formula remain invariant when swapping \(A_\) and \(A_\), hence:

$$\begin D(A_, A_) = D(A_, A_). \end$$

Proof of Property (3): Identity of Indiscernibles

We prove both directions of the biconditional statement.

(\(\Rightarrow\)) Assume \(A_ = A_\). Then all corresponding components are identical:

$$\begin \begin&\left[ \underline}},\overline}}\right] = \left[ \underline}},\overline}}\right] , \quad p_} = p_} ,\&\left[ \underline}},\overline}}\right] = \left[ \underline}},\overline}}\right] , \quad q_} = q_}. \end \end$$

This implies:

$$\begin D_ = D_ = D_ = 0 \end$$

Substituting into the distance formula yields:

$$\begin D(A_, A_) = \sqrt\left( 0 + 0 + 0 + 0 + 0\right) } = 0 \end$$

(\(\Leftarrow\)) Assume \(D(A_, A_) = 0\). From equation (1), this requires:

$$\begin \frac\left( D_^ + D_^ + D_^ + D_D_ + D_D_\right) = 0 \end$$

Using the equivalent form from equation (2):

$$\begin 2\left( \fracD_ + \fracD_\right) ^ + 2\left( \fracD_ + \fracD_\right) ^ = 0 \end$$

Since both terms are non-negative, we must have:

$$\begin \fracD_ + \fracD_ = 0 \quad \text \quad \fracD_ + \fracD_ = 0 \end$$

Solving this system gives:

$$\begin D_ = D_ = D_ = 0 \end$$

Therefore, all corresponding components of \(A_\) and \(A_\) are identical, implying \(A_ = A_\). \(\square\)