EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6110 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Exponential Intuitionistic Fuzzy Framework for Sustainable Supplier Selection in Industrial Management M. Kaviyarasu1, Luminita-Ioana Cot̂ırlă 2, Daniel Breaz3,∗, M. Rajeshwari4 1 Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, Chennai, Tamil Nadu 600062, India 2 Department of Mathematics, Technical University of Cluj Napoca, 400114, Cluj-Napoca, Romania 3 Department of Mathematics, ”1 Decembrie 1918”, University of Alba Iulia, 510009, Alba Iulia, Romania 4 Department of Mathematics, Presidency University, Bangalore 560064, India Abstract. Dealing with uncertainty and imprecision is critical in modern decision-making pro- cesses, particularly in multi-criteria situations. Exponential intuitionistic fuzzy sets (EIFS) are a sophisticated mathematical framework that extends intuitionistic fuzzy sets by integrating an exponential function, improving its capacity to adequately capture ambiguity. This study investi- gates several EIFS operations, such as EIF intersection and union, which expand conventional set operations to accommodate partial and non-membership NM degrees more flexibly. Fur- thermore, we look into crucial operations including simple difference, limited difference, disjoint sum, disjunctive sum, and Cartesian product, all of which play important roles in modelling uncertainty in decision processes. A case study on raw material purchase demonstrates how these processes might be applied. The decision-making system takes into account numerous providers, analysing aspects such as cost, quality, and delivery time with EIFS-based aggre- gation approaches. Using these activities, decision-makers may improve supplier performance, reduce risks, and optimise procurement strategies in unpredictable circumstances. 2020 Mathematics Subject Classifications: 03E72, 90B50, 90C29, 68T37 Key Words and Phrases: Fuzzy set, Exponential aggregation operators, Exponential fuzzy set, Exponential intuitionistic fuzzy set, Decision-making methods ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6110 Email addresses: kavitamilm@gmail.com (M. Kaviyarasu), luminita.cotirla@math.utcluj.ro (L.-I. Cot̂ırlă), dbreaz@uab.ro (D. Breaz), rajeshwari@presidencyuniversity.in (M. Rajeshwari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 2 of 20 1. Introduction Fuzzy Sets(FSs), a groundbreaking mathematical framework for dealing with impre- cision and uncertainty, was first presented by Zadeh in [1]. According to conventional (crisp) set theory, an element is either a member of a set or it is not (binary member- shiph (M)). By proposing FSs, in which each element has a degree of M in the interval [0, 1], Zadeh expanded on this concept. The idea of an eventual distribution is defined by Zadeh in [2] to be a fuzzy limitation that serves as an elastic constraint on the values that may be given to a variable. This definition relates to the notion of FSs that are fuzzy. Hiroshi et al suggests a fuzzy decision-making approach for multiobjective choice issues in [3]. Decision makers often prefers fuzzy connectives with the primary characteristics of unclear and unreliable solutions. The Analyst interacts with an algorithm to derive the preference pattern. Here, Zimmermann’s γ operation and their expansions are used as fuzzy connectives. Chen et al. In [4] introduce novel methods in the perspective of fuzzy set theory for solving multicriteria ambiguous problem solving. The set of factors phase methods enable the presentation of ambiguous values that stipulate the degree of satisfiability and non-satisfiability of all choice. The decision-maker can also give each criterion a varying level of priority thanks to the procedures. IFS is a popular FS that was developed by Atanassov [5]. In contrast to traditional FS theory, NM and indeterminacy degree notations are specified in addition to M degree. By expanding with several technologies, IFS may be utilised to solve problems and make decisions by taking use of their capabilities to identify ambiguity and inaccuracy. New approaches to multifaceted decision-making in an intuitionistic fuzzy context Liu pre- sented in [6]. To determine the extent to which options meet or fall short of the decision- maker’s requirements, define a measure algorithm. After that, talk about intuitionistic fuzzy point operators. According to the finest supplier in a group decision-making set- ting, Boran et al. [7] suggest combining the TOPSIS technique with intuitionistic FSs. To rate the significance of conditions and alternative solutions, the unique views within decision makers are aggregated using the intuitionistic fuzzy weighted averaging (IFWA) method. The innovative similarity metric between Atanassov’s intuitionistic FSs (AIFSs) established the transformation of unique techniques was introduced by Chen et al. in [8]. The connection metric between AIFSs was then applied to pattern recognition tasks. First, provide a novel similarity metric between Atanassov’s intuitionistic fuzzy values (AIFVs) and demonstrate a few of its characteristics. Next, The similarity measure of AIFVs is extends based on the suggested similarity measure between AIFVs. Its prop- erties are demonstrated, and examples are provided to show how the initiate similarity measure between AIFVs can overcome the shortcomings of the current similarity mea- sures. Lastly, use the AIFVs similarity metric to address issues with pattern recognition. The intuitionistic fuzzy social relational network (IFSRN) model, which includes positive, neutral, and negative relationships between the degrees which can be expressed by intu- itionistic fuzzy values (IFVs) is the basis for fuzzy query progressing methods in [9] Chen et al. Ejegwa and Agbetayo present a new similarity-distance method with a higher score M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 3 of 20 for efficiency in [10]. To demonstrate the benefits of the unique similarity-distance over comparable current techniques, an evaluation is provided. A few characteristics of the similarity-distance method are shown. Additionally, the innovative similarity-distance technique’s applicability in various cases of decision-making are investigated. In [11], Garg et al. introduce a unique algebraic structure for C− IFSs that is based on Archimedean t-norm operations, such as division, addition, multiplication, and sub- traction. A single rating may be created by combining the preferences of several experts thanks to these methods. An expanded EDAS (Evaluation Based on Distance from Av- erage Solution) approach is also used, which ranks options using defuzzification methods and weighted aggregation methods. New operators for GIFSBs with characteristics in- cluding idempotency, boundedness, monotonicity, and commutativity are introduced by Wasim et al. in [12], producing aggregated values that are in line with GIFNs. The links between these processes are thoroughly examined, providing a deep comprehension of their application. In the world of contemporary decision making especially in uncertainty and imprecision, techniques of multi-criteria decision-making play a critical role. The study by Jawad Ali [13] presents a hybrid MCDM method which uses T-spherical fuzzy sets and Aczel?Alsina prioritized aggregation operators. This holistic modeling strategy increases the complex scenario modeling of decisions, where subjective judgment and objective data are essential. It is effective in the selection of apt medical experts that provide stable and flexible ranking system. Complimenting this, another contribution by Jawad Ali complimented by Suhad Ali Osman Abdallah and N. S. Abd El-Gawaad [14] suggests a state-of-art approximation decision-analysis framework based on SWARA method in conjunction with Frank aggregation and p,q-rung orthopair fuzzy informa- tion. This model is tailored for the cases in which the decision-makers have to evaluate a number of criteria with unknown weights for the priority evaluation in the environments of uncertainty. A multi-criteria decision-making approach for assessing startup perfor- mance in the IT sector provides a real demonstration of these concepts. [15] Presents the idea of complex intuitionistic fuzzy classes, which are distinguished by their pure complex intuitionistic fuzzy M grade. Kaviyarasu and team [16] propose exponential fuzzy sets for AI-type investment decisions, considering weighted mean aggregation as a way of capturing non-linear uncertainty better. A compelling example of a pertinent application is given, together with the fundamental terminology and computations on intricate intuitionistic fuzzy classes. Using quaternion numbers as a basis Ngan et al. in [17] provide new operations and analyse the logic operations and order relations of the complex intuitionistic FS theory. Two algebraic and polar quaternion distance measurements were also covered, along with an analysis of their characteristics. Akram et al. present the CIF Hamacher weighted averaging (CIFHWA), CIF Hamacher ordered weighted averaging (CIFHOWA), CIF Hamacher weighted geometric (CIFHWG), and CIF Hamacher ordered weighted geo- metric (CIFHOWG) operators in [18]. In order to determine the phase term, Zeeshan and Khan [19] produced several fundamental conclusions and specific instances for stan- dard complex of fuzzy intersection, union, and complement functions with the same function. Sobhi and Dick [20] a novel neuro-fuzzy infrastructure built for enormous M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 4 of 20 scale learning issues, utilising complex FSs. In [21], Bilal et al. offer numerous unique operations on complex intuitionistic FSs, adding distance measure and σ-equalities. 1.1. Motivation • Traditional fuzzy approaches struggle with contradictory and inadequate data, making decision-making in uncertain situations difficult. • EIFS improve upon classic intuitionistic FSs by using exponential functions, re- sulting in a more sophisticated method to modelling uncertainty. • EIFS techniques such as intersection, union, difference, and Cartesian product facilitate the collection and analysis of ambiguous decision criteria. • EIFS-based methodologies improve supplier evaluation by taking into account unpredictable elements such as cost, quality, and delivery reliability, resulting in more successful purchasing techniques. 1.2. Novelty • Formalizes new EIFS operations, such as exponential intersection, union, and some difference operations, to extend the conventional set operations. • Disjoint and disjunctive sums and an EIF-based Cartesian product, which are specifically designed for uncertainty modeling in decision support systems. • Illustrates the applicability of EIFS by using a real-world example of raw material procurement and demonstrates the way EIFS enhances the quality of decisions under uncertainty. • Emphasizes the way EIFS-based aggregation supports supplier assessment by combining factors such as cost, quality, and delivery time, to enable more practical and risk-averse purchasing decisions. This paper is organized as: The Preliminaries of FSs and its operations are covered in section 2, The basic definitions and characteristics of EIFSs and some examples are covered in Section 3, The procedure of decision-making methodology under EIFSs are shown in Section 4, application of decision making problems are demonstrated in section 4.1 and the conclusion is in section 5 2. Preliminaries Definition 1. A FS A∗ in a universe of discourse X is characterized by a MF µA∗ which takes the value in the unit interval [0, 1] µA∗(ς̃∗) = X → [0, 1] The value of µA∗(ς̃∗) represents the grade of M of X in A∗ and is a point in [0, 1]. M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 5 of 20 Definition 2. A∗ is a FS on X, Then its complement (A∗ ′ ) is µ ′ A∗(ς̃∗) = 1− µA∗(ς̃∗). Definition 3. The union of two FSs A∗ and B∗ is with respective MFs µA∗(ς̃∗) and µB∗(ς̃∗) is a FS A∗ ∪ B∗, whose MF is related to those of A∗ and B∗ by µA∗∪B∗(ς̃∗) = ∨{µA∗(ς̃∗), µB∗(ς̃∗)} , ∀ς̃∗ ∈ X. Definition 4. The intersection of two FSs A∗ and B∗ is with respective memberships µA∗(ς̃∗) and µB∗(ς̃∗) is a FS A∗ ∩ B∗, whose MF is related to those of A∗ and B∗ by µA∗∩B∗(ς̃∗) = ∧{µA∗(ς̃∗), µB∗(ς̃∗)} , ∀ς̃∗ ∈ X. Definition 5. The usual manner of defining fuzzy simple differences between two fuzzy sets A∗ and B∗ can be expressed as A∗ − B∗ = A∗ ∩ B∗c. Definition 6. For any two FSs A∗ and B∗, the bounded difference is defined as: µA∗oB∗(ς̃∗) = ∨ [o, µA∗(ς̃∗), µB∗(ς̃∗)] where µA∗(ς̃∗) and µB∗(ς̃∗) denotes the MFto which x is a member of A∗ and B∗. Definition 7. The disjoint sum of any two FSs A∗ and B∗ on X µA∗⊗B∗(ς̃∗) = |⊗, µA∗(ς̃∗), µB∗(ς̃∗)| where µA∗(ς̃∗) and µB∗(ς̃∗) denotes the MF to which x is a member of A∗ and B∗. Definition 8. Let A∗ and B∗ be any two fuzzy sets of X then the disjunctive sum is defined as: µA∗∆B∗(ς̃∗) = ( A∗ ∩ B∗ ′ ) ∪ ( A∗ ′ ∩ B∗ ) = ( A∗ ∗ B∗ ′ ) ⊕ ( A∗ ′ ∗ B∗ ) . Definition 9. Let A∗ and B∗ be any two FSs of X then the equivalence formula is( A∗ ′ ∪ B∗ ) ∩ ( A∗ ∪ B∗ ′ ) = ( A∗ ′ ∩ B∗ ′ ) ∪ (A∗ ∩ B∗) . Definition 10. Symmetrical difference formula for two FSs A∗ and B∗ is given by( A∗ ′ ∩ B∗ ) ∪ ( A∗ ∩ B∗ ′ ) = ( A∗ ′ ∪ B∗ ′ ) ∩ (A∗ ∪ B∗) . M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 6 of 20 3. Exponential Intuitionstic Fuzzy Sets Definition 11. If X is a universe discourse and ς̃∗ be any particular element of X. The intuitionistic FS A∗ defined on X is a collection of ordered pairs, A∗ = {(ς̃∗, µA∗(ς̃∗), λA∗(ς̃∗)) |x ∈ X}, where µA∗(ς̃∗) : X → [0, 1] and λA∗(ς̃∗) : X → [0, 1] is called the MF and NMF. The degree of M and NM on elements X. 0 ≤ µA∗(ς̃∗) + λA∗(ς̃∗) ≤ 1. Definition 12. If X is a universe discourse and ς̃∗ be any particular element of X. The EIFS EA∗ defined on X is a collection of ordered pairs, EA∗ = {( ς̃∗, µA∗(ς̃∗)e −aµA∗ (ς̃∗), λA∗(ς̃∗)e −aλA∗ (ς̃∗) ) |ς̃∗ ∈ X, a > 0 } , where µA∗(ς̃∗)e −aµA∗ (ς̃∗) : X → [0, 1] is called the MF and λA∗(ς̃∗)e −aλA∗ (ς̃∗) : X → [0, 1] is called the NMF. The degree of MF is 0 ≤ µA∗(ς̃∗)e −aµA∗ (ς̃∗) + λA∗(ς̃∗)e −aλA∗ (ς̃∗) ≤ 1. Example 1. Let X = {1, 2, 3, 4, 5} be the universal set and IFMVs of X is µA(ς̃∗) = {0.9, 0.7, 0.5, 0.4, 0.3} and λA(ς̃∗) = {0.7, 0.5, 0.7, 0.3, 0.2}, the decay parameter a = 0.02. The EIF MF is given by: EA∗(ς̃∗) = ( µA∗(ς̃∗)e −aµA∗ (ς̃∗), λA∗(ς̃∗)e −aλA∗ (ς̃∗) ) . The EIF MVs EA∗(ς̃∗) = {(1, 0.884, 0.690), (2, 0.690, 0.495), (3, 0.495, 0.690), (4, 0.397, 0.298), (5, 0.298, 0.199)}. Figure 1: IEST . Definition 13. Let EA∗ = 〈 µEA∗ , λEA∗ 〉 and EB∗ = 〈 µEB∗ , λEB∗ 〉 be two EIFSs on X with their grade values given by: µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗) & λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗) M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 7 of 20 µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗) & λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗) The EIF intersection of EA∗ and EB∗ is defined as: µEA∗∩EB∗ (ς̃∗) = ∧ { µEA∗ (ς̃∗), µEB∗ (ς̃∗) } = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } (3.1) λEA∗∩EB∗ (ς̃∗) = ∨ { λEA∗ (ς̃∗), λEB∗ (ς̃∗) } = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aλB(ς̃∗) } (3.2) Similarly, the EIF union of EA∗ and EB∗ is defined as: µEA∗∪EB∗ (ς̃∗) = ∨ { µEA∗ (ς̃∗), µEB∗ (ς̃∗) } = ∨ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } (3.3) λEA∗∪EB∗ (ς̃∗) = ∧ { λEA∗ (ς̃∗), λEB∗ (ς̃∗) } = ∧ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), µB∗(ς̃∗)e −aλB(ς̃∗) } (3.4) Example 2. Let X = {ς̃∗1, ς̃∗2, ς̃∗3, ς̃∗4, ς̃∗5} be a finite universe, and let EA∗ and EB∗ be two EIFSs defined by the memberships: µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗) µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗) Now, we compute the corresponding µEA∗∪EB∗ (ς̃∗) and λEA∗∪EB∗ (ς̃∗) MNs: ς̃∗ ς̃∗1 ς̃∗2 ς̃∗3 ς̃∗5 ς̃∗1 µA(ς̃∗) 0.8 0.6 0.4 0.2 0.1 λA(ς̃∗) 0.2 0.3 0.4 0.1 0.1 µB(ς̃∗) 0.3 0.5 0.4 0.2 0.1 λB(ς̃∗) 0.6 0.5 0.5 0.6 0.7 Definition 14. Consider two EIFSs EA∗ = 〈 µEA∗ , λEA∗ 〉 and EB∗ = 〈 µEB∗ , λEB∗ 〉 where µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗) & λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗) µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗) & λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗) denotes the M and non-functions of EA∗ and EB∗. The simple difference EA∗ − EB∗ of these two EIFSs EA∗ and EB∗ is defined as: EA∗ − EB∗ = EA∗ ∩ EB∗ ′ = { µEA∗ (ς̃∗), λEA∗ (ς̃∗) } ∗ { µ ′ EB∗ (ς̃∗), λ ′ EB∗ (ς̃∗) } M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 8 of 20 = { µEA∗ (ς̃∗) ∗ µ ′ EB∗ (ς̃∗) } , { λEA∗ (ς̃∗) ∗ λ ′ EB∗ (ς̃∗) } = { µEA∗ (ς̃∗), λ ′ EB∗ (ς̃∗) } EA∗ − EB∗ = { µA∗(ς̃∗)e −aµA∗ (ς̃∗), λc B(ς̃∗)e −aλc B(ς̃∗) } Example 3. Let EA∗ = { 0.25e−a0.25 ς̃∗1 + 0.35e−a0.35 ς̃∗2 + 0.15e−a0.15 ς̃∗3 , 0.45e −a0.45 ς̃∗1 + 0.65e−a0.65 ς̃∗2 + 0.25e−a0.25 ς̃∗3 } and EB∗ = { 0.65e−a0.65 ς̃∗1 + 0.45e−a0.45 ς̃∗2 + 0.35e−a0.35 ς̃∗3 , 0.55e −a0.55 ς̃∗1 + 0.15e−a0.15 ς̃∗2 + 0.85e−a0.85 ς̃∗3 } be two two EIFSs, the simple difference is EA∗ − EB∗ = EA∗ ∩ EB∗ ′ = { 0.25e−a0.25 ς̃∗1 + 0.35e−a0.35 ς̃∗2 + 0.15e−a0.15 ς̃∗3 , 0.45e−a0.45 ς̃∗1 + 0.65e−a0.65 ς̃∗2 + 0.25e−a0.25 ς̃∗3 } ∗{ 0.35e−a0.35 ς̃∗1 + 0.55e−a0.55 ς̃∗2 + 0.65e−a0.65 ς̃∗3 , 0.45e−a0.45 ς̃∗1 + 0.85e−a0.85 ς̃∗2 + 0.15e−a0.15 ς̃∗3 } = {( 0.25e−a0.25 ς̃∗1 + 0.35e−a0.35 ς̃∗2 + 0.15e−a0.15 ς̃∗3 ) ∗ ( 0.35e−a0.35 ς̃∗1 + 0.55e−a0.55 ς̃∗2 + 0.65e−a0.65 ς̃∗3 )} ,{( 0.45e−a0.45 ς̃∗1 + 0.65e−a0.65 ς̃∗2 + 0.25e−a0.25 ς̃∗3 ∗ 0.45e−a0.45 ς̃∗1 + 0.85e−a0.85 ς̃∗2 + 0.15e−a0.15 ς̃∗3 )} = {( 0.25e−a0.25 ς̃∗1 + 0.35e−a0.35 ς̃∗2 + 0.15e−a0.15 ς̃∗3 ) , ( 0.45e−a0.45 ς̃∗1 + 0.65e−a0.65 ς̃∗2 + 0.15e−a0.15 ς̃∗3 )} . Definition 15. Consider two EIFSs EA∗ = 〈 µEA∗ , λEA∗ 〉 and EB∗ = 〈 µEB∗ , λEB∗ 〉 where µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗) & λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗) µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗) & λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗) denotes the M and non-functions of EA∗ and EB∗. The bounded difference of these two EIFSs EA∗ and EB∗ is defined as: µEA∗⋄EB∗ (ς̃∗) = ∨ [ 0, µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) ] . λEA∗⋄EB∗ (ς̃∗) = ∨ [ 0, λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aλB(ς̃∗) ] . Example 4. Let EA∗ = ( 0.22e−a0.22 ς̃∗1 + 0.82e−a0.82 ς̃∗2 + 0.36e−a0.36 ς̃∗3 , 0.35e−a0.35 ς̃∗1 + 0.52e−a0.52 ς̃∗2 + 0.46e−a0.46 ς̃∗3 ) and EB∗ = ( 0.35e−a0.35 ς̃∗1 + 0.43e−a0.43 ς̃∗2 + 0.05e−a0.05 ς̃∗3 , 0.35e−a0.35 ς̃∗1 + 0.13e−a0.13 ς̃∗2 + 0.15e−a0.15 ς̃∗3 ) be two two EIFSs. The bounded difference of these two EIFSs is: µEA∗⋄EB∗ (ς̃∗) = ( 0.07e−a0.07 ς̃∗1 + 0.39e−a0.39 ς̃∗2 + 0.31e−a0.31 ς̃∗3 ) λEA∗⋄EB∗ (ς̃∗) = ( 0e−a0 ς̃∗1 + 0.39e−a0.39 ς̃∗2 + 0.31e−a0.31 ς̃∗3 ) . M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 9 of 20 Definition 16. Let EA∗ and EB∗ be any two EIFSs the disjoint sum is defined as: µEA∗⊗EB∗ (ς̃∗) = ∣∣∣∣∣µA∗(ς̃∗)e −aµA∗ (ς̃∗) − µB∗(ς̃∗)e −aµB∗ (ς̃∗) ∣∣∣∣∣ λEA∗⊗EB∗ (ς̃∗) = ∣∣∣∣∣λA∗(ς̃∗)e −aλA∗ (ς̃∗) − λB(ς̃∗)e −aλB(ς̃∗) ∣∣∣∣∣. Example 5. Let EA∗ = ( 0.15e−a0.15 ς̃∗1 + 0.43e−a0.43 ς̃∗2 + 0.05e−a0.05 ς̃∗3 , 0.35e−a0.35 ς̃∗1 + 0.13e−a0.13 ς̃∗2 + 0.15e−a0.15 ς̃∗3 ) and EB∗ = ( 0.22e−a0.22 ς̃∗1 + 0.82e−a0.82 ς̃∗2 + 0.36e−a0.36 ς̃∗3 , 0.35e−a0.35 ς̃∗1 + 0.52e−a0.52 ς̃∗2 + 0.46e−a0.46 ς̃∗3 ) be two two EIFSs. Using the max function for calculating the phase term, the disjoint sum of theses two two EIFSs is: µEA∗⊗EB∗ (ς̃∗) = ( 0.07e−a0.07 ς̃∗1 + 0.39e−a0.39 ς̃∗2 + 0.31e−a0.31 ς̃∗3 ) λEA∗⊗EB∗ (ς̃∗) = ( 0e−a0 ς̃∗1 + 0.39e−a0.39 ς̃∗2 + 0.31e−a0.31 ς̃∗3 ) . Definition 17. Let µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗), λA∗(ς̃∗)e −aλA∗ (ς̃∗) and λB(ς̃∗)e −aλB(ς̃∗) denotes the M and non membership functions EA∗ and EB∗. Let EA∗∆EB∗ represents the disjunctive sum of EIFSs EA∗ and EB∗, as µEA∗∆EB∗ (ς̃∗) = [ µEA∗∩EB∗ ′ (ς̃∗)⊕ µEA∗ ′∩EB∗ (ς̃∗) ] = [ µA∗(ς̃∗)e −aµA∗ (ς̃∗) ∗ µ′ B∗(ς̃∗)e −aµ ′ B∗ (ς̃∗) ] ⊕ [ µ ′ A∗(ς̃∗)e −aµ ′ A∗ (ς̃∗) ∗ µB∗(ς̃∗)e −aµB∗ (ς̃∗) ] . λEA∗∆EB∗ (ς̃∗) = [ λEA∗∪EB∗ ′ (ς̃∗) ∗ λEA∗ ′∪EB∗ (ς̃∗) ] = [ λA∗(ς̃∗)e −aλA∗ (ς̃∗) ⊕ λ ′ B(ς̃∗)e −aµ ′ B∗ (ς̃∗) ] ∗ [ λ ′ A∗(ς̃∗)e −aλ ′ A∗ (ς̃∗) ⊕ λB(ς̃∗)e −aλB(ς̃∗) ] . Example 6. Suppose EA∗ = ( 0.6e−a0.6 ς̃∗1 + 0.7e−a0.7 ς̃∗2 + 0.5e−a0.5 ς̃∗3 , 0.2e −a0.2 ς̃∗1 + 0.5e−a0.5 ς̃∗2 + 0.6e−a0.6 ς̃∗3 ) and EB∗ = ( 0.3e−a0.3 ς̃∗1 + 0.4e−a0.4 ς̃∗2 + 0.7e−a0.7 ς̃∗3 , 0.4e −a0.4 ς̃∗1 + 0.3e−a0.3 ς̃∗2 + 0.5e−a0.5 ς̃∗3 ) . Then the disjunc- tive sum of these two EIFSs is define as µEA∗∆EB∗ (ς̃∗) = [ µEA∗∩EB∗ ′ (ς̃∗)⊕ µEA∗ ′∩EB∗ (ς̃∗) ] = [ µA∗(ς̃∗)e −aµA∗ (ς̃∗) ∗ µ′ B∗(ς̃∗)e −aµ ′ B∗ (ς̃∗) ] ⊕ [ µ ′ A∗(ς̃∗)e −aµ ′ A∗ (ς̃∗) ∗ µB∗(ς̃∗)e −aµB∗ (ς̃∗) ] . M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 10 of 20 µEA∗∆EB∗ (ς̃∗) = ( 0.6e−a0.6 ς̃∗1 + 0.6e−a0.6 ς̃∗2 + 0.3e−a0.3 ς̃∗3 ) ⊕ ( 0.3e−a0.3 ς̃∗1 + 0.3e−a0.3 ς̃∗2 + 0.5e−a0.5 ς̃∗3 ) . µEA∗∆EB∗ (ς̃∗) = ( 0.6e−a0.6 ς̃∗1 + 0.6e−a0.6 ς̃∗2 + 0.5e−a0.5 ς̃∗3 ) . λEA∗∆EB∗ (ς̃∗) = [ λEA∗∪EB∗ ′ (ς̃∗) ∗ λEA∗ ′∪EB∗ (ς̃∗) ] = [ λA∗(ς̃∗)e −aλA∗ (ς̃∗) ⊕ λ ′ B(ς̃∗)e −aµ ′ B∗ (ς̃∗) ] ∗ [ λ ′ A∗(ς̃∗)e −aλ ′ A∗ (ς̃∗) ⊕ λB(ς̃∗)e −aλB(ς̃∗) ] . λEA∗∆EB∗ (ς̃∗) = ( 0.7e−a0.7 ς̃∗1 + 0.7e−a0.7 ς̃∗2 + 0.5e−a0.5 ς̃∗3 ) ⊕ ( 0.8e−a0.8 ς̃∗1 + 0.5e−a0.5 ς̃∗2 + 0.5e−a0.5 ς̃∗3 ) . λEA∗∆EB∗ (ς̃∗) = ( 0.7e−a0.7 ς̃∗1 + 0.5e−a0.5 ς̃∗2 + 0.5e−a0.5 ς̃∗3 ) . Definition 18. Consider two EIFSs EA∗ = ⟨µEA∗ , λEA∗ ⟩ and EB∗ = ⟨µEB∗ , λEB∗ ⟩, where µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗), λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗) µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗), λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗) The Cartesian product of EA∗ and EB∗ is given by: EA∗ × EB∗ = {( (x, y), µEA∗×EB∗ (x, y), λEA∗×EB∗ (x, y) ) | x ∈ A, y ∈ B } where µEA∗×EB∗ (x, y) = ∧ ( µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(y)e −aµB∗ (y) ) λEA∗×EB∗ (x, y) = ∨ ( λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(y)e −aλB(y) ) Thus, the Cartesian product EA∗ ×EB∗ forms an EIFS with these M and NMfunctions. Example 7. Let the exponential parameter be a = 1, EA∗ = ( 0.8e−a0.8 ς̃∗1 + 0.6e−a0.6 ς̃∗2 , 0.3e −a0.3 ς̃∗1 + 0.4e−a0.4 ς̃∗2 ) and EB∗ = ( 0.7e−a0.7 ς̃∗1 + 0.5e−a0.5 ς̃∗2 , 0.2e −a0.2 ς̃∗1 + 0.5e−a0.5 ς̃∗2 ) . Then the Cartesian product EA∗ × EB∗ is EA × EB =  ((ς̃∗1, d1), 0.3476, 0.2222), ((ς̃∗1, d2), 0.3033, 0.3033), ((ς̃∗2, d1), 0.3293, 0.2681), ((ς̃∗2, d2), 0.3033, 0.3033)  Definition 19. A quasi-triangular norm T for EIFS is a function: T : (0, 1]× (0, 1] → [0, 1], satisfying the following conditions: (i) Boundary Condition: T(1, 1) = 1. (ii) Commutativity: T(ρ, ρ′) = T(ρ′, ρ). M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 11 of 20 (iii) Monotonicity: If ρ ≤ ρ′′ and ρ′ ≤ ρ′′′, then: T(ρ, ρ′) ≤ T(ρ′′, ρ′′′). (iv) Associativity: T(T(ρ, ρ′), ρ′′) = T(ρ,T(ρ′, ρ′′)). Theorem 1. For any three EIFSs EA∗ = ⟨µEA∗ , λEA∗ ⟩, EB∗ = ⟨µEB∗ , λEB∗ ⟩ and EC = ⟨µEC , λEC⟩ on a universe of discourse X satisfy the following i. EA∗ ∩ EB∗ = EB∗ ∩ EA∗ and EA∗ ∪ EB∗ = EB∗ ∪ EA∗ ii. EA∗ ∩ (EB∗ ∩ EC) = (EA∗ ∩ EB∗) ∩ EC and EA∗ ∪ (EB∗ ∪ EC) = (EA∗ ∪ EB∗) ∪ EC Proof. i. Let EA∗ and EB∗ be two EIFSs and µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗), µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗) λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗), and λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗),and represent their M and non membership function, respectively. EA∗ ∩ EB∗ = EA∗(ς̃∗) ∩ EB∗(ς̃∗) = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } = ∧ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µA∗(ς̃∗)e −aµA∗ (ς̃∗) } = EB∗ ∩ EA∗ EA∗ ∩ EB∗ = EA∗(ς̃∗) ∩ EB∗(ς̃∗) = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aλB(ς̃∗) } = ∨ { λB(ς̃∗)e −aλB(ς̃∗), λA∗(ς̃∗)e −aλA∗ (ς̃∗) } = EB∗ ∩ EA∗ and similar way we can prove EA∗ ∪ EB∗ = EB∗ ∪ EA∗ . ii. Let EA∗ and EB∗ be three EIFSs and µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗), µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗), µEC(ς̃∗) = µC(ς̃∗)e −aµC(ς̃∗), λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗), λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗),and λEC(ς̃∗) = λC(ς̃∗)e −aλC(ς̃∗) represent their M and non membership function, respectively. Then EA∗ ∩ (EB∗ ∩ EC) = µEA∗ (ς̃∗) ∩ ( µEA∗∩EC(ς̃∗) ) = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗),∧ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) } = (EA∗ ∩ EB∗) ∩ EC EA∗ ∩ (EB∗ ∩ EC) = λEA∗ (ς̃∗) ∩ ( λEA∗∩EC(ς̃∗) ) = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗),∨ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 12 of 20 = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) } = (EA∗ ∩ EB∗) ∩ EC and similar way we can prove EA∗ ∪ (EB∗ ∪ EC) = (EA∗ ∪ EB∗) ∪ EC . Theorem 2. For any three EIFSs EA∗ = ⟨µEA∗ , λEA∗ ⟩, EB∗ = ⟨µEB∗ , λEB∗ ⟩ and EC = ⟨µEC , λEC⟩ on a universe of discourse X satisfy the following EA∗ ∩ (EB∗ ∪EC) = (EA∗ ∩EB∗)∪ (EA∗ ∩EC) and EA∗ ∪ (EB∗ ∩EC) = (EA∗ ∪EB∗)∩ (EA∗ ∪EC) Proof. Let EA∗ and EB∗ be three EIFSs and µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗), µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗), µEC(ς̃∗) = µC(ς̃∗)e −aµC(ς̃∗), λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗), λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗),and λEC(ς̃∗) = λC(ς̃∗)e −aλC(ς̃∗) represent their M and NM function, respectively. EA∗ ∩ (EB∗ ∪ EC) = µEA∗ (ς̃∗) ∩ ( µEA∗∪EC(ς̃∗) ) = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗),∨ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) = ∨ { ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } , ∧ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} (3.5) and EA∗ ∩ (EB∗ ∪ EC) = λEA∗ (ς̃∗) ∩ ( λEA∗∪EC(ς̃∗) ) = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗),∧ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) = ∧ { ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aµB∗ (ς̃∗) } , ∨ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} (3.6) Case 1. If µEA∗ (ς̃∗) ≤ µEB∗ (ς̃∗) ≤ µEC(ς̃∗) and λEA∗ (ς̃∗) ≤ λEB∗ (ς̃∗) ≤ λEC(ς̃∗) EA∗ ∩ (EB∗ ∪ EC) = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗),∨ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) } = µA∗(ς̃∗)e −aµA∗ (ς̃∗). (3.7) (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) = ∨ { ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } , (3.8) ∧ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} = ∨ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 13 of 20 = µA∗(ς̃∗)e −aµA∗ (ς̃∗) (3.9) From equation (3.7) and (3.9) EA∗ ∩ (EB∗ ∪ EC) = (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) and EA∗ ∩ (EB∗ ∪ EC) = ∨ { λA∗(ς̃∗)e −aµA∗ (ς̃∗),∧ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aµB∗ (ς̃∗) } = λB(ς̃∗)e −aλB(ς̃∗). (3.10) (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) = ∧ { ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aλB(ς̃∗) } , (3.11) ∨ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} = ∧ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aµC(ς̃∗) } = λB(ς̃∗)e −aλB(ς̃∗) (3.12) From equation (3.10) and (3.12) EA∗ ∩ (EB∗ ∪ EC) = (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) Case 2. If µEC(ς̃∗) ≤ µEB∗ (ς̃∗) ≤ µEA∗ (ς̃∗) and λEC(ς̃∗) ≤ λEB∗ (ς̃∗) ≤ λEA∗ (ς̃∗) EA∗ ∩ (EB∗ ∪ EC) = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗),∨ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} = ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } = µB∗(ς̃∗)e −aµB∗ (ς̃∗). (3.13) (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) = ∨ { ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µB∗(ς̃∗)e −aµB∗ (ς̃∗) } , (3.14) ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) }} = ∨ { µB∗(ς̃∗)e −aµB∗ (ς̃∗), µC(ς̃∗)e −aµC(ς̃∗) } = µB∗(ς̃∗)e −aµB∗ (ς̃∗) (3.15) From equation (3.13) and (3.15) EA∗ ∩ (EB∗ ∪ EC) = (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) and EA∗ ∩ (EB∗ ∪ EC) = ∨ { λA∗(ς̃∗)e −aµA∗ (ς̃∗),∧ { λB(ς̃∗)e −aλB(ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} = ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λC(ς̃∗)e −aµC(ς̃∗) } M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 14 of 20 = λA∗(ς̃∗)e −aλA∗ (ς̃∗). (3.16) (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) = ∧ { ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λB(ς̃∗)e −aλB(ς̃∗) } , (3.17) ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λC(ς̃∗)e −aλC(ς̃∗) }} = ∧ { λA∗(ς̃∗)e −aλA∗ (ς̃∗), λA∗(ς̃∗)e −aµA∗ (ς̃∗) } = λA∗(ς̃∗)e −aλA∗ (ς̃∗) (3.18) From equation (3.16) and (3.18) EA∗ ∩ (EB∗ ∪ EC) = (EA∗ ∩ EB∗) ∪ (EA∗ ∩ EC) Theorem 3. For any two EIFSs EA∗ = ⟨µEA∗ and λEA∗ ⟩, EB∗ = ⟨µEB∗ , λEB∗ ⟩ on a universe of discourse X satisfy the following (i) (EA∗ ∩ EB∗) c = (EB∗) c ∪ (EA∗) c and (ii) (EA∗ ∪ EB∗) c = (EB∗) c ∩ (EA∗) c. Proof. (i) Let EA∗ and EB∗ be two EIFSs and µEA∗ (ς̃∗) = µA∗(ς̃∗)e −aµA∗ (ς̃∗) and µEB∗ (ς̃∗) = µB∗(ς̃∗)e −aµB∗ (ς̃∗), λEA∗ (ς̃∗) = λA∗(ς̃∗)e −aλA∗ (ς̃∗) and λEB∗ (ς̃∗) = λB(ς̃∗)e −aλB(ς̃∗) rep- resent their M and NMF , respectively. For MF (EA∗ ∩ EB∗) c = µA∗∩B∗(ς̃∗) ′ e−aµA∗∩B∗ (ς̃∗) ′ = [ 1− ∧ { µA∗(ς̃∗)e −aµA∗ (ς̃∗) ∩ µB∗(ς̃∗)e −aµB∗ (ς̃∗) }] = ∨ { 1− µA∗(ς̃∗)e −aµA∗ (ς̃∗), 1− µB∗(ς̃∗)e −aµB∗ (ς̃∗) } = ∨ { µA∗(ς̃∗) ′ e−aµA∗ (ς̃∗) ′ , µB∗(ς̃∗) ′ e−aµB∗ (ς̃∗) ′} = { µA∗(ς̃∗) ′ e−aµA∗ (ς̃∗) ′ ∪ µB∗(ς̃∗) ′ e−aµB∗ (ς̃∗) ′} = (EB∗) c ∪ (EA∗) c. For N−MF, (EA∗ ∩ EB∗) c = λA∗∩B∗(ς̃∗) ′ e−aλA∗∩B∗ (ς̃∗) ′ = [ 1− ∧ { λA∗(ς̃∗)e −aλA∗ (ς̃∗) ∩ λB(ς̃∗)e −aλB(ς̃∗) }] = ∨ { 1− λA∗(ς̃∗)e −aλA∗ (ς̃∗), 1− λB(ς̃∗)e −aλB(ς̃∗) } = ∨ { λA∗(ς̃∗) ′ e−aλA∗ (ς̃∗) ′ , λB(ς̃∗) ′ e−aλB(ς̃∗) ′} = { λA∗(ς̃∗) ′ e−aλA∗ (ς̃∗) ′ ∪ λB(ς̃∗) ′ e−aλB(ς̃∗) ′} = (EB∗) c ∪ (EA∗) c. M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 15 of 20 (ii) For M function (EA∗ ∪ EB∗) c = µA∗∪B∗(ς̃∗) ′ e−aµA∗∪B∗ (ς̃∗) ′ = [ 1− ∨ { µA∗(ς̃∗)e −aµA∗ (ς̃∗) ∪ µB∗(ς̃∗)e −aµB∗ (ς̃∗) }] = ∧ { 1− µA∗(ς̃∗)e −aµA∗ (ς̃∗), 1− µB∗(ς̃∗)e −aµB∗ (ς̃∗) } = ∧ { µA∗(ς̃∗) ′ e−aµA∗ (ς̃∗) ′ , µB∗(ς̃∗) ′ e−aµB∗ (ς̃∗) ′} = { µA∗(ς̃∗) ′ e−aµA∗ (ς̃∗) ′ ∩ µB∗(ς̃∗) ′ e−aµB∗ (ς̃∗) ′} = (EB∗) c ∩ (EA∗) c. For NMF, (EA∗ ∪ EB∗) c = λA∗∪B∗(ς̃∗) ′ e−aλA∗∪B∗ (ς̃∗) ′ = [ 1− ∨ { λA∗(ς̃∗)e −aλA∗ (ς̃∗) ∩ λB(ς̃∗)e −aλB(ς̃∗) }] = ∧ { 1− λA∗(ς̃∗)e −aλA∗ (ς̃∗), 1− λB(ς̃∗)e −aλB(ς̃∗) } = ∧ { λA∗(ς̃∗) ′ e−aλA∗ (ς̃∗) ′ , λB(ς̃∗) ′ e−aλB(ς̃∗) ′} = { λA∗(ς̃∗) ′ e−aλA∗ (ς̃∗) ′ ∩ λB(ς̃∗) ′ e−aλB(ς̃∗) ′} = (EB∗) c ∩ (EA∗) c. 4. Methodology for Decision-Making Using Exponential Intuitionstic Fuzzy Sets EIFSs offer a flexible approach to decision-making, particularly where uncertainty, imprecision, and hesitancy play a role. EIFS facilitate decision-making by integrating several sources of information using appropriate aggregation techniques. Sept 1 Suppose you have n distinct possibilities., L = {L1,L2,L3, ...Ln} in a multi- attributes A. The issue and m are various standards from the index set A = {A1,A2,A3, ...Am}. The alternatives attribute value Lr for index Fp is Esr =〈 µEsr(ς̃∗)e −α(µEsr (ς̃∗)), λEsr(ς̃∗)e −α(λEsr (ς̃∗)) 〉 , (s = 1, 2, ....n; r = 1, 2, ....m). So the EIFSs A matrix is A =  E11, ....E12, ....E1m E21, ....E21, ....E2m ... ... ... ... ... ... En1, ....En2, ....E2nm  , where, Esr = 〈 µEsr(ς̃∗)e −α(µEsr (ς̃∗)), λEsr(ς̃∗)e −α(λEsr (ς̃∗)) 〉 . M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 16 of 20 Sept 2 Assume that µEij (ς̃∗) = ∨ { µEij (ς̃∗) for fix i and j = 1, 2, 3, ....m } λEij (ς̃∗) = ∧ { λEij (ς̃∗) for fix i and j = 1, 2, 3, ....n } and thus Al max =  µE11(ς̃∗)e −α(µE11 (ς̃∗)), λE11(ς̃∗)e −α(λE11 (ς̃∗)) µE21(ς̃∗)e −α(µE21 (ς̃∗)), λE21(ς̃∗)e −α(λE21 (ς̃∗)) ... ... ... ... ... ... µEn1(ς̃∗)e −α(µEn1 (ς̃∗)), λEn1(ς̃∗)e −α(λEn1 (ς̃∗))  , (4.1) for all l = 1, 2, 3... is called max A-Matrix. Sept 3 Assume that µ ′ Eij (ς̃∗) = ∧ { µ ′ Eij (ς̃∗) for fix i and j = 1, 2, 3, ....m } λ ′ Eij (ς̃∗) = ∨ { λ ′ Eij (ς̃∗) for fix i and j = 1, 2, 3, ....n } and thus Al min =  µ ′ E11(ς̃∗)e −α(µ ′ E11 (ς̃∗)), λ ′ E11(ς̃∗)e −α(λ ′ E11 (ς̃∗)) µ ′ E21(ς̃∗)e −α(µ ′ E21 (ς̃∗)), λ ′ E21(ς̃∗)e −α(λ ′ E21 (ς̃∗)) ... ... ... ... ... ... µ ′ En1 (ς̃∗)e −α(µ ′ En1 (ς̃∗)), λ ′ En1 (ς̃∗)e −α(λ ′ En1 (ς̃∗))  , (4.2) for all l = 1, 2, 3... is called min A-Matrix. Sept 4 The average similarity measure operator(ASMO) Sl is Sl (Al max,Al min) =  max µ ′ Ei1 (ς̃∗)e −α(µ ′ Ei1 (ς̃∗)) +λ ′ Ei1 (ς̃∗)e −α(λ ′ Ei1 (ς̃∗)) 2  min µ ′ Ei1 (ς̃∗)e −α(µ ′ Ei1 (ς̃∗)) +λ ′ Ei1 (ς̃∗)e −α(λ ′ Ei1 (ς̃∗)) 2   , i = 1, 2, 3, ....n, Sl (Al max,Al min) = ( DSl1 (ς̃∗)e −αDSl1 (ς̃∗),D ′ Sl1 (ς̃∗)e −αD′ Sl1 (ς̃∗) ) Sept 5 ASMO for weight D∗ is D∗ = wrmaxDSl1 (ς̃∗)e −αDSl1 (ς̃∗)√∑ (DSl1 (ς̃∗))2 , wrmaxD′ Sl1 (ς̃∗)e −αD′ Sl1 (ς̃∗)√∑ (D′ Sl1 (ς̃∗))2  , l = 1, 2, 3..and r = 1, 2, 3....m. where wr = (1, 2, 3...wm) is the index weight vector & ∑n i=1wr = 1. Step 6 The alternatives are ranked by assessing the ASMO for weight values, with the most desirable options being chosen. M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 17 of 20 4.1. Implementation of The Proposed Methodology on a Decision Mak- ing Problem A manufacturing company must select the best supplier for raw materials based on multiple conflicting criteria such as cost (L1), quality(L2), delivery time (L3),and reliability(L4). Experts might be unsure about their judgments due to fluctuating market conditions. Some supplier performance metrics (e.g., delivery time) may have seasonal variations. The phase component in EIFS helps model such cyclic patterns. The additional exponential-valued information enhances the accuracy of ranking suppliers. By fixing the parameter (a = 0.2) for defining the prioritization of the above conflicting criteria. Sept 1 Consider three matrix A1,A2 and A3 of EIF Matrix 1: Imported Raw materials in 2018: A1 =  F1 F2 F3 F4 L1 0.3e−0.2(0.3), 0.5e−0.2(0.5) 0.7e−0.2(0.7), 0.2e−0.2(0.2) 0.4e−0.2(0.4), 0.3e−0.2(0.3) 0.4e−0.2(0.4), 0.6e−0.2(0.6) L2 0.2e−0.2(0.2), 0.5e−0.2(0.5) 0.3e−0.2(0.3), 0.5e−0.2(0.5) 0.4e−0.2(0.4), 0.8e−0.2(0.8) 0.1e−0.2(0.1), 0.2e−0.2(0.2) L3 0.1e−0.2(0.1), 0.9e−0.2(0.9) 0.4e−0.2(0.4), 0.5e−0.2(0.5) 0.3e−0.2(0.3), 0.6e−0.2(0.6) 0.3e−0.2(0.3), 0.1e−0.2(0.1) L4 0.7e−0.2(0.7), 0.1e−0.2(0.1) 0.6e−0.2(0.6), 0.3e−0.2(0.3) 0.3e−0.2(0.3), 0.2e−0.2(0.2) 0.4e−0.2(0.4), 0.4e−0.2(0.4)  Matrix 2: Imported Raw materials in 2019: A2 =  F1 F2 F3 F4 L1 0.1e−0.2(0.1), 0.3e−0.2(0.3) 0.8e−0.2(0.8), 0.2e−0.2(0.2) 0.5e−0.2(0.5), 0.3e−0.2(0.3) 0.4e−0.2(0.4), 0.6e−0.2(0.6) L2 0.5e−0.2(0.5), 0.4e−0.2(0.54) 0.4e−0.2(0.4), 0.6e−0.2(0.6) 0.2e−0.2(0.2), 0.4e−0.2(0.4) 0.1e−0.2(0.1), 0.2e−0.2(0.2) L3 0.1e−0.2(0.1), 0.8e−0.2(0.8) 0.3e−0.2(0.3), 0.6e−0.2(0.6) 0.2e−0.2(0.2), 0.3e−0.2(0.3) 0.4e−0.2(0.4), 0.1e−0.2(0.1) L4 0.7e−0.2(0.7), 0.3e−0.2(0.3) 0.4e−0.2(0.4), 0.5e−0.2(0.5) 0.1e−0.2(0.1), 0.3e−0.2(0.3) 0.5e−0.2(0.5), 0.2e−0.2(0.2)  Matrix 3: Imported Raw materials in 2020: A3 =  F1 F2 F3 F4 L1 0.6e−0.2(0.6), 0.1e−0.2(0.1) 0.6e−0.2(0.6), 0.1e−0.2(0.1) 0.4e−0.2(0.4), 0.3e−0.2(0.3) 0.5e−0.2(0.5), 0.3e−0.2(0.3) L2 0.3e−0.2(0.3), 0.4e−0.2(0.4) 0.2e−0.2(0.2), 0.3e−0.2(0.3) 0.5e−0.2(0.5), 0.6e−0.2(0.6) 0.7e−0.2(0.7), 0.1e−0.2(0.1) L3 0.4e−0.2(0.4), 0.5e−0.2(0.5) 0.6e−0.2(0.6), 0.2e−0.2(0.2) 0.4e−0.2(0.4), 0.7e−0.2(0.7) 0.1e−0.2(0.1), 0.3e−0.2(0.3) L4 0.6e−0.2(0.6), 0.2e−0.2(0.2) 0.4e−0.2(0.4), 0.5e−0.2(0.5) 0.2e−0.2(0.2), 0.3e−0.2(0.3) 0.5e−0.2(0.5), 0.4e−0.2(0.4)  We formulate the A1max,A1max and A1max matrices by using equation (4.1). A1max =  0.7e−0.2(0.7), 0.2e−0.2(0.2) 0.5e−0.2(0.5), 0.1e−0.2(0.1) 0.5e−0.2(0.5), 0.1e−0.2(0.1) 0.7e−0.2(0.7), 0.3e−0.2(0.3) A2max =  0.8e−0.2(0.8), 0.1e−0.2(0.1) 0.5e−0.2(0.5), 0.1e−0.2(0.1) 0.4e−0.2(0.4), 0.1e−0.2(0.1) 0.7e−0.2(0.7), 0.1e−0.2(0.1) & A3max =  0.7e−0.2(0.7), 0.3e−0.2(0.3) 0.7e−0.2(0.7), 0.1e−0.2(0.1) 0.6e−0.2(0.6), 0.1e−0.2(0.1) 0.6e−0.2(0.6), 0.2e−0.2(0.2)  We formulate the A1min,A1min and A1min matrices by using equation (4.2). A1min =  0.3e−0.2(0.3), 0.7e−0.2(0.7) 0.1e−0.2(0.1), 0.8e−0.2(0.8) 0.1e−0.2(0.1), 0.9e−0.2(0.9) 0.3e−0.2(0.3), 0.7e−0.2(0.7) A2min =  0.1e−0.2(0.8), 0.4e−0.2(0.3) 0.1e−0.2(0.6), 0.2e−0.2(0.6) 0.1e−0.2(0.8), 0.1e−0.2(0.7) 0.1e−0.2(0.7), 0.2e−0.2(0.5) & M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 18 of 20 A3min =  0.4e−0.2(0.4), 0.3e−0.2(0.3) 0.2e−0.2(0.2), 0.6e−0.2(0.6) 0.1e−0.2(0.1), 0.7e−0.2(0.7) 0.2e−0.2(0.2), 0.5e−0.2(0.5)  The ASMO St : t = 1, 2, 3, S1(A1max,A1min) =  0.5e−0.2(0.5), 0.35e−0.2(0.35) 0.3e−0.2(0.3), 0.45e−0.2(0.45) 0.3e−0.2(0.3), 0.5e−0.2(0.5) 0.5e−0.2(0.5), 0.5e−0.2(0.5)  , S2(A2max,A2min) =  0.45e−0.2(0.5), 0.35e−0.2(0.35) 0.3e−0.2(0.3), 0.15e−0.2(0.15) 0.25e−0.2(0.25), 0.1e−0.2(0.1) 0.4e−0.2(0.4), 0.15e−0.2(0.15)  and S3(A3max,A3min) =  0.5e−0.2(0.5), 0.3e−0.2(0.3) 0.45e−0.2(0.45), 0.35e−0.2(0.35) 0.35e−0.2(0.4), 0.4e−0.2(0.4) 0.4e−0.2(0.35), 0.35e−0.2(0.35)  Sept 5 We find ASMO of weight D∗ for the vector w = (0.2, 0.2, 0.2, 0.2) D∗ =  0.10e−2(0.5), 0.10e−2(0.3) 0.15e−2(0.45), 0.05e−2(0.15) 0.14e−2(0.4), 0.02e−2(0.1) 0.12e−2(0.5), 0.04e−2(0.15)  Sept 6 Based on D∗ values, We find that the NMS of Quality is more than the MS of all other competing criteria, and that the NMS of Quality is less than that of all other conflicting criteria. So, L2 = Quality is the important criteria for selection of raw material in a manufacturing company. By changing the fixation value (a = 0.3, 0.4, ...) we can get various results that close to 1 which means we can be able get the better solution enhanced with absolute accuracy. 4.2. Comparative Analysis EFS and EIFSare two new tools in mathematics introduced in an attempt to add up new ideas on modeling of uncertainty in real-life cases particularly when it comes to decision-making and information processing. Using exponential membership functions, EFS controls the range of the belongingness of elements to set where the smoother way of the change of the set and the better handling of non-linear fluctuations of data is provided. This methodology ensures that membership values mildly change as input changes are made in small intervals, thus increasing the decision-making sensitivity. However, the EFS is reliant only on a degree of membership and is not able to measure directly the non-membership or hesitation, thus, limiting the precision of the EFS in a higher environment with incomplete or inconsistent data. On the other hand, EFSs compliments EFS infra structure with exponential membership function/non membership function that brings to the argument hesitation degree as well. This three-component framework can assist EIFSs to describe uncertainty more satisfactorily, as it can offer a sharper and more M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 19 of 20 robust model for cases when decision-making is marred within imprecision, vagueness, and the discordant criteria. The EIFSs facilitates a more effective handling of the fuzzy data due to differentiation of the unknown information and unfavorable data. In addition, operations such as intersection, union, ordinary difference, and disjoint sum are also expanded in EIFSs model in order to preserve its expanded structure. It is notwithstanding the fact that EIFSs computations are more formidable that the system potentiality for accommodating larger uncertainties and inconsistencies and hence the resulting decisions being in the capacity of making more profound dichotomies advances its case as a very powerful tool in scenarios such as multi-criteria decision- making, the appraisal, supplier selection, and resource allocation. 5. Conclusion EIFSs are a strong expansion of traditional intuitionistic FSs that improves the capacity to express uncertainty by using exponential membership and nonmembership functions. The operations investigated, such as intersection, union, difference measures, sum operations, and Cartesian products, show their usefulness in dealing with complicated decision-making issues. The use of these processes in raw material procurement demonstrates their usefulness in assessing various providers in unpredictable settings. Organisations that use EIFS-based decision-making procedures may make more informed and balanced decisions, taking into account critical criteria such as cost, quality, and delivery time. The findings show that EIFS not only improves decision accuracy, but also provides an organised way for dealing with contradictory and insufficient information in procurement procedures. Future Works: The concept exponential intuitionistic fuzzy sets we can applied in Future Works in graph theory and decision making problems. References [1] L. A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965. [2] L. A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1):3–28, 1978. [3] Hiroshi Maeda and Shuta Murakami. A fuzzy decision-making method and its application to a company choice problem. Information Sciences, 45(2):331–346, 1988. [4] Shyi-Ming Chen and Jiann-Mean Tan. Handling multicriteria fuzzy decision-making prob- lems based on vague set theory. Fuzzy Sets and Systems, 67(2):163–172, 1994. [5] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87–96, 1986. [6] Hua-Wen Liu and Guo-Jun Wang. Multi-criteria decision-making methods based on intu- itionistic fuzzy sets. European Journal of Operational Research, 179(1):220–233, 2007. [7] Fatih Emre Boran, Serkan Genº, Mustafa Kurt, and Diyar Akay. A multi-criteria intuition- istic fuzzy group decision making for supplier selection with topsis method. Expert Systems with Applications, 36(8):11363–11368, 2009. [8] S. M. Chen and C. H. Chang. A novel similarity measure between atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Information Sciences, 291:96–114, 2015. [9] S. M. Chen, Y. Randyanto, and S. H. Cheng. Fuzzy queries processing based on intuitionistic fuzzy social relational networks. Information Sciences, 327:110–124, 2016. [10] P. A. Ejegwa and J. M. Agbetayo. Similarity-distance decision-making technique and its applications via intuitionistic fuzzy pairs. Journal of Computational and Cognitive Engi- neering, 2(1):68–74, 2022. M. Kaviyarasu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6110 20 of 20 [11] H. Garg, M. £nver, M. Olgun, et al. An extended edas method with circular intuitionistic fuzzy value features and its application to multi-criteria decision-making process. Artificial Intelligence Review, 56:3173–3204, 2023. [12] M. Wasim, A. Yousaf, H. Alolaiyan, et al. Optimizing decision-making with aggregation operators for generalized intuitionistic fuzzy sets and their applications in the tech industry. Scientific Reports, 14:16538, 2024. [13] J. Ali. Multi-criteria decision-making method based on an integrated model using t-spherical fuzzy aczel-alsina prioritized aggregation operators. Computational and Applied Mathemat- ics, 44(3):181, 2025. [14] J. Ali, S. A. O. Abdallah, and N. S. Abd El-Gawaad. Decision analysis algorithm using frank aggregation in the swara framework with p, q rung orthopair fuzzy information. Symmetry, 16(10):1352, 2024. [15] M. Ali, D. E. Tamir, N. D. Rishe, and A. Kandel. Complex intuitionistic fuzzy classes. In IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pages 2027–2034, Van- couver, BC, Canada, 2016. [16] M. Kaviyarasu, M. Alqahtani, and M. Rajeshwari. Exponential fuzzy sets and applications of ai-powered investment decision-making using the weighted mean method. European Journal of Pure and Applied Mathematics, 18(2):6050–6050, 2025. [17] Roan Thi Ngan, Le Hoang Son, Mumtaz Ali, Dan E. Tamir, Naphtali D. Rishe, and Abra- ham Kandel. Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making. Applied Soft Computing, 87:105961, 2020. [18] M. Akram, X. Peng, and A. Sattar. A new decision-making model using complex intuition- istic fuzzy hamacher aggregation operators. Soft Computing, 25:7059–7086, 2021. [19] M. Zeeshan and M. Khan. Complex fuzzy sets with applications in decision-making. Journal of Fuzzy Systems, 19(4):147–163, 2022. [20] Sayedabbas Sobhi and Scott Dick. An investigation of complex fuzzy sets for large-scale learning. Fuzzy Sets and Systems, page 108660, 2023. [21] M. Bilal, I. Ul Haq, N. Kausar, M. I. Khan, D. Pamucar, and J. El Maalouf. Multi-criteria decision-making method under the complex intuitionistic fuzzy environment. Journal of Mathematics and Computer Science, 38(3):341–370, 2025. Introduction Motivation Novelty Preliminaries Exponential Intuitionstic Fuzzy Sets Methodology for Decision-Making Using Exponential Intuitionstic Fuzzy Sets Implementation of The Proposed Methodology on a Decision Making Problem Comparative Analysis Conclusion