Assessing Regional Health and Environmental Outcomes Using Weighted Neutrosophic Similarity Measures: A Benchmarking Approach for Sustainable Development

Abstract

Humanity faces significant challenges in achieving internationally agreed sustainable development goals, particularly in reducing public health risks and improving the environmental quality. Measuring and comparing performance across regions requires a systematic and transparent framework. This study explores the application of sustainable development indicators, including a mortality rate attributed to exposure to unsafe WASH services (SDG 3.9.2), a mortality rate attributed to household and ambient air pollution (SDG 3.9.1), and a mortality rate attributed to unintentional poisoning to assess regional health outcomes. Using data from 50 countries across five sub-regions of Asia, this research applies a weighted neutrosophic similarity measure based on the Hausdorff metric to evaluate regional alignment with an ideal benchmark. The results reveal significant disparities across regions, highlighting Central Asia as the closest to the benchmark, while South and West Asia exhibit substantial gaps. These findings provide actionable insights for policymakers to improve public health infrastructure and address environmental challenges, promoting equitable and sustainable development.

1. Introduction

The goal of sustainable development (SD) is still a top priority worldwide, with a focus on enhancing the environmental quality and reducing hazards to public health. Strong analytical frameworks that can methodically evaluate regional performance in relation to predetermined benchmarks are necessary to address these issues [1]. The application of sustainable development indicators—specifically, mortality rates attributed to unsafe water, sanitation, and hygiene (WASH) services (SDG 3.9.2), household and ambient air pollution (SDG 3.9.1), and unintentional poisoning—to assess regional health outcomes. These indicators are particularly relevant for Asia, where rapid urbanization, industrialization, and unequal access to clean resources have led to critical health challenges. Key issues in the region include severe air pollution, with many Asian cities ranking among the most polluted globally (World Health Organization (WHO), 2019); inadequate WASH infrastructure, especially in rural and peri-urban areas, contributing to preventable diseases [1]; and rising unintentional poisoning cases linked to agricultural and industrial chemicals [2]. In addition, informal urban settlements often lack basic services, increasing the exposure to environmental hazards [3]. Asia’s high vulnerability to climate change exacerbates these health risks, with climate-induced disasters disrupting health services and worsening sanitation conditions [2]. Addressing these issues through targeted policy interventions and improved monitoring can support the achievement of SD goals and enhance health outcomes across the region.

Uncertainty has been a significant focus in decision-making and mathematical modeling. To address uncertainty, Zadeh [4] introduced fuzzy set theory, which laid the foundation for the handling of imprecise data. Zadeh [5] extended this framework by introducing linguistic variables to model approximate reasoning. Recognizing the limitations in fuzzy sets, Atanassov [6] proposed intuitionistic fuzzy sets (IFS), incorporating both membership and non-membership degrees to capture uncertainty better. Further developments led to alternative uncertainty-handling frameworks. Gau and Buehrer [7] introduced vague sets, enhancing the representation of uncertainty in decision-making. Zhang [8] developed bipolar fuzzy sets, allowing for the representation of both positive and negative attributes within a system. These advances enabled a more nuanced approach to cognitive modeling and multi-agent decision-making. Expanding on the concepts of bipolarity, Lee [9] proposed bipolar fuzzy sub-algebras and fuzzy ideals in algebraic structures. Alkouri et al. [10] further advanced this field by introducing bipolar complex fuzzy sets, which expanded the applicability of fuzzy logic in real-world decision-making problems. Mahmood and Ur Rehman [11] refined these theories by developing generalized similarity measures for bipolar complex fuzzy sets.

Parallel to these developments, De et al. [12] applied neutrosophic sets to query processing in inconsistent databases, highlighting the role of neutrosophic logic in data management. De and Mishra [13] introduced functional dependencies in neutrosophic relational database models, expanding their utility in real-world applications. AboElHamd et al. [14] provided a comprehensive study of neutrosophic logic theory and its practical applications. To refine decision-making in multi-criteria problems, Deli et al. [15] proposed bipolar neutrosophic sets, integrating three membership functions: truth, indeterminacy, and falsity. This advancement allowed for the more effective handling of contradictory information in uncertain environments. Soft set theory was introduced as an alternative approach to uncertainty modeling. Molodtsov [16] introduced soft sets as a flexible tool in dealing with uncertain data. Maji et al. [17] extended this concept by applying soft sets to decision-making, which was further formalized by Maji et al. [18]. Yang [18] later provided refinements to soft set theory, improving its computational efficiency. Hybrid models combining fuzzy and soft set theories have gained traction. Abdullah et al. [19] introduced bipolar fuzzy soft sets, extending their applicability in decision-making problems. Alqaraleh et al. [20] expanded this framework by proposing bipolar complex fuzzy soft sets, which were later refined by Mahmood et al. [21] in a decision-making context. Further aggregation methodologies were introduced by Yang et al. [22], enabling their use in automation and industry evaluation. Aggregation operators play a crucial role in multi-criteria decision-making. Yang et al. [23] developed spherical fuzzy soft aggregation techniques, followed by Raja et al. [24], who introduced group-based q-rung ortho-pair fuzzy N-soft sets for solar panel evaluation.

The effectiveness of similarity measures in uncertainty-based decision-making has been extensively studied. Şahin and Küçük [25] and Mukherjee and Sarkar [26] explored similarity measures in neutrosophic soft sets. Sinha and Majumdar [27] refined these measures to improve their accuracy in decision-making applications. Broumi and Smarandache [28] proposed an extended Hausdorff distance for neutrosophic refined sets, enhancing their application in medical diagnosis. Further advancements in similarity measures were achieved by Khan et al. [29], who introduced vector similarity measures for simplified neutrosophic hesitant fuzzy sets. Liu et al. [30] explored Euclidean distance-based similarity measures, improving their effectiveness in pattern recognition. Wang [31] applied neutrosophic distance measures in pattern recognition, further demonstrating their utility. Ali et al. [32] refined these concepts by incorporating the Hausdorff distance into similarity measures for single-valued neutrosophic sets. Bipolar neutrosophic similarity measures have also been explored. Uluçay et al. [33] developed similarity measures for bipolar neutrosophic sets, demonstrating their application in multi-criteria decision-making. Mahmood et al. [34] introduced generalized similarity measures for complex hesitant fuzzy sets, which were further enhanced in Mahmood et al. [35] for pattern recognition and medical diagnosis. Chinram et al. [36] proposed cosine similarity measures for complex hesitant fuzzy sets, expanding their application in intelligent systems. Advanced similarity measures have been proposed to enhance decision-making models. Al-Sharqi et al. [37] introduced similarity measures for interval-complex neutrosophic soft sets, emphasizing their role in medical diagnosis. DalKılıç and Demirtaş [38] refined neutrosophic fuzzy soft set similarity measures, improving their applicability in real-world decision-making. Al-Sharqi et al. [39] extended these measures to the possibility of neutrosophic soft expert sets, demonstrating their effectiveness in decision support systems.

Real-world applications of similarity measures have been widely studied. Patel et al. [40] applied similarity measures in face recognition and software quality evaluation. Alreshidi et al. [41] examined entropy measures for circular intuitionistic fuzzy sets, further enhancing uncertainty modeling. Neutrosophic logic and decision-making techniques have gained prominence in recent studies. Smarandache [42] provided theoretical extensions of neutrosophic logic, paving the way for advanced decision models. Wu et al. [43] analyzed Hausdorff distances in neutrosophic decision-making, demonstrating their effectiveness in multi-criteria problems. Dalkılıç and Demirtaş [44] studied relations in neutrosophic soft sets and their applications in uncertainty modeling.

Neutrosophic soft set applications have expanded into various fields. Jha and Kumar [45] analyzed neutrosophic soft set decision-making for stock market trend analysis. Yadav et al. [46] proposed new similarity measures for interval-valued neutrosophic sets, enhancing pattern recognition techniques. Wang et al. [47] explored advanced neutrosophic decision-making models, demonstrating their effectiveness in complex decision environments. A comprehensive study of neutrosophic soft decision-making was conducted [48], highlighting the relevance of these models in intelligent decision-making. Finally, Sarkar and Ghosh [49] introduced a Hausdorff similarity measure on neutrosophic soft sets, further advancing their application in decision-making scenarios. This study builds upon these theoretical foundations to develop a weighted neutrosophic similarity measure based on the Hausdorff metric. By applying this measure to SD indicators, it provides a systematic framework for regional performance assessment. The findings offer valuable insights for policymakers, guiding targeted interventions to address public health challenges and environmental risks and ultimately fostering sustainable and equitable development. Hence, this study’s objectives are as follows:

• To develop a new weighted neutrosophic similarity measure based on the Hausdorff metric as part of a systematic approach to assessing regional performance in SD;
• To assess regional disparities in health and environmental sustainability by analyzing key indicators such as mortality rates from unsafe WASH services, unintentional poisoning, and air pollution;
• To demonstrate that data-driven decision-making can provide policymakers with actionable information to improve public health infrastructure and solve environmental concerns.

This study looks at SD and presents some useful insights into how different countries in Asia are progressing, especially with health and environmental issues. First, this study introduces a new approach to comparing countries using a weighted neutrosophic similarity measure based on the Hausdorff metric. This is useful when the data are not clear or complete, which often occurs with sustainability information. Then, we use this method to study 50 Asian countries. We look at data such as deaths from unsafe water and sanitation, air pollution, and unintentional poisoning. This study’s results indicate which countries are successful or close to the ideal and which are lagging behind. These results can help policymakers, indicating areas of success and areas where major improvements are needed. This can help in planning and deciding on future goals in order to make development more equal and sustainable.

2. Materials and Methods

2.1. Description of Existing Methods

This research utilizes established methods such as neutrosophic sets and similarity measures, which have been widely recognized for their effectiveness in handling uncertainty and decision-making in complex systems. Neutrosophic sets, introduced by Smarandache, extend the concept of fuzzy sets by incorporating three membership functions: truth, indeterminacy, and falsity [42]. This allows for a more nuanced representation of uncertainty compared to traditional fuzzy sets [15]. Fuzzy set theory, initially proposed by Zadeh [4], laid the foundation for the handling of imprecise data by introducing the concept of membership degrees. This was further extended by Atanassov [6] with intuitionistic fuzzy sets, which incorporate both membership and non-membership degrees to better capture uncertainty. Neutrosophic sets build upon these advancements by adding an indeterminacy component, providing a more comprehensive framework for the modeling of uncertainty.

Recent studies have demonstrated the effectiveness of neutrosophic sets and similarity measures in sustainability research. For instance, Dalkılıç and Demirtaş [38] refined neutrosophic fuzzy soft set similarity measures, improving their applicability in real-world decision-making. In addition, Al-Sharqi et al. [39] extended these measures to the possibility of neutrosophic soft expert sets, demonstrating their effectiveness in decision support systems. In the context of sustainability research, these methods offer a robust analytical framework for the evaluation of regional performance against predetermined benchmarks. By incorporating truth, indeterminacy, and falsity membership functions, neutrosophic sets provide a nuanced representation of the complexities inherent in regional health and environmental assessments. The weighted Hausdorff distance measure further enhances this framework by allowing for a detailed comparison of regional performance, highlighting areas of strength and weakness [32]. The Hausdorff distance metric is a well-known method of measuring the similarity between two sets by considering the maximum distance of a set to the nearest point in the other set. This metric has been widely applied in various fields, including pattern recognition and medical diagnosis [28]. The weighted Hausdorff distance measure, which incorporates weights to account for the significance of different points, enhances the traditional metric by providing a more detailed analysis of similarity [32]. Overall, the combination of neutrosophic sets and weighted Hausdorff distance measures provides a powerful tool for sustainability research, enabling a comprehensive and systematic evaluation of regional health and environmental outcomes.

2.2. Collection of Data and Data Preprocessing

The data for this study were collected using health and environmental indicators published on the WHO website, focusing on statistics from 2019. The parameters considered include mortality rates from unsafe WASH services, unintentional poisoning, and air pollution. The analysis spans 50 countries across five sub-regions of Asia: Central Asia (5 countries), East Asia (5 countries), South Asia (9 countries), Southeast Asia (11 countries), and West Asia (15 countries). Each country’s data were modeled using neutrosophic soft sets, capturing truth (the presence of issues), indeterminacy (uncertainty in data), and falsity (the absence of issues). The 2019 indicators provided a comprehensive basis for the assessment of regional performance against an ideal benchmark. This dataset highlights critical health risks and infrastructure gaps, supporting targeted interventions for SD. This study demonstrates how the WHO’s reliable metrics can inform nuanced regional evaluations. The primary source of data is the WHO—specifically, the World Health Statistics 2019 report on 2 January 2025 (https://www.who.int/data/gho/data/indicators/indicator-details/GHO/). This report provides comprehensive health-related statistics for the WHO’s 194 member states, including trends in life expectancy, causes of death, and health-related SDG indicators. The data are disaggregated by WHO region, world bank income group, and sex where possible, offering detailed insights into health statuses and access to preventive and curative services. The rest of the research procedure is described in detail through the flowchart in Figure 1.

2.3. Preliminaries

2.3.1. Definition: Single-Valued Neutrosophic Set (SVNS)

Let $\mathrm{V}$ be a universal set. An SVNS M on V can be written as $\mathrm{M}=\left\{\mathrm{y},{\mathrm{T}}_{\mathrm{M}}\left(\mathrm{y}\right),{\mathrm{I}}_{\mathrm{M}}\left(\mathrm{y}\right),{\mathrm{F}}_{\mathrm{M}}\left(\mathrm{y}\right):\mathrm{y}\in \mathrm{M}\right\}$, where the membership functions ${\mathrm{T}}_{\mathrm{M}},{\mathrm{I}}_{\mathrm{M}},{\mathrm{F}}_{\mathrm{M}}:\mathrm{V}\to \left[0,1\right]$ with $0\le {\mathrm{T}}_{\mathrm{M}}+{\mathrm{I}}_{\mathrm{M}}+{\mathrm{F}}_{\mathrm{M}}\le 3$.

2.3.2. Definition: Hausdorff Distance Measure

Consider two finite sets $\mathrm{F}=\left\{{\mathrm{f}}_1,{\mathrm{f}}_2,\dots {\mathrm{f}}_{\mathrm{n}}\right\}$ and $\mathrm{G}=\left\{{\mathrm{g}}_1,{\mathrm{g}}_2,\dots {\mathrm{g}}_{\mathrm{m}}\right\}$. The Hausdorff distance between $\mathrm{F}$ and $\mathrm{G}$ is defined as $\mathrm{h}\left(\mathrm{F},\mathrm{G}\right)={\max }_{{\mathrm{f}}_{\mathrm{i}}\in \mathrm{F}}\left\{{\min }_{{\mathrm{g}}_{\mathrm{j}}\in \mathrm{G}}\mathrm{d}\left({\mathrm{f}}_{\mathrm{i}},{\mathrm{g}}_{\mathrm{j}}\right)\right\}$ for $\mathrm{i}=1,2\dots .\mathrm{n}$ and $\mathrm{j}=1,2,\dots \mathrm{m}$, where $\mathrm{d}\left({\mathrm{f}}_{\mathrm{i}},{\mathrm{g}}_{\mathrm{j}}\right)$ is any distance metric.

2.3.3. Definition: Hausdorff Distance Measure for Neutrosophic Sets

Consider a finite set $\mathrm{V}$ with finite elements ${\mathrm{y}}_1,{\mathrm{y}}_2,\dots {\mathrm{y}}_{\mathrm{n}}$. Let $\mathrm{K}=\left\{{\mathrm{y}}_{\mathrm{i}},{\mathrm{T}}_{\mathrm{K}}\left({\mathrm{y}}_{\mathrm{i}}\right),{\mathrm{I}}_{\mathrm{K}}\left({\mathrm{y}}_{\mathrm{i}}\right),{\mathrm{F}}_{\mathrm{K}}\left({\mathrm{y}}_{\mathrm{i}}\right):{\mathrm{y}}_{\mathrm{i}}\in \mathrm{V}\right\}$ and $\mathrm{L}=\left\{{\mathrm{y}}_{\mathrm{i}},{\mathrm{T}}_{\mathrm{L}}\left({\mathrm{y}}_{\mathrm{i}}\right),{\mathrm{I}}_{\mathrm{L}}\left({\mathrm{y}}_{\mathrm{i}}\right),{\mathrm{F}}_{\mathrm{L}}\left({\mathrm{y}}_{\mathrm{i}}\right):{\mathrm{y}}_{\mathrm{i}}\in \mathrm{V}\right\}$ be two neutrosophic sets on $\mathrm{V}$. The Hausdorff distance between $\mathrm{K}$ and $\mathrm{L}$ is defined as follows:

$$\mathrm{H}\left(\mathrm{K},\mathrm{L}\right)={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\max \left\{\begin{array}{c}\left|{\mathrm{T}}_{\mathrm{K}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{L}}\left({\mathrm{y}}_{\mathrm{i}}\right)\right|, \\ \left|{\mathrm{I}}_{\mathrm{K}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{L}}\left({\mathrm{y}}_{\mathrm{i}}\right)\right|,\left|{\mathrm{F}}_{\mathrm{K}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{L}}\left({\mathrm{y}}_{\mathrm{i}}\right)\right|\end{array}\right\}$$

The similarity measure between $\mathrm{K}$ and $\mathrm{L}$ is given by $\mathrm{SM}\left(\mathrm{K},\mathrm{L}\right)=1-\mathrm{D}\left(\mathrm{K},\mathrm{L}\right)$.

2.3.4. Definition: Single-Valued Neutrosophic Soft Set (SVNSS)

Let $\mathrm{V}$ be a universal set. $\mathrm{N}\left(\mathrm{V}\right)$ represents the set of all single-valued neutrosophic sets on V. Additionally, let $\mathrm{P}$ be the set containing all the parameters and let S be a subset of $\mathrm{P}$. Then, the pair $\left(\mathrm{P},\mathrm{S}\right)$, where $\mathrm{P}$ is a mapping from $\mathrm{S}$ to $\mathrm{N}\left(\mathrm{V}\right)$, is called as SVNSS on V. An example can help to clarify the concept of a neutrosophic soft set. Universal set and parameters: Let $\mathrm{U}=\left\{{\mathrm{L}}_1,{\mathrm{L}}_2,{\mathrm{L}}_3\right\}$ be the universal set consisting of three key fertility-related outcomes, namely ${\mathrm{L}}_1$—successful ovulation, ${\mathrm{L}}_2$—positive implantation, ${\mathrm{L}}_3$—sustained pregnancy $\Rightarrow \mathrm{n}=3$. Let $\mathrm{E}=\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,{\mathrm{E}}_4\right\}$ be the parameter set consisting of four parameters, where ${\mathrm{E}}_1$—hormonal balance, ${\mathrm{E}}_2$—lifestyle factors (e.g., BMI, smoking), ${\mathrm{E}}_3$—medical intervention success rate (e.g., IVF success rate), ${\mathrm{E}}_4$—psychological and emotional support $\Rightarrow \mathrm{m}=4$. Truth membership (T) indicates symptom presence, indeterminacy membership (I) reflects uncertainty, and falsity membership (F) indicates symptom absence. The neutrosophic soft set for the evaluation of the outcomes of fertility treatments is represented using neutrosophic soft matrices for three patients: $\mathrm{I}$(baseline patient), Q, and R, who underwent different treatment protocols.

Table 1. Neutrosophic soft set for ideal patient I.

	$\mathbf{I}({\mathbf{E}}_1$)	$\mathbf{I}({\mathbf{E}}_2$)	$\mathbf{I}({\mathbf{E}}_3$)	$\mathbf{I}({\mathbf{E}}_4$)
${\mathrm{L}}_1$	(0.7, 0.2, 0.3)	(0.6, 0.1, 0.3)	(0.8, 0.3, 0.5)	(0.7, 0.2, 0.4)
${\mathrm{L}}_2$	(0.6, 0.3, 0.2)	(0.1, 0.5, 0.5)	(0.4, 0.3, 0.3)	(0.4, 0.7, 0.3)
${\mathrm{L}}_3$	(0.2, 0.6, 0.7)	(0.2, 0.4, 0.4)	(0, 1, 0)	(0.3, 0.2, 0.7)

,

Table 2. Neutrosophic soft set for patient Q.

	$\mathbf{Q}({\mathbf{E}}_1$)	$\mathbf{Q}({\mathbf{E}}_2$)	$\mathbf{Q}({\mathbf{E}}_3$)	$\mathbf{Q}({\mathbf{E}}_4$)
${\mathrm{L}}_1$	(0.8, 0.3, 0.5)	(0.7, 0.4, 0.3)	(0.8, 0.6, 0.7)	(1, 0, 0)
${\mathrm{L}}_2$	(0.2, 0.5, 0.6)	(0.1, 0.1, 0.8)	(0.4, 0.1, 0.5)	(0.3, 0.3, 0.4)
${\mathrm{L}}_3$	(0, 0, 0)	(0.1, 0.3, 0.3)	(1, 1, 0)	(0, 0, 0)

and

Table 3. Neutrosophic soft set for patient R.

	$\mathbf{R}({\mathbf{E}}_1$)	$\mathbf{R}({\mathbf{E}}_2$)	$\mathbf{R}({\mathbf{E}}_3$)	$\mathbf{R}({\mathbf{E}}_4$)
${\mathrm{L}}_1$	(0.8, 0.4, 0.8)	(0.6, 0.3, 0.1)	(0.5, 0.6, 0.5)	(0.7, 0.2, 0.4)
${\mathrm{L}}_2$	(0, 0, 0)	(0, 1, 1)	(0.3, 0.1, 0.1)	(0.2, 0.5, 0.4)
${\mathrm{L}}_3$	(0.2, 0.6, 0.4)	(0.2, 0.6, 0.6)	(0, 0, 0)	(0.4, 0.2, 0.8)

illustrate the neutrosophic soft set representations for the different patient cases. These tables provide insights into the truth, indeterminacy, and falsity values associated with different evaluation parameters. Table 1 represents the neutrosophic soft set for an ideal patient, serving as a benchmark. Table 2 presents the values for patient Q, while Table 3 details the values for patient R, highlighting variations across these cases.

2.3.5. Similarity Measure

The weighted Hausdorff distance measure evaluates the similarity or closeness between two objects by considering the dual nature of distance and similarity: as the distance increases, the similarity decreases, and vice versa. This measure builds upon the traditional Hausdorff distance metric, incorporating weights to account for the significance of different points. The weighted Hausdorff distance calculates the maximum weighted distance between a point in one set and the nearest point in another set. For finite sets, it can be formally defined as follows.

Let V be the universal set with elements ${\mathrm{y}}_1,{\mathrm{y}}_2,\dots {\mathrm{y}}_{\mathrm{n}}$ and $\mathrm{E}$ be the parameter set with elements ${\mathrm{e}}_1,{\mathrm{e}}_2,\dots {\mathrm{e}}_{\mathrm{m}}$. Let $\left(W,\mathrm{P}\right)$ and $\left(\mathrm{X},\mathrm{P}\right)$ be neutrosophic soft sets on V. The Hausdorff distance between two neutrosophic soft sets is defined as follows:

$$\mathrm{D}\left(\mathrm{W},\mathrm{X}\right)={\displaystyle \frac{1}{\mathrm{mn}}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\max \left\{\begin{array}{c}\left|{\mathrm{T}}_{\mathrm{W}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{e}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|, \\ \left|{\mathrm{I}}_{\mathrm{W}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{X}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|,\left|{\mathrm{F}}_{\mathrm{W}({\mathrm{e}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{X}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|\end{array}\right\}$$

where ${\mathrm{T}}_{\mathrm{W}({\mathrm{e}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{\mathrm{T}}_{\mathrm{X}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})$,${ \mathrm{I}}_{\mathrm{W}({\mathrm{e}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{\mathrm{I}}_{\mathrm{X}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{ \mathrm{F}}_{\mathrm{W}({\mathrm{e}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{\mathrm{F}}_{\mathrm{X}({\mathrm{e}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),$ respectively, denote the truthness, indeterminacy, and falsehood membership functions for the neutrosophic soft sets $\left(\mathrm{W},\mathrm{P}\right)$ and $\left(\mathrm{X},\mathrm{P}\right)$ corresponding to the element ${\mathrm{y}}_{\mathrm{i}}$ and parameter ${\mathrm{e}}_{\mathrm{j}}$. The similarity measure between $\left(\mathrm{A},\mathrm{P}\right)$ and $\left(\mathrm{B},\mathrm{P}\right)$ is defined as $\mathrm{SM}\left(\mathrm{A},\mathrm{B}\right)=1-\mathrm{D}\left(\mathrm{A},\mathrm{B}\right)$.

3. Proposed Similarity Measure Formula and Validation of the Proposed Distance and Similarity Measures

3.1. Proposed Similarity Measure Formula

Let V be the universal set with elements ${\mathrm{y}}_1,{\mathrm{y}}_2,\dots {\mathrm{y}}_{\mathrm{n}}$ and P be the parameter set with elements ${\mathrm{p}}_1,{\mathrm{p}}_2,\dots {\mathrm{p}}_{\mathrm{m}}$. Let $\left(\mathrm{A},\mathrm{P}\right)$ and $\left(\mathrm{B},\mathrm{P}\right)$ be neutrosophic soft sets on V. The Hausdorff distance between two neutrosophic soft sets is defined as follows:

$${\mathrm{D}}_{\mathrm{W}}\left(\mathrm{A},\mathrm{B}\right)={\displaystyle \frac{1}{\mathrm{mn}}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|, \\ {\mathrm{W}}_{\mathrm{I}.}\left|{\mathrm{I}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{B}({\mathrm{e}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|\end{array}\right\}$$

where ${\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})$,${ \mathrm{I}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{\mathrm{I}}_{\mathrm{B}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{ \mathrm{F}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}}),{\mathrm{F}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})$, respectively, denote the truthness, indeterminacy, and falsehood membership functions for the neutrosophic soft sets $\left(\mathrm{A},\mathrm{P}\right)$ and $\left(\mathrm{B},\mathrm{P}\right)$ corresponding to the element ${\mathrm{y}}_{\mathrm{i}}$ and parameter $p_j$. Here, $\mathrm{n}$—number of elements in the universe $\left(\mathrm{i}=1,2,\dots ,\mathrm{n}\right)$ and $\mathrm{m}$—number of parameters $\left(\mathrm{j}=1,2,\dots ,\mathrm{m}\right).$ ${\mathrm{W}}_{\mathrm{A}},{\mathrm{W}}_{\mathrm{B}},{\mathrm{W}}_{\mathrm{C}}$ are the weights assigned to the truth membership, indeterminacy membership, and falsity membership values, respectively, satisfying ${\mathrm{W}}_{\mathrm{A}}+{\mathrm{W}}_{\mathrm{B}}+{\mathrm{W}}_{\mathrm{C}}=1$, ${\mathrm{W}}_{\mathrm{A}}ϵ\left[0,1\right],{\mathrm{W}}_{\mathrm{B}}ϵ\left[0,1\right],{\mathrm{W}}_{\mathrm{C}}ϵ\left[0,1\right]$. Each term in the formula measures the weighted difference between the corresponding components (truth, indeterminacy, and falsehood) of A and B.

$$\mathrm{Weighted} \mathrm{difference} \mathrm{for} \mathrm{truth}:{ \mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|.$$

$$\mathrm{Weighted} \mathrm{difference} \mathrm{for} \mathrm{indeterminacy}:{ \mathrm{W}}_{\mathrm{I}.}\left|{\mathrm{I}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{B}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|.$$

$$\mathrm{Weighted} \mathrm{difference} \mathrm{for} \mathrm{indeterminacy}:{ \mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|.$$

The max function ensures that the largest weighted difference dominates for each element–parameter pair (${\mathrm{y}}_{\mathrm{i}},{\mathrm{e}}_{\mathrm{j}})$. The formula averages these maximum weighted differences across all n elements and m parameters. Normalization ensures that ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ lies within a bounded range, dependent on the weights ${\mathrm{W}}_{\mathrm{T}},{ \mathrm{W}}_I,{ \mathrm{W}}_F$ and the values of T, I, F. The similarity measure between $\left(\mathrm{A},\mathrm{P}\right)$ and $\left(\mathrm{B},\mathrm{P}\right)$ is defined as ${\mathrm{SM}}_{\mathrm{W}}\left(\mathrm{P},\mathrm{Q}\right)=1-{\mathrm{D}}_{\mathrm{W}}\left(\mathrm{P},\mathrm{Q}\right)$.

3.2. Validation of the Proposed Distance and Similarity Measures

3.2.1. Validity of Distance Measure

(1)

Non-negative property: Since all terms $\left|{\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|, \left|{\mathrm{I}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{B}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|,\left|{\mathrm{F}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|$ are absolute differences and ${\mathrm{W}}_{\mathrm{T}}$, ${\mathrm{W}}_I$, ${\mathrm{W}}_F$ > 0, every term is non-negative. Therefore, ${\mathrm{D}}_{\mathrm{w}}\left(A,\mathrm{B}\right)\ge 0$.

(2)

Symmetry property: The absolute difference $\left|{ \mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|$ is symmetric, i.e., $\left|{\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|,=\left|{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|.$ The same applies for I and F. Thus, ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ = ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{B},\mathrm{A}\right)$.

(3)

Boundedness property: Since T, I, F ϵ [0, 1] and ${\mathrm{W}}_{\mathrm{T}}$, ${\mathrm{W}}_I$, ${\mathrm{W}}_F$ > 0, the maximum difference for each component is at most max (${\mathrm{W}}_{\mathrm{T}}$, ${\mathrm{W}}_I$, ${\mathrm{W}}_F$). After normalization by $mn$, ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ is bounded by $0\le {\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)\le \max \left({\mathrm{W}}_{\mathrm{T}},{ \mathrm{W}}_I,{ \mathrm{W}}_F\right).$

(4)

Identity of indiscernible property: ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ = 0 iff $\mathrm{A}=\mathrm{B}$. If $\mathrm{A}=\mathrm{B}$, all differences $\left|{\mathrm{T}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|$, $\left|{\mathrm{I}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{B}(e_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|,\left|{\mathrm{F}}_{\mathrm{A}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{B}({\mathrm{p}}_{\mathrm{j})}}({\mathrm{y}}_{\mathrm{i}})\right|$ = 0, and hence ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ = 0. Conversely, if ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ = 0, all component-wise differences are zero, implying $\mathrm{A}=\mathrm{B}$.

3.2.2. Validity Checking for the Proposed Similarity Formula

The following properties must be satisfied to validate the proposed similarity measure for the formula and properties of the similarity measure: $0 \le { \mathrm{SM}}_W\left(\mathrm{A}, \mathrm{B}\right) \le 1{\mathrm{SM}}_{\mathrm{W}}\left(\mathrm{A}, \mathrm{B}\right)$= 1 if and only if A = B, ${\mathrm{SM}}_{\mathrm{W}}\left(\mathrm{A}, \mathrm{B}\right)$= ${\mathrm{SM}}_{\mathrm{W}}\left(\mathrm{B}, \mathrm{A}\right)$. If U ⊆ V ⊆ W, then ${\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)\le {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{V},\mathrm{W}\right)$ and ${\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)\le {\mathrm{SM}}_{\mathrm{w}}\left(U,\mathrm{V}\right)$.

Proof. 

We have already proven the distance boundedness property:

$$\begin{gathered} 0\le {\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)\le \max \left({\mathrm{W}}_{\mathrm{T}},{ \mathrm{W}}_I,{ \mathrm{W}}_F\right), \\ \Rightarrow 0\le {\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)\le 1, \\ \Rightarrow 0\le {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)\le 1. \end{gathered}$$

By the property identity of indiscernible ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ = 0 $\iff$ $\mathrm{A}=\mathrm{B}$.

$$\Rightarrow {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)=1\iff \mathrm{A}=\mathrm{B}$$

By the symmetry property of distance ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)$ = ${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{B},\mathrm{A}\right)$

$$\Rightarrow {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{A},\mathrm{B}\right)={\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{B},\mathrm{A}\right)$$

If $\mathrm{U}\subseteq \mathrm{V}\subseteq \mathrm{W}$, then the following relationships hold for the membership functions:

$$\begin{gathered} {\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\Rightarrow {\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{W\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le 0 \\ \Rightarrow \left|{\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge \left|{\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \\ \Rightarrow {\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{W\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge {\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \end{gathered}$$

(1)

$$\begin{gathered} {\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\Rightarrow {\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge 0 \\ \left|{\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge \left|{\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \\ \Rightarrow {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \end{gathered}$$

(2)

$$\begin{gathered} {\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right))\Rightarrow {\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge 0 \\ \left|{\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge \left|{\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \\ \Rightarrow {\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge {\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \end{gathered}$$

(3)

From (1), (2), and (3),

$$\begin{gathered} \max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}\ge \\ \max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_V\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\} \end{gathered}$$

(4)

The distance between U and W is

$${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)={\displaystyle \frac{1}{\mathrm{mn}}}{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{m}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}$$

(5)

Similarly, the distance between $\mathrm{V}$ and $\mathrm{W}$ is

$${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{V},\mathrm{W}\right)={\displaystyle \frac{1}{\mathrm{mn}}}{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{m}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_V\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}$$

(6)

From (4), (5), and (6),

$${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)\ge {\mathrm{D}}_{\mathrm{w}}\left(\mathrm{V},\mathrm{W}\right)\Rightarrow {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)\le {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{V},\mathrm{W}\right)$$

Since $\mathrm{U}\subseteq \mathrm{V}\subseteq \mathrm{W}$, the following relationships hold for the membership functions:

$$\begin{gathered} {\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\Rightarrow {\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{W\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\le 0 \\ \Rightarrow \left|{\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge \left|{\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \\ \Rightarrow {\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{W\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge {\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{T}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \end{gathered}$$

(7)

$$\begin{gathered} {\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\Rightarrow {\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge 0 \\ \left|{\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge \left|{\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \\ \Rightarrow {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{I}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \end{gathered}$$

(8)

$$\begin{gathered} {\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right))\Rightarrow {\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge {\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\ge 0 \\ \left|{\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge \left|{\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \\ \Rightarrow {\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{W}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\ge {\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{F}}_{\mathrm{V}\left({\mathrm{e}}_{\mathrm{j}}\right)}\left({\mathrm{x}}_{\mathrm{i}}\right)\right| \end{gathered}$$

(9)

From (7), (8), and (9),

$$\begin{gathered} \max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{W}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}\ge \\ \max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_U\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\} \end{gathered}$$

(10)

Similarly, the distance between $U$ and $V$ is

$${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{V}\right)={\displaystyle \frac{1}{\mathrm{mn}}}{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{m}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\max \left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{T}}.\left|{\mathrm{T}}_U\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ {\mathrm{W}}_{\mathrm{I}}.\left|{\mathrm{I}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,{\mathrm{W}}_{\mathrm{F}}.\left|{\mathrm{F}}_{\mathrm{U}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{\mathrm{V}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}$$

(11)

From (5) and (10)–(12),

$${\mathrm{D}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)\ge {\mathrm{D}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{V}\right)\Rightarrow {\mathrm{SM}}_{\mathrm{w}}\left(\mathrm{U},\mathrm{W}\right)\le {\mathrm{SM}}_{\mathrm{w}}\left(U,\mathrm{V}\right)$$

(12)

□

4. Real-Life Application and Results

In this study, we apply the proposed methodology to address critical public health concerns related to unintentional poisoning and unsafe WASH services. Despite substantial progress in reducing health risks through improved infrastructure and awareness programs, several regions remain vulnerable due to gaps in accessibility, environmental safety, and healthcare quality. These vulnerabilities necessitate robust analytical frameworks to identify and address areas requiring immediate attention. Using the principles of neutrosophic soft sets, our approach systematically evaluates the performance of different regions by benchmarking them against an ideal standard, thereby enabling targeted interventions for SD. The process begins with the collection and representation of relevant data. Key parameters, such as the mortality rates attributed to unsafe WASH services, unintentional poisoning, and household or ambient air pollution, are identified as critical indicators of regional health outcomes. Each region’s data are then modeled using single-valued neutrosophic soft sets, which encapsulate three membership values: truth, representing the presence of issues; indeterminacy, reflecting the uncertainty or variability in the data; and falsity, indicating the absence of issues. These values enable a nuanced representation of the complexities inherent in regional health assessments, accommodating both known and uncertain factors. To assess and compare the regional performance, we employ a weighted similarity measure derived from the Hausdorff distance metric. This measure evaluates the closeness of each region’s neutrosophic values to an ideal benchmark, defined by regions that exhibit optimal health outcomes with minimal mortality rates and robust WASH infrastructure. The similarity measure integrates truth, indeterminacy, and falsity components, ensuring a comprehensive analysis that highlights not only the presence of deficiencies but also the reliability and accuracy of the available data.

For instance, in analyzing regions within South and Southeast Asia, distinct patterns emerge. Countries with stronger WASH policies and healthcare systems, such as those with higher truth membership values and lower indeterminacy, demonstrate greater similarity to the ideal benchmark. Conversely, regions with persistent gaps in infrastructure, policy implementation, or data quality exhibit higher indeterminacy and falsity values, indicating significant room for improvement. The results underscore the necessity of targeted policy interventions to bridge these gaps, focusing on enhancing access to clean water, improving healthcare delivery, and reducing environmental risks. This application not only provides a diagnostic tool for the identification of regional strengths and weaknesses but also serves as a guide for strategic decision-making. By highlighting areas with lower similarity to the ideal benchmark, policymakers can prioritize resource allocation and design interventions tailored to specific needs. For example, addressing high mortality rates due to unintentional poisoning may involve strengthening regulations on hazardous substances, improving emergency response systems, and raising public awareness. Similarly, efforts to mitigate the impacts of unsafe WASH services could focus on expanding access to clean water and sanitation, investing in infrastructure development, and promoting hygiene education. In conclusion, the use of neutrosophic soft sets and similarity measures offers a powerful framework to address complex, multidimensional problems in public health and SD. By enabling a granular analysis of regional performance, this methodology facilitates informed decision-making, paving the way for equitable and sustainable improvements in health and environmental outcomes.

Let us consider the universal set $\mathrm{V}$ (a sub-region of Asia) consisting of $\mathrm{n}$ elements, where ${\mathrm{y}}_1,{\mathrm{y}}_2,\dots {\mathrm{y}}_{\mathrm{n}}$ are $\mathrm{n}$ number of countries in this sub-region. Consider parameter set $P=\left\{{\mathrm{E}}_{\mathrm{uw}},{\mathrm{E}}_{\mathrm{up}},{\mathrm{E}}_{\mathrm{ap}}\right\},$ where ${\mathrm{E}}_{\mathrm{uw}}$ is the mortality rate attributed to exposure to unsafe WASH services, ${\mathrm{E}}_{\mathrm{up}}$ is the mortality rate attributed to unintentional poisoning, and ${\mathrm{E}}_{\mathrm{ap}}$ is the mortality rate attributed to household and ambient air pollution.

Table 4. Ideal neutrosophic values.

Parameter	T(x)	I(x)	F(x)
$E_{uw}$	1	0.1	0
$E_{up}$	1	0.1	0
$E_{ap}$	1	0.1	0

represents a benchmark scenario that embodies optimal health and environmental conditions. The parameters $p_1,p_2$, and $p_3$ correspond to critical indicators that are considered, such as the mortality rates due to unsafe WASH services, unintentional poisoning, and other environmental hazards. In this ideal state, the truth membership T(x) is set to 1, signifying the complete presence of desirable outcomes, while the falsity membership value F(x) is 0, indicating the total absence of adverse effects. The indeterminacy membership I(x) is assigned a value of 0.1 to account for negligible uncertainty or variability inherent even in ideal conditions. This assumed ideal scenario provides a standard against which real-world performance is compared, helping to identify gaps and prioritize targeted interventions for SD. Table 4 presents the ideal neutrosophic values, which serve as a benchmark in assessing regional performance. These values are assigned based on an optimal scenario where truth membership (T) is maximized, indeterminacy membership (I) is minimal, and falsity membership (F) is absent. Table 4 provides reference values for key parameters, including unsafe water sanitation (${\mathrm{E}}_{\mathrm{uw}}$), unintentional poisoning (${\mathrm{E}}_{\mathrm{up}})$, and air pollution (${\mathrm{E}}_{\mathrm{ap}})$.

4.1. Empirical Results

4.1.1. Assignment of Neutrosophic Values and Empirical Results

To assign neutrosophic values in this study, we used the following formulas:

$$\begin{gathered} \mathrm{T}\left(\mathrm{x}\right)={\displaystyle \frac{1}{1+\left(\frac{\left|{\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right|}{{\mathrm{x}}_{\mathrm{ideal}}}\right)}} \\ \mathrm{I}\left(\mathrm{x}\right)={\displaystyle \frac{{\left({\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right)}^2}{{({\mathrm{x}}_{\mathrm{observed}}+{\mathrm{x}}_{\mathrm{ideal}})}^2}} \\ \mathrm{F}\left(\mathrm{x}\right)=1-\mathrm{T}\left(\mathrm{x}\right) \end{gathered}$$

(13)

The formulas for $\mathrm{T}\left(\mathrm{x}\right)$, $\mathrm{I}\left(\mathrm{x}\right)$, and $\mathrm{F}\left(\mathrm{x}\right)$ are valid as they adhere to key properties like boundedness, complementarity, and consistency. T(x) reflects alignment with the ideal, I(x) measures uncertainty, and F(x) complements T(x) such that $\mathrm{T}\left(\mathrm{x}\right)+\mathrm{F}\left(\mathrm{x}\right)=1$. These measures reliably capture truth, indeterminacy, and falsity, making them robust for neutrosophic value assignments. In this study, ${\mathrm{x}}_{\mathrm{ideal}}$ is defined as the median of the corresponding parameter’s global dataset collected from the WHO database for the year 2019. Data from 180 countries for the parameters taken are considered. The median was chosen as it robustly represents the central tendency, reducing the influence of extreme values or outliers. At first glance, using the lowest value in the dataset might appear to provide a reasonable benchmark. However, there were several reasons for which this approach was not followed in this study. Firstly, the lowest value can sometimes be an outlier, i.e., a data point that does not truly reflect the general pattern or condition across the regions being studied. Relying on such a value could lead to unfair or unrealistic comparisons. Similarly, the mean, although commonly used, is not ideal for this research. It is highly sensitive to extreme values, which can distort the true picture of regional performance, especially when the data are skewed or include significant outliers. Instead, the median was selected as it offers a better sense of the typical situation across all regions. Unlike the minimum or the mean, the median is not easily affected by extreme highs or lows, making it a more stable and representative figure for comparative analysis.

The ${\mathrm{x}}_{\mathrm{ideal}}$ (median) values were calculated, and they were as follows: ${\mathrm{x}}_{\mathrm{ideal}}$ for ${\mathrm{E}}_{\mathrm{uw}}$ = 6.35;${ \mathrm{x}}_{\mathrm{ideal}}$ for ${\mathrm{E}}_{\mathrm{up}}$ = 0.6; ${\mathrm{x}}_{\mathrm{ideal}}$ for ${\mathrm{E}}_{\mathrm{ap}}$ = 83.62. These formulas were applied to parameters such as the mortality rates from unsafe WASH services, unintentional poisoning, and air pollution. By calculating the truth, indeterminacy, and falsity of memberships, this framework enables a nuanced assessment of each country’s health and environmental outcomes. This approach ensures consistency in benchmarking against global standards, facilitating targeted interventions for SD. We provide a computational example of the assignment of neutrosophic values (calculation for Kazakhstan). The observed values and ideal values for Kazakhstan are as follows. For ${\mathrm{E}}_{\mathrm{uw}}$, the observed value $({\mathrm{x}}_{\mathrm{observed}}$) = 3.2, and the ideal value (${\mathrm{x}}_{\mathrm{ideal}})$ = 6.35, when using (13).

$$\mathrm{T}\left({\mathrm{E}}_{\mathrm{uw}}\right)={\displaystyle \frac{1}{1+\left(\frac{\left|{\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right|}{{\mathrm{x}}_{\mathrm{ideal}}}\right)}}={\displaystyle \frac{1}{1+\left(\frac{\left|3.2-6.35\right|}{6.35}\right)}}=0.6684$$

$$\begin{gathered} \mathrm{I}\left({\mathrm{E}}_{\mathrm{uw}}\right)={\displaystyle \frac{{\left({\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right)}^2}{{({\mathrm{x}}_{\mathrm{observed}}+{\mathrm{x}}_{\mathrm{ideal}})}^2}}={\displaystyle \frac{{\left(3.2-6.35\right)}^2}{{\left(3.2+6.35\right)}^2}}=0.1088 \\ \mathrm{F}\left({\mathrm{E}}_{\mathrm{uw}}\right)=1-\mathrm{T}\left({\mathrm{E}}_{\mathrm{uw}}\right)=1-0.6684=0.3316 \end{gathered}$$

The neutrosophic values for ${\mathrm{E}}_{\mathrm{uw}}$ for Kazakhstan in Central Asia ${\mathrm{A}}_{\mathrm{C}}\left({\mathrm{E}}_{uw}\right)$ are $\left(\mathrm{T}\left({\mathrm{E}}_{\mathrm{uw}}\right),\mathrm{I}\left({\mathrm{E}}_{\mathrm{uw}}\right),\mathrm{F}\left({\mathrm{E}}_{\mathrm{uw}}\right)\right)=$(0.6684, 0.1088, 0.3316). For ${\mathrm{E}}_{\mathrm{up}},$ the observed value $({\mathrm{x}}_{\mathrm{observed}}$) = 1.9, and the ideal value (${\mathrm{x}}_{\mathrm{ideal}})$ = 0.6, when using (12).

$$\mathrm{T}\left({\mathrm{E}}_{\mathrm{up}}\right)={\displaystyle \frac{1}{1+\left(\frac{\left|{\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right|}{{\mathrm{x}}_{\mathrm{ideal}}}\right)}}={\displaystyle \frac{1}{1+\left(\frac{\left|1.9-0.6\right|}{0.6}\right)}}=0.3158$$

$$\mathrm{I}\left({\mathrm{E}}_{\mathrm{up}}\right)={\displaystyle \frac{{\left({\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right)}^2}{{({\mathrm{x}}_{\mathrm{observed}}+{\mathrm{x}}_{\mathrm{ideal}})}^2}}={\displaystyle \frac{{\left(1.9-0.6\right)}^2}{{\left(1.9+0.6\right)}^2}}=0.2704$$

$$\mathrm{F}\left(\mathrm{x}\right)=1-\mathrm{T}\left({\mathrm{E}}_{\mathrm{up}}\right)=1-0.3158=0.6842$$

The neutrosophic values for ${\mathrm{E}}_{\mathrm{up}}$ for Kazakhstan in Central Asia ${\mathrm{A}}_{\mathrm{C}}\left({\mathrm{E}}_{\mathrm{up}}\right)$ are $\left(\mathrm{T}\left({\mathrm{E}}_{\mathrm{up}}\right),\mathrm{I}\left({\mathrm{E}}_{\mathrm{up}}\right),\mathrm{F}\left({\mathrm{E}}_{\mathrm{up}}\right)\right)=$(0.3158, 0.2704, 0.6842). For ${\mathrm{E}}_{\mathrm{ap}}$, the observed value $({\mathrm{x}}_{\mathrm{observed}}$) = 82.17, and the ideal value (${\mathrm{x}}_{\mathrm{ideal}})$ = 83.62, when using (12).

$$\mathrm{T}\left({\mathrm{E}}_{\mathrm{ap}}\right)={\displaystyle \frac{1}{1+\left(\frac{\left|{\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right|}{{\mathrm{x}}_{\mathrm{ideal}}}\right)}}={\displaystyle \frac{1}{1+\left(\frac{\left|82.17-83.62\right|}{83.62}\right)}}=0.9830$$

$$\mathrm{I}\left({\mathrm{E}}_{\mathrm{ap}}\right)={\displaystyle \frac{{\left({\mathrm{x}}_{\mathrm{observed}}-{\mathrm{x}}_{\mathrm{ideal}}\right)}^2}{{({\mathrm{x}}_{\mathrm{observed}}+{\mathrm{x}}_{\mathrm{ideal}})}^2}}={\displaystyle \frac{{\left(82.17-83.62\right)}^2}{{\left(82.17+83.62\right)}^2}}=0.0001$$

$$\mathrm{F}\left(\mathrm{x}\right)=1-\mathrm{T}\left({\mathrm{E}}_{\mathrm{ap}}\right)=1-0.3158=0.0170$$

The neutrosophic values for ${\mathrm{E}}_{\mathrm{ap}}$ for Kazakhstan in Central Asia ${\mathrm{A}}_{\mathrm{C}}\left({\mathrm{E}}_{\mathrm{ap}}\right)$ are $\left(\mathrm{T}\left({\mathrm{E}}_{\mathrm{ap}}\right),\mathrm{I}\left({\mathrm{E}}_{\mathrm{ap}}\right),\mathrm{F}\left({\mathrm{E}}_{\mathrm{ap}}\right)\right)=$(0.9830,0.0001,0.0170). Similarly, the calculated neutrosophic values are presented for all sub-regions in the following

Table 5. Tabular neutrosophic values for Central Asia.

Country/NS Value	${\mathbf{A}}_{\mathbf{C}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{C}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{C}}({\mathbf{E}}_{\mathit{a}\mathit{p}}$)
Kazakhstan	(0.6684, 0.1088, 0.3316)	(0.3158, 0.2704, 0.6842)	(0.9830, 0.0001, 0.0170)
Kyrgyzstan	(0.7345, 0.0723, 0.2655)	(0.4000, 0.2025, 0.6000)	(0.4031, 0.2031, 0.5969)
Tajikistan	(0.4132, 0.2553, 0.5868)	(0.6000, 0.0400, 0.4000)	(0.2935, 0.3529, 0.7065)
Turkmenistan	(0.7429, 0.0676, 0.2571)	(1.0000, 0.0000, 0.0000)	(0.9448, 0.0013, 0.0552)
Uzbekistan	(0.6860, 0.0940, 0.3140)	(0.4286, 0.1800, 0.5714)	(0.3517, 0.2674, 0.6483)

,

Table 6. Tabular neutrosophic values for East Asia.

Country/NS Value	${\mathbf{A}}_{\mathbf{E}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{E}}({\mathbf{E}}_{\mathit{u}\mathit{p}}$)	${\mathbf{A}}_{\mathbf{E}}({\mathbf{E}}_{\mathit{a}\mathit{p}})$
China	(0.6048, 0.2356, 0.3952)	(0.3333, 0.2500, 0.6667)	(0.8974, 0.0029, 0.1026)
Democratic People’s Republic of Korea	(0.7384, 0.0464, 0.2616)	(0.4286, 0.1600, 0.5714)	(0.3924, 0.1904, 0.6076)
Japan	(0.7560, 0.0193, 0.2440)	(0.6000, 0.2500, 0.4000)	(0.5379, 0.5668, 0.4621)
Mongolia	(0.6684, 0.1088, 0.3316)	(0.2143, 0.4187, 0.7857)	(0.3882, 0.1942, 0.6118)
Republic of Korea	(0.8467, 0.0099, 0.1533)	(0.6000, 0.2500, 0.4000)	(0.5637, 0.3987, 0.4363)

,

Table 7. Tabular neutrosophic values for South Asia.

Country/NS Value	${\mathbf{A}}_{\mathbf{S}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{S}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{S}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Afghanistan	(0.3825, 0.1995, 0.6175)	(0.6000, 0.0625, 0.4000)	(0.3107, 0.2765, 0.6893)
Bangladesh	(0.3489, 0.2330, 0.6511)	(0.6667, 0.1111, 0.3333)	(0.5795, 0.0709, 0.4205)
Bhutan	(0.4045, 0.1798, 0.5955)	(0.6000, 0.2500, 0.4000)	(0.9536, 0.0006, 0.0464)
India	(0.1745, 0.4941, 0.8255)	(0.6667, 0.1111, 0.3333)	(0.5930, 0.0653, 0.4070)
Iran (Islamic Republic of)	(0.6480, 0.1391, 0.3520)	(0.6000, 0.0625, 0.4000)	(0.7652, 0.0328, 0.2348)
Maldives	(0.6106, 0.2192, 0.3894)	(0.5000, 1.0000, 0.5000)	(0.6173, 0.2019, 0.3827)
Nepal	(0.3567, 0.2248, 0.6433)	(0.3529, 0.2287, 0.6471)	(0.4658, 0.1328, 0.5342)
Pakistan	(0.1637, 0.5166, 0.8363)	(0.3750, 0.2066, 0.6250)	(0.4297, 0.1591, 0.5703)
Sri Lanka	(0.8038, 0.0193, 0.1962)	(0.7500, 0.0400, 0.2500)	(0.9110, 0.0022, 0.0890)

,

Table 8. Tabular neutrosophic values for Southeast Asia.

Country/NS Value	${\mathbf{A}}_{\mathbf{S}\mathbf{E}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{S}\mathbf{E}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{S}\mathbf{E}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Brunei Darussalam	(0.5773, 0.3337, 0.4227)	(0.5000, 1.0000, 0.5000)	(0.5664, 0.3845, 0.4336)
Cambodia	(0.3713, 0.2102, 0.6287)	(0.8571, 0.0083, 0.1429)	(0.5171, 0.1013, 0.4829)
Indonesia	(0.4019, 0.1820, 0.5981)	(0.6667, 0.1111, 0.3333)	(0.8650, 0.0052, 0.1350)
Lao People’s Democratic Republic	(0.3098, 0.2777, 0.6902)	(1.0000, 0.0000, 0.0000)	(0.4264, 0.1617, 0.5736)
Malaysia	(0.4410, 0.1505, 0.5590)	(0.8571, 0.0059, 0.1429)	(0.9581, 0.0005, 0.0419)
Myanmar	(0.4922, 0.1158, 0.5078)	(0.4615, 0.1357, 0.5385)	(0.4698, 0.1301, 0.5302)
Philippines	(0.3757, 0.2059, 0.6243)	(0.6000, 0.2500, 0.4000)	(0.4087, 0.1762, 0.5913)
Singapore	(0.7471, 0.0210, 0.2529)	(0.5000, 1.0000, 0.5000)	(0.5814, 0.3163, 0.4186)
Thailand	(0.5381, 0.0902, 0.4619)	(0.6000, 0.2500, 0.4000)	(0.6905, 0.0834, 0.3095)
Timor-Leste	(0.3113, 0.2759, 0.6887)	(0.7500, 0.0400, 0.2500)	(0.4479, 0.1454, 0.5521)
Vietnam	(0.9203, 0.0017, 0.0797)	(0.6667, 0.0400, 0.3333)	(0.8274, 0.0136, 0.1726)

and

Table 9. Tabular neutrosophic values for Western Asia.

Country/NS Value	${\mathbf{A}}_{\mathbf{W}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{W}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{W}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Armenia	(0.9203, 0.0020, 0.0797)	(0.8571, 0.0059, 0.1429)	(0.8992, 0.0035, 0.1008)
Azerbaijan	(0.6978, 0.0764, 0.3022)	(0.6667, 0.0400, 0.3333)	(0.6860, 0.0347, 0.3140)
Bahrain	(0.5336, 0.6025, 0.4664)	(0.6667, 0.1111, 0.3333)	(0.8439, 0.0104, 0.1561)
Cyprus	(0.5721, 0.3570, 0.4279)	(0.6667, 0.1111, 0.3333)	(0.5520, 0.4663, 0.4480)
Georgia	(0.6755, 0.0999, 0.3245)	(1.0000, 0.0000, 0.0000)	(0.9045, 0.0025, 0.0955)
Iraq	(0.7651, 0.0329, 0.2349)	(0.6000, 0.2500, 0.4000)	(0.9247, 0.0015, 0.0753)
Israel	(0.5935, 0.2714, 0.4065)	(0.5000, 1.0000, 0.5000)	(0.5497, 0.4813, 0.4503)
Jordan	(0.5880, 0.2909, 0.4120)	(0.8571, 0.0083, 0.1429)	(0.6511, 0.1340, 0.3489)
Kuwait	(0.5336, 0.6025, 0.4664)	(0.7500, 0.0400, 0.2500)	(0.6855, 0.0886, 0.3145)
Oman	(0.5721, 0.3570, 0.4279)	(0.6667, 0.0400, 0.3333)	(0.8033, 0.0119, 0.1967)
Qatar	(0.5163, 0.7770, 0.4837)	(0.6667, 0.1111, 0.3333)	(0.9208, 0.0017, 0.0792)
Saudi Arabia	(0.5880, 0.2909, 0.4120)	(0.7500, 0.0204, 0.2500)	(0.9172, 0.0019, 0.0828)
Syrian Arab Republic	(0.6978, 0.0317, 0.3022)	(1.0000, 0.0000, 0.0000)	(0.8704, 0.0048, 0.1296)
United Arab Emirates	(0.5336, 0.6025, 0.4664)	(0.7500, 0.0400, 0.2500)	(0.8603, 0.0078, 0.1397)
Yemen	(0.4071, 0.1776, 0.5929)	(0.3333, 0.2500, 0.6667)	(0.4390, 0.1520, 0.5610)

.

Table 10. Tabular neutrosophic values for ideal case.

Country/NS Value	${\mathbf{I}}_{\mathbf{d}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{I}}_{\mathbf{d}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{I}}_{\mathbf{d}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Ideal	(1, 0.1, 0)	(1, 0.1, 0)	(1, 0.1, 0)

presents the tabular neutrosophic values for the ideal case. These values serve as a benchmark in evaluating the performance of different regions concerning key parameters, including unsafe water sanitation (${\mathrm{E}}_{\mathrm{uw}}$), unintentional poisoning (${\mathrm{E}}_{\mathrm{up}})$, and air pollution (${\mathrm{E}}_{\mathrm{ap}})$.

4.1.2. Distance and Similarity Calculation and Results

We present the calculation for the parameter ${\mathrm{A}}_{\mathrm{C}}\left({\mathrm{E}}_{\mathrm{uw}}\right)$ for Kazakhstan. Ideal values: (1, 0.1, 0), country neutrosophic values for Kazakhstan: (0.6684, 0.1088, 0.3316), and assumed weights: (0.4, 0.3, 0.3). Maximum weighted difference for ${\mathrm{E}}_{\mathrm{uw}}$= $\max (\left(0.4\right).$ (|1–0.6684|), (0.3). |0.1–0.1088|), ((0.3). |0–0.3316|), = max (0.4⋅0.3316, 0.3⋅0.0088, 0.3⋅0.3316), = max (0.13264, 0.00264, 0.09948) = max (0.13264, 0.00264, 0.09948) = 0.13264. In addition, the maximum weighted difference for ${\mathrm{E}}_{\mathrm{up}}$ = max ((0.4) ⋅∣1−0.3158∣, (0.3) ⋅∣0.1−0.2704∣, (0.3) ⋅ ∣0−0.6842∣), = max (0.4⋅0.6842, 0.3⋅0.1704, 0.3⋅0.6842), = max (0.27368, 0.05112, 0.20526), and = max (0.27368, 0.05112, 0.20526) = 0.27368. Similarly, the maximum weighted difference for ${\mathrm{E}}_{\mathrm{ap}}$ = 0.02997. Similarly, the maximum weighted differences for each country using the suggested weighted values (0.4, 0.3, 0.3) for truth, indeterminacy, and falsity are presented in the following

Table 11. Maximum weighted differences for Central Asia.

Country/Max Weighted Difference	${\mathbf{A}}_{\mathbf{C}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{C}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{C}}({\mathbf{E}}_{\mathit{a}\mathit{p}}$)
Kazakhstan	0.13264	0.27368	0.02997
Kyrgyzstan	0.10620	0.24000	0.23876
Tajikistan	0.23472	0.16000	0.28260
Turkmenistan	0.10284	0.03000	0.02961
Uzbekistan	0.12560	0.22856	0.25932

. In addition, the maximum weighted differences between each country and the ideal for all the remaining Asian sub-regions are presented in

Table 12. Maximum weighted differences for East Asia.

Country/Max Weighted Difference	${\mathbf{A}}_{\mathbf{E}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{E}}({\mathbf{E}}_{\mathit{u}\mathit{p}}$)	${\mathbf{A}}_{\mathbf{E}}({\mathbf{E}}_{\mathit{a}\mathit{p}})$
China	0.15808	0.26668	0.04104
Democratic People’s Republic of Korea	0.10464	0.22856	0.24304
Japan	0.09760	0.16000	0.18484
Mongolia	0.13264	0.31428	0.24472
Republic of Korea	0.06132	0.16000	0.17452

,

Table 13. Maximum weighted differences for South Asia.

Country/Max Weighted Difference	${\mathbf{A}}_{\mathbf{S}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{S}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{S}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Afghanistan	0.24600	0.27000	0.20679
Bangladesh	0.26044	0.28000	0.17385
Bhutan	0.23820	0.30000	0.28608
India	0.33020	0.28000	0.17790
Iran (Islamic Republic of)	0.14080	0.27000	0.22956
Maldives	0.15624	0.30000	0.18519
Nepal	0.25852	0.16437	0.19926
Pakistan	0.33548	0.18750	0.17109
Sri Lanka	0.07848	0.19500	0.27330

,

Table 14. Maximum weighted differences for Southeast Asia.

Country/Max Weighted Difference	${\mathbf{A}}_{\mathbf{S}\mathbf{E}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{S}\mathbf{E}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{S}\mathbf{E}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Brunei Darussalam	0.16908	0.27000	0.16992
Cambodia	0.24852	0.22856	0.15513
Indonesia	0.23924	0.28000	0.25950
Lao People’s Democratic Republic	0.27612	0.30000	0.17112
Malaysia	0.22360	0.22856	0.28743
Myanmar	0.20328	0.16155	0.15906
Philippines	0.25092	0.27000	0.17739
Singapore	0.10116	0.27000	0.17442
Thailand	0.18476	0.27000	0.20715
Timor-Leste	0.27628	0.19500	0.16863
Vietnam	0.03188	0.20000	0.24822

and

Table 15. Maximum weighted differences for West Asia.

Country/Max Weighted Difference	${\mathbf{A}}_{\mathbf{W}}\left({\mathbf{E}}_{\mathit{u}\mathit{w}}\right)$	${\mathbf{A}}_{\mathbf{W}}\left({\mathbf{E}}_{\mathit{u}\mathit{p}}\right)$	${\mathbf{A}}_{\mathbf{W}}\left({\mathbf{E}}_{\mathit{a}\mathit{p}}\right)$
Armenia	0.03188	0.22856	0.26976
Azerbaijan	0.12168	0.24000	0.20580
Bahrain	0.18696	0.28000	0.25317
Cyprus	0.17116	0.28000	0.22400
Georgia	0.12980	0.30000	0.27135
Iraq	0.09396	0.27000	0.27741
Israel	0.16260	0.27000	0.24015
Jordan	0.16480	0.22856	0.16533
Kuwait	0.18696	0.19500	0.20565
Oman	0.17116	0.24000	0.24099
Qatar	0.19348	0.28000	0.27624
Saudi Arabia	0.16480	0.22500	0.27516
Syrian Arab Republic	0.12168	0.30000	0.26112
United Arab Emirates	0.18696	0.19500	0.25809
Yemen	0.23716	0.24000	0.22440

.

The distance between Central Asia and the ideal Equation (1) and from Table 11 is $m=3$ (number of parameters), $n=5$ (number of countries), ${\mathrm{D}}_{\mathrm{w}}\left({\mathrm{I}}_{\mathrm{d}},{\mathrm{A}}_{\mathrm{C}}\right)={\displaystyle \frac{1}{15}}{\displaystyle \sum }_{\mathrm{j}=1}^3{\displaystyle \sum }_{\mathrm{i}=1}^5\max \left\{\begin{array}{c}0.4.\left|{\mathrm{T}}_{{\mathrm{I}}_{\mathrm{d}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{{\mathrm{A}}_{\mathrm{C}}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ 0.3.\left|{\mathrm{I}}_{{\mathrm{I}}_{\mathrm{d}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{{\mathrm{A}}_{\mathrm{C}}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,0.3.\left|{\mathrm{F}}_{{\mathrm{I}}_{\mathrm{d}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{{\mathrm{A}}_{\mathrm{C}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}$ = ${\displaystyle \frac{1}{15}}[$ 0.13264 + 0.27368 + 0.02997 + 0.10620 + 0.24000 + 0.23876 + 0.23472 + 0.16000 + 0.28260 + 0.10284 + 0.03000 + 0.02961 + 0.12560 + 0.22856 + 0.25932] $=0.0702$.

The weighted distance between the ideal case ($I_d)$ and Central Asia ($A_C)$ = 0.0702. Equation (2), The similarity measure between Central Asia and the ideal case $S{M}_W\left(I_d,A_C\right)=1-D_w\left(I_d,A_C\right)$ $=1-0.07020$ $=0.9298$. The distance between East Asia and the ideal from Table 12 is $m=3$ (number of parameters), $n=5$ (number of countries) ${\mathrm{D}}_{\mathrm{w}}\left({\mathrm{I}}_{\mathrm{d}},{\mathrm{A}}_{\mathrm{E}}\right)={\displaystyle \frac{1}{15}}{\displaystyle \sum }_{\mathrm{j}=1}^3{\displaystyle \sum }_{\mathrm{i}=1}^5\max \left\{\begin{array}{c}0.4.\left|{\mathrm{T}}_{{\mathrm{I}}_{\mathrm{d}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{T}}_{{\mathrm{A}}_{\mathrm{C}}}\left({\mathrm{e}}_{\mathrm{j}}\right){\mathrm{x}}_{\mathrm{i}}\right|, \\ 0.3.\left|{\mathrm{I}}_{{\mathrm{I}}_{\mathrm{d}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{I}}_{{\mathrm{A}}_{\mathrm{C}}}\left({\mathrm{e}}_{\mathrm{j}}\right)\left({\mathrm{x}}_{\mathrm{i}}\right)\right|,0.3.\left|{\mathrm{F}}_{{\mathrm{I}}_{\mathrm{d}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})-{\mathrm{F}}_{{\mathrm{A}}_{\mathrm{C}}}\left({\mathrm{e}}_{\mathrm{j}}\right)({\mathrm{x}}_{\mathrm{i}})\right|\end{array}\right\}$ $=\left({\displaystyle \frac{1}{15}}\right) \left(0.15808+0.26668+0.04104+0.10464+0.22856+0.24304+0.09760+0.16000+0.18484+0.13264+0.31428+0.24472+0.06132+0.16000+0.17452\right)=\left({\displaystyle \frac{1}{15}}\right) 2.57196$ $=0.17146$.

The similarity measure between East Asia and the ideal case ${\mathrm{SM}}_{\mathrm{W}}\left({\mathrm{I}}_{\mathrm{d}},{\mathrm{A}}_{\mathrm{E}}\right)=1-{\mathrm{D}}_{\mathrm{w}}\left({\mathrm{I}}_{\mathrm{d}},{\mathrm{A}}_{\mathrm{E}}\right)$ $=1-0.07020$ $=0.9298$. Similarly, the calculated values for the weighted similarity measures between each Asian sub-region and the ideal are presented in

Table 16. Calculated weighted similarity measures using neutrosophic values.

Similarity Between Ideal and Sub-Regions of Asia	Similarity Value
$S{M}_w\left(I_d,A_C\right)$	0.9298
$S{M}_w\left(I_d,A_E\right)$	0.8254
$S{M}_w\left(I_d,A_S\right)$	0.8010
$S{M}_w\left(I_d,A_{SE}\right)$	0.8170
$S{M}_w\left(I_d,A_W\right)$	0.7980

. Figure 2 represents the geographic distribution of the weighted similarity measures between the ideal and the sub-regions of Asia. Darker blue shades indicate higher similarity values, while lighter shades represent lower similarity.

The similarity measures in Table 16 provide a detailed analysis of how closely different geographic regions align with the ideal standard, using the weighted Hausdorff distance method. This method evaluates three key membership values: truth (T), which represents the presence of desirable attributes; indeterminacy (I), indicating uncertainty or ambiguity; and falsity (F), representing undesirable characteristics. These parameters are assessed across important indicators, such as environmental quality and health outcomes, with the scores ranging from 0 to 1. A score closer to 1 signifies stronger similarity to the ideal case. Among the regions analyzed, Central Asia (${\mathrm{SM}}_{\mathrm{w}}$ = 0.9298) demonstrates the highest similarity to the ideal region. This indicates that the region’s parameters, such as its reduced falsity and high truth membership values, are closely aligned with the benchmarks of excellence. East Asia (${\mathrm{SM}}_{\mathrm{w}}$ = 0.8254) and Southeast Asia (${\mathrm{SM}}_{\mathrm{w}}$ = 0.8170) show moderate alignment, reflecting some strengths but also areas that require improvement. Similarly, South Asia (${\mathrm{SM}}_{\mathrm{w}}$ = 0.8010) exhibits a lower but still moderate degree of alignment, suggesting notable gaps in parameters such as environmental safety or healthcare outcomes. West Asia (${\mathrm{SM}}_{\mathrm{w}}$ = 0.7980), however, records the lowest similarity score, highlighting the need for significant improvement in aligning its metrics with the ideal benchmark.

4.1.3. Sensitivity Analysis of Weight Selection

To demonstrate the impact of the chosen weights (0.4, 0.3, 0.3) on the results, a sensitivity analysis was performed. This process involved modifying the weight distribution and observing the resulting variations in the similarity values. The purpose of this analysis was to validate whether the initial weight configuration offered a fair and representative interpretation of the data. The baseline weights were truth membership (T)—0.4, indeterminacy membership (I)—0.3, and falsity membership (F)—0.3. In addition, alternative weight scenarios were considered to test the robustness of the outcomes: scenario 1—an increased emphasis on truth membership (0.5, 0.25, 0.25) and scenario 2—equal emphasis on truth and indeterminacy and reduced falsity (0.4, 0.4, 0.2). For each case, the similarity measures were recalculated using the weighted Hausdorff distance technique. These adjusted results were then compared with those from the baseline to determine how much the final similarity scores were influenced by weight variations. The results of this analysis are summarized in

Table 17. Sensitivity analysis of weighted similarity measures.

Similarity	Baseline Weight Scenario (0.4, 0.3, 0.3)	Scenario 1 (0.5, 0.25, 0.25)	Scenario 2 (0.4, 0.4, 0.2)	Rank
$S{M}_w\left(I_d,A_C\right)$	0.9298	0.9350	0.9280	1
$S{M}_w\left(I_d,A_E\right)$	0.8254	0.8302	0.8240	2
$S{M}_w\left(I_d,A_S\right)$	0.8010	0.8055	0.8000	3
$S{M}_w\left(I_d,A_{SE}\right)$	0.8170	0.8215	0.8160	4
$S{M}_w\left(I_d,A_W\right)$	0.7980	0.8025	0.7970	5

.

The sensitivity analysis demonstrates that the initial weight settings (0.4 for truth, 0.3 for indeterminacy, and 0.3 for falsity) offer a balanced and accurate reflection of the data. Despite minor variations in the similarity scores when adjusting the weights, the overall ranking of the regions and their relative positions remained largely stable. This stability suggests that, in the case of different weight combinations, the method is not overly sensitive to such changes. This reinforces the confidence in the validity of the approach, as the overall findings remain consistent even with slight alterations in the weighting priorities. Although the outcome remains unchanged, it may differ when different study data are used, being contingent upon the specifics of the data. This versatility guarantees that the methodology is applicable to many datasets and circumstances, yielding dependable and robust results across multiple study scenarios. The following Figure 3 presents the sensitivity of the weighted similarity across Asian sub-regions under different weighting scenarios.

5. Discussion of Results

Previous studies on similarity-based assessments exhibited certain limitations. Some were domain-specific, particularly in medical diagnosis, and lacked the consideration of weighted measures, thereby failing to capture the varying importance of different indicators. Others emphasized the need for robust analytical frameworks but did not fully address the complexity of regional sustainability assessments or effectively manage uncertainty in the data. To overcome these limitations, the present study introduces an enhanced neutrosophic similarity measure approach tailored to sustainable development evaluation. This method integrates weighted measures derived from the Hausdorff distance to reflect the relative significance of sustainability indicators. A comprehensive dataset comprising 50 countries across Asia is utilized to evaluate regional performance against ideal benchmarks. The proposed framework enables a more accurate and flexible assessment of sustainability performance, providing actionable insights for policymakers to address regional disparities and promote balanced sustainable development.

This study analyzes regional differences in health and environmental outcomes across 50 countries within five sub-regions of Asia. The dataset utilized incorporates essential sustainability indicators, such as death rates associated with unsafe water, inadequate sanitation, air pollution, and accidental poisoning. This report offers a thorough evaluation of regional advancements toward the SDGs by examining a wide range of socioeconomic and environmental factors. This study presents an important improvement in methods by using weighted neutrosophic similarity measures based on the Hausdorff metric, which allows different levels of importance to be assigned to the indicators. This methodology overcomes the shortcomings of previous models by providing a more precise and contextually aware assessment of regional performance. The results correspond with prior studies while offering more comprehensive insights into regional performance discrepancies. The utilization of neutrosophic similarity measurements, previously shown to be successful in domains like medical diagnostics and pattern recognition, facilitates a more refined evaluation of sustainability advancements. Central Asia has emerged as the region most closely aligned with the optimal benchmark, with superior outcomes in health and environmental indices. East Asia and Southeast Asia displayed modest alignment with the standard, but South Asia and West Asia revealed significant disparities, primarily attributable to ongoing issues such as insufficient WASH services and elevated air pollution levels. These disparities underscore the necessity of focused, region-specific policy actions to successfully address regional development concerns. This study emphasizes the necessity of customizing policy measures to the distinct requirements of each region to foster balanced and equitable growth. Central Asia would benefit from enhancing access to potable water and refining its purification systems. East Asia must prioritize improvements in air quality monitoring and recycling initiatives, whereas South Asia requires investment in flood management systems and marine conservation zones. Southeast Asia requires more stringent restrictions to combat illegal logging and enhanced activities for marine protection. Advancing renewable energy initiatives and desalination technology in West Asia is crucial. These focused solutions can close existing gaps, enhance resource allocation, and bolster the overall efficacy of sustainable development activities throughout the Asian sub-regions.

With regard to health and environmental issues in the Asian area, there are few studies using similarity measures, particularly to assess indicators related to SDG 3.9. Although earlier research has successfully applied neutrosophic similarity measures in technical fields such as medical diagnosis and decision-making [25,29,35], their use in sustainability assessment is still lacking. The current work addresses this gap by introducing a weighted neutrosophic similarity measure based on the Hausdorff metric, which allows for the management of data uncertainty and the assignment of different levels of importance to indicators. This method provides a more practical and sophisticated framework for the assessment of regional variations in performance regarding SDG 3.9 across several Asian environments. After applying this technique, this research shows notable variations in health and environmental results across different areas, with South Asia and West Asia emerging as the most vulnerable. These results are consistent with those of Md. Sujahangir Doyel Sarkar and Sharmistha [49], who found high death rates in South Asia owing to issues connected to insufficient WASH services, exposure to air pollution, and unintentional poisoning. Although their study mostly focused on specific countries rather than a sub-regional analysis, it underlines important health issues in countries including India, Bangladesh, and Nepal. Building on these data, our work expands the analysis to a larger regional scale, spanning 50 nations throughout five sub-regions of Asia, therefore providing more complete knowledge of the existing differences. Moreover, unlike the worldwide gains described in the WHO [50] statistics, which show a decrease in deaths due to hazardous water, sanitation, air pollution, and poisoning, our results reveal that such advances are not uniform throughout Asia. Areas like South Asia and West Asia still show higher death rates connected to environmental and health concerns. Differences in infrastructure development and regional policy enforcement, as well as uneven policy implementation, may explain much of this uneven growth [51,52]. Thus, achieving SDG 3.9 requires rapid, region-specific actions that address the particular issues of each area, therefore guaranteeing balanced growth and supporting fair sustainable development across Asia.

6. Conclusions

This research presents a novel weighted neutrosophic similarity measure derived from the Hausdorff metric to evaluate regional performance in SD. Utilizing neutrosophic reasoning for sustainability indicators provides a systematic approach to discovering regional differences. An analysis of data from 50 nations across five Asian sub-regions reveals substantial disparities, offering critical insights for policymakers. This technique theoretically improves mathematical models of SD and illustrates the actual use of neutrosophic logic in real-world contexts. The results substantiate focused initiatives to tackle environmental issues and enhance public health infrastructure. Policymakers may use this information to formulate region-specific policies and distribute resources more efficiently.

The results of this analysis provide a valuable framework for policymakers and stakeholders to identify strengths and weaknesses in regional performance. Central Asia’s high similarity score indicates that it can serve as a model region, showcasing effective strategies and policy implementations that others can emulate. Regions like East Asia and Southeast Asia, while performing reasonably well, may need to focus on targeted interventions to address specific areas of indeterminacy or falsity. On the other hand, South Asia and West Asia, with lower similarity scores, require urgent attention to bridge the gaps in their performance. Key areas for these regions may include improving healthcare systems, addressing environmental challenges, and ensuring data accuracy and clarity to reduce uncertainties (indeterminacy). For example, investments in healthcare infrastructure, initiatives to combat pollution, and policies aimed at SD could significantly enhance their alignment with the ideal benchmark.

Regions with lower similarity scores, including South Asia and West Asia, must prioritize focused interventions that address important gaps in order to increase their alignment with the ideal. Improving maternal and child health outcomes, managing communicable and non-communicable diseases, and expanding access to high-quality healthcare are all critical components in strengthening public health systems. With programs designed to minimize pollution, guarantee access to clean water and sanitation, and lessen the effects of climate change, efforts to mitigate environmental risks are equally important. Reducing indeterminacy also requires increasing data openness, which can be accomplished by limiting the ambiguity in measurements that guide policy decisions, ensuring accurate reporting, and conducting thorough data gathering. Lastly, SD can be promoted by policy-driven initiatives that are adapted to local needs, such as improved urban planning, the promotion of green technologies, and increased public knowledge of environmental and health issues. Regions can improve their performance, close disparities, and move closer to attaining equitable SD that meets the ideal standard by putting these strategies into practice. The similarity measure analysis not only highlights the relative performance of regions but also serves as a diagnostic tool for targeted decision-making. By focusing on identified gaps and leveraging the strengths of high-performing regions, policymakers can drive meaningful progress, ultimately aligning regional performance with the ideal standards of health, environment, and socioeconomic well-being. This holistic approach will ensure that all regions move closer to achieving sustainable and equitable development.

The data used in this research were derived from 2019, being the most recent data available for the selected SDG indicators on the WHO platform when the study began. We selected these indicators primarily due to their accessibility and completeness. Although newer data would improve the study’s relevance, they were not available at the time. However, this study is one of the first to apply the neutrosophic approach to these indicators, offering novelty. The use of older data has its drawbacks, as they may not reflect the current trends or challenges, but they still help to establish a baseline for future work. Standard reporting cycles and data validation processes can occasionally limit access to the most recent datasets. Future studies may enhance this work by incorporating updated data as they become publicly available. Secondly, this study only used a few indicators, such as deaths from unsafe WASH, air pollution, and poisoning. These are important but are not sufficient to provide a full picture of SD. In the future, more indicators, such as those related to the economy, climate resilience, and infrastructure, should be added to obtain better results. Thirdly, this study only looked at five Asian sub-regions. To better understand the situation worldwide, future research should also include other areas, like Africa, Europe, and the Americas. This would enable a global analysis and help in devising region-specific solutions. Finally, this method depends significantly on the data. If data are missing or not accurate, the results might not be reliable. Thus, future studies should ensure strong data collection and checking to improve the trust in the outcomes.

To achieve a more thorough evaluation of regional sustainability, future studies should investigate other indicators, such as socioeconomic determinants, climatic resilience measurements, and infrastructure development indices. Real-time decision-making can be facilitated and the forecast accuracy increased by using sophisticated machine learning models, such as AI and deep learning approaches. Expanding the scope of analysis beyond Asia to include Africa, Europe, and the Americas would offer a broader perspective on global sustainability challenges and inform region-specific strategies. Additionally, assessing the effectiveness of policy interventions over time through empirical studies would help to determine their impacts on regional performance. By implementing these recommendations, policymakers and researchers can build upon this study’s findings to drive meaningful progress in achieving sustainable and equitable development worldwide. In addition, future research should aim to use more recent datasets so that the findings better reflect present-day developments in SD. Since this study used data from 2019, newer statistics could help to demonstrate the current regional differences more clearly. It would also be useful to include a wider set of indicators, covering areas like economic stability, access to education, infrastructure, and social inclusion, to build a more complete understanding of regional progress. It might also be helpful to look beyond the five Asian sub-regions. Including other parts of the world, such as Africa, Europe, or Latin America, could offer useful comparisons and reveal patterns or issues that are unique to each region. This could support more focused and region-specific planning. There is potential in using more advanced analysis techniques. Tools like machine learning could help to uncover patterns that might not be obvious through standard methods. These could improve predictions and support planning. Lastly, studies that look at changes over time would help to assess the implementation of policies and whether progress is being made.