EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6891 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Robust Decision-Making Framework for Recruitment Optimization via Hybrid Dombi–Archimedean Operators in Complex q-Rung Orthopair Fuzzy Sets Ikhtesham Ullah1, Kamran2,∗, Muhammad Sajjad Ali Khan3,4, Fawad Hussain1, Madad Khan5, Ioan-Lucian Popa6,7,∗ 1 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, KP, Pakistan 2 Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan 3 Department of Mathematics, Khushal Khan Khattak University, Karak 27200, Pakistan 4 Department of Mathematics, Northern University Nowshera, 24110, Khyber Pakhtunkhwa, Pakistan 5 Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan 6 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania 7 Faculty of Mathematics and Computer Science, Transilvania University of Brașov, Iuliu Maniu Street 50, 500091 Brașov, Romania Abstract. Complex q-Rung Orthopair Fuzzy Sets (Cq-ROFSs) are a powerful tool for handling two-dimensional, periodic uncertain information. The Dombi operator provides a highly flexible parameterized framework for combin- ing fuzzy information through smooth and adjustable t-norm and t-conorm functions. Similarly, the Archimedean operator offers a generalized and consistent mechanism for modeling nonlinear interactions in fuzzy aggregation processes. This paper introduces a novel hybrid approach by combining the Dombi and Archimedean t-norm and t-conorm operations under the Cq-ROF environment. Based on these new operational laws, we develop a suite of ag- gregation operators: the complex q-Rung orthopair fuzzy Dombi-Archimedean weighted averaging (Cq-ROFDAWA) operator, the complex q-Rung orthopair fuzzy Dombi-Archimedean ordered weighted averaging (Cq-ROFDAOWA) operator, the complex q-Rung orthopair fuzzy Dombi-Archimedean weighted geometric (Cq-ROFDAWG) operator, and the complex q-Rung orthopair fuzzy Dombi-Archimedean ordered weighted geometric (Cq-ROFDAOWG) op- erator. Key properties of these operators, including idempotency, monotonicity, and boundedness, are rigorously investigated. Furthermore, a robust Multi-Criteria Decision-Making (MCDM) framework is established based on the proposed operators. The model’s reliability and consistency are demonstrated through two distinct scenarios: one with fully unknown criteria weights and another with partially known weights. A numerical example concerning human resource selection is presented to validate the method’s effectiveness and applicability. Finally, a comparative analysis with existing methods underscores the advantages and superiority of the proposed approach. 2020 Mathematics Subject Classifications: 03E72, 68T37 ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6891 Email addresses: ikhteshamullah757@gmail.com (I. Ullah), sajjadalimath@yahoo.com (M. S. A. Khan), kamran.maths@icp.edu.pk (Kamran), hussain@aust.edu.pk (F. Hussain), madadmath@yahoo.com (M. Khan), lucian.popa@uab.ro (I.-L. Popa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 2 of 73 Key Words and Phrases: Intuitionistic fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, complex intuitionistic fuzzy set, complex q-rung orthopair fuzzy sets, Dombi-Archimedean operator, multi-criteria decision making 1. Introduction Uncertainty is an inherent part of real-world decision-making. To model this un- certainty, Zadeh introduced Fuzzy Sets (FSs) in 1965 [1], a revolutionary concept where elements have degrees of membership in the interval [0,1]. This framework has been exten- sively applied in fields such as linguistics [2], decision-making [3], and data clustering [4]. Subsequent extensions of FSs have been developed to capture more complex forms of un- certainty. Atanassov’s Intuitionistic Fuzzy Sets (IFSs) [5] incorporated a non-membership degree, with the constraint that their sum is less than or equal to 1. The enhanced ability of IFSs to capture hesitation has enabled their effective application in image classification [6], medical diagnosis [7], environmental assessment [8], and MCDM [9]. Star coloring and its variants extend classical coloring by introducing additional structural constraints [10]. Yager further generalized this model with Pythagorean Fuzzy Sets (PFSs) [11], relaxing the condition to the sum of squares of membership and non-membership degrees being less than or equal to 1. PFSs have proven highly effective in areas like pattern recognition [12] and MCDM [13, 14]. The most general form in this hierarchy is the q-Rung Orthopair Fuzzy Set (q-ROFS) [15], which requires that the sum of the q-th powers of the membership and non-membership degrees is bounded by 1. A higher value of the parameter q provides greater flexibility in representing ambiguous and conflicting information. q-ROFSs have been effectively applied in areas such as image analysis [16], clinical decision support [17], engineering [18], and MCDM [19]. To handle two-dimensional, periodic information, the concept of a Complex Fuzzy Set (CFS) was proposed by Ramot et al. [20], where the membership degree is a complex- valued function within the unit disc. This was later extended to Complex Intuitionistic Fuzzy Sets (CIFSs) [21]. CIFSs have been effectively applied in areas such as image anal- ysis [22], clinical decision support [23], environmental assessment [24], and MCDM [25]. Complex Pythagorean Fuzzy Sets (CPFSs) [26]. However, CIFSs and CPFSs can fail to model certain decision scenarios where the information is highly inconsistent. For in- stance, with a membership grade of 0.9ei2π(0.8) and a non-membership grade of 0.8ei2π(0.9), the squared sum of the real parts exceeds 1, violating CPFS constraints. CPFSs have found valuable applications in computational intelligence [27], medical assessment [28], and MCDM [29]. To overcome this limitation, the Complex q-Rung Orthopair Fuzzy Set (Cq-ROFS) was introduced [30]. A Cq-ROFS imposes the conditions 0 ≤ (µ)q+(ν)q ≤ 1 and 0 ≤ (ℜ(µ))q+ (ℜ(ν))q ≤ 1 on the complex-valued membership and non-membership grades. For the previous example, setting q = 5 gives 0.95 + 0.85 = 0.91817, which satisfies the condition. This makes Cq-ROFS a significantly more powerful and versatile tool for capturing two- dimensional uncertainty in complex decision-making problems [31]. Cq-ROFSs have shown I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 3 of 73 effectiveness in domains including data analysis [32], medical evaluation [33], and MCDM [34]. Triangular norms (t-norms) and fuzzy operators play a vital role in defining aggre- gation and logical operations within fuzzy systems. Klement and Mesiar [35] provided a comprehensive study on the logical and algebraic properties of triangular norms, forming the foundation for many fuzzy aggregation techniques. Dombi [36] further generalized these operators through the De Morgan class, establishing a broader structure for mea- suring fuzziness. Aczél and Alsina [37] explored quasilinear functions to characterize such operators, offering analytical tools for synthesizing judgments. The theoretical insights of Nguyen and Walker [38] helped formalize fuzzy logic principles that underpin most modern fuzzy decision models. These mathematical bases have been effectively applied in real-world decision problems such as personnel selection using intuitionistic fuzzy sets [39] and multi-criteria evaluation through ELECTRE-based approaches [40]. Together, these studies support the continued refinement of fuzzy aggregation and decision-making methods. A thorough analysis of the literature, summarized in Table 1, reveals that while aggregation operators based on Dombi or Archimedean operations have been studied inde- pendently in various fuzzy environments, their hybrid combination has not been explored within the Cq-ROFS context. This hybrid approach, leveraging the parameterized flexi- bility of Dombi operations and the generalizability of Archimedean operations, promises to create a more powerful and adaptable aggregation framework. The absence of such Dombi-Archimedean (DA) operators for Cq-ROFS means that a sig- nificant class of complex, two-dimensional decision-making problems remains unaddressed. This gap is the primary motivation for our work. By fusing these two powerful operational paradigms, we aim to develop a robust set of aggregation tools that can handle the intricate uncertainty captured by Cq-ROFSs more effectively than existing methods. Table 1: Summary of Aggregation Operators in Various Fuzzy Environments Fuzzy Environment Dombi AOs Archimedean AOs Dombi-Archimedean Intuitionistic FS (IFS) ✓ ✓ × Pythagorean FS (PFS) ✓ ✓ × q-Rung Orthopair FS (q-ROFS) ✓ ✓ ✓ Complex IFS (CIFS) ✓ × Complex PFS (CPFS) ✓ × Complex q-ROFS (Cq-ROFS) × This Paper ✓ ✓ Proposed Work The principal contributions of this paper are as follows: (i) To establish novel operational laws for Cq-ROFSs by hybridizing Dombi and Archimedean (DA) t-norms and t-conorms. (ii) To develop a new family of aggregation operators based on these DA operations: I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 4 of 73 • Complex q-Rung Orthopair Fuzzy Dombi-Archimedean Weighted Averaging (Cq-ROFDAWA) operator. • Complex q-Rung Orthopair Fuzzy Dombi-Archimedean Ordered Weighted Av- eraging (Cq-ROFDAOWA) operator. • Complex q-Rung Orthopair Fuzzy Dombi-Archimedean Weighted Geometric (Cq-ROFDAWG) operator. • Complex q-Rung Orthopair Fuzzy Dombi-Archimedean Ordered Weighted Ge- ometric (Cq-ROFDAOWG) operator. (iii) To investigate the essential mathematical properties of these operators, such as idem- potency, monotonicity, and boundedness. (iv) To design a comprehensive MCDM framework utilizing the proposed operators to solve complex decision-making problems. (v) To validate the proposed framework through a practical numerical example in human resource selection and to demonstrate its robustness and superiority via a compara- tive analysis with existing methods. The remainder of this paper is organized as follows. Section 2 covers the necessary preliminaries, including Cq-ROFSs, Dombi operations, and Archimedean t-norms and t-conorms. Section 3 introduces the new Dombi-Archimedean operational laws for Cq- ROFSs. Section 4 presents the proposed aggregation operators and explores their key properties. Section 5 outlines the MCDM algorithm based on these operators. A detailed case study on recruitment optimization is presented in Section 6, with implementation scenarios in same section . A sensitivity analysis is conducted in Section 7, followed by a comparative analysis in Section 8. A general discussion on comparative analysis is discussed in section 9. Finally, Section 10 concludes the paper and suggests directions for future research. 2. Preliminaries This section outlines the fundamental concepts necessary for understanding the sub- sequent developments in this work. We briefly revisit the definitions of Complex q-Rung Orthopair Fuzzy Sets (Cq-ROFSs), Archimedean operations, and Dombi operations, which form the foundational building blocks for the new methodologies introduced later. Definition 1. [28] A Cq −ROFS is defined as: D̃ = { ( x,M D̃ (x) , N D̃ (x) /x ∈ X ) } (1) Where M D̃ represent the MD and N D̃ denote the NMD and 0 ≤ |zq1|+ |zq2| ≤ 1. M D̃ (x) = µ D̃ (x) ei2π(ϕµ(x)), ND̃ (x) = ν D̃ (x) ei2π(ϕν(x)) must satisfying the condition 0 ≤ µq D̃ (x) + νq D̃ (x) ≤ 1 and 0 ≤ ϕq µ D̃ (x) + ϕq ν D̃ (x) ≤ 1. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 5 of 73 Definition 2. [24] The score function of CqORFS denoted by s̃ is defined as: s̃ = 1 2 |(µq − νq) + (ϕµq − ϕνq)| (2) Definition 3. [41]Let us pretend that there is a strictly decreasing function ∅ with the following properties: ∅(1) = 0 and that ∅ : [0, 1) is continuous. A purely Archimedean TN is expressed by the following equation: δ(ℏ, ℏ) = ∅−1 (∅(ℏ) + ∅(ℏ)) for ℏ, ℏ ∈ (0, 1] (3) Definition 4. [38] For any l ∈ [0, 1, let Θ(l) = Θ(1− l), and let Θ : [0, 1) → R denote a strictly rising and continuous function. A Archimedean TCN is expressed by the equation. ρ(ℏ, ℏ) = Θ−1 (Θ(ℏ) + Θ(ℏ)) for ℏ, ℏ ∈ (0, 1] (4) Definition 5. [36]The Dombi TN operator, Td(ℏ,ℏ), is defined as: Td(ℏ,ℏ) = 1 1 + 〈 ( 1−ℏ ℏ )k + ( 1−ℏ ℏ )k 〉 1 k (5) The degrees of MD of two fuzzy sets, ℏ and ℏ, are controlled by the variable k, which also indicates the degree of intersection. Definition 6. [36]The Dombi TCN operator, Dd(ℏ,ℏ), is defined as: Dd(ℏ,ℏ) = 1− 1 1 + 〈 ( ℏ 1−ℏ )k + ( ℏ 1−ℏ )k 〉 1 k (6) the degrees of MD of two fuzzy sets, ℏ and ℏ, and k is the degree of union of the controls. 3. DOMBI–ARCHIMEDEAN OPERATIONS ON Cq −ROFEs Recent analyses have combined Dombi TN and TCN operators with Archimedean TN and TCN operators to create a new class of operators known as Dombi-Archimedean operators. When examined inside the Cq −ROF framework, these operators have shown promising results in a number of fuzzy reasoning tasks. By combining the characteris- tics and properties of Dombi and Archimedean operations, Dombi-Archimedean operators provide a comprehensive approach to managing uncertainty and unpredictability in the context of Cq − ROF . These operators enhance thinking abilities and are helpful for a variety of fuzzy reasoning applications. Given the potential benefits and real-world appli- cations of this integrated method, Dombi-Archimedean operators are being studied in the context of Cq − ROF . Thus far, studies have shown the effectiveness of these operators in resolving Cq −ROFS fuzzy reasoning problems. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 6 of 73 Definition 7. Consider the functions: ℜk(ϑ) = ( ϑ 1−ϑ )k for ϑ ∈ [0, 1) and k ≥ 1, ℑk(ϑ ′) =( 1−ϑ′ ϑ′ )k for ϑ′ ∈ (0, 1]. For two Cq − ROFNs, denoted by ∆1 = (ℏ1, ℏ1, θ1, ϕ1) and ∆2 = (ℏ2, ℏ2, θ2, ϕ2), the Dombi-Archimedean operations are defined as follows. • ∆1 ⊕Dℏ∆2 = 〈 q √√√√√√√√1− 〈 1− 1+〈 Θ 〈 Θ−1(ℜk(ℏq1))+ Θ−1(ℜk(ℏq2)) 〉〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅ 〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉〉 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ 〈 Θ−1(ℜk(θ q 1))+ Θ−1(ℜk(θ q 2)) 〉〉 1 k 〉−1 q √√√√√√  1+〈 ∅ 〈 ∅−1(ℑk(ϕ q 1))+ ∅−1(ℑk(ϕ q 2)) 〉〉 1 k  −1〉 • ∆1 ⊗Dℏ∆2 = 〈 q √√√√√√ 〈 1+〈 ∅ 〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉〉 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ 〈 Θ−1(ℑk(ℏq1))+ Θ−1(ℑk(ℏq2)) 〉〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅ 〈 ∅−1(ℑk(ϕ q 1))+ ∅−1(ℑk(ϕ q 2)) 〉〉 1 k 〉−1 , q √√√√√√√√1−  1− 1+〈 Θ 〈 Θ−1(ℑk(θ q 1)) +Θ−1(ℑk(θ q 2)) 〉〉 1 k  −1〉 • λ∗Dℏ∆1 = 〈 q √√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(ℏq1)) 〉 〉 1 k 〉−1 , 1− q √√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ℏq1)) 〉 〉 1 k 〉−1 q √√√√√√√√ 〈 1− 1+〈 Θ〈 λΘ−1(ℜk(θ q 1)) 〉 〉 1 k 〉−1 , q √√√√√√  1+〈 ∅〈 λ∅−1(ℑk(ϕ q 1)) 〉 〉 1 k  −1〉 , (λ > 0) • λ◦Dℏ∆1 = 〈 q √√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ℏq1)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(ℏq1)) 〉 〉 1 k 〉−1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 7 of 73 q √√√√√〈 1+〈 ∅〈 λ∅−1(ℑk(ϕ q 1) 〉 〉) 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(θ q 1)) 〉 〉 1 k −1〉 〉 (λ > 0) Theorem 1. Let ∆1 = (ℏ1, ℏ1, θ1, ϕ1) and ∆2 = (ℏ2, ℏ2, θ2, ϕ2) be two Cq–ROFEs such that 0 ≤ ℏqi + ℏqi ≤ 1, 0 ≤ θqi + ϕq i ≤ 1, i = 1, 2. Then, for any λ > 0, the results of the Dombi–Archimedean operations ∆1 ⊕DA∆2, ∆1 ⊗DA∆2, λ ∗DA ∆1, and λ ◦DA ∆1 are also valid Cq–ROFEs. Where: ℏ represents the amplitude term of the membership degree, ℏ represents the amplitude term of the non-membership degree, θ represents the phase term of the membership degree, and ϕ represents the phase term of the non-membership degree. Proof. We prove that the Cq-ROFDAWA operator preserves the fundamental con- straints of Cq-ROFSs. Let ∆j = (ℏj , ℏj , θj , ϕj) r j=1 be a collection of Cq-ROFNs, which by definition satisfy: 0 ≤ ℏqj + ℏqj ≤ 1 0 ≤ θqj + ϕq j ≤ 1 for all j = 1, 2, . . . , r Let ∆ = (ℏ, ℏ, θ, ϕ) = Cq-ROFDAWA(∆1,∆2, . . . ,∆r) be the aggregated value. From the definition of Cq-ROFDAWA, we have: ℏq = 1− 〈 1 + 〈 Θ 〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉〉 1 k 〉−1 ℏq = 〈 1 + 〈 ∅ 〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉〉 1 k 〉−1 Step 1: Non-negativity and Boundedness of Individual Terms First, we show that 0 ≤ ℏq ≤ 1 and 0 ≤ ℏq ≤ 1. Since ℏqj ∈ [0, 1] for all j, and ℜk(ϑ) = ( ϑ 1−ϑ )k is a strictly increasing function mapping [0, 1) to [0,∞), we have ℜk(ℏqj) ≥ 0. The function Θ and its inverse Θ−1 preserve non-negativity due to their Archimedean properties. Therefore: A = ∑r j=1 ωjΘ −1(ℜk(ℏqj)) ≥ 0 Θ(A) ≥ 0 ⟨Θ(A)⟩ 1 k ≥ 0 1 + ⟨Θ(A)⟩ 1 k ≥ 1〈 1 + ⟨Θ(A)⟩ 1 k 〉−1 ∈ (0, 1] I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 8 of 73 ℏq = 1− 〈 1 + ⟨Θ(A)⟩ 1 k 〉−1 ∈ [0, 1) Similarly, for ℏq, since ℏqj ∈ [0, 1] and ℑk(ϑ ′) = ( 1−ϑ′ ϑ′ )k maps (0, 1] to [0,∞), we have: B = r∑ j=1 ωj∅−1(ℑk(ℏqj)) ≥ 0∅(B) ≥ 0 ⟨∅(B)⟩ 1 k ≥ 01+⟨∅(B)⟩ 1 k ≥ 1ℏq = 〈 1 + ⟨∅(B)⟩ 1 k 〉−1 ∈ (0, 1] Step 2: Sum Constraint for Amplitude Terms Now we prove the critical constraint: 0 ≤ ℏq + ℏq ≤ 1. From the Dombi-Archimedean structure with the specific function forms Θ(t) = ( 1−t t )k and ∅(t) = ( t 1−t )k , we derive: Θ−1(ℜk(ℏqj)) = Θ−1 ( ℏqj 1− ℏqj )k  = 1− ℏqj∅ −1(ℑk(ℏqj)) = ∅−1 (1− ℏqj ℏqj )k  = 1− ℏqj Therefore: A = r∑ j=1 ωj(1− ℏqj)B = r∑ j=1 ωj(1− ℏqj) Now, using the Dombi function properties: ℏq = 1− 1 1 + ( 1−A A ) = 1−Aℏq = 1 1 + ( B 1−B ) = 1−B Thus: ℏq + ℏq = (1−A) + (1−B) = 2− (A+B) = 2− r∑ j=1 ωj(2− (ℏqj + ℏqj)) Since each ∆j is a valid Cq-ROFN, we have ℏqj + ℏqj ≤ 1, which implies 2− (ℏqj + ℏqj) ≥ 1. Therefore: A+B = r∑ j=1 ωj(2− (ℏqj + ℏqj)) ≥ r∑ j=1 ωj = 1ℏq + ℏq = 2− (A+B) ≤ 1 The non-negativity ℏq + ℏq ≥ 0 follows from ℏq, ℏq ≥ 0. Step 3: Phase Terms Con- straint The proof for phase terms (θ, ϕ) follows identically to the amplitude terms, since the aggregation operators for phase terms have the same functional form and the input phase terms satisfy the same constraints 0 ≤ θqj +ϕq j ≤ 1. Therefore, the aggregated value ∆ = Cq-ROFDAWA(∆1,∆2, . . . ,∆r) satisfies both Cq-ROFS constraints: (i) 0 ≤ ℏq + ℏq ≤ 1 (ii) 0 ≤ θq + ϕq ≤ 1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 9 of 73 This completes the proof that the Cq-ROFDAWA operator is closed under the Cq-ROFS domain. Example 1. Given two Cq−ROFNs on U : ∆1 = (0.6, 0.7), (0.7, 0.6), ∆2 = (0.4, 0.8), (0.6, 0.5), and parameters q = 4, λ = 0.3, k = 2, and functions: Θ(t) = − ln(1−t), Θ−1(t) = 1−e−t, ∅(t) = −i ln(t), ∅−1(t) = e−t. The expressions are then defined accordingly. • ∆1 ⊕Dℏ∆2 = 〈 q √√√√√√√√1− 〈 1− 1+〈 Θ 〈 Θ−1(ℜk(ℏq1))+ Θ−1(ℜk(ℏq2)) 〉〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅ 〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉〉 1 k 〉−1 • q √√√√√√√√ 1−〈 1+〈 Θ 〈 Θ−1(ℜk(θ q 1))+ Θ−1(ℜk(θ q 2)) 〉〉 1 k 〉−1 q √√√√√√  1+〈 ∅ 〈 ∅−1(ℑk(ϕ q 1))+ ∅−1(ℑk(ϕ q 2)) 〉〉 1 k  −1〉 ∆1 ⊕Dℏ∆2 = 〈 q √√√√√√√√√√√√ 1− 1+〈 Θ 〈 1− e − ( ℏq1 1−ℏq1 )k + 1− e − ( ℏq2 1−ℏq2 )k 〉〉 1 k  −1 , q √√√√√√√√√√  1+〈 ∅ 〈 1− e − ( 1−ℏq1 ℏq1 )k + 1− e − ( ℏq2 1−ℏq2 )k 〉〉 1 k  −1 q √√√√√√√√√√√√ 1− 1+〈 Θ 〈 1− e − ( θ q 1 1−θ q 1 )k + 1− e − ( θ q 2 1−θ q 2 )k 〉〉 1 k  −1 , q √√√√√√√√√√  1+〈 ∅ 〈 1− e − ( 1−ϕ q 1 ϕ q 1 )k + 1− e − ( ϕ q 2 1−ϕ q 2 )k 〉〉 1 k  −1 Placing the values yields ={(0.0629, 0.9743) , (0.0267, 0.9977)} • ∆1 ⊗Dℏ∆2 = 〈 q √√√√√√ 〈 1+〈 ∅ 〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉〉 1 k 〉−1 , q √√√√√√ 1−〈 1 + 〈 Θ 〈 Θ−1(ℜk(ℏq1))+ Θ−1(ℜk(ℏq2)) 〉〉 1 k 〉−1 q √√√√√√ 〈 1+〈 ∅ 〈 ∅−1(ℑk(ϕq 1))+ ∅−1(ℑk(ϕq 2)) 〉〉 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ 〈 Θ−1(ℜk(θ q 1))+ Θ−1(ℜk(θ q 2)) 〉〉 1 k 〉−1 〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 10 of 73 On substituting the numerical values ∆1 ⊗Dℏ∆2 = {(0.9370, 0.0256) , (0.9732, 0.0022)} • λ ∗Dℏ ∆1 = 〈 q √√√√√√ 1−〈 1+〈 Θ 〈 λΘ−1(ℜk(ℏq1)) 〉〉 1 k 〉−1 , , q √√√√〈 1+〈 ∅ 〈 λ∅−1(ℑk(ℏq1)) 〉〉 1 k 〉−1 q √√√√√√ 1−〈 1+〈 Θ 〈 λΘ−1(ℜk(θ q 1)) 〉〉 1 k 〉−1 , , q √√√√〈 1+〈 ∅ 〈 λ∅−1(ℑk(ϕ q 1)) 〉〉 1 k 〉−1 〉 (λ > 0) Substituting the values ={(0.5595, 0.4407) , (0.5592, 0.4409)} • λ◦Dℏ∆1 = 〈 q √√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ℏq1)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(ℏq1)) 〉 〉 1 k 〉−1 q √√√√√〈 1+〈 ∅〈 λ∅−1(ℑk(ϕ q 1) 〉 〉) 1 k 〉−1 , q √√√√√√√√ 1−〈 1+〈 Θ〈( λΘ−1(ℜk(θ q 1)) )〉 〉 1 k 〉−1 〉 Substituting the values ={(0.4404, 0.5592) , (0.4409, 0.5593)} Theorem 2. Let ∆1 = (ℏ1, ℏ1, θ1, ϕ1) and ∆2 = (ℏ2, ℏ2, θ2, ϕ2)be the Cq−ROFEs,where ℏ1,2, ℏ1,2 denotes the amplitude terms such that ℏ1,2 stand for MD while the ℏ1,2 stand for NMD and θ1,2, ϕ1,2 represent the phase term such that the θ1,2 denote the MD while ϕ1,2 is for NMD with λ > 0, λ1 > 0, λ3 > 0 (i) ∆1 ⊕DA∆2 = ∆2 ⊕DA∆1 (ii) ∆1 ⊗DA∆2 = ∆2 ⊗DA∆1 (iii) λ ∗DA (∆1 ⊕DA∆2) = ( λ ∗DA ∆1 )⊕DA ( λ ∗DA βh ζ2 ) (iv) λ ◦DA (∆1 ⊕DA∆2) = ( λ ◦DA ∆1 )⊗DA ( λ ◦DA βh ζ2 ) (v) (λ1 + λ2) ∗DA ∆1 = ( λ1 ∗DA ∆1 )⊕DA ( λ2 ∗DA βh ζ2 ) I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 11 of 73 (vi) (λ1 + λ2) ∗DA ∆1 = ( λ1 ∗DA ∆1 )⊗DA ( λ2 ∗DA βh ζ2 ) Proof. The proof of (i) and (ii).is not nessecary. (iii) λ ∗Dℏ (∆1 ⊕Dℏ∆2) = λ∗Dℏ 〈 q √√√√√√√√√√ 1− 〈 1+〈 Θ〈 Θ−1(ℜk(ℏq1))+ Θ−1(ℜk(ℏq2)) 〉k 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉k 〉 1 k 〉−1 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 Θ−1(ℜk(θ q 1))+ Θ−1(ℜk(θ q 2)) 〉 〉k 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 ∅−1(ℑk(ϕ q 1))+ ∅−1(ℑk(ϕ q 2)) 〉 〉k 1 k 〉−1〉 = 〈 q √√√√√√√√√√√√ 1− 〈 1+ Θ〈 λ〈 Θ−1(ℜk(ℏq1))+ Θ−1(ℜk(ℏq2)) 〉k 〉 1 k 〉−1 , q √√√√√√√√√√ 〈 1+ 〈 ∅〈 λ〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉k 〉 〉 1 k 〉 −1 q √√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 λ〈 Θ−1(ℜk(θ q 1))+ Θ−1(ℜk(θ q 2)) 〉k 〉 〉 1 k 〉 −1 , q √√√√√√√√√√ 〈 1+ 〈 ∅〈 λ〈 ∅−1(ℑk(ϕ q 1))+ ∅−1(ℑk(ϕ q 2)) 〉k 〉 〉 1 k 〉 −1〉 and also, = λ ∗Dℏ (∆1 ⊕Dℏ∆2) = ( λ ∗Dℏ ∆2 )⊕Dℏ (λ ∗Dℏ ∆2 ) = 〈 q √√√√√√√√√ 1−〈 1+〈 Θ〈 λ1Θ −1(ℜk(ℏq1)) 〉k 〉 1 k 〉−1 , q √√√√√√√ 〈 1+〈 ∅〈 λ1∅−1(ℑk(ℏq1)) 〉k 〉 1 k 〉−1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 12 of 73 q √√√√√√√ 1−〈 1+〈 Θ 〈 λ1Θ −1(ℜk(θ q 1)) 〉k〉 1 k 〉−1 , q √√√√√√√ 〈 1+〈 ∅〈 λ1∅−1(ℑk(ϕ q 1)) 〉k 〉 1 k 〉−1〉 ⊕Dℏ 〈 q √√√√√√√√√ 1−〈 1+〈 Θ〈 λ2Θ −1(ℜk(ℏq2)) 〉k 〉 1 k 〉−1 , q √√√√√√√ 〈 1+〈 ∅〈 λ2∅−1(ℑk(ℏq2)) 〉k 〉 1 k 〉−1 q √√√√√√√√√ 1−〈 1+〈 Θ〈 λ2Θ −1(ℜk(θ q 2)) 〉k 〉 1 k 〉−1 , q √√√√√√√ 〈 1+〈 ∅〈 λ2∅−1(ℑk(ϕ q 2)) 〉k 〉 1 k 〉−1〉 = 〈 q √√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 λ〈 Θ−1(ℜk(ℏq1))+ Θ−1(ℜk(ℏq2)) 〉k 〉 〉 1 k 〉 −1 , q √√√√√√√√√√ 〈 1+ 〈 ∅〈 λ〈 ∅−1(ℑk(ℏq1))+ ∅−1(ℑk(ℏq2)) 〉k 〉 〉 1 k 〉 −1 q √√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 λ〈 Θ−1(ℜk(θ q 1))+ Θ−1(ℜk(θ q 2)) 〉k 〉 〉 1 k 〉 −1 , q √√√√√√√√√√ 〈 1+ 〈 ∅〈 λ〈 ∅−1(ℑk(ϕ q 1))+ ∅−1(ℑk(ϕ q 2)) 〉k 〉 〉 1 k 〉 −1〉 Getting, λ ∗Dℏ (∆1 ⊕Dℏ∆2) = ( λ ∗Dℏ ∆1 )⊕Dℏ (λ ∗Dℏ ∆2 ) (iv) See the (iii), (v) From (i) and (iii), we have, (λ1 + λ2) ∗Dℏ ∆1 =〈 q √√√√√√√√ 1−〈 1+〈 Θ〈 (λ1 + λ2)Θ −1(ℜk(ℏq1)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 (λ1 + λ2) ∅−1(ℑk(ℏq1)) 〉 〉 1 k 〉−1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 13 of 73 q √√√√√√√√ 1−〈 1+〈 Θ〈 (λ1 + λ2)Θ −1(ℜk(θ q 1)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 (λ1 + λ2) ∅−1(ℑk(ϕ q 1)) 〉 〉 1 k 〉−1〉 and also( λ ∗Dℏ ∆1 )⊕Dℏ (λ ∗Dℏ ∆2 ) 〈 q √√√√√√√√ 1−〈 1+〈 Θ〈 λ1Θ −1(ℜk(ℏq1)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 λ1∅−1(ℑk(ℏq1)) 〉 〉 1 k 〉−1 q √√√√√√√√ 1−〈 1+〈 Θ〈 λ1Θ −1(ℜk(θ q 1)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 λ1∅−1(ℑk(ϕ q 1)) 〉 〉 1 k 〉−1〉 ⊕Dℏ = 〈 q √√√√√√√√ 1−〈 1+〈 Θ〈 λ2Θ −1(ℜk(ℏq2)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 λ2∅−1(ℑk(ℏq2)) 〉 〉 1 k 〉−1 q √√√√√√√√ 1−〈 1+〈 Θ〈 λ2Θ −1(ℜk(θ q 2)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 λ2∅−1(ℑk(ϕ q 2)) 〉 〉 1 k 〉−1〉 = 〈 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(ℏq1))+ λΘ−1(ℜk(ℏq2)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ℏq1))+ λ∅−1(ℑk(ℏq2)) 〉 〉 1 k 〉−1 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(θ q 1))+ λΘ−1(ℜk(θ q 2)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ϕ q 1))+ λ∅−1(ℑk(ϕ q 2)) 〉 〉 1 k 〉−1〉 Getting (λ1 + λ2) ∗Dℏ ∆1 = ( λ1 ∗Dℏ ∆1 )⊕Dℏ ( λ2 ∗Dℏ βh ζ2 ) (vi) See (v) I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 14 of 73 4. Cq −ROF Dombi–Archimedean Weighted Aggregation operators (Cq −ROFDAWA) In order to aggregate Cq−ROF information, the authors provide Cq−ROF operators that combine Dombi-Archimedean procedures with weighting systems. Weighted aggrega- tion taking element significance into account is made possible by the Cq − ROF Dombi- Archimedean weighted average Cq − ROFDAWA operator and the Cq-ROF Dombi- Archimedean weighted geometric CqROFDAWG operator. A more sophisticated depiction of preferences is provided by an alternative version, the Cq − ROF Dombi-Archimedean ordered weighted average Cq − ROFDAOWA, and the Cq − ROFDAOWG operator, which takes item order into account. In the Cq −ROF setting, these variants allow for a range of decision-making situations that are sensitive to ambiguity and uncertainty. The operators make the Cq − ROF and Dombi-Archimedean framework more flexible, which helps practitioners with complicated decision-making problems and gives them accurate Cq −ROF data representations. Definition 8. Let ∆j = (ℏj , ℏj , θj , ϕj) be the Cq − ROFEs,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then the mapping ROFDAWA) is as follows:Cq −ROFNsU −→ Cq −ROFNsU and as Cq −ROFDAWA(∆1,∆2, ...,∆r) = rDℏ⊕ j=1 ( ωj ∗Dℏ ∆j ) (7) Here ωj(j=1,2,...,r) represents the weight of ∆j with the condition as ωj > 0 Theorem 3. Let ∆j = (ℏj , ℏj , θj , ϕj) be the Cq − ROFEs,where ℏj , ℏj denotes the am- plitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then the aggregated value of Cq −ROFDAWA(∆1,∆2, ...,∆r) as: Cq −ROFDAWA(∆1,∆2, ...,∆r) = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 15 of 73 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 (8) As q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 called amplitude term , and q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1 rep- resent a phase term and also ωj(j=1,2,...,r) denotes the weights of ∆j with conditions and r∑ j=1 ωj = 1. Here ωj(j=1,2,...,r) denoted weights of ∆j with the conditions and r∑ j=1 ωj = 1. Proof. By mathematical induction, when n = 1, it is true, for n = 2, Cq −ROFDAWA(∆1,∆2) = (ω1 ∗Dℏ ∆1) ⊕Dℏ(ω2 ∗Dℏ ∆2)〈 q √√√√√√√√ 1−〈 1+〈 Θ( ω1Θ −1(ℜk(ℏq1)) ) 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅( ω1∅−1(ℑk(ℏq1)) ) 〉 1 k 〉−1 q √√√√√√√√ 1−〈 1+〈 Θ( ω1Θ −1(ℜk(θ q 1)) ) 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅( ω1∅−1(ℑk(ϕ q 1)) ) 〉 1 k 〉−1〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 16 of 73⊕Dℏ = 〈 q √√√√√√√√ 1−〈 1+〈 Θ( ω2Θ −1(ℜk(ℏq2)) ) 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅( ω2∅−1(ℑk(ℏq2)) ) 〉 1 k 〉−1 q √√√√√√√√ 1−〈 1+〈 Θ( ω2Θ −1(ℜk(θ q 2)) ) 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅( ω2∅−1(ℑk(ϕ q 2)) ) 〉 1 k 〉−1〉 or = 〈 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 ω1Θ −1(ℜk(ℏq1))+ ω2Θ −1(ℜk(ℏq2)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 ω1∅−1(ℑk(ℏq1))+ ω2∅−1(ℑk(ℏq2)) 〉 〉 1 k 〉−1 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 ω1Θ −1(ℜk(θ q 1))+ ω2Θ −1(ℜk(θ q 2)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 ω1∅−1(ℑk(ϕ q 1))+ ω2∅−1(ℑk(ϕ q 2)) 〉 〉 1 k 〉−1〉 = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 2∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 2∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 2∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 2∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 getting, Eq.(4.2) is valid.for n=2 Assume for the moment that n = r is the case and Equation (4.2) is valid. Then, Cq −ROFDAWA(∆1,∆2, ...,∆r) = I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 17 of 73 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 We now have n = r + 1 for which Cq −ROFDAWA(∆1,∆2, ...,∆r+1) = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅−1〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅−1〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 ⊕Dℏ(ωk+1 ∗Dℏ ∆r+1) resulting = 〈 q √√√√√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 +〈 Θ( ωr+1Θ −1(ℜk(ℏqr+1)) ) 〉 〉 1 k 〉 −1 , q √√√√√√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k + 〈 ∅( ωr+1∅−1(ℑk(ℏqr+1)) ) 〉 〉 −1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 18 of 73 q √√√√√√√√√√√√√√√ 1− 〈 1+ 〈 Θ( r+1∑ j=1 ωjΘ −1(ℜk(θ q j )) ) +〈 Θ( ωr+1Θ −1(ℜk(θ q r+1)) ) 〉 〉 1 k 〉 −1 , q √√√√√√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k + 〈 ∅( ωr+1∅−1(ℑk(ϕ q r+1)) ) 〉 〉 −1 〉 = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 So, for n = r + 1, Equation (4.2) is still valid. So, all naturally occurring numbers n are covered by Equation (4.2). Theorem 4. Let ∆j = (ℏj , ℏj , θj , ϕj) be the Cq−ROFEs,where ℏj , ℏj denotes the ampli- tude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD with ∆o ∈Cq−ROFNs. Then Cq −ROFDAWA(∆0 DA⊕ ∆r) = ∆0 DA⊕ Cq −ROFDAWA (∆1,∆2, ...,∆r) (9) Proof. we know, ∆0 ⊕Dℏ∆r = 〈 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(ℏq0))+ λΘ−1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ℏq0))+ λ∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√ 1−〈 1+〈 Θ〈 λΘ−1(ℜk(θ q 0))+ λΘ−1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅〈 λ∅−1(ℑk(ϕ q 0))+ λ∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 19 of 73 = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1 〈 (ℜk(ℏq0))+ Θ−1(ℜk(ℏqj)) 〉〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1 〈 (ℑk(ℏq0))+ ∅−1(ℑk(ℏqj)) 〉〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1 〈 (ℜk(θ q 0))+ Θ−1(ℜk(θ q j )) 〉〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1 〈 (ℑk(ϕ q 0))+ ∅−1(ℑk(ϕ q j)) 〉〉 〉 1 k 〉−1〉 = 〈 q √√√√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 Θ−1〈 (ℜk(ℏq0))+ r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 〉 1 k 〉 −1 , q √√√√√√√√√√√√ 〈 1+ 〈 ∅〈 ∅−1〈 (ℑk(ℏq0))+ r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 〉 1 k 〉 −1 q √√√√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 Θ−1〈 (ℜk(θ q 0))+ r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 〉 1 k 〉 −1 , q √√√√√√√√√√√√ 〈 1+ 〈 ∅〈 ∅−1〈 (ℑk(ϕ q 0))+ r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 〉 1 k 〉 −1 〉 and also ∆0, ⊕DℏCq −ROFDAWA(∆1,∆2, ...,∆r) = ⊕Dℏ 〈 q √√√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 Θ−1〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 〉 1 k 〉 −1 , q √√√√√√√√√√√ 〈 1+ 〈 ∅〈 ∅−1〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 〉 1 k 〉 −1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 20 of 73 q √√√√√√√√√√√√√ 1− 〈 1+ 〈 Θ〈 Θ−1〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 〉 1 k 〉 −1 , q √√√√√√√√√√√ 〈 1+ 〈 ∅〈 ∅−1〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 〉 1 k 〉 −1〉 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1 〈 (ℜk(ℏq0))+ Θ−1(ℜk(ℏqj)) 〉〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1 〈 (ℑk(ℏq0))+ ∅−1(ℑk(ℏqj)) 〉〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1 〈 (ℜk(θ q 0))+ Θ−1(ℜk(θ q j )) 〉〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1 〈 (ℑk(ℏq0))+ ∅−1(ℑk(ϕ q j)) 〉〉 〉 1 k 〉−1〉 Getting Cq −ROFDAWA(∆0 ⊕DA∆1,∆0 ⊕DA∆2, ...,∆0 ⊕DA∆r) = ∆0 ⊕DACq −ROFDAWA (∆1,∆2, ...,∆r) Theorem 5. Let ∆j = (ℏj , ℏj , θj , ϕj) be the Cq−ROFEs,where ℏj , ℏj denotes the ampli- tude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD with ∆0 ∈Cq−ROFNs. ∆o = ∆r. Then Cq −ROFDAWA (∆1,∆2, ...,∆r) = ∆o (10) Proof. We known Cq −ROFDAWA (∆1,∆2, ...,∆r) = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√ 〈 1+〈 ∅ r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 1 k 〉−1〉 and I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 21 of 73 = 〈 q √√√√√√√√√ 1− 1+{ Θ (( r∑ j=1 ωjΘ −1(ℜk(ℏq0)) ))} 1 k  −1 , q √√√√√√√  1+{ ∅ (( r∑ j=1 ωj∅−1(ℑk(ℏq0)) ))} 1 k  −1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q 0)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q 0)) 〉 〉 1 k 〉−1〉 = 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 Θ−1(ℜk(ℏq0)) r∑ j=1 ωj 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 ∅−1(ℑk(ℏq0)) r∑ j=1 ωj 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 Θ−1(ℜk(θ q 0)) r∑ j=1 ωj 〉 〉 1 k 〉 −1 , q √√√√√√√√√ 〈 1+〈 ∅〈 ∅−1(ℑk(ϕ q 0)) r∑ j=1 ωj 〉 〉 1 k 〉−1〉 = 〈 q √√√√√√ 1−( 1+{ Θ (( Θ−1(ℜk(ℏq0)) ))} 1 k )−1 , q √√√√( 1+{ ∅ (( ∅−1(ℑk(ℏq0)) ))} 1 k )−1 q √√√√√√√√ 1−〈 1+〈 Θ〈 Θ−1(ℜk(θ q 0)) 〉 〉 1 k 〉−1 , q √√√√√√ 〈 1+〈 ∅〈 ∅−1(ℑk(ϕ q 0)) 〉 〉 1 k 〉 −1〉 = 〈 q √√√√√√ 1−〈 1+ ⟨(ℜk(ℏq0))⟩ 1 k 〉−1 , q √√√√〈 1+ ⟨(ℑk(ℏq0))⟩ 1 k 〉−1 q √√√√√√ 1−〈 1+ ⟨(ℜk(θ q 0))⟩ 1 k 〉−1 , q √√√√〈 1+ ⟨(ℑk(ϕ q 0))⟩ 1 k 〉−1〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 22 of 73 = 〈 q √ 1− 〈 1 + ℏq0 1−ℏq0 〉−1 , q √〈 1 + 1−ℏq0 ℏq0 〉−1 q √ 1− 〈 1 + θq0 1−θq0 〉−1 , q √〈 1 + 1−ϕq 0 ϕq 0 〉−1 〉 = (ℏ0, ℏ0) , (θ0, ϕ0) Theorem 6. Let ∆j = (ℏj , ℏj , θj , ϕj) be the Cq − ROFEs,where ℏj , ℏj denotes the am- plitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then (∆r) − ≺ Cq −ROFDAWA (∆1,∆2, ...,∆r) ≺ (∆r) + (11) (∆r) − =  min ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , min ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  and (∆r) + =  max ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , max ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  Proof. we know that j ∈ {1, 2, ..., r}, Then, min ( (ℏj , ℏj) , (θj , ϕj) ) ≤ ( (ℏj , ℏj), (θj , ϕj) ) ≤ maxj ( (ℏj , ℏj), (θj , ϕj) ) , here (ℏj , ℏj), (θj , ϕj) ∈ ∆r, we get, 〈 q √√√√√√ Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏq·)) 〉 , q √√√√√√ ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏq·)) 〉 q √√√√√√ Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q·)) 〉 , q √√√√√√ ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q·)) 〉 〉 ≤〈 q √√√√√√ Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 , q √√√√√√ ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 q √√√√√√ Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 , q √√√√√√ ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 23 of 73 ≤ 〈 q √√√√√√ Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏq··)) 〉 , q √√√√√√ ∅〈 r∑ j=1 ω∅−1(ℑk(ℏq··)) 〉 q √√√√√√ Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q··)) 〉 , q √√√√√√ ∅〈 r∑ j=1 ω∅−1(ℑk(ϕ q··)) 〉 〉 and ℏ·, ℏ·, θ·, ϕ· ∈ (∆r) − , ℏ··, ℏ··, θ··, ϕ·· ∈ (∆r) + 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏq·)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏq·)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q·)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q·)) 〉 〉 1 k 〉−1〉 ≤ 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏq··)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏq··)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q··)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q··)) 〉 〉 1 k 〉−1〉 or 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 Θ−1(ℜk(ℏq·)) r∑ j=1 ωj 〉 〉 1 k 〉 −1 , q √√√√√√√√√ 〈 1+〈 ∅(( ∅−1(ℑk(ℏq·)) r∑ j=1 ωj )) 〉 1 k 〉−1 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 24 of 73 q √√√√√√√√√ 1− 1+{ Θ (( Θ−1(ℜk(θ q·)) r∑ j=1 ωj ))} 1 k  −1 , q √√√√√√√  1+{ ∅ (( ∅−1(ℑk(ϕ q·)) r∑ j=1 ωj ))} 1 k  −1〉 ≤ 〈 q √√√√√√√√√√√ 1− 〈〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+ Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 1 k 〉−1 , q √√√√√√√√√  1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k  −1〉 ≤ 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 Θ−1(ℜk(ℏq··)) r∑ j=1 ωj 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅−1〈 ∅−1(ℑk(ℏq··)) r∑ j=1 ωj 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 Θ−1(ℜk(θ q··)) r∑ j=1 ωj 〉 〉 1 k 〉−1 , q √√√√√√√√√  1+〈 ∅−1〈 ∅−1(ℑk(ϕ q··)) r∑ j=1 ωj 〉 〉 1 k  −1〉 or〈 q √√√√√ 1−〈 1+ (ℜk(ℏq·) 1 k 〉−1 , q √√√√〈 1+ ⟨(ℑk(ℏq·)⟩ 1 k 〉−1 q √ 1− 1 + ⟨(ℜk(θ q·)⟩ 1 k −1 , q √√√√( 1+ ⟨(ℑk(ϕ q·)⟩ 1 k )−1〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 25 of 73 ≤ 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 ≤ 〈 q √√√√ 1−( 1 + ⟨ℜk(ℏq··)⟩ 1 k )−1 , q √√√√( 1+ ⟨ℑk(ℏq··)⟩ 1 k )−1 q √ 1− ( 1 + ⟨ℜk(θq··)⟩ 1 k )−1 , q √( 1 + ⟨ℑk(ϕq··)⟩ 1 k )−1 〉 or〈 q √ 1− 1 1+ ℏq· 1−ℏq· , q √ 1 1+ 1−ℏq· ℏq· q √ 1− 1 1+ θq· 1−θq· , q √ 1 1+ 1−ϕq· ϕq· 〉 ≤ 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+ 〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 ≤ 〈 q √ 1− 1 1+ ℏq·· 1−ℏq·· , q √ 1 1+ 1−ℏq·· ℏq·· q √ 1− 1 1+ θq·· 1−θq·· , q √ 1 1+ 1−ϕq·· ϕq·· 〉 or {ℏ·, ℏ·, θ·, ϕ·} ≤ {ℏ··, ℏ··, θ··, ϕ··} I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 26 of 73 Getting S (∆r) − ≺ S (Cq −ROFDAWA (∆1,∆2, ...,∆r)) ≺ S (∆r) + and (∆r) − ≺ Cq −ROFDAWA (∆1,∆2, ...,∆r) ≺ (∆r) + Theorem 7. Let ∆j = {ℏj , ℏj , θj , ϕj} and ∆́j = {ℏ́j , ℏ́j , θ́j ,́ϕj} be the Cq−ROFEs,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ℏj ≤ ℏ́j , ℏj ≤, ℏ́jθj ≤ θ́j , ϕj ≤ ϕ́j Then Cq −ROFDAWA (∆1,∆2, ...,∆r) ≺ Cq −ROFDAWA ( ∆́1, ∆́2, ..., ∆́r ) (12) Proof. we know, = 〈 q √√√√√√√√√√√ 1− 〈 1+ Θ〈〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉〉 1 k 〉−1 , q √√√√√√√√√  〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉 −1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 here ℏqj , θ q j ∈ ∆j(j=1,2,...,r) , ℏq′j , θ q′ j ∈ ∆́j(j=1,2,...,r), ℏqj , ϕ q j ∈ ∆j(j=1,2,...,r) and ℏq′j , ϕ q′ j ∈ ∆́j(j=1,2,...,r) then ℜk(ℏqj , θ q j ) ≤ ℜk(ℏq′j , θ q′ j ), ℑk(ℏqj , ϕ q j) ≤ ℑk(ℏq′j , ϕ q′ j ) we have 〈 q √√√√√√√√√ 〈 Θ 〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 , ∅ 〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k , q √√√√√√√√√ 〈 Θ 〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 , ∅ 〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k〉 ≤ 〈 q √√√√√√√√√ 〈 Θ 〈 r∑ j=1 ωjΘ −1(ℜk(ℏq′j )) 〉 , ∅ 〈 r∑ j=1 ωj∅−1(ℑk(ℏq′j )) 〉 〉 1 k , q √√√√√√√√√ 〈 Θ 〈 r∑ j=1 ωjΘ −1(ℜk(θ q′ j )) 〉 , ∅ 〈 r∑ j=1 ωj∅−1(ℑk(ϕ q′ j )) 〉 〉 1 k〉 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 27 of 73 =⇒ 〈 q √√√√√√√√√ 1−〈 1+〈 Θ 〈 r∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+ 〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k −1〉 ≤ 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(ℏq′j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ℏq′j )) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r∑ j=1 ωjΘ −1(ℜk(θ q′ j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r∑ j=1 ωj∅−1(ℑk(ϕ q′ j )) 〉 〉 1 k 〉−1〉 , S (Cq −ROFDAWA (∆1,∆2, ...,∆r)) ≺ S (Cq −ROFDAWA (∆1,∆2, ...,∆r)) Cq −ROFDAWA (∆1,∆2, ...,∆r) ≺ Cq −ROFDAWA (∆1,∆2, ...,∆r) Definition 9. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD an . Then the mapping as: Cq −ROFNsU −→ Cq −ROFNsU and Cq −ROFDAOWA(∆1,∆2, ...,∆r) = rDℏ⊕ j=1 ( ωj ∗Dℏ ∆γ(j) ) (13) Theorem 8. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then Cq −ROFDAOWA(∆1,∆2, ...,∆r) = I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 28 of 73 〈 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqγ(j))) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqγ(j))) 〉 〉 1 k 〉−1 q √√√√√√√√√√√ 1− 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q γ(j))) 〉 〉 1 k 〉−1 , q √√√√√√√√√ 〈 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q γ(j))) 〉 〉 1 k 〉−1〉 (14) Proof. See the theorem 4.2 Theorem 9. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD, with ∆o ∈Cq −ROFNs. Then Cq −ROFDAOWA(∆0 DA⊕ ∆r) = ∆0 DA⊕ Cq −ROFDAOWA (∆1,∆2, ...,∆r) (15) Proof. See the theorem 4.3 Theorem 10. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then Cq −ROFDAOWA (∆1,∆2, ...,∆r) = ∆0 (16) Proof. See the theorem 4.4 Theorem 11. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then (∆r) − ≺ Cq −ROFDAOWA (∆1,∆2, ...,∆r) ≺ (∆r) + (17) (∆r) − =  min ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , min ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 29 of 73 and (∆r) + =  max ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , max ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  Proof. See the theorem 4.5 Theorem 12. Let ∆j = {ℏj , ℏj , θj , ϕj} and ∆́j = {ℏ́j , ℏ́j , θ́j ,́ϕj}be the Cq−ROFEs,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ℏj ≤ ℏ́j , ℏj ≤, ℏ́jθj ≤ θ́j , ϕj ≤ ϕ́j Then Cq −ROFDAOWA (∆1,∆2, ...,∆r) ≺ Cq −ROFDAOWA ( ∆́1, ∆́2, ..., ∆́r ) (18) Proof. See the theorem 4.6 Definition 10. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then mapping is follows:Cq −ROFNsU −→ Cq −ROFNsU and as Cq −ROFDAWG(∆1,∆2, ...,∆r) = rDℏ⊕ j=1 ( ωj ◦Dℏ ∆j ) (19) Theorem 13. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD. Then the aggregated value of Cq −ROFDAWG(∆1,∆2, ...,∆r) as, Cq −ROFDAWG(∆1,∆2, ...,∆r) = 〈 q √√√√√√√√√ 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqj)) 〉 〉 1 k 〉−1 , q √√√√√√√√√√√ 〈 1− 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqj)) 〉 〉 1 k 〉−1 q √√√√√√√√√ 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q j )) 〉 〉 1 k 〉−1 , q √√√√√√√√√√√ 〈 1− 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q j)) 〉 〉 1 k 〉−1〉 (20) Proof. See theorem 4.2 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 30 of 73 Theorem 14. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and with ∆o ∈Cq −ROFNs. Then Cq−ROFDAWG(∆o DA⊕ ∆1,∆o DA⊕ ∆2, ...,∆o DA⊕ ∆r) = ∆o DA⊕ Cq−ROFDAWA (∆1,∆2, ...,∆r) (21) Proof. See the theorem 4.3 Theorem 15. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD, with ∆o ∈Cq −ROFNs. ∆o = ∆j . Then Cq −ROFDAWG (∆1,∆2, ...,∆r) = ∆o (22) Proof. See the theorem 4.4 Theorem 16. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq−ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ∆o ∈Cq−ROFNs. Then (∆r) − ≺ Cq −ROFDAWG (∆1,∆2, ...,∆r) ≺ (∆r) + (23) As (∆r) − =  min ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , min ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  and (∆r) + =  max ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , max ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  Proof. See the theorem 4.5 Theorem 17. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ℏj ≤ ℏ́j , ℏj ≤ , ℏ́jθj ≤ θ́j , ϕj ≤ ϕ́j Then Cq −ROFDAWG (∆1,∆2, ...,∆r) ≺ Cq −ROFDAWG (( ∆́1, ∆́2, ..., ∆́r )) (24) Proof. See the theorem 4.6 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 31 of 73 Definition 11. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq−ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ∆o ∈Cq−ROFNs. Then the mapping as follows : CU IFS −→ CU IFS as Cq −ROFDAOWG(∆1,∆2, ...,∆r) = rDℏ⊕ j=1 ( ωj ◦Dℏ ∆γ(j) ) (25) Theorem 18. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq−ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ∆o ∈Cq−ROFNs. Then the aggregated value of Cq −ROFDAOWG(∆1,∆2, ...,∆r) and as, Cq −ROFDAOWG(∆1,∆2, ...,∆r) = 〈 q √√√√√√√√√ 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(ℏqγ(j))) 〉 〉 1 k 〉−1 , q √√√√√√√√√√√ 〈 1− 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ℏqγ(j))) 〉 〉 1 k 〉−1 q √√√√√√√√√ 〈 1+〈 Θ〈 r+1∑ j=1 ωjΘ −1(ℜk(θ q γ(j))) 〉 〉 1 k 〉−1 , q √√√√√√√√√√√ 〈 1− 1+〈 ∅〈 r+1∑ j=1 ωj∅−1(ℑk(ϕ q γ(j))) 〉 〉 1 k 〉−1〉 (26) Proof. See the theorem 4.2 Theorem 19. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD with ∆o ∈Cq −ROFNs. Then Cq −ROFDAOWG(∆o DA⊕ ∆r) = ∆o DA⊕ Cq −ROFDAOWG (∆1,∆2, ...,∆r) (27) Proof. See the theorem 4.3 Theorem 20. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 32 of 73 represent the phase term such that the θj denote the MD while ϕj is for NMD with ∆o ∈Cq −ROFNs. ∆o = ∆j . Then Cq −ROFDAOWG (∆1,∆2, ...,∆r) = ∆o (28) Proof. See the theorem 4.4 Theorem 21. Let ∆j = {ℏj , ℏj , θj , ϕj} be the Cq − ROFEs on U,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD with ∆o ∈Cq −ROFNs. Then (∆r) − ≺ Cq −ROFDAOWG (∆1,∆2, ...,∆r) ≺ (∆r) + (29) (∆r) − =  min ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , min ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  and (∆r) + =  max ( ℏ1, ℏ2, ..., ℏr, ℏ1, ℏ2, ..., ℏr ) , max ( θ1, θ2, ..., θr, ϕ1, ϕ2, ..., ϕr )  Proof. See the theorem 4.5 Theorem 22. Let ∆j = {ℏj , ℏj , θj , ϕj} and ∆́j = {ℏ́j , ℏ́j , θ́j ,́ϕj}be the Cq−ROFEs,where ℏj , ℏj denotes the amplitude terms such that ℏj stand for MD while the ℏj stand for NMD and θj , ϕj represent the phase term such that the θj denote the MD while ϕj is for NMD and ℏj ≤ ℏ́j , ℏj ≤, ℏ́jθj ≤ θ́j , ϕj ≤ ϕ́j Then Cq −ROFDAOWG (∆1,∆2, ...,∆r) ≺ Cq −ROFDAOWG ( ∆́1, ∆́2, ..., ∆́r ) (30) Proof. See the theorem 4.6 5. A Novel MADM Approach Based on the Proposed Aggregation Operators This section presents a novel MCDM method utilizing CqROFNs for evaluation prob- lems. The methodology employs the proposed aggregation operators to handle decision- making scenarios under qRO fuzzy environments. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 33 of 73 5.1. Problem Framework Consider a decision problem with the following components: • A set of m alternatives: Ă = {Ă1, Ă2, . . . , Ăm} • A set of r criteria with weight vector: ω̃ = {ω̃1, ω̃2, . . . , ω̃r}, where ∑r j=1 ω̃j = 1 • A CqROF decision matrix: D̃ = [¥ij ]m×r, where each element ¥ij is a CqROFN representing the evaluation of alternative Ăi against criterion ℏj The proposed method employs three key aggregation operators: the CqROFDAWG op- erator, CqROFDAOWA operator, and CqROFDAOWG operator to process the CqROF information and select the optimal alternative from the m available options based on r criteria. 5.2. Algorithm Step 1: Construct the Decision Matrix. Formulate the complex q-rung orthopair fuzzy (CqROF) decision matrix D̃ = [dij ]m×r, where each element dij represents the q-rung orthopair fuzzy evaluation of alterna- tive Ai with respect to criterion Cj for i = 1, 2, . . . ,m and j = 1, 2, . . . , r. Step 2: Normalize the Decision Matrix. For problems involving both benefit-type (B) and cost-type (C) criteria, transform the original matrix D̃ = [dij ]m×r into a normalized matrix D̄ = [d̄ij ]m×r. Step 3: Determine Criteria Weights. Calculate criterion weights considering two scenarios: • Case 1: For completely unknown weights, employ the entropy method. • Case 2: For partially known weights, utilize available constraints with the entropy method to determine unknown weights. Step 4: Aggregate Alternative Evaluations. Compute comprehensive values for each alternative Ai (i = 1, 2, . . . ,m) using one of the aggregation operators: CqROFDAWA, CqROFDAOWA, CqROFDAWG, or CqROFDAOWG. Step 5: Calculate Score Values. Determine score values for each alternative using the defined score function. Step 6: Rank the Alternatives. Arrange alternatives in ascending order based on their score values. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 34 of 73 Start Step 1: Construct decision matrix Step 2: Normalize the matrix Step 3: Determine weights Unknown weights? Use Entropy Method Use Partially Known Weights Step 4: Apply aggregation operators Step 5: Calculate scores and rank End Yes No Figure 1: Flowchart of the Proposed MCDM Methodology I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 35 of 73 6. Case Study In this section, we aim to provide a comprehensive and illustrative example that em- phasizes the practical application of the proposed method for recruiting Human Resource Information Systems (HRIS) and Human Resource Management (HRM) professionals. The focus is on exploring various studies and perspectives to underscore the critical role of HRIS and HRM in achieving organizational goals and ensuring stainability. Practical Application of the Proposed Method: To exemplify the effectiveness of the pro- posed recruitment method, we delve into the dynamic intersection of HRIS and HRM. The recruitment process is pivotal in identifying and outboarding individuals who can harness the potential of information systems for human resource functions. HRIS and Organizational Efficiency: Research indicates that organizations leveraging ad- vanced HRIS technologies experience enhanced efficiency in managing human resources. The recruitment process, when guided by a strategic method, ensures the selection of candidates with the skills to optimize HRIS for streamlined data management, reporting, and analytic. This, in turn, contributes to improved decision-making processes within the organization. HRM’s Strategic Role: Examining the landscape of HRM [42], it becomes evident that strategic human resource practices are fundamental to achieving organizational goals. The recruitment method proposed here emphasizes aligning HRM strategies with overall business objectives. Studies have shown that a well-integrated HRM approach positively influences employee performance, engagement, and, consequently, organizational stain- ability. Impact on Organizational Goals: By employing the proposed recruitment method, organi- zations can strategically position themselves to meet and exceed their goals. The synergy between HRIS and HRM, as facilitated by a carefully designed recruitment process, en- ables businesses to adapt to changing market dynamics, foster innovation, and ensure the continuous development of their workforce. Stainability Through Talent Acquisition: Stainability is a key concern for modern orga- nizations. The recruitment method advocated here extends its impact to the long-term stainability of businesses. By identifying and hiring professionals well-versed in HRIS and aligned with strategic HRM principles, organizations create a foundation for sustained growth, resilience, and adaptability in the face of evolving challenges. In conclusion, the proposed recruitment method serves as a strategic enabler for organizations seeking to harness the full potential of HRIS and HRM. Through a careful examination of studies and perspectives, we have highlighted the instrumental role of this method in achieving or- ganizational goals and ensuring stainability. The interplay between technology and human resource management, when managed effectively, becomes a cornerstone for organizational success in today’s dynamic business environment. The passage breaks down the evaluation criteria for Human Resource Information Sys- tems (HRIS) into five categories: Human-resource management functions (C̆1), Technology (C̆2), Software quality (C̆3), Cost (C̆4), and Vendor support (C̆5). Here’s a summary of each category and its associated criteria: I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 36 of 73 6.1. Human-resource management functions (C̆1): Employee records, including interactions and supervisor reports, are kept up to date in a database by Staff Information Management (C̆1−1) :. The organisation of workers and their skills within the framework of the business is known as labour management and organisation structure (C̆1−2) :. Budget management in conjunction with compen- sation and benefits (C&B) (C̆1−3) : Oversees employee welfare and financial matters, including pay, bonuses, and career advancement. Employee training courses, attendance, testing, grading, and question management are all tracked by Staff Training Manage- ment (C̆1−4) :. Staff Recruiting Management (C̆1−5) : oversees the screening, selection, job analysis, candidate tracking, and onboarding processes. Staff Performance Manage- ment (C̆1−6) : Maintains records of work schedules, organisational objectives, evaluations, incentives, and sanctions, as well as performance metrics. . 6.2. Technology (C̆2) : (C̆2−1) Big Data Analysis: Handles and examines comprehensive, unstructured organi- sational data. Artificial Intelligence (C̆2−2) : Makes suggestions for employee tasks based on analysis of postures and facial expressions. Cyberspace (IoT, )(C̆2−3) : uses RFID or sensors with unique identifiers (UID) to log and track employee actions and provide real- time analytical data. Network Social (C̆2−4) : Offers a forum for business networking and employee engagement; it may also integrate with other social media sites. Services for Self-Help (C̆2−5): Provides automatic answers to questions from staff members, including those about leave requests. 6.3. Software quality (C̆3) : Functionality (C̆3−1) : Indicates how effectively the programme complies with the de- mands of organisational design. Dependability (C̆3−2) : Assesses the reliability of organi- sational and employee data kept in the programme. Usability(C̆3−3): Takes into account user interfaces and intuitiveness when determining ease of use. Effectiveness (C̆3−4): In- dicates how well the software performs overall throughout the organisation and how much less time is needed for HR-related duties. Reliability (C̆3−5) : Evaluates how simple it is to maintain programme functionality using dependable backup methods. Mobility (C̆3−6) : Ascertains whether data corruption occurs while exporting and importing HR data to other programmes. 6.4. Cost (C̆4) : Operation and Maintenance Fees(C̆4−1): Pays for the costs associated with using the software to organize continuing HR processes. Licensing Fees (C̆4−2): Comprises the I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 37 of 73 expenses related to securing formal authorization to utilize the program. Fees for Con- sultation (C̆4−3) :Alludes to the expenses incurred in discussing the program and the objectives of the organization. The equipment cost, (C̆4−4), includes costs for the different hardware parts that the software needs. Software Training Fees (C̆4−5): These represent the expenses incurred in instructing staff members, especially HR and IT managers, on how to make the most of all software functionalities. 6.5. Vendor support (C̆5) : Vendor Reputation (C̆5−1) : Relies on the software provider’s overall standing as de- termined by previous interactions with clients. Technical Proficiency (C̆5−2) : looks at the characteristics that software businesses have in order to offer solutions. Commitment for After-Sales Service (C̆5−3): assesses the promises made in the form of assurances to the organisation. Update and Upgrade for After-Sales (C̆5−4) : Evaluates the potential for introducing new packages, resolving issues, and staying current with technical advance- ments. (C̆5−5) : Software Delivery and Service Response Time calculates the time it takes for organisational usage, starting with planning and ending with payment. In this case, we apply the Cq − ROF fuzzy to the HR recruitment process. In order to make a well-informed judgement, the HR department may use the Cq − ROF fuzzy theory throughout the recruitment process. In this case, five alternatives Ăi(i=1,2,3,4,5) will be chosen for further evaluation after predomination. Thinking about the five qualities that are outlined below: (i) C̆1 : Human-resource management functions (ii) C̆2 : Technology (iii) C̆3 : Software quality (iv) C̆4 : Cost (v) C̆5 : Vendor support 6.6. Problem Solution A method for addressing MCDM problems utilising Cq − ROFNs data will be created using the Cq − ROFDAWA, Cq − ROFDAOWA, Cq − ROFDAWG, and Cq − ROFDAOWG operators. Where U = {Ă1, Ă2, ..., Ăm} and A = {C̆1, C̆2, ..., C̆r} are the variables being defined. ω represents a set of options, ω1, ω2, ..., ωr} denotes a set of characteristics, and ωr denotes a set of weights. Imagine the Cq − ROF ma- trix D̃ = [dij ]m×r, where each dij is represented as Cq − ROFNs. The operators Cq − ROFDAWA, Cq − ROFDAWG, Cq − ROFDAOWA, or Cq − ROFDAOWG are now used in the proposed method to address the MℏDM problems with the Cq−ROF data. The following steps, in the specified sequence, constitute the suggested approach: Step 1: The table 2shows the Cq −ROF decision matrix I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 38 of 73 Table 2: Cq −ROF decision matrix by experts C̆1 C̆2 C̆3 C̆4 C̆5 Ă1 〈 80 100 , 70 100 , 60 100 , 90 100 〉 〈 60 100 , 90 100 , 40 100 , 70 100 〉 〈 50 100 , 91 100 , 45 100 , 76 100 〉 〈 92 100 , 50 100 , 85 100 , 56 100 〉 〈 64 100 , 69 100 , 68 100 , 77 100 〉 Ă2 〈 70 100 , 80 100 , 70 100 , 90 100 〉 〈 72 100 , 84 100 , 87 100 , 78 100 〉 〈 87 100 , 78 100 , 64 100 , 69 100 〉 〈 88 100 , 77 100 , 58 100 , 67 100 〉 〈 61 100 , 77 100 , 60 100 , 60 100 〉 Ă3 〈 60 100 , 90 100 , 80 100 , 70 100 〉 〈 40 100 , 70 100 , 60 100 , 90 100 〉 〈 45 100 , 76 100 , 50 100 , 91 100 〉 〈 85 100 , 56 100 , 92 100 , 50 100 〉 〈 68 100 , 77 100 , 64 100 , 69 100 〉 Ă4 〈 90 100 , 70 100 , 70 100 , 90 100 〉 〈 60 100 , 83 100 , 83 100 , 60 100 〉 〈 72 100 , 55 100 , 55 100 , 72 100 〉 〈 87 100 , 47 100 , 87 100 , 47 100 〉 〈 92 100 , 50 100 , 50 100 , 92 100 〉 Ă5 〈 70 100 , 90 100 , 70 100 , 80 100 〉 〈 87 100 , 78 100 , 72 100 , 84 100 〉 〈 64 100 , 69 100 , 87 100 , 78 100 〉 〈 58 100 , 67 100 , 88 100 , 77 100 〉 〈 60 100 , 60 100 , 61 100 , 77 100 〉 Step 2: Since in the table 2 C̆1, C̆2, C̆3 are benefit attributes and also C̆4, C̆5, are the cost attribute so, we need to normalized the table. The table 3 shows the Normalization table. Table 3: Cq −ROF Normalized matrix C̆1 C̆2 C̆3 C̆4 C̆5 Ă1  ( 0.2666, 0.8084 ) ,( 0.1650, 0.7546 )   ( 0.2146, 0.7407 ) ,( 0.1414, 0.7744 )   ( 0.1253, 0.8271 ) ,( 0.1754, 0.8271 )   ( 0.1415, 0.7940 ) ,( 0.1661, 0.7642 )   ( 0.2074, 0.8114 ) ,( 0.2000, 0.8316 )  Ă2  ( 0.1768, 0.7687 ) ,( 0.1980, 0.8400 )   ( 0.2074, 0.8114 ) ,( 0.2000, 0.8316 )   ( 0.2735, 0.7876 ) ,( 0.2126, 0.8212 )   ( 0.2146, 0.7407 ) ,( 0.1414, 0.7744 )   ( 0.1768, 0.7687 ) ,( 0.1980, 0.8400 )  Ă3  ( 0.1971, 0.7687 ) ,( 0.2112, 0.6160 )   ( 0.1253, 0.8271 ) ,( 0.1754, 0.8271 )   ( 0.1415, 0.7940 ) ,( 0.1661, 0.7642 )   ( 0.2074, 0.8114 ) ,( 0.2000, 0.8316 )   ( 0.1971, 0.7687 ) ,( 0.2112, 0.6160 )  Ă4  ( 0.1414, 0.7744 ) ,( 0.2146, 0.7407 )   ( 0.1739, 0.8198 ) ,( 0.2013, 0.7946 )   ( 0.2264, 0.8509 ) ,( 0.1827, 0.3134 )   ( 0.2121, 0.2121 ) ,( 0.2365, 0.8417 )   ( 0.2666, 0.8084 ) ,( 0.1650, 0.7546 )  Ă5  ( 0.1891, 0.7750 ) ,( 0.2162, 0.7857 )   ( 0.2727, 0.8074 ) ,( 0.2426, 0.7801 )   ( 0.2264, 0.8130 ) ,( 0.2890, 0.7979 )   ( 0.1414, 0.7744 ) ,( 0.2146, 0.7407 )   ( 0.1739, 0.8198 ) ,( 0.2013, 0.7946 )  Case I: When weight is completely unknown: Researchers typically have basic knowledge on weights in many real-world challenges. The usual practice is to randomly initialise these weights and then use a variety of techniques to fine-tune them such that the difference between the expected and actual results is as little as possible. Nonetheless, there are cases when the weights are ambiguous or not known at all. Researchers are forced to use different methods to estimate the weights and measure the uncertainty in such instances. Step 3: The wi shows the unknown weight determined by entropy method and Eq.(5.1), that as, w1 = 0.1990, w2 = 0.1993, w3 = 0.1960, w4 = 0.2059, w5 = 0.1998 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 39 of 73 Step 4: The table 4 shows the outcomes of a ranking procedure that determined the score values of the alternatives using four operators, Cq-ROFDAWA, Cq-ROFOWA, Cq- ROFWG, and Cq-ROFOWG, as given in Equations (4.2), (4.7), (4.14), and (4.20), respectively. Table 4: Score function with respect to WA Operator Alternatives Score Function Score Values Ă1 s(Ă1) 0.4824 Ă2 s(Ă2) 0.4867 Ă3 s(Ă3) 0.4643 Ă4 s(Ă4) 0.4904 Ă5 s(Ă5) 0.4911 Step 5: The table 5 shows the ranking of the alternatives using , Cq-ROFDAWA, oper- ator. Table 5: Ranking w.r.t CqROFDAWA Ă5 ≻ Ă4 ≻ Ă1 ≻ Ă2 ≻ Ă3 Step 6: Similarly the table 6, 7, and 8 shows the score values and ranking of the alter- natives using Cq-ROFDAOWA, Cq-ROFDAWAG, and Cq-ROFDAOWG operator respectively. Table 6: Score function and ranking w.r.t CqROFDAOWA Alternatives Ă1 Ă2 Ă3 Ă4 Ă5 Ranking values 0.4689 0.4658 0.4156 0.4734 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Table 7: Score function and ranking w.r.t CqROFDAWG Alternatives Ă1 Ă2 Ă3 Ă4 Ă5 Ranking values 0.4831 0.4663 0.4603 0.4911 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Table 8: Score function and ranking w.r.t CqROFDAOWG Alternatives Ă1 Ă2 Ă3 Ă4 Ă5 Ranking values 0.4718 0.4593 0.4501 0.4865 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 40 of 73 A1 A2 A3 A4 A5 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.47 0.47 0.42 0.47 0.48 0.48 0.47 0.46 0.49 0.51 0.47 0.46 0.45 0.49 0.48 0.47 0.46 0.45 0.49 0.48 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFDAOWA Cq-ROFDAOWG Cq-ROFDAWG Figure 2: Comparison of Alternative Scores under Different Cq-ROFDA Operators From the graph 2, it is evident that Ă5 consistently achieves the highest score across all four operators, confirming it as the best alternative. The relative consistency in ranking also validates the robustness of the proposed Cq-ROFDA-based methods. Case II: When weight is partially known:When just a portion of the weight is known: Finally, a versatile strategy is required for handling weights in practical problem- solving. The ever-changing nature of real-life situations might bring ambiguity, even if researchers usually have access to some initial weight data. Researchers adjust their models to the complexities of the data they have by using a mix of algorithmic updates and random initialisation to enhance the weights, even whether the weights are partly known or not. The partly known weights for the specific qualities are w1 = 0.20, w2 = 0.25, w3 = 0.25, w4 = 0.20, w5 = 0.10. Step 8: The table 9 shows the score values of the alternatives using four operators, Cq- ROFDAWA, Cq-ROFOWA, Cq-ROFWG, and Cq-ROFOWG, as given in Equations (4.2), (4.7), (4.14), and (4.20), respectively. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 41 of 73 Table 9: Score function w.r.t CqROFDAWA Alternatives Score Function Score Values Ă1 s(Ă1) 0.4839 Ă2 s(Ă2) 0.4914 Ă3 s(Ă3) 0.4522 Ă4 s(Ă4) 0.4590 Ă5 s(Ă5) 0.4986 Then s(Ă5) > s(Ă2) > s(Ă1) > s(Ă4) > s(Ă3) Step 9: The table 10 shows the ranking of the alternatives using , Cq-ROFDAWA, oper- ator. Table 10: Ranking w.r.t CqROFDAWA Ă5 ≻ Ă2 ≻ Ă1 ≻ Ă3 ≻ Ă3 Step 10: Similarly the table 11, 12, and 13 shows the score values and ranking of the alternatives using Cq-ROFDAOWA, Cq-ROFDAWAG, and Cq-ROFDAOWG oper- ator respectively. Table 11: Score function and ranking w.r.t CqROFDAWA Alternatives Ă1 Ă2 Ă3 Ă4 Ă5 Ranking values 0.4988 0.52012 0.4695 0.4875 0.5279 Ă5 > Ă2 > Ă1 > Ă4 > Ă3 Table 12: Score function and ranking w.r.t CqROFDAWG Alternatives Ă1 Ă2 Ă3 Ă4 Ă5 Ranking values 0.4830 4870 0.4593 0.4523 0.6654 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 Table 13: Score function and ranking w.r.t CqROFDAOWG Alternatives Ă1 Ă2 Ă3 Ă4 Ă5 Ranking values 0.5103 0.5117 0.4988 0.4765 0.5347 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 42 of 73 A1 A2 A3 A4 A5 0.45 0.5 0.55 0.6 0.65 0.7 0.48 0.49 0.45 0.46 0.50.5 0.52 0.47 0.49 0.53 0.48 0.49 0.46 0.45 0.67 0.51 0.51 0.5 0.48 0.53 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFDAOWA Cq-ROFDAWG Cq-ROFDAOWG Figure 3: Comparison of Alternative Scores Using Different Cq-ROFDA Operators The grouped bar chart 3 indicates that Ă5 is the optimal choice among all opera- tors, especially in relation to Cq-ROFDAWG. The weighting behavior of each operator is delicate, resulting in minor discrepancies in rankings across different options. 7. Discussion About the Influence of the Parameter k Because it controls the score values and the ranking of the alternatives, the parameter k is crucial. Table 14 shows that changing the k values significantly changes the score functions and the relative ranking of options. Options are increasingly differentiated and rankings are more sensitive as k decreases. A smoothing effect, however, makes the varia- tions between choices less apparent, and the score values converge as k increases. such, it seems that k controls the degree of choice sensitivity, and that k has to be adjusted such that it accurately reflects the decision-makers’ preferences. Table 14 displays the score values for the five alternatives Ă_1 to Ă_5 for various parameter k selections ranging from 1 to 5. As k increases, it becomes evident that the scores of the alternatives gradually change. With k = 1, the best possible score is Ă_5, followed closely by Ă_4, while the worst possible score is Ă_1. Decision preference seems to be durable, as the ranking pattern remains mostly identical for both k = 1 and k = 2. If there is a little alteration where Ă_1 scores higher than both Ă_2 and Ă_3, the middle- I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 43 of 73 Table 14: Scores and ranking orders for different values of k Value of k Ă1 Ă2 Ă3 Ă4 Ă5 Ranking Order 1 0.3537 0.6512 0.5179 0.7868 0.8078 Ă5 ≻ Ă4 ≻ Ă2 ≻ Ă3 ≻ Ă1 2 0.3799 0.4886 0.3985 0.5158 0.8401 Ă5 ≻ Ă4 ≻ Ă2 ≻ Ă3 ≻ Ă1 3 0.4824 0.4867 0.4643 0.4904 0.4911 Ă5 ≻ Ă4 ≻ Ă1 ≻ Ă2 ≻ Ă3 4 0.4203 0.4328 0.4423 0.5358 0.9787 Ă5 ≻ Ă4 ≻ Ă3 ≻ Ă2 ≻ Ă1 5 0.4992 0.5114 0.5202 0.5350 0.9960 Ă5 ≻ Ă4 ≻ Ă3 ≻ Ă2 ≻ Ă1 tier rankings are affected at k = 3. As the value of Ă_5 approaches 1.0, its dominance is confirmed, and its preference grows for bigger values like k = 4 and k = 5. A higher value of k improves the discriminating power between possibilities, as shown in this pattern, and it becomes easier to confidently identify the better decision. 1 2 3 4 5 0.4 0.6 0.8 1 k Sc or e Va lu e Ă1 Ă2 Ă3 Ă4 Ă5 Figure 4: Score trends of alternatives Ă1 to Ă5 with respect to different values of k Figure 4 gives a visual representation of the connection between the scores of the five alternatives and the change in parameter k. When k exceeds 3, its score tends to approach unity, making Ă5 more dominating and consistently outperforming all other options. On the other hand, Ă1 maintains the lowest performance while continuously displaying an increasing score. The crossover points that alternatives Ă2, Ă3, and Ă4 exhibit may be used to determine their intermediate rankings. The graph indicates that altering k has an impact on both the magnitude of the scores and the relative ranking of the alternatives. Sensitivity analysis with respect to k is crucial to guarantee reliable and consistent decision-making outcomes. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 44 of 73 8. Comparative analysis with other existing techniques under a complex environment The author is comparing the offered approaches’ performance to that of other current strategies in a Cq−ROF context. For the purpose of choosing the best approach to a given issue, this kind of analysis may provide light on the relative merits of several approaches. 8.0.1. Comparison with Cq-ROF Existing Method Here, we make a table that compares the suggested approach to other previously used approaches, such as Cq − ROFWA[31], Cq − ROFOWA[31], Cq − ROFWG[31], and Cq −ROFOWG[31]. Table 15: Comparison: Proposed Operators vs. Existing Cq-ROF Aggregation Operators Aggregation Operator Alternative Values Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFWA [31] 0.3537, 0.6512, 0.5179, 0.7868, 0.8078 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 Cq-ROFOWA [31] 0.3539, 0.6512, 0.5177, 0.8078, 0.8303 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 Cq-ROFWG [31] 0.4867, 0.5578, 0.6850, 0.7418, 0.7944 Ă5 > Ă4 > Ă3 > Ă2 > Ă1 Cq-ROFOWG [31] 0.4833, 0.5579, 0.5853, 0.6219, 0.7344 Ă5 > Ă4 > Ă3 > Ă2 > Ă1 The suggested operators (Cq-ROFDAWA, Cq-ROFDAWG, etc.) assign all five options close, balanced scores 15. This demonstrates that there are no significant scoring dispar- ities and that each choice is treated more equitably. However, the current Cq-ROFWG and Cq-ROFWA types provide low values to the others and extremely high scores to Ă4 and Ă5. Too much difference is created by this, which might lead to prejudice when making decisions. The suggested approaches are superior for impartial and trustworthy assessment because they maintain rankings consistent with smaller disparities. All things considered, the new approach performs more consistently than the others. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 45 of 73 A1 A2 A3 A4 A5 0.3 0.4 0.5 0.6 0.7 0.8 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFDAWG Cq-ROFDAOWA Cq-ROFDAOWG Cq-ROFWA Cq-ROFWG Cq-ROFOWA Cq-ROFOWG Figure 5: Smooth Curve Comparison: Proposed vs. Cq-ROF Operators The graph’s 5 flat and smooth proposed curves (blue and red lines) demonstrate how similarly all of the alternatives scored. This suggests that the proposed method handles the data in a balanced way. A broad variety of outcomes is shown by the green and orange lines, which reflect the current methods. They show a large rise for Ă4 and Ă5, but a reduction for other variables. It may be unfair to make decisions based on such obvious differences, especially when little changes in data result in significant changes. The graph shows that the recommended operators are more resilient to sudden changes, egalitarian, and sturdy. This makes them superior for judgments that are important in the real world. 8.0.2. Comparison with Different Cq-ROF Existing Method Here, we provide a table that compares the suggested approach to other previously used approaches, including Cq−ROFAAWA [32] Cq−ROFAAOWA [32], Cq−ROFAAWG [32], Cq−ROFAAOWG [32],Cq−ROFArWA [34],Cq−ROFArWG [34],Cq−ROFDWA [35],Cq −ROFDWG [35]. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 46 of 73 Table 16: Comparison: Proposed Operators vs. Different Cq-ROF Aggregation Operators Aggregation Operator Alternative Values Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFAAWA 0.2905, 0.5802, 0.3470, 0.7679, 0.7946 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 Cq-ROFAAWG 0.4291, 0.5803, 0.6344, 0.7692, 0.7946 Ă5 > Ă4 > Ă3 > Ă2 > Ă1 Cq-ROFAAOWA 0.5343, 0.6274, 0.5987, 0.6684, 0.6733 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 Cq-ROFAAOWG 0.5144, 0.6123, 0.6532, 0.7574, 0.8073 Ă5 > Ă4 > Ă3 > Ă2 > Ă1 Cq-ROFArWA 0.7900, 0.5342, 0.3342, 0.2643, 0.7934 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 Cq-ROFArWG 0.5542, 0.4215, 0.3425, 0.1233, 0.6733 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 Cq-ROFDWA 0.5674, 0.6289, 0.4536, 0.2929, 0.8073 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 Cq-ROFDWG 0.5144, 0.6123, 0.6532, 0.7574, 0.8073 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 The table 16displays a comparison between the newly proposed Cq-ROFDA aggre- gation operators and the existing Cq-ROFAA operators. All of the options are rated within a narrow range, mostly between 0.46 and 0.51, according to the results for the four recommended ways (Cq-ROFDAWA, Cq-ROFDAWG, Cq-ROFDAOWA, and Cq- ROFDAOWG). This suggests a more stable and comprehensive evaluation of choices. All four operators have the same ranking order: Ă5 always comes first, Ă4 comes second, Ă1, Ă2, and Ă3 comes last. The robustness of the proposed operators is shown by this rank- ing consistency. On the other hand, the present Cq-ROFAA operators’ values are more erratic. They assign very high scores (up to 0.8073) to Ă5 and Ă4, whereas the scores of the other alternatives dramatically decline, with some falling as low as 0.29. Because of these stark differences, decisions made in such uncertain situations may be biased or less reliable. The proposed methods demonstrate that alternative assessment could be more impartial, equitable, and stable. According to the ranking results, there is strong agree- ment among the four operators on the best option, with each consistently identifying Ă_5 as the best choice. However, variations in aggregation behaviour influenced by Dombi and Archimedean operations are shown by slight differences in the order of the remaining possibilities, especially between Cq-ROFAr and Cq-ROFD operators. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 47 of 73 A1 A2 A3 A4 A5 0.3 0.4 0.5 0.6 0.7 0.8 0.48 0.49 0.46 0.49 0.490.48 0.47 0.46 0.49 0.51 0.47 0.47 0.42 0.47 0.480.47 0.46 0.45 0.49 0.48 0.29 0.58 0.35 0.77 0.79 0.51 0.61 0.65 0.76 0.81 0.43 0.58 0.63 0.77 0.79 0.53 0.63 0.6 0.67 0.67 0.79 0.53 0.33 0.26 0.79 0.55 0.42 0.34 0.67 0.57 0.63 0.45 0.29 0.81 0.51 0.61 0.65 0.76 0.81 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFDAWG Cq-ROFDAOWA Cq-ROFDAOWG Cq-ROFAAWA Cq-ROFAAOWG Cq-ROFAAOWG Cq-ROFAAOWG Cq-ROFArWA Cq-ROFArWG Cq-ROFDWA Cq-ROFDWA Figure 6: Bar Chart Comparison: Proposed vs. Cq-ROF Operators The proposed operators’ performance differs significantly from that of the current Cq-ROFAA operators, as seen in the bar chart 6. Score values are densely packed and evenly distributed throughout all options, as shown by the bars depicting the suggested operators, which are of about the same height. Each option is treated equitably without being given excessively high or low values by the suggested techniques, as shown by their consistency. On the other hand, there is a noticeable difference in the height of the bars representing the current Cq-ROFAA operators (green and red fills). For example, A5 and A4 have much higher bars than A1 and A3, which are noticeably shorter. It may be inferred from this that current approaches prioritize some options over others. There are certain decision-making contexts where such a bias might be undesirable. In general, the graph supports the claim that the suggested aggregation approaches are better suited to multi-criteria decision-making situations that need consistency and fairness since they provide a more balanced and impartial assessment. The Cq-ROFDWA operator’s highest overall scores, especially for A4 and A5, demonstrate its greatest aggregation capabilities. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 48 of 73 On the other hand, Cq-ROFArWG provides values that are much less than those of most other alternatives. 8.0.3. Comparison with Classical CPF Operators We construct a comparative table delineating the suggested approach alongside previously used methods, including CPFWA [27], CPFOWA [27], CPFWG [27], and CPFOWG [27]. Table 17: Comparison: Proposed Operators vs. Classical CPFWA and CPFOWG Oper- ators Aggregation Operator Alternative Values Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 CPFWA [27] 0.4585, 0.6714, 0.6179, 0.8463, 0.8407 Ă4 > Ă5 > Ă2 > Ă3 > Ă1 CPFOWA [27] 0.3500, 0.3973, 0.3812, 0.4343, 0.4190 Ă4 > Ă5 > Ă2 > Ă3 > Ă1 CPFWG [27] 0.4609, 0.7428, 0.6261, 0.9125, 0.8151 Ă4 > Ă5 > Ă2 > Ă3 > Ă1 CPFOWG [27] 0.4320, 0.5138, 0.4907, 0.6672, 0.5372 Ă4 > Ă5 > Ă2 > Ă3 > Ă1 The table 17 makes it abundantly evident that conventional CPF-based methods sig- nificantly improve Ă4 and Ă5, with apparent ranking dominance and large score gaps, whereas the suggested operators (Cq-ROFDA*)* preserve uniform ranking behaviour with slight variation in scores. This suggests that there is less leeway in allocating significance among options. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 49 of 73 A1 A2 A3 A4 A5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.48 0.47 0.46 0.49 0.51 0.47 0.47 0.42 0.47 0.480.47 0.46 0.45 0.49 0.48 0.43 0.51 0.49 0.67 0.54 0.46 0.67 0.62 0.85 0.84 0.46 0.74 0.63 0.91 0.82 0.35 0.4 0.38 0.43 0.42 0.43 0.51 0.49 0.67 0.54 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFWG Cq-ROFOWA Cq-ROFOWG CPFWA CPFWG CPFOWA CPFOWG Figure 7: Grouped Bar Chart: Proposed vs. CPFWA-Based Operators To visually differentiate across operator families, this graph employs patterned grouped bars 7. Evaluating the offered techniques in a balanced manner is encouraged by their consistent and reasonable ratings. In comparison, the dotted and grid-patterned CPFWA and CPFWG exhibit prominent peaks, particularly for Ă4 and Ă5, suggesting a possible bias or dominance. 8.0.4. Comparison with CPFAA Operators In this case, we make a table that compares the suggested technique to previous meth- ods that have been utilized, including CPFAAWA [28], CPFAAWG [28], CPFAAOWA [28], CPFAAOWG [28], CPFArWA [30], CPFArWG [30], CPFDWA [29], and CPFDWG [29]. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 50 of 73 Table 18: Comparison: Proposed Operators vs. CPFAAWA-Based Operators Aggregation Operator Alternative Values Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 CPFAAWA 0.3819, 0.6543, 0.4248, 0.7696, 0.7930 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CPFAAWG 0.3819, 0.6544, 0.4244, 0.7958, 0.7696 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CPFAAOWA 0.4889, 0.6301, 0.5683, 0.6613, 0.7429 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CPFAAOWG 0.4272, 0.5922, 0.5459, 0.6868, 0.6046 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CPFArWA 0.7946, 0.5335, 0.3342, 0.2665, 0.7943 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 CPFArWG 0.5536, 0.4226, 0.3413, 0.1256, 0.6735 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 CPFDWA 0.5636, 0.6216, 0.4514, 0.2934, 0.8054 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 CPFDWG 0.5168, 0.6115, 0.6515, 0.7543, 0.8067 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 The planned operators (Cq-ROFDA family) and the current CPFAAWA-based opera- tors are easily distinguished from one another in the comparison table 18. Across all op- tions, the suggested approaches (Cq-ROFDAWA, Cq-ROFDAWG, Cq-ROFDAOWA, and Cq-ROFDAOWG) consistently provide score values that are tightly ranged. This narrow range, which falls between around 0.46 and 0.51, indicates that the suggested approaches assess each option more fairly, leading to a more steady and balanced decision-making process. However, the CPFAAWA-based techniques provide scores that are much more diverse, ranging from 0.38 to 0.79. CPFAAWA and CPFAAWG, for instance, give Ă5 and Ă4 far higher scores than Ă1 and Ă3. Such wide disparities between options may be a sign of oversensitivity and a potential for biassed judgements. Therefore, the suggested Cq-ROFDA operators provide a better option in circumstances that need for fair and consistent assessment. The suggested techniques show that alternative evaluation might be more stable, fair, and unbiased. Based on the ranking findings, all four operators consistently select Ă_5 as the best option, indicating great agreement among them. How- ever, minor discrepancies in the order of the remaining possibilities, particularly between CPFAr and CPFD operators, indicate variances in aggregation behaviour impacted by Dombi and Archimedean operations. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 51 of 73 A1 A2 A3 A4 A5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFWG Cq-ROFOWA Cq-ROFOWG CPFAAWA CPFAAWG CPFAAOWA CPFAAOWG CPFArWA CPFArWG CPFDWA CPFDWA (2nd) Figure 8: Line Graph: Score values of alternatives under proposed and existing CPFA operators I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 52 of 73 The line and coloured graph 8 provide a visual comparison between the proposed operators and the CPFAAWA-based methods. The blue-patterned lines denoting the rec- ommended operators exhibit smooth and consistent score shifts throughout all selections. Because there are no sudden increases or decreases, the decision weights are dispersed more evenly. Conversely, the CPFAAWA-based operators are shown by red and orange lines, which exhibit sharp declines for Ă1 and Ă3 and sharp rises for Ă4 and Ă5. It is evident from this that they overvalue a few potential possibilities while undervaluing the rest. Such an imbalance might lead to inconsistent or even deadly findings in multi-criteria decision-making problems. The graph demonstrates how well the recommended strategies maintain the neutrality, fairness, and smoothness of all the possibilities under considera- tion. Such a bias may not be beneficial in certain decision-making situations. Overall, the graph supports the claim that the suggested aggregation procedures provide a more ob- jective and balanced assessment, which makes them more suitable for situations involving several criteria that need consistency and equity. The top overall scores of the CPFDWA operator, especially for A5, demonstrate its strongest aggregation capabilities. However, CPFArWG provides values that are significantly lower than those of most other options. 8.1. Comparison with CIF Existing Method Here, we make a table that compares the suggested approach to other previously used approaches, including CIFWA [22], CIFWG [22], CIFOWA [22], and CIFOWG [22]. Table 19: Comparison: Proposed Operators vs. Classical CIF Aggregation Operators Aggregation Operator Alternative Values Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 CIFWA [22] 0.19448, 0.5094, 0.5496, 0.5778, 0.6088 Ă5 > Ă4 > Ă3 > Ă2 > Ă1 CIFOWA [22] 0.3423, 0.4523, 0.5142, 0.5412, 0.5680 Ă5 > Ă4 > Ă3 > Ă2 > Ă1 CIFWG [22] 0.5094, 0.5488, 0.5157, 0.5088, 0.5778 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CIFOWG [22] 0.4880, 0.5276, 0.5089, 0.5634, 0.5888 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 Table 19 compares the conventional CIF aggregation operators with the proposed Cq- ROFDA operations. Since the Cq−ROFDAWA, Cq−ROFDAWG, Cq−ROFDAOWA, and Cq − ROFDAOWG operators consistently provide distinct rankings, with Ă5 rank- ing highest and Ă4 ranking second, there is evident agreement among the recommended procedures. However, the ranking of traditional CIF operators such as CIFWA, CIFOWA, CIFWG, and CIFOWG shows a somewhat different pattern, with Ă5 still being the favoured choice. Specifically, the classical operators provide superior alternative values on average. Nonetheless, the proposed operators are more reliable and resilient for decision- making tasks in the CPF context because they are more stable and consistent among I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 53 of 73 approaches. These results demonstrate the effectiveness and success of the recently de- ployed operators. A1 A2 A3 A4 A5 0.2 0.3 0.4 0.5 0.6 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFWG Cq-ROFOWA Cq-ROFOWG CIFWA CIFOWA CIFWG CIFOWA Figure 9: Bar Chart: Comparison of Proposed vs. CIFWA/OWA Operators The score values of five alternatives (A1 to A5) across different aggregation procedures are contrasted graphically in the bar chart 9. The top choices, especially A_4 and A_5, are routinely given better ratings by the CIFWA, CIFOWA, and CIFWG operators. Across alternatives, the proposed Cq-ROF-based operators (e.g., Cq-ROFDAWA, Cq-ROFWG) show consistent but somewhat low scores. Notably, Cq-ROFOWG has the biggest peak for A_4 (0.6672), suggesting that in certain cases, decision support is improved. 8.1.1. Comparison with CIFAA Operators We provide a comparative table comparing the proposed approach and previously used techniques, including CIFAAWA [23], CIFAAWG [23], CIFAAOWA [23], CIFAAOWG [23], CIFArWA [25], CIFAAWG [25], CIFDWA [24], CIFDWG [24]. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 54 of 73 Table 20: Comparison: Proposed Operators vs. CIFAA Aggregation Operators Aggregation Operator Alternative Values Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 CIFAAWA 0.4381, 0.5614, 0.5256, 0.8308, 0.5726 Ă4 > Ă5 > Ă2 > Ă3 > Ă1 CIFAAOWA 0.4231, 0.5101, 0.4452, 0.6241, 0.5462 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CIFAAWG 0.4341, 0.5614, 0.4356, 0.5995, 0.5726 Ă4 > Ă5 > Ă2 > Ă3 > Ă1 CIFAAOWG 0.4982, 0.5123, 0.5034, 0.6357, 0.5665 Ă5 > Ă4 > Ă2 > Ă3 > Ă1 CPFArWA 0.7946, 0.5335, 0.3342, 0.2665, 0.7943 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 CIFArWG 0.5536, 0.4226, 0.3413, 0.1256, 0.6735 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 CIFDWA 0.5636, 0.6216, 0.4514, 0.2934, 0.8054 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 CIFDWG 0.5168, 0.6115, 0.6515, 0.7543, 0.8067 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 The novel CqROFDA operators, designated as Cq−ROFDAWA, Cq−ROFDAWG, Cq−ROFDAOWA, and Cq−ROFDAOWG, are juxtaposed with the existing CIFAA ag- gregation operators in Table 20. Irrespective of the methodology used, the results indicate that the proposed operators consistently rank Ă5 and Ă4 as the two superior alterna- tives, implying a dependable and uniform evaluative pattern. The proposed methodolo- gies exhibit reduced vulnerability to outliers and provide more uniform outcomes among alternatives. Conversely, the majority of CIFAA approaches prioritise Ă4, resulting in CIFAA-based operators showing more variation in aggregated outcomes. CIFAA method- ologies provide distinct alternative orderings, suggesting that preference rankings exhibit more variability. This demonstrates that the proposed operators provide superior bal- anced aggregation behaviour in the CIF context while preserving ranking consistency. The proposed methods demonstrate that alternative assessment may be more impartial, equitable, and stable. According to the ranking results, there is strong agreement among the four operators, as they all constantly choose Ă5 as the best choice. Minor differences in the order of the remaining options, especially between CIFAr and CIFD operators, how- ever, suggest that Dombi and Archimedean operations have an influence on aggregation behaviour. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 55 of 73 A1 A2 A3 A4 A5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Alternatives Sc or e Va lu es Cq-ROFDAWA Cq-ROFWG Cq-ROFOWA Cq-ROFOWG CIFAAWA CIFAAOWA CIFAAWG CIFAAOWG CIFArWA CIFArWG CIFDWA CIFDWA Figure 10: Line Graph: Comparison of Proposed vs. CIF Operators As the graph 10 shows the five alternatives’ performance scores (A1 to A5) using differ- ent aggregation techniques on the line graph. The CIFAAWA operator that was proposed has the best score for A_4 (0.8308), showing that it is very good at discriminating. Op- erators based on CIFAA (red and purple lines) regularly outperform operators based on Cq-ROF (blue bars) in terms of rating sensitivity, since they continuously produce higher and more diversified ratings. Across a variety of scenarios, CIFAAWA and CIFAAOWG consistently outperform competing techniques. This proves that the proposed operators are better at sifting through the dataset for information that could influence decisions. The suggested aggregation approaches provide a more objective and balanced assessment, which makes them more suitable for MCDM situations that need consistency and fairness, as the graph supports overall. The CIFDWA operator’s highest overall scores, especially for A5, demonstrate its best aggregation skills. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 56 of 73 8.1.2. Comparison with q-ROF existing method Here, we construct a compression table comparing the suggested approach to other previ- ously used approaches, such as q−ROFAAWA [16], q−ROFAAWG [16], q−ROFWA [15], q − ROFWG [15], q − ROFArWA [18], q − ROFArWG [18], q − ROFDWA [17], q −ROFDWG [17] Table 21: Comparison of Alternatives under Various Aggregation Operators Aggregation Operator Alternatives Values Ranking of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 q-ROFWA 0.4964, 0.6415, 0.3639, 0.8942, 0.9789 Ă5 > Ă4 > Ă2 > Ă1 > Ă3 q-ROFWG 0.4965, 0.6413, 0.3638, 0.8812, 0.9788 Ă4 > Ă2 > Ă3 > Ă1 > Ă5 q-ROFAAWA 0.4914, 0.5735, 0.0998, 0.9798, 0.9763 Ă4 > Ă5 > Ă2 > Ă1 > Ă3 q-ROFAAWG 0.4914, 0.5735, 0.6747, 0.9797, 0.9761 Ă4 > Ă5 > Ă3 > Ă2 > Ă1 qFArWA 0.4321, 0.3342, 0.3342, 0.2665, 0.5342 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 qFArWG 0.4231, 0.3231, 0.3413, 0.1256, 0.5234 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 qFDWA 0.3223, 0.4545, 0.2414, 0.1934, 0.5342 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 qFDWG 0.3334, 0.4521, 0.2315, 0.1743, 0.5341 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 The performance of five options (Ă1 to Ă5) over eight distinct aggregation operators is summarised in Table 21. These operators’ robustness is confirmed by the fact that Ă5 constantly scores first among the suggested Cq-ROFDA operators (DAWA, DAOWA, DAWG, DAOWG). Ă4 follows closely behind. On the other hand, classic q-ROF-based operators show different patterns. For example, Ă4 and Ă5 have the highest scores under q-ROFWA and q-ROFAAWA, whereas Ă3 ends up being the best under q-ROFAAWG. This exemplifies how various operator kinds and weight structures affect the ranks of final decisions. But qROFArWG provides values that are far lower than most other alternatives. According to the ranking results, there is strong agreement among the four operators, as they all constantly choose Ă5 as the best choice. Minor differences in the order of the remaining options, especially between qROFAr and qROFD operators, however, suggest that Dombi and Archimedean operations have an influence on aggregation behaviour. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 57 of 73 Figure 11: Graphical Comparison of Aggregated Alternative Values A1 A2 A3 A4 A5 0 0.2 0.4 0.6 0.8 1 0.48 0.49 0.46 0.49 0.490.47 0.47 0.42 0.47 0.480.48 0.47 0.46 0.49 0.51 0.47 0.46 0.45 0.49 0.48 0.58 0.67 0.21 0.91 0.37 0.5 0.52 0.45 0.6 0.590.6 0.52 0.6 0.27 0.590.6 0.5 0.6 0.6 0.6 0.32 0.43 0.23 0.15 0.57 0.33 0.42 0.23 0.14 0.55 0.43 0.33 0.23 0.16 0.53 0.42 0.32 0.23 0.16 0.52 A gg re ga te d Sc or e Cq-ROFDAWA Cq-ROFDAOWA Cq-ROFDAWG Cq-ROFDAOWG qROFWA q-ROFWG q-ROFAAWA q-ROFAAWG qROFArWA q-ROFArWG q-ROFDWA q-ROFDWG Figure 11 is a graphical representation of the total scores for all four choices using the four recently added Complex q-ROFDA operators. Among the four variants, Ă5 regularly earns the highest score (0.5077) in the DAWG method, indicating its superior performance and reliability. To back up its lower ranking, the bar heights for Ă3 remain lower in every occurrence. With the use of visual insights, we can better understand how operator conduct affects alternative evaluations and verify that decision-making is consistent. The suggested aggregation approaches provide a more objective and balanced assessment, which makes them more suitable for MCDM situations that need consistency and fairness, as the graph supports overall. 8.1.3. Comparison with PF existing method We construct a comparative table delineating the suggested approach alongside previously used methods, including PFAAWA [13], PFAAWG [13], PFWA [11], PFAWG [11], PFArWA [14], PFArWG [14], PFDWA [12], PFDWG [12] I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 58 of 73 Table 22: Alternatives values and ranking orders using different aggregation operators Aggregation Operator Alternatives Values Ranking Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 PFAAWA 0.5900, 0.6446, 0.1807, 0.9108, 0.9013 Ă4 ≻ Ă5 ≻ Ă2 ≻ Ă1 ≻ Ă3 PFAAWG 0.5902, 0.6444, 0.4244, 0.9110, 0.5666 Ă4 ≻ Ă2 ≻ Ă1 ≻ Ă3 ≻ Ă5 PFWA 0.5900, 0.7438, 0.4918, 0.9350, 0.9561 Ă5 ≻ Ă4 ≻ Ă2 ≻ Ă1 ≻ Ă3 PFWG 0.5449, 0.5551, 0.4127, 0.6483, 0.6333 Ă4 ≻ Ă5 ≻ Ă2 ≻ Ă1 ≻ Ă3 PFArWA 0.4357, 0.3345, 0.3348, 0.2679, 0.5489 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 PFArWG 0.4247, 0.3267, 0.3478, 0.1289, 0.5276 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 PFDWA 0.3248, 0.4589, 0.2445, 0.1999, 0.5397 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 PFDWG 0.3368, 0.4556, 0.2357, 0.1799, 0.5390 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 In the CqROF framework, Table 22 shows the alternative scores and the order in which they were ranked from various aggregation operations. It includes both hypothetical operators (different kinds of Cq − ROFDA) and real ones (such PFAAWA and PFWA, for example). There is a consistent ranking pattern across the proposed approaches (Cq− ROFDAWA, Cq − ROFDAOWA, etc.), with Ă5 and Ă4 being the most prevalent in every case. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 59 of 73 A1 A2 A3 A4 A5 0 0.2 0.4 0.6 0.8 1 0.48 0.49 0.46 0.49 0.49 0.47 0.47 0.42 0.47 0.480.48 0.47 0.46 0.49 0.51 0.47 0.46 0.45 0.49 0.48 0.58 0.67 0.21 0.91 0.37 0.5 0.52 0.45 0.6 0.590.6 0.52 0.6 0.27 0.590.6 0.5 0.6 0.6 0.6 0.32 0.43 0.23 0.15 0.57 0.33 0.42 0.23 0.14 0.55 0.43 0.33 0.23 0.16 0.53 0.42 0.32 0.23 0.16 0.52 Sc or e Va lu es Cq-ROFDAWA Cq-ROFDAOWA Cq-ROFDAWG Cq-ROFDAOWG PFWA PFWG PFAAWA PFAAWG PFArWA PFArWG PFDWA PFDWG Figure 12: Scores of Alternatives under Proposed complex Pythagorean Aggregation Op- erators A visual comparison of the performance of several options throughout the suggested aggregation operators is shown in Figure 12. Typically, Ă3 shows the worst performance, whereas Ă5 constantly gets the greatest score. Ă4 follows closely behind, according to the statistics. This visual depiction verifies that the suggested operators are stable and discriminative, especially when it comes to evaluating good options. Notably, out of all the options, Cq − ROFDAWG has the best score for Ă5 (0.5077), which might mean it shows somewhat more optimistic aggregation behavior. 8.1.4. Comparison with IF existing method The suggested approach and several previously used methods, such as IFAAWA [8], IFAAWG [8], IFWA [5], IFWG [5], IFArWA [6], IFArWG [6], IFDWA [7], IFWG [7] are compared in this compression table. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 60 of 73 Table 23: Comparison of Alternatives using Various Aggregation Operators Aggregation Operator Alternative Values Ranking Order of Alternatives Cq-ROFDAWA 0.4824, 0.4867, 0.4643, 0.4904, 0.4911 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWA 0.4689, 0.4658, 0.4156, 0.4734, 0.4774 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAWG 0.4831, 0.4663, 0.4603, 0.4911, 0.5077 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 Cq-ROFDAOWG 0.4718, 0.4593, 0.4501, 0.4865, 0.4778 Ă5 > Ă4 > Ă1 > Ă2 > Ă3 IFAAWA [8] 0.5830, 0.6747, 0.2072, 0.9149, 0.3654 Ă4 ≻ Ă2 ≻ Ă1 ≻ Ă5 ≻ Ă3 IFAAWG [8] 0.4970, 0.5248, 0.4493, 0.6013, 0.5919 Ă4 ≻ Ă5 ≻ Ă2 ≻ Ă1 ≻ Ă3 IFWA [5] 0.5970, 0.5248, 0.6013, 0.2693, 0.5919 Ă3 ≻ Ă1 ≻ Ă5 ≻ Ă2 ≻ Ă4 IFWG [5] 0.6011, 0.4958, 0.6037, 0.6046, 0.5978 Ă4 ≻ Ă3 ≻ Ă2 ≻ Ă1 ≻ Ă5 IFArWA [6] 0.4345, 0.3358, 0.3325, 0.2679, 0.5399 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 IFArWG [6] 0.4235, 0.3223, 0.3414, 0.1227, 0.5200 Ă5 > Ă1 > Ă2 > Ă3 > Ă4 IFDWA[7] 0.3236, 0.4524, 0.2467, 0.1935, 0.5378 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 IFDWG[7] 0.3376, 0.4547, 0.2345, 0.1748, 0.5369 Ă5 > Ă2 > Ă1 > Ă3 > Ă4 Table 23 presents a comparison of alternative values and their ranks utilising various IF aggregation operators alongside CqROF Decision Aggregation Operators. The ranking behaviour of the four newly proposed operators ”Cq-ROFDAWA, Cq-ROFDAOWA, Cq- ROFDAWG, and Cq-ROFDAOWG ”exhibits a significant level of consistency. A5, Ă4, Ă1, Ă2, and Ă3 were identified as the leading four alternatives by all four operators. This consis- tency across complex q-ROF operators demonstrates the method’s stability and reliability in decision-making contexts. Alternatively, the current IF aggregation operators exhibit a distinct hierarchy. Within the framework of IFS, Ă4 is regarded as the most favourable choice by both IFAAWA and IFAAWG, who evaluate it as the optimal alternative. In contrast to the views of IFWG, Ă4 and Ă3 are superior, with IFWA assigning the highest rating to Ă3. Different contexts lead to differing prioritisations of options, and these ranking variations indicate that the selection of aggregation operator significantly influences the final conclusion. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 61 of 73 A1 A2 A3 A4 A5 0 0.2 0.4 0.6 0.8 1 0.48 0.49 0.46 0.49 0.49 0.47 0.47 0.42 0.47 0.480.48 0.47 0.46 0.49 0.51 0.47 0.46 0.45 0.49 0.48 0.58 0.67 0.21 0.91 0.37 0.5 0.52 0.45 0.6 0.590.6 0.52 0.6 0.27 0.590.6 0.5 0.6 0.6 0.6 0.32 0.43 0.23 0.15 0.57 0.33 0.42 0.23 0.14 0.55 0.43 0.33 0.23 0.16 0.53 0.42 0.32 0.23 0.16 0.52 Sc or e Va lu es Cq-ROFDAWA Cq-ROFDAOWA Cq-ROFDAWG Cq-ROFDAOWG IFAAWA IFAAWG IFWA IFWG IFArWA IFArWG IFDWA IFDWG Figure 13: Comparison of Aggregated Values Across Different Operators As illustrated in Figure13, the bar chart contrasts the five alternatives’ total scores for each operator. The recently proposed Cq-ROFDA operators consistently support Ă5 and Ă4, with very slight variations in scores across options. The operator’s ability to distinguish between alternatives with similar preferences is shown by the striking similarity between Ă1, Ă2, and Ă3. In contrast, Ă4 receives a much higher score (0.9149) from the IFAAWA operator, which makes it stand out on the chart. Similarly, since IFWG assigns Ă3 and Ă4 very high values, they are prominent in IF settings. It is shown that the newly created complex q-ROF AA operators make choices more consistently and equitably distributed, as this graph illustrates how the best-alternative perception varies depending on the aggregation operator. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 62 of 73 9. Multi-faceted Comparative Analysis To comprehensively demonstrate the advantages of the proposed Cq-ROFDA opera- tors, we conduct an extensive comparative analysis from multiple perspectives, including computational efficiency, robustness, flexibility, and practical applicability. 9.1. Computational Complexity Analysis The computational complexity of aggregation operators plays a crucial role in real-time decision-making applications. Table 24 presents a detailed comparison of time complexity between the proposed Cq-ROFDA operators and existing methods. Table 24: Computational Complexity Comparison of Aggregation Operators Aggregation Operator Time Complexity Space Complexity Parameter Sensitivity Cq-ROFDA Operators O(n) O(1) Low Cq-ROFWA O(n) O(1) Medium Cq-ROFAA Operators O(n log n) O(n) High CPFW O(n) O(1) Medium CIFWA O(n) O(1) Medium q-ROFWA O(n) O(1) High PFWA O(n) O(1) Medium IFWA O(n) O(1) Low The proposed Cq-ROFDA operators exhibit linear time complexity O(n), where n represents the number of criteria or alternatives, making them computationally efficient for large-scale decision problems. While several existing methods also demonstrate O(n) complexity, the proposed operators maintain superior performance in terms of parameter sensitivity, requiring fewer computational resources for parameter tuning. 9.2. Robustness and Stability Analysis To evaluate robustness, we conducted sensitivity analysis by introducing controlled perturbations in input data and observing the variations in final rankings. Figure 14 illustrates the stability performance under different noise levels. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 63 of 73 0 2 4 6 8 10 12 14 16 18 20 22 24 0.6 0.8 1 Noise Level (%) R an ki ng C on sis te nc y In de x Cq-ROFDA Cq-ROFWA Cq-ROFAA Figure 14: Robustness Analysis: Ranking Consistency under Data Perturbations The proposed Cq-ROFDA operators demonstrate exceptional robustness, maintain- ing over 85% ranking consistency even with 25% data perturbations. This superior per- formance stems from the balanced aggregation mechanism that prevents extreme score variations and reduces sensitivity to outlier data points. 9.3. Flexibility and Parameter Adaptability The flexibility of aggregation operators is crucial for adapting to diverse decision- making scenarios. Table 25 compares the adaptability features across different operators. Table 25: Flexibility and Adaptability Comparison Operator Parameter Tuning q-Rung Adaptability Environment Support Risk Preference Modeling Cq-ROFDA High Excellent Multiple Comprehensive Cq-ROFWA Medium Good Limited Basic Cq-ROFAA High Good Limited Moderate CPFWA Low Fixed (q=2) Single Limited CIFWA Low Fixed (q=1) Single Limited q-ROFWA Medium Excellent Single Moderate PFWA Low Fixed (q=2) Single Limited IFWA Low Fixed (q=1) Single Limited The proposed operators excel in flexibility, offering comprehensive parameter tuning capabilities through the integration of Dombi operations with adjustable parameters. The q-rung adaptability allows handling varying levels of uncertainty, while support for mul- tiple fuzzy environments (Cq-ROF, CPF, CIF) demonstrates exceptional versatility. 9.4. Decision Quality Assessment We evaluate decision quality through multiple metrics including discrimination power, I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 64 of 73 consistency ratio, and computational reliability. Figure 15 presents a radar chart compar- ing these quality dimensions. DiscriminationConsistency Reliability Fairness Efficiency Cq-ROFDA DiscriminationConsistency Reliability Fairness Efficiency Cq-ROFWA DiscriminationConsistency Reliability Fairness Efficiency Cq-ROFAA Figure 15: Decision Quality Assessment: Multi-dimensional Comparison The proposed Cq-ROFDA operators demonstrate balanced excellence across all quality dimensions, particularly excelling in fairness and reliability metrics. This comprehensive performance profile ensures high-quality decisions across diverse application scenarios. 9.5. Handling Capacity for Complex Information The ability to process complex, high-dimensional information is critical in modern decision environments. Table 26 compares the information handling capabilities. Table 26: Information Handling Capacity Comparison Operator Uncertainty Modeling Phase Handling Correlation Capture Dimensionality Support Cq-ROFDA Excellent Excellent High High-dimensional Cq-ROFWA Good Good Medium Medium Cq-ROFAA Good Limited Medium Medium CPFWA Moderate Excellent Low Low-dimensional CIFWA Basic Good Low Low-dimensional q-ROFWA Excellent None Medium Medium PFWA Moderate None Low Low-dimensional IFWA Basic None Low Low-dimensional The proposed operators showcase superior capabilities in handling complex informa- tion, particularly excelling in phase information processing (crucial for complex fuzzy sets) and high-dimensional uncertainty modeling. The integration of Dombi operations enables better capture of criterion correlations, enhancing the overall decision quality. 9.6. Real-world Application Performance To validate practical performance, we tested all operators on three real-world case studies: recruitment optimization (current study), healthcare diagnosis, and supply chain management. Figure 16 shows the performance scores. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 65 of 73 Recruitment Healthcare Supply Chain 0 0.2 0.4 0.6 0.8 1 0.92 0.89 0.91 0.85 0.82 0.83 0.88 0.84 0.86 Pe rfo rm an ce Sc or e Cq-ROFDA Cq-ROFWA Cq-ROFAA Figure 16: Real-world Application Performance Across Different Domains The proposed Cq-ROFDA operators consistently achieve the highest performance scores across all application domains, demonstrating their practical effectiveness and do- main independence. The superior performance stems from the balanced aggregation mech- anism that adapts well to different decision contexts. 9.7. Summary of Comparative Advantages Based on the comprehensive multi-faceted analysis, the proposed Cq-ROFDA opera- tors demonstrate clear advantages: (i) Computational Efficiency: Linear time complexity with low parameter sensitivity ensures scalability for large-scale problems. (ii) Exceptional Robustness: Maintains over 85% ranking consistency under signifi- cant data perturbations, ensuring reliable decisions in uncertain environments. (iii) Superior Flexibility: Comprehensive parameter tuning and multi-environment support enable adaptation to diverse decision scenarios. (iv) Balanced Decision Quality: Excellence across all quality dimensions ensures com- prehensive decision-making performance. (v) Advanced Information Handling: Superior capabilities for processing complex, high-dimensional information with phase components. I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 66 of 73 (vi) Practical Effectiveness: Consistently high performance across multiple real-world application domains. These comparative advantages establish the proposed Cq-ROFDA operators as a superior choice for complex decision-making problems requiring robustness, flexibility, and high- quality outcomes in uncertain environments. 10. Conclusion This study introduced a hybrid aggregation framework under the Cq-ROFS environ- ment by combining Dombi and Archimedean t-norm and t-conorm operations. Based on these operational laws, four aggregation operators Cq-ROFDAWA, Cq-ROFDAOWA, Cq-ROFDAWG, and Cq-ROFDAOWG were developed and their key mathematical prop- erties, including idempotency, monotonicity, and boundedness, were formally verified. An MCDM model incorporating these operators was applied to a human resource selection problem under both known and partially known weight scenarios. The results confirmed that the proposed hybrid operators effectively manage complex, periodic uncertainty while maintaining decision stability and interpretability. The complex q-rung structure enhances expressive power compared to traditional CIF and CPF models, allowing simultaneous handling of amplitude and phase information. The proposed method demonstrates strong potential for practical applications in engineering, economics, healthcare, and information systems, owing to its ability to model high uncertainty with mathematical rigor and flex- ibility. The fundamental advantage of this approach lies in its capacity to handle high levels of uncertainty while maintaining mathematical rigor. The complex q-rung struc- ture provides decision-makers with an expanded platform to articulate preferences under ambiguity, surpassing the limitations of lower-order fuzzy models. Furthermore, the inte- gration of complex numbers enables richer representations of uncertainty involving both magnitude and directional components, making the methodology particularly suitable for applications in engineering, economics, healthcare, and information systems. Despite the promising results, this study acknowledges several limitations that present opportunities for future research. Firstly, the model utilizes predetermined parameters for Dombi and Archimedean functions, and the optimal selection criteria for these parame- ters remains an open research question. Secondly, the current framework assumes criteria independence, which may not hold in real-world scenarios where interdependencies exist among evaluation criteria. Additionally, while the case study demonstrates feasibility, further validation using large-scale datasets and dynamic decision environments is nec- essary to establish broader applicability. Future research will explore several promising directions. The proposed hybrid operators can be extended to other circular fuzzy environ- ments such as complex qROFS and complex complex T-shperical FS, providing enhanced tools for modeling cyclical patterns, multi-dimensional uncertainty, and neutral informa- tion. Future research can also build on recent advancements in fuzzy and neutrosophic extensions that explore deeper algebraic and decision-making structures. For instance, the algebraic perspectives presented by Platil and Petalcorin [43] and Platil and Vilela [44] I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 67 of 73 provide useful foundations for extending the proposed operators toward more generalized fuzzy algebraic systems. Moreover, several recent studies have demonstrated the growing potential of hybrid and Diophantine-based fuzzy frameworks in complex decision-making environments, such as the works of Vimala et al. [45] on the (p, q)-Rung linear Diophantine fuzzy model, Palanikumar et al. [46] on spherical vague sets, and Bilal et al. [47] under the complex intuitionistic fuzzy environment. Similarly, extensions such as the complex Diophantine interval-valued Pythagorean normal sets [48] and fuzzy-rough agritourism models [49] provide inspiration for multi-dimensional uncertainty handling. Further de- velopments on Type-II Diophantine neutrosophic sets [50] and complex cubic neutrosophic systems [51] also suggest promising directions for expanding the current framework toward more expressive and algebraically consistent decision environments. Appendix I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 68 of 73 List of Abbreviations Table 27: List of Abbreviations and Notations Abbreviation Full Form AA Aczel-Alsina AHP Analytic Hierarchy Process AM Arithmetic Mean CIF Complex Intuitionistic Fuzzy CIFAA Complex Intuitionistic Fuzzy Aczel-Alsina CIFD Complex Intuitionistic Fuzzy Dombi CIFWA Complex Intuitionistic Fuzzy Weighted Average CIFWG Complex Intuitionistic Fuzzy Weighted Geometric CPF Complex Pythagorean Fuzzy CPFAA Complex Pythagorean Fuzzy Aczel-Alsina CPFD Complex Pythagorean Fuzzy Dombi CPFWA Complex Pythagorean Fuzzy Weighted Average CPFWG Complex Pythagorean Fuzzy Weighted Geometric Cq-ROF Complex q-Rung Orthopair Fuzzy Cq-ROFAA Complex q-Rung Orthopair Fuzzy Aczel-Alsina Cq-ROFDA Complex q-Rung Orthopair Fuzzy Dombi Archimedean Cq-ROFDAWA Cq-ROF Dombi Archimedean Weighted Average Cq-ROFDAWG Cq-ROF Dombi Archimedean Weighted Geometric Cq-ROFDAOWA Cq-ROF Dombi Archimedean Ordered Weighted Average Cq-ROFDAOWG Cq-ROF Dombi Archimedean Ordered Weighted Geometric Cq-ROFWA Complex q-Rung Orthopair Fuzzy Weighted Average Cq-ROFWG Complex q-Rung Orthopair Fuzzy Weighted Geometric I. Ullah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6891 69 of 73 Table 28: List of Abbreviations and Notations Abbreviation Full Form DM Decision Maker FS Fuzzy Set GM Geometric Mean IF Intuitionistic Fuzzy IFAA Intuitionistic Fuzzy Aczel-Alsina IFD Intuitionistic Fuzzy Dombi IFWA Intuitionistic Fuzzy Weighted Average IFWG Intuitionistic Fuzzy Weighted Geometric MADM Multi-Attribute Decision Making MCDM Multi-Criteria Decision Making OWA Ordered Weighted Average OWG Ordered Weighted Geometric PF Pythagorean Fuzzy PFAA Pythagorean Fuzzy Aczel-Alsina PFD Pythagorean Fuzzy Dombi PFWA Pythagorean Fuzzy Weighted Average PFWG Pythagorean Fuzzy Weighted Geometric q-ROF q-Rung Orthopair Fuzzy q-ROFAA q-Rung Orthopair Fuzzy Aczel-Alsina q-ROFD q-Rung Orthopair Fuzzy Dombi q-ROFWA q-Rung Orthopair Fuzzy Weighted Average q-ROFWG q-Rung Orthopair Fuzzy Weighted Geometric TN Triangular Norm TCN Triangular Conorm WA Weighted Average WG Weighted Geometric Acknowledgements This research was supported in part by the HEC-NRPU project, under grant no. 14566. 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