EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5891 ISSN 1307-5543 – ejpam.com Published by New York Business Global Einstein Aggregation Operators with Cubic Fermatean Fuzzy Sets Aliya Fahmi1, Aziz Khan2, Thabet Abdeljawad2,∗, Muhammad Arshad Shehzad Hassan3, D.K.Almutairi4 1 Department of Mathematics and Statistics, Faculty of Sciences, The University of Faisalabad, Pakistan 2 Department of Mathematics and Sciences, Prince Sultan University, P.O.Box 66833, 11586 Riyadh, Saudi Arabia 3 Department of Electrical Engineering, The University of Faisalabad, Faisalabad, Pakistan 4 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, 11952 AlMajmaah, Saudi Arabia Abstract. The selection of optimal locations for electric vehicle (EV) charging stations is a com- plex multi-criteria decision-making (MCDM) problem influenced by quantitative and qualitative factors. The inherent uncertainty in these evaluations further complicates the decision-making pro- cess. This study introduces an integrated decision-making framework based on Cubic Fermatean Fuzzy Sets (CFFS) to address these challenges. We extend the work of three novel aggregation operators including Cubic Fermatean Einstein Fuzzy Weighted Averaging (CFEFWA), Cubic Fer- matean Einstein Fuzzy Ordered Weighted Averaging (CFEFOWA), and Cubic Fermatean Einstein Fuzzy Weighted Averaging (CFEFHWA) to enhance decision accuracy in uncertain environments. Additionally, we propose innovative Einstein aggregation operations to integrate decision data effectively. The proposed framework is applied to a real-world EV charging station selection case study. Comparative analysis with existing methodologies demonstrates that our approach improves decision consistency and reliability. The results indicate that the proposed method en- hances the robustness of decision-making by effectively handling uncertainty and providing more precise rankings for optimal site selection. These findings underscore the potential of CFFS-based MCDM approaches in complex decision-making scenarios, offering valuable insights for sustainable EV infrastructure planning. 2020 Mathematics Subject Classifications: 03E72, 90B50, 90C29, 68T37 Key Words and Phrases: Fuzzy sets, Multi-Criteria Decision Making, Cubic fermatean fuzzy set, Einstein ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5891 Email addresses: aliyafahmi@gmail.com (A. Fahmi), akhan@psu.edu.sa (A. Khan), tabdeljawad@psu.edu.sa (T. Abdeljawad) 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 (2) (2025), 5891 2 of 27 1. Introduction Support for fossil fuel-based transportation negatively affects both the environment [1–5] and the economy. Speaking of the weakening of fossil vigor and ecological concerns, requires finding viable replacements for conservative vehicles. Electric vehicles (EVs) and their charging positions are a flattering essential answer to these glitches [6]. Zhang [7] offered an innovative ranking technique for PFNs. Xu et al. [8] presented the geomet- ric aggregation operators. Xu [9] introduced the intuitionistic fuzzy weighted averaging operator. Zadeh [10] introduced the fuzzy set. Parvathi [11] introduced the intuition- istic fuzzy set. Garg et al. [12] proposed the Fermatean fuzzy Yager hybrid weighted geometric operators. Several Fermatean fuzzy aggregation operators were proposed by Aydemir et al. [13]. Chen [14] proposed the bridge selection of the construction method of a multiple-criteria decision-making problem. Chen [15] proposed the ultimate priority orders of the alternative. Chen [16] proposed the various forms of preference structures of the criterion weights using incomplete data in decision-making. A few methods for re- solving Pythagorean fuzzy multiple attribute decision-making problems were developed by Wu et al. [17]. A likelihood-based linear assignment model for multiple criteria decision analysis in the interval-valued intuitionistic fuzzy environment was developed by Wang et al. [18]. In their proposal, Akram et al. [19] addressed how to handle Fermatean fuzzy data in MADM issues. Introducing the intuitionistic fuzzy set was Atanassov [20]. Yager [21] introduced the membership grades for Pythagoreans. Yang [22] introduced the managerial implications of the user-choice paradigm. Yang et al. [23] conducted a socioe- conomic analysis was conducted in a proportionately reluctant, hazy setting. A Decision Support System (DSS) was examined by Sarabi et al. [24] to choose the best logistic service provider among the three services of the businesses. A Fermatean fuzzy TOPSIS method was developed by Senapati et al. [25] to address the problem of multiple crite- ria decision-making. Senapati et al. [26] developed a Fermatean fuzzy weighted product model to solve the multi-criteria decision-making problem. The Fermatean fuzzy weighted average and Fermatea fuzzy weighted geometric operators were proposed by Senapati et al. [27]. Large-scale data consideration by Shahraki et al. [28]. The features and appli- cations of the proposed metrics in pattern recognition were proposed by Singh et al. [29]. The single-valued neutrosophic sets of the MULTIMOORA technique were proposed by Stanujkic et al. [30]. Stanujkic et al. [31] proposed single-valued bipolar fuzzy numbers of the MULTIMOORA method. Aliya et al. [32] proposed the VIKOR method to solve the MCDM method based on triangular cubic fuzzy numbers. The generalized triangular cubic linguistic hesitant fuzzy hybrid averaging operators were first presented by Amin et al. [33]. The TrCFEHWA operator was developed by Aliya et al. [34]. The triangular cubic fuzzy hybrid aggregation was proposed by Aliya et al. [35]. Different features of these operators were proposed by Aliya et al. [36], who also deduced the link between the proposed operators [37]. The ideal location for EV charging stations has been the subject of extensive research. For instance, training has focused heavily on reducing controller loss overall, bus dominance, and the attractiveness of the sending scheme’s dependability through the Distribution Network Operator tilt [8, 11, 38–41]. While Wang et al. [18] and A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 3 of 27 Deb et al. [42] have promoted the development and installation of charging stations, Turk et al. [43] emphasized the need of taking EV weight and its effects on delivery schedules into account. Furthermore, the theoretical foundation of fuzzy set-based decision-making has been expanded by noteworthy investigations [7], such as those conducted by Zhang and Xu [44] and Zeng et al. [45]. To address the uncertain multi-criteria decision analysis difficulties, Rani et al. [46] were introduced. The suggested 3,4-quasirung fuzzy weighted averaging (geometric) operators are used by Seikh et al. [47] and have been applied in various fields [27, 48–52]. Numerous dynamic problems for existence, uniqueness, and nu- merical simulations had their foundation established by Khan et al. [53]. The uniqueness and Hyers-Ulam stability of the solutions to the mABC-FDEs system are examined by Khan et al. [54]. Lagrange’s interpolation was introduced by Khan et al. [3] to build computationally efficient numerical algorithms for the fractal-fractional waterborne illness model. The Hyers-Ulam stability of the considered equation was presented by Khan et al. [2] and other fileds [4, 5, 38]. Das et al. [55] proposed a method for solving group decision- making problems (DMPs) using an intuitionistic fuzzy parameterized (IFP) intuitionistic multi-fuzzy N-soft set. Das et al. [56] introduced an effective water pollution rating sys- tem, the fuzzy multi-criteria decision-making (FMCDM) model, to calculate a relative weighted water pollution score (-score) for assessing pollution levels. Their approach is applied to evaluate the water quality indices of the Haora River in Tripura, India. Das et al. [57] introduced a modified adjustable GDMM for solving FSS-based GDMPs, re- placing absolute scores with weighted average ratings to enhance stability and feasibility. Das et al. [58] proposed and defined novel concepts, including the root mean square dif- ference operator (RMSDO), the root mean square difference score matrix (RMSDSM), and the weighted score. Das et al. [59] proposed an approach for solving real-life group decision-making problems (DMPs) using fuzzy parameterized intuitionistic fuzzy soft sets. In advance of earlier research, there is still a lack of information in the literature about the thorough integration of sophisticated fuzzy set theories with transport and electrical network considerations. More specifically, not enough research has been done on the use of cubic Fermatean fuzzy sets in the framework for choosing the location of EV charging stations. Comparing CFFSs to typical interval-valued fuzzy sets, they provide a more ac- curate and nuanced assessment of possible sites because they better capture uncertainty and flexibility. To close this gap, this study suggests a unique framework that improves the precision and dependability of EV charging station site selection by fusing CFFSs with multi-criteria decision-making techniques. The rest of the paper is structured as follows: Section 2 defines the operational laws and cubic Fermatean fuzzy numbers. Section 3 describes the aggregation operators using CFF data. Section 4 details the multi-criteria decision-making process using the CFF classical. Section 5 offers a case study of the EV charging position location selection problem, representative of the practical implementa- tion of the proposed framework. Section 6 deliberates the comparative examination. In section 7, we define the discussion. Finally, Section 8 delivers the conclusion. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 4 of 27 1.1. Contribution In this subsection, we define the contribution given below • Developed the idea of Cubic Fermatean Fuzzy Sets to deliver a more nuanced symbol of uncertainty and multi-criteria decision-making in the setting of EV charging station assortment. • To develop the three aggregation operators CFEFWA, CFEFOWA and CFEFHWA op- erators. • Offered innovative Einstein aggregation acts to effectually aggregate decision data within the CFFS outline, speaking of complexity and indecision of multi-criteria executive. 1.2. Motivation The motivation of this article is a demarcation of the Cubic Fermatean Fuzzy Sets. • The goal of this research is to provide more reliable tools for aggregating and evaluating decision data, which enhance the decision-making process correctness and dependability, creating new Einstein aggregation operators specifically designed for CFFS. • To maximize charging station usefulness, minimize operating costs, and guarantee broad accessibility, all of which promote environmental sustainability and economic efficiency, they must be located efficiently. • The process of choosing where to put EV charging stations is intrinsically complicated and involves several considerations, including financial considerations, infrastructure pre- paredness, user convenience, and geographic coverage. • The interplay of these components is often too complex for traditional decision-making techniques to handle, particularly when the decisions involve both objective measurements and subjective judgments. Advanced techniques that can successfully integrate and bal- ance these many requirements are needed. • Practical applications of advanced fuzzy set theories to real-world issues are becoming more and more in demand. The necessity to implement and test novel approaches in the context of choosing locations for EV charging stations is what spurred this research. • The research intends to give tangible evidence of the advantages and practical benefits of the proposed methods by comparing them with existing ways and using a case study to demonstrate their efficacy. 1.3. Objective The objective of the research is given below • To develop the CFS and operational laws • To develop three aggregation operators including CFEFWA, CFEFOWA, and CFEFHWA operators. • To develop the proposed method based on CFEFWA operators. • To develop a numerical example. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 5 of 27 2. CFFN and operational laws on Einstein Definition 1. The fixed set C and the A is defined in A =  ⟨[E− A (u), E+ A (u)], EA(u)⟩, ⟨[L− A(u), L+ A(u)], LA(u)⟩  , the IVFFS is  [E− A (u), E+ A (u)], [L− A(u), L+ A(u)]  and the FFS is [EA(u), LA(u)], IVFFS represent the MED and FFS presented the NOMED, and [ [E− A (u), E + A (u)], [L− A(u), L + A(u)] ] ∈ [0, 1], [EA(u), LA(u)] ∈ [0, 1]. The IVFFS and FFS A is said to be CFFN. Definition 2. [37] Let a1 = { ⟨[H− 1 , H+ 1 ], H1⟩, ⟨[R− 1 , R + 1 ], R1⟩ } and a2 = { ⟨[H− 2 , H+ 2 ], H2⟩, ⟨[R− 2 , R + 2 ], R2⟩ } be two CFEFNs and λ > 0, then a1 ⊕ a2 =  〈[ 3 √ (H− 1 )3+(H− 2 )3 1+(H− 1 )3(H− 2 )3 , 3 √ (H+ 1 )3+(H+ 2 )3 1−(H+ 1 )3(H+ 2 )3 ] , H1H2 3 √ 1+(1−H1)3(1−H2)3 〉 ,〈[ 3 √ (R− 1 )3+(R− 2 )3 1+(R− 1 )3(R− 2 )3 , 3 √ (R+ 1 )3+(R+ 2 )3 1−(R+ 1 )3(R+ 2 )3 ] , R1R2 3 √ 1+(1−R1)3(1−R2)3 〉  ; a1 ⊗ a2 =  〈 [ H− 1 H− 2 3 √ 1+(1−H− 1 )3(1−H− 2 )3 , H+ 1 H+ 2 3 √ 1+(1−H+ 1 )3(1−H+ 2 )3 ] , 3 √ (H1)3+(H2)3 1+(H1)3(H2)3 〉 , 〈 [ R− 1 R− 2 3 √ 1+(1−R− 1 )3(1−R− 2 )3 , R+ 1 R+ 2 3 √ 1+(1−R+ 1 )3(1−R+ 2 )3 ] , 3 √ (R1)3+(R2)3 1+(R1)3(R2)3 〉  ; λa1 =  〈 [ 3 √ (1+H− 1 )3)λ−(1−H− 1 )3)λ (1+H− 1 )3)λ+(1−H− 1 )3)λ , 3 √ (1+H+ 1 )3)λ−(1−H+ 1 )3)λ (1+H+ 1 )3)λ+(1−H+ 1 )3)λ ] , 3√2(H1)λ 3 √ (2−H1)3)λ+(H1)3)λ 〉 , 〈 [ 3 √ (1+R− 1 )3)λ−(1−R− 1 )3)λ (1+R− 1 )3)λ+(1−R− 1 )3)λ , 3 √ (1+R+ 1 )3)λ−(1−R+ 1 )3)λ (1+R+ 1 )3)λ+(1−R+ 1 )3)λ ] , 3√2(R1)λ 3 √ (2−R1)3)λ+(R1)3)λ 〉  ; aλ1 =  〈 [ 3√ λ(H− 1 )λ 3 √ ((2−H− 1 )3)λ+((H− 1 )3)λ , 3√ λ(H+ 1 )λ 3 √ ((2−H+ 1 )3)λ+((H+ 1 )3)λ ] , 3 √ ((1+H1)3)λ−((1−H1)3)λ ((1+H1)3)λ+((1−H1)3)λ 〉 , 〈 [ 3√ λ(R− 1 )λ 3 √ ((2−R− 1 )3)λ+((R− 1 )3)λ , 3√ λ(R+ 1 )λ 3 √ ((2−R+ 1 )3)λ+((R+ 1 )3)λ ] , 3 √ ((1+R1)3)λ−((1−R1)3)λ ((1+R1)3)λ+((1−R1)3)λ 〉  A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 6 of 27 Definition 3. [37] The CFFNs are a = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } , then the score function δ is define as:δ = {[(H−)3+(H−+)3−(H)3][(R−)3+(R−+)3−(R)3]} 6 . Definition 4. [37] The CFFNs are a = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } , then the accuracy function Γ is define as:Γ = {[(H−)3+(H−+)3+(H)3][(R−)3+(R−+)3+(R)3]} 6 .. 3. Einstein-based aggregating operator for CFFNs In this section, CFEFWA, CFEFOWA, CFEFHWA operators are provided. 3.1. CFEFWA operator Definition 5. The family of CFFNs are cj = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } and the weight vector is G = (G1, G2, ..., Gn) T with Gj ∈ [0, 1] and n∑ j=1 Gj = 1. Then CFEFWA(c1, c2, ..., cn) = n⊕ j=1 Gjcj is said CFEFWA operator. Theorem 1. The collection of CFFNs are aj = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } and the weight vector is L = (L1, L2, ..., Ln) T with Lj ∈ [0, 1] and n∑ j=1 Lj = 1. Then it is said CFEFWA operator and CFEFWA(a1, a2, ..., an) = 〈  3 √√√√√√√ n∏ j=1 ((1+H− j )3)L− n∏ j=1 (1−H− j )3)L n∏ j=1 ((1+H− j )3)L+ n∏ j=1 ((1−H− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+H+ j )3)L− n∏ j=1 ((1−H+ j )3)L n∏ j=1 ((1+H+ j )3)L+ n∏ j=1 ((1−H+ j )3)L  , 3√2 n∏ j=1 (Hj) L 3 √√√√ n∏ j=1 ((2−Hj)3)L+ n∏ j=1 ((Hj)3)L 〉 , 〈  3 √√√√√√√ n∏ j=1 ((1+R− j )3)L− n∏ j=1 ((1−R− j )3)L n∏ j=1 ((1+R− j )3)L+ n∏ j=1 (1−R− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+R+ j )3)L− n∏ j=1 ((1−R+ j )3)L n∏ j=1 ((1+R+ j )3)L+ n∏ j=1 ((1−R+ j )3)L  , 3√2 n∏ j=1 (Rj) L 3 √√√√ n∏ j=1 ((2−Rj)3)L+ n∏ j=1 ((Rj)3)L 〉  A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 7 of 27 Proof. Since n is true and n = 1 L1a1 =  〈 [ 3 √ ((1+H− 1 )3)L1−(1−H− 1 )3)L1 ((1+H− 1 )3)L1+(1−H− 1 )3)L1 , 3 √ ((1+H+ 1 )3)L1−((1−H+ 1 )3)L1 ((1+H+ 1 )3)L1+((1−H+ 1 )3)L1 ] , 3√2(H1)L1 3 √ (2−H1)3)L1+(H1)3)L1 〉 , 〈 [ 3 √ ((1+R− 1 )3)L1−(1−R− 1 )3)L1 ((1+R− 1 )3)L1+(1−R− 1 )3)L1 , 3 √ ((1+R+ 1 )3)L1−((1−R+ 1 )3)L1 ((1+R+ 1 )3)L1+((1−R+ 1 )3)L1 ] , 3√2(R1)L1 3 √ ((2−R1)3)L1+((R1)3)L1 〉  L2a2 =  〈 [ 3 √ ((1+H− 2 )3)L2−(1−H− 2 )3)L2 ((1+H− 2 )3)L2+(1−H− 2 )3)L2 , 3 √ ((1+H+ 2 )3)L2−((1−H+ 2 )3)L2 ((1+H+ 2 )3)L2+((1−H+ 2 )3)L2 ] , 3√2(H2)L2 3 √ (2−H2)3)L2+(H2)3)L2 〉 , 〈 [ 3 √ ((1+R− 2 )3)L2−(1−R− 2 )3)L2 ((1+R− 2 )3)L2+(1−R− 2 )3)L2 , 3 √ ((1+R+ 2 )3)L2−((1−R+ 2 )3)L2 ((1+R+ 2 )3)L2+((1−R+ 2 )3)L2 ] , 3√2(R2)L2 3 √ ((2−R2)3)L2+((R2)3)L2 〉  L1a1 ⊕ L2a2 =  〈 [ 3 √ ((1+H− 1 )3)L1−(1−H− 1 )3)L1 ((1+H− 1 )3)L1+(1−H− 1 )3)L1 , 3 √ ((1+H+ 1 )3)L1−((1−H+ 1 )3)L1 ((1+H+ 1 )3)L1+((1−H+ 1 )3)L1 ] , 3√2(H1)L1 3 √ (2−H1)3)L1+(H1)3)L1 〉 , 〈 [ 3 √ ((1+R− 1 )3)L1−(1−R− 1 )3)L1 ((1+R− 1 )3)L1+(1−R− 1 )3)L1 , 3 √ ((1+R+ 1 )3)L1−((1−R+ 1 )3)L1 ((1+R+ 1 )3)L1+((1−R+ 1 )3)L1 ] , 3√2(R1)L1 3 √ ((2−R1)3)L1+((R1)3)L1 〉  ⊕  〈 [ 3 √ ((1+H− 2 )3)L2−(1−H− 2 )3)L2 ((1+H− 2 )3)L2+(1−H− 2 )3)L2 , 3 √ ((1+H+ 2 )3)L2−((1−H+ 2 )3)L2 ((1+H+ 2 )3)L2+((1−H+ 2 )3)L2 ] , 3√2(H2)L2 3 √ (2−H2)3)L2+(H2)3)L2 〉 , 〈 [ 3 √ ((1+R− 2 )3)L2−(1−R− 2 )3)L2 ((1+R− 2 )3)L2+(1−R− 2 )3)L2 , 3 √ ((1+R+ 2 )3)L2−((1−R+ 2 )3)L2 ((1+R+ 2 )3)L2+((1−R+ 2 )3)L2 ] , 3√2(R2)L2 3 √ ((2−R2)3)L2+((R2)3)L2 〉  Since n = k CFEFWA(a1, a2, ..., an) = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 8 of 27 〈  3 √√√√√√√√ k∏ j=1 (1+H− j )3)L− k∏ j=1 (1−H− j )3)L k∏ j=1 (1+H− j )3)L+ k∏ j=1 (1−H− j )3)L , 3 √√√√√√√√ k∏ j=1 (1+H+ j )3)L− k∏ j=1 (1−H+ j )3)L k∏ j=1 (1+H+ j )3)L+ k∏ j=1 (1−H+ j )3)L  , 3√2 k∏ j=1 (Hj) L 3 √√√√√ k∏ j=1 (2−Hj)3)L+ k∏ j=1 (Hj)3)L 〉 , 〈  3 √√√√√√√√ k∏ j=1 (1+R− j )3)L− k∏ j=1 (1−R− j )3)L k∏ j=1 (1+R− j )3)L+ k∏ j=1 (1−R− j )3)L , 3 √√√√√√√√ k∏ j=1 (1+R+ j )3)L− k∏ j=1 (1−R+ j )3)L k∏ j=1 (1+R+ j )3)L+ k∏ j=1 (1−R+ j )3)L  , 3√2 k∏ j=1 (Rj) L 3 √√√√√ k∏ j=1 (2−Rj)3)L+ k∏ j=1 (Rj)3)L 〉  Since n = k + 1 CFEFWA(a1, a2, ..., an) = 〈  3 √√√√√√√√ k+1∏ j=1 (1+H− j )3)L− k+1∏ j=1 (1−H− j )3)L k+1∏ j=1 (1+H− j )3)L+ k+1∏ j=1 (1−H− j )3)L , 3 √√√√√√√√ k+1∏ j=1 (1+H+ j )3)L− k+1∏ j=1 (1−H+ j )3)L k+1∏ j=1 (1+H+ j )3)L+ k+1∏ j=1 (1−H+ j )3)L  , 3√2 k+1∏ j=1 (Hj) L 3 √√√√√k+1∏ j=1 (2−Hj)3)L+ k+1∏ j=1 (Hj)3)L 〉 , 〈  3 √√√√√√√√ k+1∏ j=1 (1+R− j )3)L− k+1∏ j=1 (1−R− j )3)L k+1∏ j=1 (1+R− j )3)L+ k+1∏ j=1 (1−R− j )3)L , 3 √√√√√√√√ k+1∏ j=1 (1+R+ j )3)L− k+1∏ j=1 (1−R+ j )3)L k+1∏ j=1 (1+R+ j )3)L+ k+1∏ j=1 (1−R+ j )3)L  , 3√2 k+1∏ j=1 (Rj) L 3 √√√√√k+1∏ j=1 (2−Rj)3)L+ k+1∏ j=1 (Rj)3)L 〉  Theorem 2. (Idempotency):If Z̃V = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } for all j = 1, 2, 3, ..., n, A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 9 of 27 then CFEFWA(ZV,ZV, ZV, ..., ZV ) = ZV. Proof. Zv = ZV are equal to 〈  3 √√√√√√√√√√√ ((1+H− j )3) n∑ j=1 Lj −((1+H− j )3) n∑ j=1 Lj ((1+H− j )3) n∑ j=1 Lj +((1+H− j )3) n∑ j=1 Lj , 3 √√√√√√√√√√√ ((1+H+ j )3) n∑ j=1 Lj −((1+H+ j )3) n∑ j=1 Lj ((1+H+ j )3) n∑ j=1 Lj +((1+H+ j )3) n∑ j=1 Lj  , 3√2((Hj) 3) n∑ j=1 Lj 3 √√√√√√√ (2−H3 j ) n∑ j=1 Lj +(H3 j ) n∑ j=1 Lj 〉 , 〈  3 √√√√√√√√√√√ ((1+R− j )3) n∑ j=1 Lj −((1+R− j )3) n∑ j=1 Lj ((1+R− j )3) n∑ j=1 Lj +((1+R− j )3) n∑ j=1 Lj , 3 √√√√√√√√√√√ ((1+R+ j )3) n∑ j=1 Lj −((1+R+ j )3) n∑ j=1 Lj ((1+R+ j )3) n∑ j=1 Lj +((1+R+ j )3) n∑ j=1 Lj  , 3√2((Rj) 3) n∑ j=1 Lj 3 √√√√√√√ (2−R3 j) n∑ j=1 Lj +(R3 j) n∑ j=1 Lj 〉   〈 [ 3 √ ((1+H− j )3)−((1+H− j )3) ((1+H− j )3)+((1+H− j )3) , 3 √ ((1+H+ j )3)−((1+H+ j )3) ((1+H+ j )3)+((1+H+ j )3) ] , 3√2((Hj) 3) 3 √ (2−H3 j )+(H3 j ) 〉 , 〈 [ 3 √ ((1+R− j )3)−((1+R− j )3) ((1+R− j )3)+((1+R− j )3) , 3 √ ((1+R+ j )3)−((1+R+ j )3) ((1+R+ j )3)+((1+R+ j )3) ] , 3√2((Rj) 3) 3 √ (2−R3 j)+(R3 j) 〉  Theorem 3. (Boundedness):If C− = min(d1, d2, ..., dn), C + = max(d1, d2, ..., dn), then C− ≤CFEFWA(d1, d2, ..., dn) ≤ C+. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 10 of 27 3.2. CFEFOWA operator Definition 6. The gathering of CFFNs are cj = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } and the weight vector is G = (G1, G2, ..., Gn) T with Gj ∈ [0, 1] and n∑ j=1 Gj = 1. Then CFEFOWA(c1, c2, ..., cn) = n⊕ j=1 Gjcj is said CFEFOWA operator. Theorem 4. The collection of CFFNs are aj = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } and the weight vector is L = (L1, L2, ..., Ln) T with Lj ∈ [0, 1] and n∑ j=1 Lj = 1. Then it is said CFEFOWA operator and CFEFOWA(a1, a2, ..., an) = 〈  3 √√√√√√√ n∏ j=1 ((1+H− j )3)L− n∏ j=1 (1−H− j )3)L n∏ j=1 ((1+H− j )3)L+ n∏ j=1 ((1−H− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+H+ j )3)L− n∏ j=1 ((1−H+ j )3)L n∏ j=1 ((1+H+ j )3)L+ n∏ j=1 ((1−H+ j )3)L  , 3√2 n∏ j=1 (Hj) L 3 √√√√ n∏ j=1 ((2−Hj)3)L+ n∏ j=1 ((Hj)3)L 〉 , 〈  3 √√√√√√√ n∏ j=1 ((1+R− j )3)L− n∏ j=1 ((1−R− j )3)L n∏ j=1 ((1+R− j )3)L+ n∏ j=1 (1−R− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+R+ j )3)L− n∏ j=1 ((1−R+ j )3)L n∏ j=1 ((1+R+ j )3)L+ n∏ j=1 ((1−R+ j )3)L  , 3√2 n∏ j=1 (Rj) L 3 √√√√ n∏ j=1 ((2−Rj)3)L+ n∏ j=1 ((Rj)3)L 〉  Theorem 5. (Idempotency):If Ỹ = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } for all k = 1, 2, 3, ..., n, then CFEFOWA(Y, Y, Y, ..., Y ) = Y. Theorem 6. (Boundedness):If Z− = min(b1, b2, ..., bn), Z + = max(b1, b2, ..., bn), then Z− ≤CFEFOWA(b1, b2, ..., bn) ≤ Z+. 3.3. CFEFHWA operator Definition 7. The gathering of CFFNs are zj = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } and the weight vector is G = (G1, G2, ..., Gn) T with Gj ∈ [0, 1] and n∑ j=1 Gj = 1 and associated vector is A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 11 of 27 G = (G1, G2, ..., Gn) T with Gj ∈ [0, 1] and n∑ j=1 Gj = 1. Then CFEFHWA(z1, z2, ..., zn) = n⊕ j=1 Gjzj is said CFEFHWA operator. Theorem 7. The collection of CFFNs are aj = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } and the weight vector is L = (L1, L2, ..., Ln) T with Lj ∈ [0, 1] and n∑ j=1 Lj = 1. Then it is said CFEFHWA operator and CFEFHWA(a1, a2, ..., an) = 〈  3 √√√√√√√ n∏ j=1 ((1+H− j )3)L− n∏ j=1 (1−H− j )3)L n∏ j=1 ((1+H− j )3)L+ n∏ j=1 ((1−H− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+H+ j )3)L− n∏ j=1 ((1−H+ j )3)L n∏ j=1 ((1+H+ j )3)L+ n∏ j=1 ((1−H+ j )3)L  , 3√2 n∏ j=1 (Hj) L 3 √√√√ n∏ j=1 ((2−Hj)3)L+ n∏ j=1 ((Hj)3)L 〉 , 〈  3 √√√√√√√ n∏ j=1 ((1+R− j )3)L− n∏ j=1 ((1−R− j )3)L n∏ j=1 ((1+R− j )3)L+ n∏ j=1 (1−R− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+R+ j )3)L− n∏ j=1 ((1−R+ j )3)L n∏ j=1 ((1+R+ j )3)L+ n∏ j=1 ((1−R+ j )3)L  , 3√2 n∏ j=1 (Rj) L 3 √√√√ n∏ j=1 ((2−Rj)3)L+ n∏ j=1 ((Rj)3)L 〉  Theorem 8. (Idempotency):If c̃ = { ⟨[H−, H+], H⟩, ⟨[R−, R+], R⟩ } for all k = 1, 2, 3, ..., n, then CFEFHWA(Y, Y, Y, ..., Y ) = Y. Theorem 9. (Boundedness):If F− = min(d1, d2, ..., dn), F + = max(d1, d2, ..., dn), then F− ≤CFEFHWA(d1, d2, ..., dn) ≤ F+. 4. MCDM technique on CFF Step 1:Describe the CFF decision matrix Step 2:Describe the CFEFWA operator and L = (L1, L2, ..., Ln) . CFEFWA(d1, d2, ..., dn) = A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 12 of 27 〈  3 √√√√√√√ n∏ j=1 ((1+H− j )3)L− n∏ j=1 (1−H− j )3)L n∏ j=1 ((1+H− j )3)L+ n∏ j=1 ((1−H− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+H+ j )3)L− n∏ j=1 ((1−H+ j )3)L n∏ j=1 ((1+H+ j )3)L+ n∏ j=1 ((1−H+ j )3)L  , 3√2 n∏ j=1 (Hj) L 3 √√√√ n∏ j=1 ((2−Hj)3)L+ n∏ j=1 ((Hj)3)L 〉 , 〈  3 √√√√√√√ n∏ j=1 ((1+R− j )3)L− n∏ j=1 ((1−R− j )3)L n∏ j=1 ((1+R− j )3)L+ n∏ j=1 (1−R− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+R+ j )3)L− n∏ j=1 ((1−R+ j )3)L n∏ j=1 ((1+R+ j )3)L+ n∏ j=1 ((1−R+ j )3)L  , 3√2 n∏ j=1 (Rj) L 3 √√√√ n∏ j=1 ((2−Rj)3)L+ n∏ j=1 ((Rj)3)L 〉  Step 3:Describe the CFEFWA operator and L = (L1, L2, ..., Ln) . CFEFWA(d1, d2, ..., dn) = 〈  3 √√√√√√√ n∏ j=1 ((1+H− j )3)L− n∏ j=1 (1−H− j )3)L n∏ j=1 ((1+H− j )3)L+ n∏ j=1 ((1−H− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+H+ j )3)L− n∏ j=1 ((1−H+ j )3)L n∏ j=1 ((1+H+ j )3)L+ n∏ j=1 ((1−H+ j )3)L  , 3√2 n∏ j=1 (Hj) L 3 √√√√ n∏ j=1 ((2−Hj)3)L+ n∏ j=1 ((Hj)3)L 〉 , 〈  3 √√√√√√√ n∏ j=1 ((1+R− j )3)L− n∏ j=1 ((1−R− j )3)L n∏ j=1 ((1+R− j )3)L+ n∏ j=1 (1−R− j )3)L , 3 √√√√√√√ n∏ j=1 ((1+R+ j )3)L− n∏ j=1 ((1−R+ j )3)L n∏ j=1 ((1+R+ j )3)L+ n∏ j=1 ((1−R+ j )3)L  , 3√2 n∏ j=1 (Rj) L 3 √√√√ n∏ j=1 ((2−Rj)3)L+ n∏ j=1 ((Rj)3)L 〉  Step 4:Provide the score function. Step 5:Find the ordering. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 13 of 27 5. Case study In this section, four modernizer middles have been painstaking at the twitch opinions, which are full by A opinion and finally second hand by D opinion. Respectively opinion has been examined by their conservation antiquity and recent peak weight opinion. Mod- ernizer conservation antiquity and recent peak load opinion can exemplify the greatest of their makings. This investigation efforts to generate a distinct peep to identify opinions, which are essential for decisive the site of EV charging positions, that container help to shape a joining with the network strategy. In the next units, the future solution for the decisive organization of EV charging positions will be expounded with instances. Ankara municipal electricity network is unique in the actual cases to education, representative significant limits, when the network must be prepared to stock electric vehicles and load- ings. As validated in the works, the groups of charging positions coldness can alteration excellence of control, service, cost, and EVs skill receipt ratio concerning memberships of civilization. numerous substances, such as the network substructure and current Evs’ successful round, strongminded the variety of charging positions. Moreover, if the network does not use charging positions, some sites should be gritty by additional stipulations. These can be diverse in each network and be contingent on the network’s disorder. Ac- cording to the hitherto-stated data and Ankara metropolitan electricity substructure, the distance of network stations and power volume are critical opinions in the organization of the electric vehicle charge position theme. A city’s transport routes are dissimilar from the way of the power broadcast network and each site has exact power volume particulars. In the first step, creators must choose sites like opinions or seats, which have the lowermost coldness to electricity stations and are appropriate for transport schemes. The goalmouth is to degree energy and duty for supply and request. For instance, two dissimilar opinions in Ankara have been measured by the A and B modifier middles. Both are on the chief and packed roads, but the A modernizer middle’s disorder brands its additional working than the B modifier midpoint. This is because the modernizer midpoint is adjacent to the center where the main transport stations, such as buses and Pullmans, are contained. Whereas 1-2 km from canter, metro, stations, and location of modifier A and B have nu- merous allowed seats, space, schools, hospitals, and ministries. It appears that in this part, many vehicles are stationary, and this can be potentially beneficial. In contrast, the B transformer midpoint does not have personal choices, and because of this, the B mod- ernizer midpoint misplaces its importance in shaping a charging position. Site EV charge position opinions must be branded by sensible detachments such as 30, 35, 40, 45, and 50 km, contingent on part and substructure disorder. In the stated circumstance, the network situations were analysed, based on the modernizer rank, even though there remained no pre-defined storing midpoint or prepared charge position. This designates that modernizer stipulations could deliver potential data to assess the probability of charging positions [50]. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 14 of 27 5.1. Numerical example EV charging can be classified into four main categories: slow, fast, rapid, and ultra- quick. These represent the power outputs and, thus, charging speeds that are possible to charge an electric vehicle (EV). FGNEX 60-160kW DC Fast Charger Fast charging of pure electric vehicles is the main use for the DC charging pile, an isolated DC charging pile focused on product safety performance. With an IP 55 protection grade and an environ- mental protection design, these charging heaps are built for outdoor floor types and include corrosion-proof, waterproof, and dust-proof capabilities. The product’s modular design in- tegrates the car connector, human-machine interface, charger, communication, and billing elements into a single cabinet, making it easy to install, troubleshoot, and maintain. The items may be used in large parking lots, residential areas, malls, hospitals, transfer sta- tions, airports, docks, parks, and other places. FGNEX 60-180kW Multi-Standard DC Fast Charger Wuxi Fgnex Technology Co., Ltd. developed a particular kind of DC EV charging station known as an EV Charging Station to increase the safety of its products. To provide the maximum charging power from any charging station, this charging system solution’s flexible combination and free power distribution characteristics can be adjusted to the needs of the customer. The product’s modular design idea combined the vehicle connector, human-machine interface, charger, communication, and billing parts into a sin- gle cabinet to enable straightforward installation, debugging, operation, and maintenance, among other benefits. The items may be used in large parking lots, residential areas, malls, hospitals, transfer stations, airports, docks, parks, and other places. FGNEX 300-600kW Split Type Pantograph System With a focus on product safety, Wuxi Fgnex Technology Co., Ltd. developed the DC split pantograph HK-300-BE1. Mostly, all-electric bus quick charging is done with it. This range of outdoor split-type goods has an IP54 protection cer- tification and is designed to be waterproof, dustproof, anti-corrosion, and environmentally friendly. This pantograph has combined the charging interface, location, communication, invoicing, monitoring, and other features into a single cabinet by using a modular design approach. It could be used by the bus station or a large parking lot, for example. FGNEX 300kw Split Type Fan Cooling DC Charging System To increase the safety of its products, Wuxi Fgnex Technology developed a particular kind of DC EV charging station known as an EV Charging Station. This charging system solution’s free power distribution and flexible combination features allow it to be customized to the needs of the user to provide the maximum charging power possible from any type of charging station. The product’s modular design idea incorporated the vehicle connector, charger, human-machine interface, communication, and billing sections into one cabinet to promote easy installation and de- bugging, simple operation and maintenance. The goods may be used in airports, docks, parks, shopping centers, hospitals, hospitals, large parking lots, and other venues, quick, ultra-rapid, fast, and swift. Step 1:Explain Tables 1 and 2 of the CFF Decision Matrix.. CFF decision matrix table 1. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 15 of 27 S F R U PS1  〈 [0.1, 0.3], 0.2 〉 , 〈 [0.2, 0.4], 0.3 〉   〈 [0.02, 0.05], 0.04 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉   〈 [0.01, 0.14], 0.2 〉 , 〈 [0.31, 0.37], 0.6 〉  PS2  〈 [0.02, 0.05], 0.04 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.1, 0.3], 0.2 〉 , 〈 [0.2, 0.4], 0.3 〉   〈 [0.01, 0.14], 0.2 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉  PS3  〈 [0.01, 0.14], 0.2 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.02, 0.05], 0.04 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.1, 0.3], 0.2 〉 , 〈 [0.2, 0.4], 0.3 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉  PS4  〈 [0.01, 0.14], 0.2 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉   〈 [0.02, 0.05], 0.04 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.1, 0.3], 0.2 〉 , 〈 [0.2, 0.4], 0.3 〉  CFF decision matrix table 2. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 16 of 27 S F R U PS1  〈 [0.11, 0.13], 0.12 〉 , 〈 [0.2, 0.4], 0.3 〉   〈 [0.12, 0.15], 0.14 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.21, 0.23], 0.22 〉 , 〈 [0.21, 0.34], 0.4 〉   〈 [0.31, 0.34], 0.22 〉 , 〈 [0.31, 0.37], 0.6 〉  PS2  〈 [0.31, 0.34], 0.22 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.1, 0.3], 0.2 〉 , 〈 [0.2, 0.4], 0.3 〉   〈 [0.01, 0.14], 0.2 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉  PS3  〈 [0.01, 0.14], 0.2 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.02, 0.05], 0.04 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.31, 0.34], 0.22 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉  PS4  〈 [0.31, 0.34], 0.22 〉 , 〈 [0.31, 0.37], 0.6 〉   〈 [0.11, 0.13], 0.1 〉 , 〈 [0.21, 0.34], 0.4 〉   〈 [0.02, 0.05], 0.04 〉 , 〈 [0.1, 0.3], 0.1 〉   〈 [0.31, 0.34], 0.22 〉 , 〈 [0.31, 0.37], 0.6 〉  Step 2:Describe the CFEFWA operator and ξ = (0.26, 0.21, 0.25, 0.28) . CFEFWA operator table 3. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 17 of 27 S F R U PS1  〈 [0.1001, 0.1983], 0.1092 〉 , 〈 [0.2123, 0.4098], 0.3045 〉   〈 [0.1452, 0.1675], 0.1124 〉 , 〈 [0.1231, 0.3011], 0.1154 〉   〈 [0.2561, 0.2343], 0.2032 〉 , 〈 [0.2671, 0.3034], 0.0914 〉   〈 [0.3091, 0.3344], 0.2092 〉 , 〈 [0.3041, 0.3127], 0.6163 〉  PS2  〈 [0.3031, 0.3014], 0.2052 〉 , 〈 [0.3081, 0.3047], 0.6129 〉   〈 [0.1231, 0.3453], 0.9872 〉 , 〈 [0.2125, 0.4097], 0.3026 〉   〈 [0.0001, 0.1004], 0.2123 〉 , 〈 [0.3001, 0.3097], 0.6098 〉   〈 [0.1091, 0.1023], 0.1347 〉 , 〈 [0.2001, 0.3094], 0.4931 〉  PS3  〈 [0.0131, 0.1134], 0.2359 〉 , 〈 [0.3001, 0.3037], 0.6124 〉   〈 [0.0212, 0.0556], 0.0411 〉 , 〈 [0.1034, 0.3012], 0.1104 〉   〈 [0.3051, 0.3084], 0.2082 〉 , 〈 [0.3121, 0.3097], 0.6912 〉   〈 [0.1091, 0.1973], 0.1123 〉 , 〈 [0.2091, 0.3344], 0.4654 〉  PS4  〈 [0.3061, 0.3074], 0.2122 〉 , 〈 [0.3051, 0.3057], 0.6981 〉   〈 [0.1011, 0.1033], 0.117 〉 , 〈 [0.2051, 0.3415], 0.4951 〉   〈 [0.0122, 0.0045], 0.0054 〉 , 〈 [0.0121, 0.3453], 0.5671 〉   〈 [0.3001, 0.3944], 0.2342 〉 , 〈 [0.3671, 0.3745], 0.6956 〉  Step 3:Describe the CFEFWA operator and ξ = (0.26, 0.21, 0.25, 0.28) . CFEFWA operator table 4. CFEFWA operator table 4 PS1 [⟨[0.1201, 0.2983], 0.1492⟩ , ⟨[0.2003, 0.4698], 0.3645⟩] PS2 [⟨[0.4531, 0.5014], 0.2452⟩ , ⟨[0.7081, 0.6047], 0.7129⟩] PS3 [⟨[0.5101, 0.6134], 0.1359⟩ , ⟨[0.3231, 0.1237], 0.1124⟩] PS4 [⟨[0.3461, 0.1674], 0.3422⟩ , ⟨[0.8951, 0.0457], 0.0081⟩] Step 4: Find the score function Z1 = 0.0189, Z2 = 0.4556, Z3 = 0.1425, Z4 = 0.3909. Step 5: The ranking is Z2 > Z4 > Z3 > Z1 and best is Z2. 6. Comparsion technique with existing method The suggested method’s comparison with a different strategy based on FFSs [26] and IVFF data [46] is shown with cases in order to validate and establish its feasibility. Ad- ditional strategies are exceptional instances of our technique strategy, which is based on CFFN to the same example. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 18 of 27 6.1. FF approach using the current methodology Step 1:Table 5 in the list below represents the FF decision matrix. S F R U PS1 [0.1, 0.3] [0.12, 0.14] [0.9, 0.11] [0.1, 0.3] PS2 [0.12, 0.13] [0.4, 0.6] [0.31, 0.33] [[0.112, 0.313] PS3 [0.34, 0.36] [0.12, 0.13] [0.4, 0.6] [0.554, 0.556] PS4 [0.3004, 0.3006] [0.34, 0.36] [0.12, 0.13] [0.4, 0.6] Step 2:Define (0.1, 0.2, 0.3, 0.5) and the FFWA operator. Using the provided formula now FFWA operator table 6 PS1 [0.1024, 0.3145] PS2 [0.0134, 0.2156] PS3 [0.0991, 0.3353] PS4 [0.0201, 0.3043] Step 3:Provide score function Z1 = 0.0105, Z2 = 0.4859, Z3 = 0.1454, Z4 = 0.3349. Step 4:The ranking is Z2 > Z4 > Z3 > Z1 and best is Z2. 6.2. IVFF number using the current technique Step 1:In Table 7, the IVFF choice matrix is provided. Table 7 of the IVFF choice matrix S F R U PS1  [0.1, 0.3], [0.2, 0.4]   [0.02, 0.05], [0.1, 0.3]   [0.11, 0.13], [0.21, 0.34]   [0.01, 0.14], [0.31, 0.37]  PS2  [0.11, 0.13], [0.21, 0.34]   [0.1, 0.3], [0.2, 0.4]   [0.02, 0.05], [0.1, 0.3]   [0.1, 0.3], [0.2, 0.4]  PS3  [0.02, 0.05], [0.1, 0.3]   [0.11, 0.13], [0.21, 0.34]   [0.1, 0.3], [0.2, 0.4]   [0.02, 0.05], [0.1, 0.3]  PS4  [0.1, 0.3], [0.2, 0.4]   [0.01, 0.14], [0.31, 0.37]   [0.02, 0.05], [0.1, 0.3]   [0.11, 0.13], [0.21, 0.34]  Step 2:Describe the IVFFWA operator and ξ = (0.26, 0.21, 0.25, 0.28) . IVFFWA operator table 8. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 19 of 27 IVFFWA operator table 8 PS1 [ [0.1211, 0.3123], [0.2123, 0.4123] ] PS2 [ [0.1011, 0.1312], [0.2111, 0.3234] ] PS3 [ [0.1102, 0.2305], [0.1098, 0.3123] ] PS4 [ [0.1231, 0.3234], [0.6752, 0.4125] ] Step 3: Find the score function Z1 = 0.1102, Z2 = 0.5851, Z3 = 0.2004, Z4 = 0.4352. Step 4:The ranking is Z2 > Z4 > Z3 > Z1 and best is Z2. Different existing way Table 9 Techniques Score function Ordering Final Ordering MCDM [45]  η1 = 0.0014, η2 = 0.7755, η3 = 0.1811, η4 = 0.3711   η2 > η4 > η3 > η1   η2 > η4 > η3 > η1  IFN [9]  η1 = 0.1014, η2 = 0.4234, η3 = 0.1112, η4 = 0.3723   η2 > η4 > η3 > η1   η2 > η4 > η3 > η1  FFY operators [12]  η1 = 0.0055, η2 = 0.1456, η3 = 0.0804, η4 = 0.1193   η2 > η4 > η3 > η1   η2 > η4 > η3 > η1  IVIFSS operators [14]  η1 = 0.0126, η2 = 0.2255, η3 = 0.0712, η4 = 0.1456   η2 > η4 > η3 > η1   η2 > η4 > η3 > η1  Third Column: The items or criteria are ranked in the Organization column based on the scores indicated in the Score Function column. The performance levels or priorities that the approach has assigned are accurately represented here. Significance: Understanding the relative ranks of the criteria for each technique depends on this column. It reflects the instantaneous result of the scoring system and shows a clear hierarchical order. Analysis: Study how each practise arranges the criteria. For example, if System A places η2 first and η1 last, this indicates a strong preference for η2 over η1. The comparison between different techniques’ orderings reveals their relative priorities and approaches. 4th Column: Final Ordering Goal: Following the smearing of the system’s score function and any additional processing or aggregation processes, the Final Ordering column provides the final ranking. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 20 of 27 It represents the ultimate choice or front-runner because of the early score. Significance: This column is essential to comprehending the study’s outcome. It illustrates how the combined ranking or final product is determined by the scores and ordering strategies applied. This is the result that would guide choices. Analysis: Consider how crosswise approaches are associated with the final order- ings. Variations in the final orderings can indicate how various approaches have affected the position process. This comparison helps understand which approach, based on the criteria, offers a better-desired outcome. Ordering (3rd Column): This column reproduces the position of criteria founded on the scores provided in the Score Function column. It signifies the preliminary preparation or arranging of criteria by each system. For example, in the MCDM technique, the ordering of η2 > η4 > η3 > η1 shows a higher partiality for η2 over other criteria. Final Ordering (4th Column): The final ordering consolidates the rankings derived from the ordering column to present a definitive preference list. This is the final output after applying the technique’s methodology. For example, the final ordering for MCDM remains η2 > η4 > η3 > η1, consistent with the initial ordering, reflecting that the technique maintains the same prioritization in its final output. 6.3. Superior CFFSs are considered superior to IVFSs for several reasons: Richer Information Content: While IVFSs use intervals to represent membership de- grees, CFFSs extend this by also incorporating non-membership degrees as intervals. This dual interval approach provides a more comprehensive picture of the uncertainty. Better Handling of Ambiguity: CFFSs can model situations where the degree of mem- bership and non-membership are not only uncertain but also interdependent. This ca- pability is crucial for more accurately capturing the nuances of human reasoning and decision-making. Advanced Aggregation Operators: CFFSs enable the development of more sophisticated aggregation operators, such as the Dombi aggregation operators. These operators can combine information more effectively, leading to better decision-making outcomes. To improve biomedical waste management by using advanced aggregation operators within a Fermatean fuzzy framework [51]. This research combines interval-valued Fer- matean fuzzy sets with Dombi aggregation operators and the SWARA-based PROMETHEE II method to handle the uncertainty and complexity of biomedical waste management [51]. To identify the best renewable energy sources in India by using a sophisticated decision- making framework [47]. 6.4. Advantages of Einstein Aggregation Operators Improved Flexibility: Einstein aggregation operators provide a higher degree of flex- ibility compared to traditional aggregation operators. They can model a wider range of fuzzy interactions, making them suitable for complex decision-making scenarios. A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 21 of 27 Enhanced Precision: By leveraging the properties of Einstein t-norm and t-conorm, these operators allow for more precise manipulation of fuzzy sets. This precision is crucial in scenarios where the exact nature of the fuzzy interactions significantly impacts the decision outcome. Better Handling of Uncertainty: Einstein aggregation operators are particularly effec- tive in dealing with high levels of uncertainty and imprecision. They offer a robust frame- work for combining fuzzy information, which is essential for multi-criteria decision-making processes. Compatibility with Advanced Fuzzy Sets: These operators are well-suited for advanced fuzzy sets such as cubic Fermatean fuzzy sets and q-Rung orthopair fuzzy sets, enabling their effective use in complex decision-making problems. 6.5. Why Einstein Aggregation Operators Defined Einstein aggregation operators are defined to offer a novel approach to combining fuzzy information, especially in the context of decision-making processes that involve uncertainty and imprecision. These operators are based on the Einstein t-norm and t-conorm, which are mathematical functions used to generalize the concept of intersection and union in fuzzy logic. The primary motivation behind defining Einstein aggregation operators is to enhance the flexibility and applicability of fuzzy set theory in real-world scenarios, particularly those requiring nuanced handling of uncertainty. 7. Discussion In this section, we have proposed the advantages and Limitations or weaknesses. 7.1. Advantage • Uses Cubic Fermatean fuzzy sets to effectively manage high levels of ambiguity and imprecision. • Presents new Einstein aggregation operators that restore the consistency and weight of the totaled choice statistics. • Lessens bias and prejudice by allowing a wide humanoid to preside over the decision- making process. • Completes advanced fuzzy set model and aggregation approaches with more reliable and unbiased outcomes. • Provides a more thorough and adaptable strategy for making decisions in ambiguous situations. Since Einstein aggregation operators are more flexible and adaptive than the Bonferroni mean, which focuses primarily on averaging, or Dombi operations, which rely on strict mathematical formulations, they are particularly effective in fuzzy and intuitionistic fuzzy environments. Their nonlinear structure allows for better representation of interactions between criteria, which ensures that the aggregated values retain important information, reducing potential distortions that may arise in other aggregation methods. However, A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 22 of 27 Einstein aggregation operators offer a significant advantage in handling uncertainty and complex interrelations among fuzzy parameters. Furthermore, by being stable even while working with extremely ambiguous or imprecise data, Einstein aggregation operators im- prove the dependability of decision-making procedures. They are more suited for situations when conventional approaches would not be adequate because of their capacity to capture the fuzziness that exists naturally in real-world issues. Einstein operators are a popular option for complex group decision-making scenarios because they balance computational simplicity and robustness when compared to other aggregation strategies. 7.2. Limitation • The system of requirements has significant computational assets and periods, which could be a limitation for real submissions. • The accuracy of the system is highly dependent on the excellence and precision of the exertion data. • The effort of the cubic Fermatean fuzzy Einstein aggregation operator’s strength attitude challenges in application and understanding for experts not familiar with liberal fuzzy set theory. • The technique’s strength meets difficulties when applied to larger and extra compound decision-making states, perhaps cautioning its scalability. 8. Conclusion Selecting the best location for EVs demonstrates a proactive role in promoting the expansion of the electric vehicle system. We have suggested a Cubic Fermatean fuzzy ap- proach using traditional Einstein aggregation operators in CFFS contexts for EV growth. Some Einstein operators have been industrialized under a Cubic Fermatean fuzzy envi- ronment thanks to the discovery of algebraic operators. The technique has been used in an exemplary case study of the EV site assortment problem with CFFSs setting to ver- ify its potential and efficacy. The current study involves an assessment index procedure that includes setting, budget, social, and skill elements of criteria for EV site candidates. The criteria’s four elements are comprised of facts from experts, research intelligence, and writings. The suggested outline contrasts with current techniques using CFFS settings to validate the fallouts. The application procedure and the results obtained demonstrate the potential for a unified technique to be established in an industry with erratic data and a wide range of valuation criteria sizes. Nonetheless, several potential boundaries need to be assessed in the follow-up work. These include: (i) the study mishandles the contract with multilayered EV location assortment problematic with interdependent criteria; (ii) the study uses slow, unreliable data; and (iii) additional sustainability criteria may be complex in the valuation directory scheme. The method that is being described will be useful in determining the most suitable option in situations with several criteria and uncertainty. Future developments should include training, more EV sites, and more easily maintained A. Fahmi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5891 23 of 27 criteria. In addition, additional training will focus on ideal constraints that account for yield growth, degree of gratification and request uncertainty, customer behavior assessment during the charging phase, and charging station profiles that account for topographical and chronological data of circulation current. Furthermore, the proposed methodology will be extended to address additional challenges within the context of CFFS, such as low-carbon sustainable supplier assortment, medical hotel interference, e-learning website assessment, and others. In the future scope, our technique could also be practical to another exten- sion of fuzzy sets, such as quasirung fuzzy sets. Multiple attribute decision-making based on 3, 4-quasirung fuzzy sets offers a vigorous framework for management uncertainty and intricacy in decision-making situations (refer to [49] for thorough practices and requests). Including these sets might enhance the litheness and applicability of our method in many areas. Acknowledgements DKA extends appreciation to the Deanship of Postgraduate Studies and Scientific Re- search at Majmaah University for supporting this research work. Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and the support through TAS research lab. 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