Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2645 https://internationalpubls.com Multi-Criteria Decision Analysis for Wastewater Management using Neutrosophic Hypersoft Sets Mythili S1 and Arokialancy A2 1Asst Prof, Department of Mathematics, Nirmala college for women, Coimbatore (TN), India., mythiliaugustin@gmail.com 2Asst Prof, Department of Mathematics, Nirmala college for women, Coimbatore (TN), India., aarokia.lancy@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: The Neutrosophic Hypersoft Set (NHSS) introduces a parameterized family structure capable of handling intricate sub-attributes of parameters, thereby extending the expressive power of the Neutrosophic Soft Set. This study applies the NHSS framework to a multi-criteria decision-making problem in the context of wastewater management. By incorporating uncertainty, indeterminacy, and sub-parameter dependencies, the proposed approach provides a more robust and flexible decision-making tool compared to existing models. The TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method is integrated with NHSS to provide a structured ranking of alternatives. A case study involving three management strategies Recycling System, Advanced Treatment Plant, and Waste Reduction Program is used to demonstrate the applicability of the approach. Keywords: Neutrosophic hypersoft set, wastewater management, multi-criteria decision- making, TOPSIS, neutrosophic topology. 1. Introduction In 1999, Molodtsov [3] introduced the concept of soft sets, establishing a novel mathematical framework for handling uncertainties in decision-making. Building upon this foundation, Abbas, Murtaza, and Smarandache [1] introduced fundamental operations on hypersoft sets and the concept of a hypersoft point. Subsequently, Sagvan Y. Musa and Baravan A. [6] advanced the theory by proposing the notion of hypersoft topological spaces. Further extending this concept, Smarandache [8] generalized soft sets into hypersoft sets by replacing single-argument functions with multi-argument functions defined over the Cartesian product of parameter sets. This enhancement significantly improves the flexibility and applicability of soft sets, particularly in complex decision-making scenarios. Saeed, Ahsan, and Siddique [4] conducted an in-depth study of the foundational elements of hypersoft set theory. To address decision problems involving uncertainty and vagueness, Zadeh [10] introduced fuzzy sets. Building on this, Shu-Jen Chen and Ching-Lai Hwang [7] extended the concept of fuzzy multiple attribute decision-making (MADM) and proposed an enhanced version of the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method. Their contributions are crucial for accurately capturing uncertainty and incompleteness in decision processes. By utilizing a newly developed score function, a structured decision-making framework was introduced to address the complexities inherent in MADM problems. To validate the effectiveness of the proposed approach, a numerical example is provided, applying the TOPSIS method to evaluate and select a wastewater management system. This application demonstrates the model’s capacity to support critical mailto:mythiliaugustin@gmail.com mailto:aarokia.lancy@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2646 https://internationalpubls.com decisions in environmental management. The TOPSIS technique, originally developed by Hwang and Yoon in 1987, was theoretically presented by Yoon [9] and later refined by Hwang et al. in 1993. This method identifies the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS) to evaluate alternatives based on their shortest Euclidean distance from the ideal. Each criterion is considered for either maximization or minimization depending on its nature. TOPSIS ranks alternatives according to their closeness to the ideal solution. The best alternative receives a rank of one, indicating optimality, while the least preferred alternative receives a rank near zero. All other alternatives are assigned intermediate values, ensuring a comprehensive and relative assessment. This systematic approach enables decision-makers to effectively identify the most suitable option among multiple alternatives. 2 Preliminaries Definition 2.1 ([2]). Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Let A ⊆ E. A pair (FA, E) is called a soft set over U, where FA is a mapping given by, FA : A → P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. Definition 2.2. [6]. A pair (F, A 1 × A 2 × ... × A n ) is called a hypersoft set over U, where F is a mapping given by F : A 1 × A 2 × ... × A n → P(U). Simply, we write the symbol E for E 1 ×E 2 ×...× E n , and for the subsets of E : the symbols A for A 1 × A 2 × ... × A n , and B for B 1 × B 2 × ... × B n . Clearly, each element in A, B and E is an n-tuple element. We can represent a hypersoft set (F,A) as an ordered pair, (F,A) = {(α, F(α)) :α ∈ A}. Definition 2.3. [6]. Fuzzy Set (FS) AFS = { 𝑥(𝑇A(𝑥)), 𝑥 ∈ 𝒰 }, where 𝑇A: U ⟶ P([0, 1]) is the membership degree of the generic element x with respect to the set A, and P([0, 1]) is the powerset of [0, 1], is called a Fuzzy Set. Definition 2.4. [6]. Given a universal set U, a Neutrosophic set A is characterized by a truth-membership function TA, an indeterminacy-membership function IA, and a falsity-membership function FA. For an element x in U: A(x) = {TA(x), IA(x), FA(x)}, where: • TA(x) represents the degree of truth of x in A, • IA(x) represents the degree of indeterminacy of x in A, • FA(x) represents the degree of falsity of x in A. Definition 2.5 [1]: A neutrosophic hypersoft set (NHSS) in a universe U with respect to a set of attributes A and their sub-attributes Bi is defined as: 𝑁𝐻𝑆𝑆 = {(𝑥, {(𝐸𝑖 , 𝑇𝐸𝑖 (𝑥), 𝐼𝐸𝑖 (𝑥), 𝐹𝐸𝑖 (𝑥) | 𝐸𝑖 ∈ 𝐵𝑖)}/𝑥 ∈ 𝑈)}, where x is an element of the universe U, Ei represents a sub-attribute in the set of sub-attributes Bi of the attribute A and TEi(x), IEi(x) and FEi(x) denote the degrees of truth, indeterminacy, and falsity, respectively, for the element x with respect to the sub-attribute Ei. 3. Methodology Step 1: Define alternatives: R (Recycling System), T (Advanced Treatment Plant), W (Waste Reduction Program) Step 2: Define criteria: E (Environmental Impact), C (Cost), Ef (Efficiency), A (Community Acceptance) Step 3: Construct the NHSS decision matrix with values of the form (T, I, F) for each alternative under each criterion. Step 4: Aggregate the neutrosophic values into scalar scores using a linear model Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2647 https://internationalpubls.com Step 5: Use scalar scores in the TOPSIS method: • Normalize the decision matrix • Construct the weighted normalized matrix • Identify ideal and anti-ideal solutions • Compute distances to ideal solutions • Calculate relative closeness • Rank alternatives 4. Case study Waste water management is an essential aspect of environmental sustainability and public health. Decision-making in Waste water management involves evaluating multiple criteria, such as environmental impact, cost, efficiency, and community acceptance. Traditional decision-making methods often fall short in addressing the multifaceted nature of water waste management, which involves multiple criteria and inherent uncertainties. In Waste water management, decision-makers must consider various factors, such as environmental impact, cost, efficiency, and community acceptance. Each of these criteria can be subjective and influenced by uncertainty, making it challenging to arrive at an optimal decision. To overcome these challenges, advanced decision- making frameworks that can handle complexity and uncertainty are required. Neutrosophic sets extend the concept of fuzzy sets by introducing three components: truth (T), indeterminacy (I), and falsity (F). This extension allows for a more nuanced representation of uncertainty, accommodating the inherent vagueness and indeterminacy in real-world scenarios. Hypersoft sets further generalize soft sets by considering multi-attribute evaluations, providing a robust framework for decision-making in complex environments. In this study, we apply the TOPSIS method within the Neutrosophic hypersoft set framework to evaluate and rank different waste water management alternatives. The alternatives considered are: 1. Recycling System (R) 2. Advanced Treatment Plant (T) 3. Waste Reduction Program (W) These alternatives are assessed based on four key criteria: 1. Environmental Impact (E) 2. Cost (C) 3. Efficiency (Ef) 4. Community Acceptance (A) Consider three alternatives for waste water management as mentioned in the previous section. Step 1: Define membership functions: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2648 https://internationalpubls.com Define the neutrosophic membership functions for each alternative with respect to each criterion. We have 3 values for each criterion truth, indeterminacy and falsity. Table 4.1 Alternative Environmental Impact (E) (T,I,F) Cost (C) (T,I,F) Efficiency (T,I,F) Community Acceptance (T,I,F) Recycling System (R) (0.8, 0.1, 0.1) (0.7, 0.2, 0.1) (0.9, 0.05, 0.05) (0.6, 0.3, 0.2) Advanced Treatment Plant (T) (0.9, 0.05, 0.05) (0.5, 0.3, 0.2) (0.8, 0.1, 0.1) (0.7, 0.2, 0.1) Waste Reduction Program (W) (0.7, 0.2, 0.1) (0.9, 0.05, 0.05) (0.6, 0.3, 0.2) (0.8, 0.1, 0.1) Step 2: Here we define criteria weights, that use AHP to determine the weights for each criterion. After pairwise comparisons and consistency assume the weights as, • Environmental Impact: 0.4 • Cost: 0.2 • Efficiency: 0.3 • Community Acceptance: 0.1 Step 3: Next, we construct the neutrosophic decision matrix. The decision matrix with membership values and weight is Table 4.2 Alternative Environmental Impact (E) (0.4) Cost (C) (0.2) Efficiency (0.3) Community Acceptance (0.1) Recycling System (R) (0.8, 0.1, 0.1) (0.7, 0.2, 0.1) (0.9, 0.05, 0.05) (0.6, 0.3, 0.2) Advanced Treatment Plant (T) (0.9, 0.05, 0.05) (0.5, 0.3, 0.2) (0.8, 0.1, 0.1) (0.7, 0.2, 0.1) Waste Reduction Program (W) (0.7, 0.2, 0.1) (0.9, 0.05, 0.05) (0.6, 0.3, 0.2) (0.8, 0.1, 0.1) Step 4: Now, apply TOPSIS for neutrosophic hypersoft sets. Calculate the neutrosophic normalized Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2649 https://internationalpubls.com decision matrix using the formula Table 4.3: TRUTH VALUES Alternative Environmental Impact (E) Cost (C) Efficiency Community Acceptance Recycling System (R) 0.5657 0.5774 0.6782 0.4619 Advanced Treatment Plant (T) 0.6364 0.4122 0.6027 0.5385 Waste Reduction Program (W) 0.4944 0.7416 0.4520 0.6154 Table 4.4: INDETERMINACY VALUES Alternative Environmental Impact (E) Cost (C) Efficiency Community Acceptance Recycling System (R) 0.4082 0.5164 0.1526 0.8018 Advanced Treatment Plant (T) 0.2041 0.7745 0.3052 0.5345 Waste Reduction Program (W) 0.8165 0.1291 0.9158 0.2673 Table 4.5: FALSITY VALUES Alternative Environmental Impact (E) Cost (C) Efficiency Community Acceptance Recycling System (R) 0.6667 0.4082 0.3333 0.5774 Advanced Treatment Plant (T) 0.3333 0.8165 0.6667 0.5774 Waste Reduction Program (W) 0.6667 0.2041 0.6667 0.5774 The weighted normalized decision matrix using the formula, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2650 https://internationalpubls.com Alternative Environmental Impact (E) Cost (C) Efficiency Community Acceptance Recycling System (R) T-0.2263 I-0.1632 F-0.2666 T-0.1155 I-0.1032 F-0.0816 T-0.2035 I-0.0457 F-0.0999 T-0.0462 I-0.0801 F-0.0577 Advanced Treatment Plant (T) T-0.2546 I-0.0816 F-0.1333 T-0.0824 I-0.1549 F-0.1633 T-0.1808 I-0.0915 F-0.2000 T-0.0538 I-0.0534 F-0.0577 Waste Reduction Program (W) T-0.1977 I-0.3266 F-0.2666 T-0.1483 I-0.0258 F-0.0408 T-0.1356 I-0.2747 F-0.2000 T-0.0615 I-0.0267 F-0.0577 Step 5: Calculate the ideal (A*) and negative-ideal (A^-) solutions for the Truth(T), Indeterminacy(I) and Falsity(F) values using the formula Table 4.7 Recycling System(R) A* 0.2546 0.1483 0.2035 0.0615 A- 0.1978 0.0824 0.1356 0.0462 Advanced Treatment Plant(T) A* 0.3266 0.1549 0.2747 0.0801 A- 0.0816 0.0258 0.0457 0.0262 Waste Reduction Program(W) A* 0.2666 0.1633 0.2000 0.0577 A- 0.1333 0.0408 0.0999 0.0577 Step 6: Calculate the distance of each alternative from D* and D- using Euclidean distance for each alternative from the positive and negative ideal solutions for Truth(T), Indeterminacy(I) and Falsity(F) using the formula, Table 4.8 Alternatives Truth Indeterminacy Falsity 0.02960 0.21851 0.09172 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2651 https://internationalpubls.com 0.03420 0.09049 0.13496 0.06647 0.27927 0.13333 0.05890 0.13193 0.13252 0.06141 0.13195 0.12251 0.06595 0.29744 0.14332 Step 7: Calculate the Relative closeness for Truth(T), Indeterminacy(I) and Falsity(F) values as follows, Truth(T) Indeterminacy(I) Falsity(F) Step 8: Ranking the Alternatives: Based on the relative closeness values for Truth(T), Indeterminacy(I) and Falsity(F) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2652 https://internationalpubls.com ➢ For Truth(T) values, the Recycling System (R) is the best alternative with a relative closeness of 0.5360 ➢ For Indeterminacy(I) and Falsity(F) values, the Recycling System (R) are the least alternative with a relative closeness of 0.2928 and 0.3984 In the context of neutrosophic hypersoft sets, the Recycling System(R) is the best alternative for the water waste management system when considering the given criteria and weights. This conclusion is derived from its highest relative closeness values in terms of truth, indeterminacy and falsity indicating its overall superior performance and reliability in managing water waste effectively. 5. Conclusion The Recycling System (R) emerged as the best alternative for managing waste water, scoring the highest relative closeness to the ideal solution. This indicates that, given the criteria and their respective weights, the Recycling System (R) provides the most balanced and optimal solution, effectively minimizing environmental impact while maintaining cost-efficiency, operational efficiency, and community acceptance. This decision supports the implementation of the Recycling System (R) as the preferred choice for a sustainable waste water management system. References [1] Abbas, M.; Murtaza, G.; Smarandache, F. Basic operations on hypersoft sets and hypersoft point. Neutrosophic Sets Syst. 2020,35, 407-421. [2] Maji P.K, Neutrosophic Soft Set, Annals of Fuuzy Mathematics and Informatics, 2013, 5(1), 157- 168. [3] Moldstov D “Soft Set Theory- first results”, Computers an Mathematics with applications, 1999, vol.37(4-7), pp, 19-31 [4] Saeed, M., Ahsan, M. Siddique, M.; Ahmad, M. A study of the fundamentals of hypersoft set theory. Inter.J. Sci. Eng. Res. 2020, 11. [5] Saqlain.M, Moin M, Jafar, Saeed, Smarandache F, Aggregate operators of NHS, Neutrosophic Sets Syst, 2020, 32, 294-306. [6] Sagvan Y. Musa, Baravan A. Asaad, Hypersoft Topological Spaces, Neutrosophic Sets and Systems, Vol. 49, 2022 401 [7] Shu-Jen Chen & Ching-Lai Hwang, Fuzzy Multiple Attribute Decision Making Methods, Springer- Verlag Berlin Heidelberg, Chapter 5, 1992., 375 , 289-486. [8] Smarandache, F. Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic Sets Syst. 2018, 22, 168-170. [9] Yoon K.P., Hwang C.-L, Multiple Attribute Decision Making: An Introduction, International Educational and Professional Publisher, 1995, Vol. 104 [10] Zadeh. L. A., Fuzzy sets, Information and control, 8(1965), 338-353.