EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6227 ISSN 1307-5543 – ejpam.com Published by New York Business Global Triangular Intuitionistic Fuzzy Frank Aggregation for Efficient Renewable Energy Project Selection A. Fahmi1, A. Hashmi2, Aziz Khan3, Aiman Mukheimer3, Thabet Abdeljawad3,5,∗, Rajermani Thinakaran4 1 Department of Mathematics and Statistics, Faculty of Sciences, The University of Faisalabad, Pakistan 2 Faculty of Management Sciences, The University of Central Punjab, Lahore City, Pakistan 3 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia 4 Faculty of Data Science and Information Technology, INTI International University, Negeri Sembilan, Malaysia 5 Department of Mathematics and Applied Mathematics Sefako Makgatho Health Sciences University Garankuwa, Medusa 0204, South Africa Abstract. Optimizing renewable energy project selection presents a complex challenge that demands intelligent decision-making under conditions of uncertainty. In multi-criteria decision-making (MCDM), the need to balance numerous conflicting factors makes these techniques invaluable for effective project evaluation and selection. This paper introduces a novel Triangular Intuitionistic Fuzzy Frank (TIFF) framework, which integrates triangular intuitionistic fuzzy averaging and geometric aggregation operators to enhance decision-making in renewable energy project assessment. We develop several new aggrega- tion operators, including: Triangular Intuitionistic Fuzzy Frank Weighted Averaging (TIFFWA), Ordered Weighted Averaging (TIFFOWA), Hybrid Averaging (TIFFHA), Weighted Geometric (TIFFWG), Or- dered Weighted Geometric (TIFFOWG), and Hybrid Geometric (TIFFHG). These operators, built upon the Frank t-norm and t-conorm, enable more accurate and adaptive evaluations by effectively managing varying levels of uncertainty. In addition, novel scoring and precision functions are introduced to further refine the decision-making process, yielding more reliable outcomes. A step-by-step methodology is pre- sented for applying the TIFF approach to renewable energy project selection, providing clear guidance for practical implementation. To validate the method, a numerical case study is conducted, demonstrating the superior performance of the TIFF framework compared to existing techniques. The results under- score the method’s efficiency, adaptability, and practical value as a robust tool for optimizing renewable energy project decisions under uncertainty. 2020 Mathematics Subject Classifications: 03E72, 90B50, 62C86 Key Words and Phrases: Fuzzy set, Multi-attribute decision making, Triangular Fuzzy Frank Aggre- gation, energy efficiency 1. Introduction The drive for sustainable living and growing environmental concerns have increased the need for energy efficiency in buildings, particularly smart houses [1–19]. The development of AI technology has opened up new avenues for energy consumption optimization. However, con- ventional optimization techniques are severely hampered by the complexity of building systems, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6227 Email addresses: aliyafahmi@gmail.com (A. Fahmi), arshiahashmee@gmail.com (A. Hashmi), akhan@psu.edu.sa (A. Khan), tabdeljawad@psu.edu.sa (T. Abdeljawad), rajermani.thina@newinti.edu.my (R. Thinakaran) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 2 of 37 which include varying occupancy levels and erratic weather. Current methods, such as those based on [20] static models or heuristic algorithms, frequently struggle to adjust to changes in the environment in real time, which restricts their capacity to achieve consistent and ideal energy use [21]. Fuzzy logic was the foundation of many AI-driven techniques for building en- ergy optimization. Although these approaches were successful in controlling uncertainty, they usually depended on oversimplified models that were unable to fully represent the intricacy of smart home environments [22–32]. Systems such as fuzzy-based decision frameworks for as- sessing HVAC system performance [14] and the adaptive neuro-fuzzy inference system for load forecasting [33] had trouble managing uncertainty in real-time energy management, especially when taking dynamic factors like weather or human behavior into account [33–44]. Figure 1 is efficient renewable energy project selection given as The concept of a fuzzy set, which can address the issue of uncertainty in various contexts, was developed by Zadeh [16]. Sets with degrees of membership are called FSs, and member- ship functions with values in the interval [0, 1] are allowed according to FS theory. However, because there could be some reluctant degree, it might not always hold in real life that the degree of nonmembership function is equal to one minus membership function. The intuition- istic FS was initially proposed by Atanassov [45]. Truth and falsity grades are assigned to the constituents of IFSs. The degree of hesitancy about an element’s truth and falsity grades within a set must be represented using IFS. IFSs are used in numerous real-world scenarios to solve issues. One example of this is when we toss a coin; there are two possible outcomes, head or tail. Expert opinions were represented using the Basic Uncertain Information (BUI) technique, and in a group decision-making context, these opinions were combined using aggre- gation operations [29]. The coin will show either way at a time, but not both ways at once. The theory of Pythagorean fuzzy sets to address these types of issues [5]. Because PyFS and IFS have the same structure but different conditions, PyFS is the improved version of IFSs that have overcome their limitations. IFS and PyFS are often closely related. Compared to IFSs, PyFS allows us to measure uncertainty more precisely and adequately. Aggregation operators [28, 40, 43, 46], similarity measures [1, 29, 44, 47], decision-making approaches [12, 48, 49], and various sorts of procedures was a few examples of uses of FS and its extensions. By clustering fuzzy c-numbers, Xu and Li [10] prophesied the reversion of a fuzzy time sequence. This study offers a class of fuzzy clustering procedures specifically tailored for processing fuzzy data [42]. Fuzzy c-number clusterings are novel algorithms designed to handle different kinds of fuzzy data A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 3 of 37 more efficiently, fuzzy number forms such as conventional [50–58], trapezoidal, LR-type, and triangular fuzzy numbers [14]. Akram et al. [59, 60] introduced several PFS-imitable applica- tions. A thorough case study on choosing medical subject experts was used to test an integrated MCDM algorithm that was created using the suggested operators and the combined criterion weight determination model [46, 61–63]. As a result, it can represent the relative relevance of the supplied Pythagorean fuzzy argument and its ordered position. Several interval-valued Fur- thermore, we provide some interval-valued Pythagorean fuzzy point weighted averaging (IVPF- PWA) operators that can modify the degree of the aggregated arguments with a parameter by combining the interval-valued Pythagorean fuzzy point operators with the IVPFWA operator. In order to address multi-attribute group decision-making under interval-valued Pythagorean fuzzy information, Rahman et al. [4] presented an operator. The value and compatibility of the discussed methodologies and decision support systems were examined [1, 2]. The general- ized IVPNFWG operator were presented by Yang et al. [14] to aggregate IVPNF information. Petchimuthu et al. [47], a new concept of Pythagorean neutrosophic normal interval-valued weighted averaging (PNSNIVWA), Pythagorean neutrosophic normal interval-valued weighted geometric (PNSNIVWG), and generalized Pythagorean neutrosophic normal interval-valued weighted averaging (GPNSNIVWA) as well as generalized Pythagorean neutrosophic normal interval-valued weighted geometric (GPNSNIVWG) were discussed by Palanikumar et al. [43] and Peng et al.[44]. Figure 2 gives methods for enhancing as Based on the consistency of the InPLPR, Wang et al. [7] created a decision-making method that includes estimating missing data, enhancing consistency, and evaluating the options. An incomplete probabilistic linguistic term set was presented by Liu et al. [41]. The distinction between absolute and relative knowledge distances in the structural characteristics of hierarchical clustering was examined by Lian et al. [? ]. Anusha et al. [62] covered the hybridizations in addition to providing an extension of the MSM operators and their requests based on q-rung probabilistic dual hesitant fuzzy sets. Depending on the evaluation values of each choice, it is frequently possible to examine many possibilities to arrive at a comprehensive assessment result, for example by using MADM. A unique approach to selecting robotic systems for homogenous group DM was presented by Bairagi [64]. These operators were utilized to devise a method for handling group decision-making with CPF infor- mation, introduced the Artificial intelligence applied in DM to choose a maintenance approach and other research [7–12, 16, 40, 42, 65]. Figure 3 gives the as MCDM method below is This manuscript is structured as follows: Section 2 introduces the concept of IFSs. Section 3 presents operational rules for TIFSs, including algebraic and Frank operational laws. In Section 4, we propose the TIFFWA, TIFFOWA, TIFFHWA, TIFFWG, TIFFOWG and TIFFHWG operators. Section 5 outlines an optimized MCDM process for the TIF model. Section 6, define the case study and a comparative analysis. Finally, Section 7 concludes the study. 1.1. Contribution of study The application of triangular intuitionistic Fuzzy Frank Aggregation Operators (TIFFAO) has significantly advanced the efficient selection of renewable energy projects by effectively addressing uncertainty and optimizing project evaluations. The main contributions of this study are as follows: (a) Operational laws and triangular intuitionistic Fuzzy Numbers (TIFN) are defined to provide a solid mathematical basis for selecting renewable energy projects. (b) A novel accuracy and scoring function is introduced to improve the precision of decision- making processes in renewable energy project evaluations. (c) Various TIFN aggregation operators are introduced, such as TIFFWA, TIFFOWA, TIFFHWA, TIFFWG, TIFFOWG and TIFFHWG, tailored for Multi-Criteria Decision-Making (MCDM) in renewable energy project selection. These operators enhance the adaptability and A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 4 of 37 effectiveness of evaluating projects under dynamic conditions. (d) The TIFFAO framework efficiently processes inconsistent and diverse data, leading to more accurate selection of renewable energy projects, with better forecasting and cost man- agement. This method is particularly useful in evaluating projects where factors like budget, environmental impact, and technological feasibility vary. (e) The TIFFAO approach excels in managing multi-dimensional evaluations for renewable energy projects, where various factors such as energy efficiency, sustainability, and stakeholder needs must be balanced. By optimizing the selection process, this method minimizes uncertain- ties and maximizes the long-term benefits of the chosen projects. This study offers a robust methodology that significantly enhances the selection process for renewable energy projects, providing a more adaptive, efficient, and data-driven approach to making informed decisions. 1.2. Motivation This study focuses on the challenge of selecting efficient renewable energy projects, providing a foundation for future work in the development of intelligent decision-making systems for energy project evaluation. The triangular intuitionistic Fuzzy Set (TIFS) framework has been designed to allow energy A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 5 of 37 project evaluators to offer insights more easily. By covering a wider range of information un- certainty, preventing the loss of critical data when converting qualitative project characteristics into quantitative information. The case study presented can be adapted for use by energy project selection teams, helping them improve decision-making processes through structured evaluation. This method provides a clear path from expert opinions to actionable project selection solutions. Renewable energy project evaluation data is often complex, uncertain, and incomplete due to varying environmental conditions, technological advancements, and stakeholder considera- tions. Conventional selection methods struggle with such uncertainty. The triangular intuition- istic Fuzzy Frank Aggregation Operators (TIFFAO) effectively manage and integrate this data, significantly improving the reliability of project selection decisions and enhancing the overall efficiency of renewable energy project evaluation systems. 1.3. Novelty In this article, we aim to design the following: i. To define advanced operational laws for triangular intuitionistic Fuzzy Frank statistics, extending traditional operational laws to effectively address the complexity of renewable energy project selection and assess its mathematical properties. ii. To introduce innovative aggregation operators, such as triangular intuitionistic Fuzzy Frank Aggregation Operators, specifically tailored for optimizing the selection process of re- newable energy projects, improving decision-making in dynamic evaluation scenarios. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 6 of 37 iii. To propose a Multi-Criteria Decision-Making (MCDM) technique using triangular in- tuitionistic fuzzy sets (TIF) that helps evaluate various factors influencing renewable energy project selection. iv. To apply the proposed methodology to solve a real-world problem in renewable energy project selection, demonstrating its practical value and efficiency in real-world decision-making scenarios. v. To validate the robustness of the proposed approach through a sensitivity analysis, assessing how different influencing factors impact the overall project selection process. The abbreviation of table 1 is written below. Abbreviations Full Name IFFNs Intuitionistic fuzzy frank numbers TIFN Triangular intuitionistic fuzzy number TIFFAO Triangular intuitionistic fuzzy frank aggregation operator TIFFWA Triangular intuitionistic fuzzy frank weighted average TIFFOWA Triangular intuitionistic fuzzy frank ordered weighted average TIFFHWA Triangular intuitionistic fuzzy frank hybrid weighted average TIFFWG Triangular intuitionistic fuzzy frank weighted geometric TIFFOWG Triangular intuitionistic fuzzy frank ordered weighted geometric TIFFHWG Triangular intuitionistic fuzzy frank hybrid weighted geometric 2. Basic ideas Definition 1. [16] Considering that Φ ̸= X and let is use a fuzzy set γ = { 〈 x, µγ(x) 〉 : x ∈ X } . An element x in X is represented by the membership function, µγ(x) is a mapping from X to [0, 1]. Definition 2. [31] Let a1 = [L1, κ1] and a2 = [L2, κ2] be the IFSs based on frank operators and λ > 0, then a1 ⊕ a2 = 1 − log ( 1+ (β1−L1 −1)(β1−L2 −1) β−1 ) β , log ( 1+ (βk2 −1)(βκ2 −1) β−1 ) β  ; a1 ⊗ a2 = log ( 1+ (β L− 1 −1)(βL2 −1) β−1 ) β , 1 − log ( 1+ (β1−K1 −1)(β1−κ2 −1) β−1 ) β  ; aλ 1 = log ( 1+ (βL1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−κ1 −1)λ (β−1)λ−1 ) β  ; λa1 = 1 − log ( 1+ (β1−L1 −1)λ (β−1)λ−1 ) β , log ( 1+ (β1−κ1 −1)λ (β−1)λ−1 ) β  . Definition 3. [11] Let a = [ς, χ] be the IFSs, then the score function is a = ςα − χα. Definition 4. [11] Let a = [ς, χ] be the IFSs, then the accuracy function is a = ςα + χα. 3. TIFN and operational laws on Frank The section address the operational laws of the frank t-norm and t-conorm. The frank operational laws are a collection of axioms that control how operations in TIF logic, such as t-norms and t-conorms. Definition 5. Let a1 = [ [C1, D1, E1], [G1, H1, L1] ] and a2 = [ [C2, D2, E2], [G2, H2, L2] ] be two TIFFNs and λ > 0, then A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 7 of 37 a1 ⊕ a2 =   1 − log ( 1+ (β1−C1 −1)(β1−C2 −1) β−1 ) β , 1 − log ( 1+ (β1−D1 −1)(β1−D2 −1) β−1 ) β , 1 − log ( 1+ (β1−E1 −1)(β1−E2 −1) β−1 ) β  ,  log ( 1+ (βG1 −1)(βG2 −1) β−1 ) β , log ( 1+ (βH1 −1)(βH2 −1) β−1 ) β , log ( 1+ (βL1 −1)(βL2 −1) β−1 ) β   ; a1 ⊗ a2 =   log ( 1+ (βC1 −1)(βC2 −1) β−1 ) β , log ( 1+ (βD1 −1)(βD2 −1) β−1 ) β , log ( 1+ (βE1 −1)(βE2 −1) β−1 ) β  ,  1 − log ( 1+ (β1−G1 −1)(β1−G2 −1) β−1 ) β , 1 − log ( 1+ (β1−H1 −1)(β1−H2 −1) β−1 ) β , 1 − log ( 1+ (β1−L1 −1)(β1−L2 −1) β−1 ) β   ; aλ 1 =   log ( 1+ (βC1 −1)λ (β−1)λ−1 ) β , log ( 1+ (βD1 −1)λ (β−1)λ−1 ) β , log ( 1+ (βE1 −1)λ (β−1)λ−1 ) β  ,  1 − log ( 1+ (β1−G1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−H1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−L1 −1)λ (β−1)λ−1 ) β   ; λa1 =   1 − log ( 1+ (β1−C1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−D1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−E1 −1)λ (β−1)λ−1 ) β  ,  log ( 1+ (β1−G1 −1)λ (β−1)λ−1 ) β , log ( 1+ (β1−H1 −1)λ (β−1)λ−1 ) β , log ( 1+ (β1−L1 −1)λ (β−1)λ−1 ) β   Definition 6. The TIFNs are a = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] , score function I is defined as:I = ⟨[Cj+Dj+Ej ]−[Gj+Hj+Lj ]⟩ 6 . Definition 7. The TIFNs are a = [ Cj , Dj , Ej , Gj , Hj , Lj ] , accuracy function M is defined as:M = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 8 of 37 ⟨[Cj+Dj+Ej ]+[Gj+Hj+Lj ]⟩ 6 . TIFFWA, TIFFOWA, TIFFHA, TIFFWG, TIFFOWG, and TIFFHG are the aggregation operators shown in Figure 4. The six suggested aggregation operations for processing decision- making data in the form of Triangular Intuitionistic Fuzzy Numbers (TIFNs) within the Frank t-norm framework are depicted in this image. These operators were created especially to combine several criteria or professional judgments while taking into account the degrees of membership, non-membership, and hesitancy that are present in intuitionistic fuzzy settings. Figure 4 is given as below 4. TIFNs aggregation operator based on frank This section presents the TIFFWA, TIFFOWA, TIFFWHA, TIFFWG, TIFFOWG and TIFFHG operators new methods for TIFNs with several noteworthy characteristics based on frank operators. 4.1. TIFFWA operator Definition 8. Let kj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the gathering of TIFNs and u = (u1, u2, ..., um)T is the weight vector with uj ∈ [0, 1] and m∑ j=1 uj = 1. Then TIFFWA (k1, k2, ..., km) = m⊕ j=1 ujkj is said TIFFWA operator. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 9 of 37 Theorem 1. The collection of TIFNs are aj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and λ = (λ1, λ2, ..., λn)T is the weight vector with λj ∈ [0, 1] and n∑ j=1 λj = 1. Then it is said TIFFWA operator and TIFFWA(a1, a2, ..., an) =  1 − log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   . Proof. Since n = 1, a1λ1 =  1 − log ( 1+ (β1−C1 −1)λj (β−1)λj −1 ) β , 1 − log ( 1+ (β1−D1 −1)λj (β−1)λj −1 ) β , 1 − log ( 1+ (β1−E1 −1)λj (β−1)λj −1 ) β  , log ( 1+ (β1−G1 −1)λj (β−1)λj −1 ) β , log ( 1+ (β1−H1 −1)λj (β−1)λj −1 ) β , log ( 1+ (β1−L1 −1)λj (β−1)λj −1 ) β   ; a2λ2 =  1 − log ( 1+ (β1−C2 −1)λj (β−1)λj −1 ) β , 1 − log ( 1+ (β1−D2 −1)λj (β−1)λj −1 ) β , 1 − log ( 1+ (β1−E2 −1)λj (β−1)λj −1 ) β  , log ( 1+ (β1−G2 −1)λj (β−1)λj −1 ) β , log ( 1+ (β1−H2 −1)λj (β−1)λj −1 ) β , log ( 1+ (β1−L2 −1)λj (β−1)λj −1 ) β   n = k A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 10 of 37  1 − log 1+ k∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ k∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ k∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ k∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ k∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ k∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   n = k + 1  1 − log 1+ k+1∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ k+1∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ k+1∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ k+1∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ k+1∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ k+1∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   Theorem 2. (Idempotency):If Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] for all N = 1, 2, 3, ..., m, then TIFFWA(Z̃V , Z̃V , Z̃V , ..., Z̃V ) = Z̃V . A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 11 of 37 Proof. Since Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ]   1 − log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   =   1 − log 1+ (β 1−Cj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , 1 − log 1+ (β 1−Dj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , 1 − log 1+ (β 1−Ej −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β  ,  log 1+ (β 1−Gj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , log 1+ (β 1−Hj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , log 1+ (β 1−Lj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β   =  1 − log ( 1+ (β 1−Cj −1)1 (β−1)1−1 ) β , 1 − log ( 1+ (β 1−Dj −1)1 (β−1)1−1 ) β , 1 − log ( 1+ (β 1−Ej −1)1 (β−1)1−1 ) β  ,log ( 1+ (β 1−Gj −1)1 (β−1)1−1 ) β , log ( 1+ (β 1−Hj −1)1 (β−1)1−1 ) β , log ( 1+ (β 1−Lj −1)1 (β−1)1−1 ) β   =  [ 1 − log(1+(β1−Cj −1)1) β , 1 − log(1+(β1−Dj −1)1) β , 1 − log(1+(β1−Ej −1)1) β ] ,[ log(1+(β1−Gj −1)1) β , log(1+(β1−Hj −1)1) β , log(1+(β1−Lj −1)1) β ]  A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 12 of 37 Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] Theorem 3. (Boundedness):If Y = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be a collection of TIFSs. If Y − = min(Y1, Y2, ..., Ym), Y + = max(Y1, Y2, ..., Ym), then Y − ≤ TIFFWA(Y1, Y2, ..., Ym) ≤ Y +. Proof. Y − = min(Y1, Y2, ..., Ym) = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and Y + = max(Y1, Y2, ..., Ym) = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] Since minj(Cj) ≤maxj(Cj);minj(Dj) ≤maxj(Dj);minj(Ej) ≤maxj(Ej) and minj(Gj) ≤maxj(Gj);minj(Hj) ≤maxj(Hj);minj(Lj) ≤maxj(Lj) Which implies that 1 − log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β ≥ 1 − log 1+ n∏ j=1 (β 1−(min Cj )−1)λj (β−1)λj −1  β ; 1 − log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β ≥ 1 − log 1+ n∏ j=1 (β 1−(min Dj )−1)λj (β−1)λj −1  β ; 1 − log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β ≥ 1 − log 1+ n∏ j=1 (β 1−(min Ej )−1)λj (β−1)λj −1  β and log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β ≥ log 1+ n∏ j=1 (β 1−(max Gj )−1)λj (β−1)λj −1  β ; log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β ≥ log 1+ n∏ j=1 (β 1−(max Hj )−1)λj (β−1)λj −1  β ; log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β ≥ log 1+ n∏ j=1 (β 1−(max Lj )−1)λj (β−1)λj −1  β TIFFWA(Y1, Y2, ..., Ym) = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 13 of 37 S(Y ) = [[Cj+Dj+Ej ]−[Gj+Hj+Lj ]] 6 ≤ [[maxj Cj+maxj Dj+maxj Ej ]−[minj Gj+minj Hj+minj Lj ]] 6 = S(Y −) S(Y ) ≤ S(Y −) and S(Y ) = [[Cj+Dj+Ej ]−[Gj+Hj+Lj ]] 6 ≥ [[maxj Cj+maxj Dj+maxj Ej ]−[minj Gj+minj Hj+minj Lj ]] 6 = S(Y +) S(Y ) ≥ S(Y +) which implies that TIFFWA(Y1, Y2, ..., Ym) = Y − and TIFFWA(Y1, Y2, ..., Ym) = Y + 4.2. TIFFOWA operator Definition 9. Let kj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the gathering of TIFNs and u = (u1, u2, ..., um)T is the weight vector with uj ∈ [0, 1] and m∑ j=1 uj = 1. Then TIFFOWA (k1, k2, ..., km) = m⊕ j=1 ujkj is said TIFFOWA operator. Theorem 4. The collection of TIFNs are aj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and λ = (λ1, λ2, ..., λn)T is the weight vector with λj ∈ [0, 1] and n∑ j=1 λj = 1. Then it is said TIFFOWA operator and TIFFOWA(a1, a2, ..., an) = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 14 of 37  1 − log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   . Proof:This proof is the same as in Theorem 1 Theorem 5. (Idempotency):If Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] for all N = 1, 2, 3, ..., m, then TIFFOWA(Z̃V , Z̃V , Z̃V , ..., Z̃V ) = Z̃V . Proof:This proof is the same as in Theorem 2 Theorem 6. (Boundedness):If Y − = min(Z̃V 1, Z̃V 2, ..., Z̃V m), Y + = max(Z̃V 1, Z̃V 2, ..., Z̃V m), then Y − ≤ TIFFOWA(Z̃V 1, Z̃V 2, ..., Z̃V m) ≤ Y +. Proof:This proof is the same as in Theorem 3. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 15 of 37 4.3. TIFFHWA operator Definition 10. Let kj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the gathering of TIFNs and u = (u1, u2, ..., um)T is the weight vector with uj ∈ [0, 1] and m∑ j=1 uj = 1. Then TIFFHWA (k1, k2, ..., km) = m⊕ j=1 ujkj is said TIFFHWA operator. Theorem 7. The collection of TIFNs are aj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and λ = (λ1, λ2, ..., λn)T is the weight vector with λj ∈ [0, 1] and n∑ j=1 λj = 1. Then it is said TIFFHWA operator and TIFFHWA(a1, a2, ..., an) =  1 − log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   . Proof:This proof is the same as in Theorem 1 A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 16 of 37 Theorem 8. (Idempotency):If Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] for all R = 1, 2, 3, ..., m, then TIFFHWA(Z̃V , Z̃V , Z̃V , ..., Z̃V ) = Z̃V . Proof:This proof is the same as in Theorem 2. Theorem 9. (Boundedness):If Y − = min(Z̃V 1, Z̃V 2, ..., Z̃V m), Y + = max(Z̃V 1, Z̃V 2, ..., Z̃V m), then Y − ≤ TIFFHWA(Z̃V 1, Z̃V 2, ..., Z̃V m) ≤ Y +. Proof:This proof is the same as in Theorem 3. 4.4. TIFFWG operator Definition 11. Let kj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the gathering of TIFNs and u = (u1, u2, ..., um)T is the weight vector with uj ∈ [0, 1] and m∑ j=1 uj = 1. Then TIFFWG (k1, k2, ..., km) = m⊗ j=1 k uj j is said TIFFWG operator. Theorem 10. The collection of TIFNs are aj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and λ = (λ1, λ2, ..., λn)T is the weight vector with λj ∈ [0, 1] and n∑ j=1 λj = 1. Then it is said TIFFWG operator and TIFFWG(a1, a2, ..., an) = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 17 of 37  log 1+ n∏ j=1 (β Cj −1)λ (β−1)λ−1  β , log 1+ n∏ j=1 (β Dj −1)λ (β−1)λ−1  β , log 1+ n∏ j=1 (β Ej −1)λ (β−1)λ−1  β  ,  1 − log 1+ n∏ j=1 (β 1−Gj −1)λ (β−1)λ−1  β , 1 − log 1+ n∏ j=1 (β 1−Hj −1)λ (β−1)λ−1  β , 1 − log 1+ n∏ j=1 (β 1−Lj −1)λ (β−1)λ−1  β   . Proof. Since n = 1, a1λ1 =  log ( 1+ (βC1 −1)λ1 (β−1)λ1−1 ) β , log ( 1+ (βD1 −1)λ1 (β−1)λ1−1 ) β , log ( 1+ (βE1 −1)λ1 (β−1)λ1−1 ) β  , 1 − log ( 1+ (β1−G1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−H1 −1)λ (β−1)λ−1 ) β , 1 − log ( 1+ (β1−L1 −1)λ (β−1)λ−1 ) β   ; a2λ2 =  log ( 1+ (βC2 −1)λ2 (β−1)λ2−1 ) β , log ( 1+ (βD2 −1)λ2 (β−1)λ2−1 ) β , log ( 1+ (βE2 −1)λ2 (β−1)λ2−1 ) β  , 1 − log ( 1+ (β1−G2 −1)λ2 (β−1)λ2−1 ) β , 1 − log ( 1+ (β1−H2 −1)λ2 (β−1)λ2−1 ) β , 1 − log ( 1+ (β1−L2 −1)λ2 (β−1)λ2−1 ) β   n = k A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 18 of 37  log 1+ k∏ j=1 (β Cj −1)λ (β−1)λ−1  β , log 1+ k∏ j=1 (β Dj −1)λ (β−1)λ−1  β , log 1+ k∏ j=1 (β Ej −1)λ (β−1)λ−1  β  ,  1 − log 1+ k∏ j=1 (β 1−Gj −1)λ (β−1)λ−1  β , 1 − log 1+ k∏ j=1 (β 1−Hj −1)λ (β−1)λ−1  β , 1 − log 1+ k∏ j=1 (β 1−Lj −1)λ (β−1)λ−1  β   n = k + 1 A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 19 of 37  log 1+ k+1∏ j=1 (β Cj −1)λ (β−1)λ−1  β , log 1+ k+1∏ j=1 (β Dj −1)λ (β−1)λ−1  β , log 1+ k+1∏ j=1 (β Ej −1)λ (β−1)λ−1  β  ,  1 − log 1+ k+1∏ j=1 (β 1−Gj −1)λ (β−1)λ−1  β , 1 − log 1+ k+1∏ j=1 (β 1−Hj −1)λ (β−1)λ−1  β , 1 − log 1+ k+1∏ j=1 (β 1−Lj −1)λ (β−1)λ−1  β   Theorem 11. (Idempotency):If Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] for all L = 1, 2, 3, ..., m, then TIFFWG(Z̃V , Z̃V , Z̃V , ..., Z̃V ) = Z̃V . Proof. Since Z̃V = [[Cj , Dj , Ej ], [Gj , Hj , Lj ]] A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 20 of 37  log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  1 − log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   =   log 1+ (β 1−Cj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , log 1+ (β 1−Dj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , log 1+ (β 1−Ej −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β  ,  1 − log 1+ (β 1−Gj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , 1 − log 1+ (β 1−Hj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β , 1 − log 1+ (β 1−Lj −1) n∑ j=1 λj (β−1) n∑ j=1 λj −1  β   =  log ( 1+ (β 1−Cj −1)1 (β−1)1−1 ) β , log ( 1+ (β 1−Dj −1)1 (β−1)1−1 ) β , log ( 1+ (β 1−Ej −1)1 (β−1)1−1 ) β  ,1 − log ( 1+ (β 1−Gj −1)1 (β−1)1−1 ) β , 1 − log ( 1+ (β 1−Hj −1)1 (β−1)1−1 ) β , 1 − log ( 1+ (β 1−Lj −1)1 (β−1)1−1 ) β   =  [ log(1+(β1−Cj −1)1) β , log(1+(β1−Dj −1)1) β , log(1+(β1−Ej −1)1) β ] ,[ 1 − log(1+(β1−Gj −1)1) β , 1 − log(1+(β1−Hj −1)1) β , 1 − log(1+(β1−Lj −1)1) β ]  Z̃V = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 21 of 37 Theorem 12. (Boundedness):If Y = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be a collection of TIFSs. If Y − = min(Y1, Y2, ..., Ym), Y + = max(Y1, Y2, ..., Ym), then Y − ≤ TIFFWG(Y1, Y2, ..., Ym) ≤ Y +. Proof. Y − = min(Y1, Y2, ..., Ym) = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and Y + = max(Y1, Y2, ..., Ym) = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] Since minj(Cj) ≤maxj(Cj);minj(Dj) ≤maxj(Dj);minj(Ej) ≤maxj(Ej) and minj(Gj) ≤maxj(Gj);minj(Hj) ≤maxj(Hj);minj(Lj) ≤maxj(Lj) Which implies that log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β ≥ log 1+ n∏ j=1 (β 1−(min Cj )−1)λj (β−1)λj −1  β ; log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β ≥ log 1+ n∏ j=1 (β 1−(min Dj )−1)λj (β−1)λj −1  β ; log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β ≥ log 1+ n∏ j=1 (β 1−(min Ej )−1)λj (β−1)λj −1  β and 1-log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β ≥ 1 − log 1+ n∏ j=1 (β 1−(max Gj )−1)λj (β−1)λj −1  β ; 1-log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β ≥ 1 − log 1+ n∏ j=1 (β 1−(max Hj )−1)λj (β−1)λj −1  β ; 1-log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β ≥ 1 − log 1+ n∏ j=1 (β 1−(max Lj )−1)λj (β−1)λj −1  β TIFFWG(Y1, Y2, ..., Ym) = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] S(Y ) = [[Cj+Dj+Ej ]−[Gj+Hj+Lj ]] 6 ≤ [[maxj Cj+maxj Dj+maxj Ej ]−[minj Gj+minj Hj+minj Lj ]] 6 = S(Y −) S(Y ) ≤ S(Y −) and A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 22 of 37 S(Y ) = [[Cj+Dj+Ej ]−[Gj+Hj+Lj ]] 6 ≥ [[maxj Cj+maxj Dj+maxj Ej ]−[minj Gj+minj Hj+minj Lj ]] 6 = S(Y +) S(Y ) ≥ S(Y +) which implies that TIFFWG(Y1, Y2, ..., Ym) = Y − and TIFFWG(Y1, Y2, ..., Ym) = Y + 4.5. TIFFOWG operator Definition 12. Let kj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the gathering of TIFNs and the weight vector is g = (g1, g2, ..., gm)T with gj ∈ [0, 1] and m∑ j=1 gj = 1. Then TIFFOWG (k1, k2, ..., km) = m⊗ j=1 k gj j is said TIFFOWG operator. Theorem 13. Let aj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the collection of TIFNs and the weight vector is λ = (λ1, λ2, ..., λn)T with λj ∈ [0, 1] and n∑ j=1 λj = 1. Then it is said TIFFOWG operator and TIFFOWG(a1, a2, ..., an) = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 23 of 37  log 1+ n∏ j=1 (β Cj −1)λ (β−1)λ−1  β , log 1+ n∏ j=1 (β Dj −1)λ (β−1)λ−1  β , log 1+ n∏ j=1 (β Ej −1)λ (β−1)λ−1  β  ,  1 − log 1+ n∏ j=1 (β 1−Gj −1)λ (β−1)λ−1  β , 1 − log 1+ n∏ j=1 (β 1−Hj −1)λ (β−1)λ−1  β , 1 − log 1+ n∏ j=1 (β 1−Lj −1)λ (β−1)λ−1  β   . Proof:This proof is the same as in Theorem 10. Theorem 14. (Idempotency):If B̃U = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] for all r = 1, 2, 3, ..., m, then TIFFOWG(BU, BU, BU, ..., BU) = BU. This proof is the same as in Theorem 11. Theorem 15. (Boundedness):If Y − = min(f1, f2, ..., fm), Y + = max(f1, f2, ..., fm), then Y − ≤ TIFFOWG(f1, f2, ..., fm) ≤ Y +. This proof is the same as in Theorem 12. 4.6. TIFFHWG operator Definition 13. The gathering of TIFNs are fj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] and the weight vector is u = (u1, u2, ..., un)T with uj ∈ [0, 1] and n∑ j=1 uj = 1, the associated vector is u = (u1, u2, ..., un)T A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 24 of 37 with uj ∈ [0, 1] and n∑ j=1 uj = 1. Then TIFFHWG (f1, f2, ..., fn) = m⊗ j=1 f uj j is said TIFFHWG operator. Theorem 16. Let aj = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] be the collection of TIFNs and the weight vector is λ = (λ1, λ2, ..., λn)T with λj ∈ [0, 1] and n∑ j=1 λj = 1. Then it is said TIFFHWG operator and TIFFHWG(a1, a2, ..., an) =  log 1+ n∏ j=1 (β Cj −1)λ (β−1)λ−1  β , log 1+ n∏ j=1 (β Dj −1)λ (β−1)λ−1  β , log 1+ n∏ j=1 (β Ej −1)λ (β−1)λ−1  β  ,  1 − log 1+ n∏ j=1 (β 1−Gj −1)λ (β−1)λ−1  β , 1 − log 1+ n∏ j=1 (β 1−Hj −1)λ (β−1)λ−1  β , 1 − log 1+ n∏ j=1 (β 1−Lj −1)λ (β−1)λ−1  β   . Proof: This proof is the same as in Theorem 10. Theorem 17. (Idempotency):If à = [ [Cj , Dj , Ej ], [Gj , Hj , Lj ] ] for all R = 1, 2, 3, ..., m, then TIFFHWG(Ã, Ã, Ã, ..., Ã) = Ã. Proof:This proof is the same as in Theorem 11. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 25 of 37 Theorem 18. (Boundedness):If Y − = min(f1, f2, ..., fn), Y + = max(f1, f2, ..., fn), then Y − ≤ TIFFHWG(f1, f2, ..., fn) ≤ Y +. Proof:This proof is the same as in Theorem 12. 5. Proposed technique based on TIFAA operator triangular fuzzy C-mean clustering algorithm Triangular Fuzzy clustering algorithms are used to group people according to shared charac- teristics or habits, a process known as user profiling. Users can have their membership degrees a measure of how much they belong to each cluster assigned to them thanks to triangular fuzzy clustering. This strategy is especially helpful in situations where users may simultaneously dis- play traits from several categories. This is a simple overview of how triangular fuzzy clustering for user profiling could be used. Compile pertinent user information that can be utilized for profiling. The demographic data, browsing history, purchasing patterns, and interactions with a website or application are some examples of this data. Choose the characteristics or features that will be utilized to the clustering process. These attributes ought to accurately reflect the traits of the users and have a bearing on the process of profiling. For the given problem, choose a triangular fuzzy clustering algorithm that is suitable triangular Fuzzy C-Means. Step 1:Describe the TIF decision matrix Step 2:Describe the TIFFWA operator and λ = (λ1, λ2, ..., λn) . TIFFWA(A1, A2, ..., An) = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 26 of 37  1 − log 1+ n∏ j=1 (β 1−Cj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Dj −1)λj (β−1)λj −1  β , 1 − log 1+ n∏ j=1 (β 1−Ej −1)λj (β−1)λj −1  β  ,  log 1+ n∏ j=1 (β 1−Gj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Hj −1)λj (β−1)λj −1  β , log 1+ n∏ j=1 (β 1−Lj −1)λj (β−1)λj −1  β   Step 3:The TIFCM algorithms work on TIFFWA operator U = [Uic]c=1...c i=1...n Step 4:Using k-means clusting value is defined as cluster’s parameters. Step 5:Calculate the cluster center to find the centroid Mj =  n∑ j=1 (Cj)α n∑ j=1 (Dj)α n∑ j=1 (Ej)αξj n∑ j=1 (Cj)α n∑ j=1 (Dj)α n∑ j=1 (Ej)α , n∑ j=1 (Gj)α n∑ j=1 (Hj)α n∑ j=1 (Lj)αυj n∑ j=1 (Gj)α n∑ j=1 (Hj)α n∑ j=1 (Lj)α  n =Number of alternatives or decision makers; α =Weighting exponent to adjust sensitivity of the aggregation; Cj , Dj , Ej ,Lower, middle, and upper bounds of the membership degree for alternative Gj , Hj , Lj ,Lower, middle, and upper bounds of the non-membership degree for alternative ξj =Weight or importance level assigned j−th membership value υj =Weight or importance level assigned to the j-th non-membership value Step 6:Find out the distance of each point from the centroid fj = ⟨∥C− − rj∥ + ∥D− − rj∥ + ∥E− − rj∥ , ∥G − rj∥ + ∥H − rj∥ + ∥L − rj∥⟩ Step 7:Calculate the score function ⟨[Cj+Dj+Ej ]−[Gj+Hj+Lj ]⟩ 6 Step 8:Find the ranking. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 27 of 37 6. Case history Recognize the goal of selecting the most efficient renewable energy projects. Effective project selection requires a well-structured review process, based on data-driven insights, focusing on en- ergy generation potential, sustainability, and cost-effectiveness (not just assumptions or guess- work). This requires proper planning and the implementation of strategies to evaluate and choose optimal projects. Instead of focusing solely on project failures, engage in a constructive conversation about how to improve the selection process. Prioritize development, sustainability, and cost-effectiveness over simply criticizing the system. Share findings and recommendations for enhancing the selection process and suggest next steps for more efficient renewable energy projects. If any of these issues seem familiar, it’s because these challenges often arise in renew- able energy project selection, leading to higher costs, wasted resources, or missed energy goals. While technology in renewable energy can help optimize energy production, many projects strug- gle to identify and address inefficiencies in energy generation. These small inefficiencies, if ignored, often escalate into larger, harder-to-manage issues. For example, if energy generation or cost projections deviate unexpectedly, after conducting a thorough analysis, it may be due to inefficient system design, outdated technologies, or inaccurate forecasting models. In your role as a project manager or energy consultant, you might suggest upgrading outdated systems or adjusting project parameters to improve performance. However, do you act on these issues early, or, like many, do you allow minor inefficiencies to escalate into bigger problems that could be harder to address later? Just like medical symptoms serve as early warnings, ineffi- ciencies in renewable energy projects act as warning signs that need immediate attention. If not treated early, these inefficiencies can lead to higher costs, system overload, or even project failure. Addressing issues early enables quick fixes and prevents significant consequences. PID1: Renewable Energy Systems use advanced technologies like AI and machine learning to optimize project performance by analyzing real-time data and predicting future energy produc- tion. These systems improve project efficiency by monitoring resource usage, optimizing energy output, and adjusting project settings based on environmental patterns. PID2: Integration of solar, wind, and energy storage solutions within renewable energy projects introduces complex dynamics. By managing energy from these diverse sources, projects can balance energy production, reduce waste, and enhance sustainability. Efficiently integrating these systems can increase self-sufficiency and minimize reliance on external energy sources. PID3: Energy Monitoring and Analytics are key to identifying inefficiencies in renewable energy projects. With data-driven insights, project managers can optimize energy generation, pinpoint areas of waste, and make informed decisions about how to maximize energy production. These insights empower project teams to adjust operational strategies and minimize inefficien- cies. Step 1:Explain the table 2 and 3 TIF decision matrix. Table 2 of the TIF decision matrix. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 28 of 37 Su Ra Ch PID1  [0.1, 0.2, 0.3], [0.02, 0.04, 0.5]   [0.11, 0.21, 0.33], [0.024, 0.044, 0.11]   [0.21, 0.41, 0.53], [0.14, 0.24, 0.31]  PID2  [0.21, 0.41, 0.53], [0.14, 0.24, 0.31]   [0.1, 0.2, 0.3], [0.02, 0.04, 0.5]   [0.11, 0.21, 0.33], [0.024, 0.044, 0.11]  PID3  [0.21, 0.41, 0.53], [0.14, 0.24, 0.31]   [0.11, 0.21, 0.33], [0.024, 0.044, 0.11]   [0.1, 0.2, 0.3], [0.02, 0.04, 0.5]  TIF decision matrix table 3. Su Ra Ch PID1  [0.11, 0.12, 0.13], [0.1, 0.2, 0.12]   [0.1, 0.2, 0.3], [0.03, 0.04, 0.5   [0.21, 0.22, 0.23], [0.1, 0.3, 0.22]  PID2  [0.1, 0.2, 0.3], [0.03, 0.04, 0.5   [0.11, 0.12, 0.13], [0.1, 0.2, 0.12]   [0.21, 0.22, 0.23], [0.1, 0.3, 0.22]  PID3  [0.21, 0.22, 0.23], [0.1, 0.3, 0.22]   [0.1, 0.2, 0.3], [0.03, 0.04, 0.5   [0.11, 0.12, 0.13], [0.1, 0.2, 0.12]  Step 2:Describe the TIFFWA operator and ξ = (0.26, 0.21, 0.25) . TIFFWA operator is in table 4.TIFFWA operator table 4. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 29 of 37 Su Ra Ch PID1  [0.1001, 0.3698, 0.1478] [0.1983, 0.1258 0.1092   [0.1452, 0.7541, 0.9451] [0.1675, 0.2587 0.1124   [0.2561, 0.0026, 0.4598 [0.2343, 0.4587, 0.2032]  PID2  [0.3031, 0.7412, 0.9871] [0.3014, 0.3974, 0.2052]   [0.1698, 0.3012, 0.1231, [0.3453, 0.0198 0.9872   [0.0001, 0.3698, 0.3014 [0.1004, 0.9014, 0.2123  PIPS3  [0.0541, 0.0131, 0.3214], [0.7412, 0.1134, 0.2359]   [0.3211, 0.0123, 0.0212], [0.4569, 0.0556, 0.0411   [0.3698, 0.4569, 0.3051], [0.3124, 0.3084, 0.2082]  Step 3:The TFCM algorithms work on TIFFWA operator U = [Uic]c=1...c i=1...n. TFCM algorithms work on TIFFWA operator in table 5. TFCM algorithms table 5. Su Ra Ch PID1 0.4 0.2 0.1 PID2 0.02 0.12 0.03 PID3 0.01 0.23 0.01 Step 4:Using k-means clusting value is defined as cluster’s parameters in table 6. k-means clusting table 6 Su Ra Ch PID1  [0.1232, 0.2563, 0.3698], [0.1258, 0.0987, 0.2587]   [0.1232, 0.2563, 0.3698], [0.1258, 0.0987, 0.2587]   [0.1209, 0.1163, 0.9898], [0.8518, 0.0967, 0.1477]  PID2  [0.1963, 0.2258, 0.3741], [0.1753, 0.3577, 0.9877]   [0.1942, 0.2753, 0.3148], [0.1559, 0.7534, 0.2741]   [0.0002, 0.0143, 0.4568], [0.1008, 0.0904, 0.2014]  PID3  [0.4432, 0.6963, 0.3448], [0.1148, 0.0753, 0.2369]   [0.1753, 0.2159, 0.3149], [0.1157, 0.0741, 0.2358]   [0.1154, 0.2258, 0.3354], [0.1145, 0.0985, 0.2147]  Step 5:Calculate the cluster center to find the centroid C1 = 0.2345, C2 = 0.1034, C3 = 0.3456. Step 6:Find out the distance of each point from the centroid in table 7. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 30 of 37 Centroid table 7 PID1 [ [0.1256, 0.1016, 0.1852], [0.9874, 0.1984, 0.1397] ] PID2 [ [0.1036, 0.1298, 0.1987], [0.9012, 0.1387, 0.1989] ] PID3 [ [0.1003, 0.1009, 0.0002], [0.0694, 0.9874, 0.3219] ] Step 7:Calculate the score function DH1 = 0.0091, DH2 = 0.6412, DH3 = 0.7056. Step 8:Find the ranking DH3 > DH2 > DH1 and DH3 is the best. Score function of figure 5 is different ranking. because Figure 5’s ranking seems to alter depending on the score function. This disparity results from different evaluation standards being applied: DH3 is the first ranking, DH2 is the second ranking and DH1 is the last ranking. Figure 5 is given as below 6.1. Comparsion technique with existing way Different existing ways are written below in Table 8. Table 8 Existing techniques. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 31 of 37 Methods Average operator Geometric operator MCDM [46]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  Similarity measure [20]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  T-SFS operators [47]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  Aggregation operators [5]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  Hamacher operators [46]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  Particle swarm [50]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  Intuitionistic operators [11]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  IFFPA operators [31]  DH3 > DH2 > DH1   DH3 > DH2 > DH1  6.2. Validity way In this subsection, we present the proposed method for validating the aggregating operator, as shown in Table 9. Ways Average operator Geometric operator hybrid Proposed technique ✓ ✓ ✓ Muirhead mean-based 2-tuple linguistic [48] ✓ ✓ ✓ pythagorean fuzzy TOPSIS method [60] ✓ ✓ ✓ pythagorean Dombi method [59] ✓ ✓ ✓ Complex Pythagorean fuzzy information [49] ✓ ✓ ✓ PLTM operators [32] ✓ ✓ ✓ Table 9 presents a comparative overview of different decision-making techniques based on their use of various aggregation operators—namely, average, geometric, and hybrid operators. The methods listed include both the proposed technique and several existing approaches from the literature. The " √ " (check mark) symbol in the table indicates that the corresponding method utilizes that specific type of aggregation operator. For example, if a method has a √ under "Average Operator," it means that the method applies an average-based aggregation approach. This table is intended to highlight the versatility and comprehensiveness of the proposed technique in comparison to existing methods, demonstrating its capability to support all three types of aggregation operators. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 32 of 37 6.3. Results and discussion There are several reasons why the proposed method is superior to traditional approaches in renewable energy project selection. Conventional models typically rely on fixed, static data and fail to accommodate the dynamic nature of energy markets, fluctuating resource availability, and evolving environmental conditions. To improve the effectiveness of current energy project evaluation systems, integrating TIFF aggregation operators can process complex, uncertain data such as fluctuating renewable energy production, varying demand, and technology performance metrics. Provide comprehensive guidelines for energy project evaluators to efficiently use systems based on TIFF aggregation operators. Stress the importance of accurate data interpretation and highlight the advantages of triangular intuitionistic fuzzy methods for optimizing the selection of renewable energy projects. To validate the reliability and practicality of TIFF aggregation operators in real-world sce- narios, conduct pilot evaluations of renewable energy projects across various contexts. Gather critical feedback and data during these evaluations to refine and improve the methodology before large-scale implementation. Collaborate with regulatory authorities to ensure that the integration of TIFF aggregation operators in renewable energy project selection complies with all necessary standards and regu- lations. 6.4. Advantages There are numerous important benefits to using triangular intuitionistic Fuzzy Frank Aggre- gation Operators in decision-making for selecting renewable energy projects. • A clear demonstration of Multi-Criteria Decision-Making (MCDM) evidence in a non- verbal manner using the TIF method. The TIF method has proven to be essential for clarifying uncertain and incomplete project evaluation results. • TIFF aggregation operators successfully integrate imprecise and heterogeneous data, en- hancing the accuracy of renewable energy project selection forecasts. This results in more precise project evaluations, particularly in complex scenarios with fluctuating energy generation, vari- able demand, and diverse environmental conditions. • TIFFAO enables consistent decision-making across various project selection scenarios by combining multiple criteria in a more advanced way. As a result, the reliability of selecting optimal energy projects is increased, and variability in our is minimized. • MCDM becomes more transparent and easier to understand due to TIFFO’s structured approach to data aggregation. This promotes more confident project selection decisions and helps stakeholders better comprehend the rationale behind the final selections. • Due to their expertise in renewable energy project evaluation, their ability to make practical judgments, and the effectiveness of the evaluation processes, energy project specialists are often required to guide the optimization of renewable energy systems. Current methods rely heavily on expert judgment to interpret and select viable energy projects. 6.5. Sensitive study We define the sensitive study in this subsection, which is represented in table 10 below. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6227 33 of 37 Value DH1 DH2 DH3 ∆ = 0, DHi 0.0112 0.8967 0.2545 Preference order ranking 3 1 2 ∆ = 0.1, DHi 0.0145 0.6066 0.4014 Preference order ranking 3 1 2 ∆ = 0.2, DHi 0.0123 0.6234 0.4124 Preference order ranking 3 1 2 ∆ = 0.3, DHi 0.0101 0.6765 0.4873 Preference order ranking 3 1 2 ∆ = 0.4, DHi 0.0077 0.8345 0.2098 Preference order ranking 3 1 2 ∆ = 0.5, DHi 0.0145 0.4987 0.2012 Preference order ranking 3 1 2 ∆ = 0.6, DHi 0.1021 0.2345 0.1987 Preference order ranking 3 1 2 ∆ = 0.7, DHi 0.1129 0.3409 0.2983 Preference order ranking 3 1 2 ∆ = 0.8, DHi 0.1983 0.7654 0.5672 Preference order ranking 3 1 2 7. Conclusion In this section, we introduce triangular intuitionistic fuzzy data based Frank Aggregation Operators applied to the selection of renewable energy projects. The operational laws are defined along with the corresponding score and accuracy functions. This study uses the TIFFWA, TIFFOWA, TIFFHWA, TIFFWG, TIFFOWG and TIFFHWG operators within the Multi- Criteria Decision-Making (MCDM) approach to address challenges in renewable energy project selection. It is demonstrated that these operators enable the MCDM technique to effectively dif- ferentiate between various renewable energy alternatives, offering flexibility in making decisions related to project selection and energy efficiency. These operators exhibit essential properties such as commutativity, idempotency, boundedness, associativity, and monotonicity. When com- bined with clustering techniques like the triangular Fuzzy C-means algorithm, these aggregation operators enhance the accuracy and computational efficiency of the project selection process, making it possible to cluster data effectively for more reliable decision-making. In the near future, we plan to incorporate artificial intelligence into renewable energy project selection. This includes utilizing neural networks, automation, data analysis, and virtual assis- tants to enhance decision-making processes related to project assessment. Additionally, we aim to expand the current approach by incorporating Generalized triangular Cubic Fuzzy Frank Aggre- gation Operators and triangular Cubic Fuzzy Frank Geometric Aggregation Operators, allowing for more advanced, adaptive, and efficient systems for selecting renewable energy projects. 8. Compliance with Ethical Standards The authors declare that there is no conflict of interests regarding the publication of this paper. Compliance with Ethical Standards: This study is not supported by any source or any organizations. Ethical approval: This article does not contain any studies with human participants or ani- mals performed by any of the authors. A. Fahmi et al. / Eur. J. Pure Appl. 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