Fuzzy graph based machine learning optimization for permeable pavement systems in smart cities of Thoothukudi

Abstract

Frequent hydrometeorological variations have increased the frequency of flooding, posing major risks to infrastructure and public safety. Fuzzy graph theory relies on fundamental concepts such as synchronising and graph domination problems, which have enormous applications in everyday life. In this study, we introduce a novel optimization framework that integrates machine learning with Bipolar Intuitionistic Fuzzy Graphs (BIFG) to optimize stormwater management through permeable pavement systems in the smart cities of Thoothukudi district, India. Bipolar Intuitionistic Fuzzy Graph based Machine Learning Optimization (BIFG-MLO) is used to characterise the ambiguous interactions between metropolitan zones affected by heavy rains, using both positive and negative membership values to indicate drainage effectiveness and waterlogging intensity. Graph dominant concepts and efficient edge detection are used to identify key zones that require infrastructure improvement. Using predictive analytics and a multi-criteria optimization method that ranks smart cities based on dominance data, permeable pavement installation can be prioritised. The results indicate that P&T Colony and Kovilpatti are the most vulnerable cities, making them exemplary choices for implementing permeable pavement. Based on performance evaluation parameters such as accuracy, precision, recall, and F1 score, the proposed model outperforms traditional methods in forecasting flood-prone areas. To improve waterlogging prediction and strategically install permeable pavement systems in Thoothukudi, India’s smart city efforts, this study introduces a hybrid optimization framework that employs Bipolar Intuitionistic Fuzzy Graph with Machine Learning Optimization (BIFG-MLO). This framework offers a scalable decision-support system for sustainable urban development in flood-prone locations.

Introduction

Flooding is the deadliest natural disaster in India and accounts for over 40% of all fatalities caused by catastrophic weather events. River channels lose part of their carrying capacity due to erosion and silting of the river beds, which exacerbates flooding. Thoothukudi Smart City focuses on pathways, access to water, garbage disposal, and a strong IT infrastructure in order to keep the city healthy and prosperous. On December 17, 2023, floods in Thoothukudi had recorded 950 mm, or approximately a metre, of rain, devastating communities, killing several people, and departing many more stranded on the tops of buildings, relying on meals that were dropped by air and waiting for deliverance. Despite the fact that the Thoothukudi smart cities flood mitigation strategy was introduced on May 31, 2022, to lessen water stagnation, inadequate project planning results in failure. There is a lot of unpredictability in flood catastrophe management. Decision-making and disaster management are just a few of the applications that use bipolar intuitionistic fuzzy graphs (BIFGs) with predictive analytics to explain imprecise or uncertain data. Bipolar intuitionistic fuzzy graphs (BIFG), a more sophisticated version of fuzzy graphs, can be essential for integrating ambiguous or speculative data in an assortment of applications. Such notions may be analysed ambiguously when there is more unpredictability regarding the edges and vertices of a crisp graph. Machine learning (ML) and fuzzy graph theory serve as strong tools for water management strategy modelling and optimization. City planners can determine which locations are most susceptible to waterlogging by using fuzzy graphs to describe smart cities, where vertices represent urban zones and edges capture relationships like topography, drainage connectivity, and water accumulation. In real-world situations where environmental data may be ambiguous or lacking, fuzzy logic’s ability to handle ambiguity and partial truth is essential. Domination in Fuzzy Graphs with optimized machine learning has been suggested to be an ideal method for detecting the relationship between the contributing elements in a complicated framework. Bipolar Intuitionistic Fuzzy Graph with Machine Learning Optimization (BIFG-MLO) has significant implications for surveillance systems, particularly when accuracy and precision measurements are employed to evaluate performance. Some innovative techniques have been highlighted to describe the aforementioned concepts if the relevant edges are missing.

Graphs are used to show relationships between different objects. Fuzzy graphs were introduced to incorporate fuzziness into such relations. Currently, graphs are being used for knowledge representation across a wide range of application development. To investigate the barriers in designing distinguishable ambiguous graph concepts, Alnaser et al.¹ focused on fuzzy sets and the connections between them. To determine how to calculate the edge domination number, Selvam et al.² present the idea of strong edge dominance in fuzzy. In their overview of this paper, Bozhenyuk et al.³ sought to elucidate the notions of a dominance set as an integral of the IFG and a minimal intuitionistic dominating vertex subset of an IFG. To determine the dominating vertex edge set in bipolar fuzzy graphs, Uma Mageshwari et al.⁴ exploited strong edges. Nazeer et al.⁵ put forward an entirely novel idea for fuzzy incidence graphs’ (FIGs’) precise dominance. Noppakaew et al.⁶ have investigated the concept of dominance in a picture fuzzy graph. This idea was put forth with the help of certain essential concepts explored in this manuscript, including weak, strong, and strongest image fuzzy influence pairings. To maximise the overall success of creating sustainable economic growth, Rehman et al.⁷ developed an innovative optimization technique to describe and resolve facility point issues by implementing the Clique Covering application of FIGs for an entrepreneurial cooperation. Deva⁸ noted that the most and least vulnerable parts of Chennai are located in flood-risk areas, which are determined using a BIFG-based decision-making technique. By establishing novel graph measurements and attributes that apply traditional dominance ideas to multi-valued uncertainty models, Waheed et al.⁹ revealed the complete domination principle in Bipolar Picture Fuzzy Graphs (BPPFGs). The terms status and architecturally equivalent of bipolar picture fuzzy graphs (BPPFGs) were incorporated in this research to illustrate precisely the domination in bipolar picture fuzzy graphs that can be used to interact with others. Shao et. al.¹⁰ presents a novel kind of ambiguous graph and explain concerning how it may be used in making decisions. Bipolar intuitionistic fuzzy graphs (BIFG) with precisely integrated domination were the subject of pertinent instances presented by Mujeeburahman et al.¹¹. Mahapatra et al.¹² presents a novel method of measuring centrality using a neutrosophic graph that makes use of fuzzy theory. In specifically, the neutrosophic fuzzy theory is employed since it incorporates additional information like true values, falsity, and indeterminacy. In the course of highlighting the special features of these proposed aggregation operators, Nadeem et al.¹³ explore the potential uses of intuitionistic fuzzy sets (IFS) with rough sets in the context of sparse data and look into their use in the multi-attribute decision-making domain. Through the use of the graphical characteristics of interval-valued intuitionistic fuzzy sets (IVIFSs) to address partial ignorance, Misbah Rasheed et al.¹⁴ intend to create a framework for a deeper comprehension of two-way formal concepts and their hierarchical representation in concept lattices. Using the structural features of quasi-divisor graphs, Muhammad Nadeem et al.¹⁵ created a few topological indices based on degree, distance, and degree-distance. In order to figure out this unpredictability, Mahapatra et al.¹⁶ emphasise on applying fuzzy theory, notably linguistic Z-graphs, revealing more thorough insights than fuzzy graphs. Rupkumar et al.¹⁷ offer a novel approach to measuring centrality in networks where relationships between nodes are expressed with uncertainty using linguistic Z-numbers.

Masoud Kayhanian et al.¹⁸ reviewed and compiled the key conclusions of these group investigations about the assessment of clogging in permeable pavement surfaces. In the context of irrigation decision-making, Srinivasa Raju et al.¹⁹ use the concrete technique of fuzzy, data envelope analysis (DEA) to determine the performance metrics such as TPR, FPR, and specificity, particularly when dealing with imprecise or ambiguous data. To enhance the irrigation system, Regulwar et al.²⁰ employ fuzzy multi-objective optimisation models to generate confusion matrices for metric calculation, which are among the accomplished outcomes. The fuzzy optimization model offers a scalable framework for transportation system design and functions well for infrastructure networks with inconsistent or missing data. In order to generate confusion matrices and calculate the pertinent metrics, Yan Liu et al.²¹ emphasise the significance of managing the statistical data. The study demonstrates the integration of evolutionary algorithms with a hybrid adaptive neuro-fuzzy inference system (ANFIS), wherein Ghordoyee et al.²² generate confusion matrices and calculate classification metrics such as accuracy, precision, recall, specificity, FPR, and TPR by comparing the model outputs to ground truth or benchmark values. Mohammed N. Singer et al.²³ discusses several evolving aspects for water quality improvements, innovative investigations that include recycled aggregates, and other lessons learned and future research directions in the area of Permeable Pavement System. Surya Dewi et al.²⁴ reviewed the use of porous asphalt, and their analysis highlighted that these permeable pavements significantly outperform conventional pavements in reducing flooding by allowing better infiltration and lowering peak runoff rates. Kiran Tota et al.²⁵ provide a comprehensive evaluation of permeable pavement systems as sustainable stormwater management solutions within the framework of Sustainable Drainage Systems. Zhang et al.²⁶ provide a thorough analysis of the implementation of optimum performance methods based on fuzzy logic in the planning and construction of metropolitan buildings. According to Chen et al.²⁷, machine learning algorithms greatly improve decision-making by spotting trends, predicting urban dynamics, and facilitating prompt solutions to challenging urban issues. Jones et al.²⁷ concentrate on optimising the design and functionality of permeable pavement systems in urban settings through the application of data-driven modelling methodologies. A comparison of traditional and modern pavement design techniques is presented by Kurniawan et al.²⁹, with a focus on metropolitan areas that are prone to flooding.

The purpose of the paper is to improve the design, implementation, and management of permeable pavement systems in urban environments by creating and utilising an advanced optimization framework based on fuzzy graph-based machine learning techniques. The framework can handle uncertainties and missing data that are frequently present in urban hydrological systems by using fuzzy logic and graph-based machine learning techniques. This results in more precise predictions and better ways to alleviate waterlogging in places. By optimizing the installation of permeable pavement, the study helps to reduce urban floods and the resulting financial losses. Because it incorporates data-driven decision-making under unpredictable conditions, the use of an intelligent optimisation model for porous pavement constitutes a significant leap in urban stormwater management. The initiative is driven by an urgent desire to solve the growing difficulties of floods and waterlogging in smart cities, where impermeable surfaces and population density make them more susceptible to environmental problems.

Using an analogous construction paradigm, it analyses the relationships between various aspects and pinpoints the most notable differences. This study clarifies waterlogging areas, including their sources, impacts, types, and measures to prevent them. When compared to other graphs, bipolar intuitionistic fuzzy graphs have opened up a more effective method for handling ambiguity. As a result, the dominant concepts used in bipolar intuitionistic fuzzy graphs have been addressed in this study. The application of swarm optimization with dominance idea is beneficial when integrating these optimization algorithms to multi-objective optimization (MOO), in which several antagonistic goals must be optimized in tandem, as it is necessary to optimize the distinct contradictions in many real-world problems. The process deals with the topic of catastrophe preparedness, which has lately affected some Thoothukudi district locations. Thus, in order to prevent harm to the city, permeable pavement initiatives aim to improve drainage systems in smart cities by protecting them from floods. It represents smart cities as a bipolar intuitionistic fuzzy network, with each city acting as a vertex and its neighbouring cities acting as edges. The E-edge in the BIFG displays the most precise relationships between the neighbouring cities. By employing fuzzy graph domination problems, a well-designed stormwater harvesting technique can be used to mitigate the effects of storm occurrences and control areas that become swamped due to cyclones and rainfall. The Systems tracking enhances greatly from the application of BIFG-MLO because the technique provides accurate, precise, and reliable monitoring in uncertain and intricate contexts. This work presents a computationally sophisticated, locally-contextual, uncertainty-aware approach that combines machine learning and fuzzy graph theory to optimise permeable pavement design and management in the particular infrastructure and climate of Thoothukudi smart city. In order to achieve more dependable and sustainable urban water management, this strategy would bring the field closer to true smart infrastructure management, which incorporates real, imperfect, and various data sources. This model would contribute in achieving the sustainable development in Thoothukudi’s smart cities by reducing the impact of tropical cyclones.

Among these smart cities, traumatic and torrential rain caused chaos in some of the cities, are given below:

• 1. Ottapidaram

• 2. Sawyerpuram

• 3. Kovilpatti

• 4. Kandasamypuram

• 5. Tuticorin Melur

• 6. P&T Colony

• 7. National Highway 45B

• 8. Nazareth

Preliminaries

Definition 2.1¹

Given a set X, a fuzzy subset µ is a function µ:X → [0,1], which assigns to each element $x \in X$, a membership value $\mu \left(x\right)$ in the interval [0,1]. This represents the degree to which $x$ belongs to the fuzzy subset. A function v: Χ x Χ $\to \left[\text{0,1}\right]$ is a fuzzy relation if it satisfies the condition:

$v\left(x,y\right)= v\left(y,x\right)$ for all $x,y \in X$.

This result indicates that the least of their membership values in the fuzzy subset μ could not be attained by the significance of the relationship between x and y.

Definition 2.2¹

A Bipolar Fuzzy Graph is defined as a triple: $G=(V, A,B)$

where:

V is the set of nodes (vertices).

$A=({\mu }_{A}^{P}, {\mu }_{A}^{N})$ is a bipolar fuzzy set on V, where:

${\mu }_{A}^{P} :V\to [\text{0,1}]$ represents the degree of positive membership of a node.

${\mu }_{A}^{N} :V\to [\text{0,1}]$ represents the degree of negative membership of a node.

$B=({\mu }_{B}^{P}, {\mu }_{B}^{N})$ is a bipolar fuzzy relation on the edge set E ⊂ V × V, where:

${\mu }_{B}^{P}\left(xy\right)$ gives the positive membership of an edge.

${\mu }_{B}^{N}(xy)$ gives the negative membership of an edge.

For any edge $xy\in E$:

The minimum of the positive node membership, ${\mu }_{B}^{P}\left(xy\right)\le \left({\mu }_{A}^{P}\left(x\right),{\mu }_{A}^{P}\left(y\right)\right)$ must not be surpassed by the positive edge membership. At least the greatest number of the negative node memberships has to be incorporated into the negative edge membership: ${\mu }_{B}^{N}(xy) \ge max({\mu }_{A}^{N}\left(x\right),{\mu }_{A}^{N}\left(y\right))$. According to these criteria, an edge cannot be “weaker” than a dominant node on the basis of negativity or “stronger” than the least significant node in regards of positivity. The example for bipolar fuzzy graph is shown in Fig. 1,

Fig. 1

Bipolar fuzzy graph.

Definition 2.3³

An intuitionistic fuzzy graph (IFG) is of the form G = $(V,E)$ where

$V = \left\{ {v_{1} , v_{2} , \ldots v_{n} } \right\}$ such that µ₁$:V \to \left[ {0,1} \right]$ and $\gamma_{1} : V \to \left[ {0,1} \right]$ denote the degree of membership and non-membership of the element $v_{i} \varepsilon V$ respectively and $0 \le$ µ₁ (for every $v_{i} \varepsilon V$ (i = 1, 2 …. n),

$E \subseteq V X V$ where µ₂$:V\to [\text{0,1}]$ and $\gamma_{2} :{ }V \to \left[ {0,1} \right]$ are such that

$${\upmu }_{2} \left( {v_{i} ,v_{j} } \right){ } \le {\upmu }_{1} (v_{i} ){ }.{{\upmu }}_{1} (v_{j} )$$

$$\gamma_{2} \left( {v_{i} ,v_{j} } \right){ } \le \gamma_{1} (v_{i} ){ }.{ }\gamma_{1} (v_{j} )$$

$$0 \le {\upmu }_{2} \left( {v_{i} ,v_{j} } \right){ } + \gamma_{2} \left( {v_{i} ,v_{j} } \right) \le 1$$

for every $\left( {v_{i} ,v_{j} } \right) \in E$ where, means the ordinary multiplication.

Consider the intuitionistic fuzzy network $G = \left( {V, E} \right)$ where the vertex set be $V = \left\{ {v_{1} ,v_{2} ,v_{3} ,v_{4} ,v_{5} ,v_{6} } \right\}$ and the edge set $E = \left\{ {v_{1} v_{2} , v_{2} v_{3} , v_{1} v_{3} ,v_{3} v_{4} , v_{3} v_{5} ,v_{4} v_{5} ,v_{5} v_{6} ,v_{3} v_{6} } \right\}$. The Intuitionistic fuzzy graph of G with S of E is shown below in Fig. 2,

$$S = \{ { }\left( {v_{1} v_{2} ,{ }0.4{ },0.3} \right),\left( {v_{2} v_{3} ,0.4,0.2} \right),\left( {v_{1} v_{3} ,0.5,0.2} \right),\left( {v_{3} v_{4} ,0.5,0.2} \right),\left( {{ }v_{3} v_{5} ,0.3,0.2} \right),\left( {{ }v_{4} v_{5} ,0.3,0.2} \right),\left( {{ }v_{5} v_{6} ,0.5,0.3} \right),\left( {{ }v_{3} v_{6} ,0.2,0.2} \right),$$

Fig. 2

Intuitionistic fuzzy graph.

In this intuitionistic fuzzy graph, each vertex $v_{i}$ represents a specific region or urban zone in the study area. Each vertex is associated with a pair of values $\mu_{1} \left( {v_{i} } \right), \gamma_{1} \left( {v_{i} } \right)$, where $\mu_{1} \left( {v_{i} } \right),$ denotes the degree of effective drainage, and $\gamma_{1} \left( {v_{i} } \right)$ represents the degree of vulnerability or inefficiency. These values satisfy the condition $\mu_{1} \left( {v_{i} } \right) + \gamma_{1} \left( {v_{i} } \right) \le 1$, modeling both certainty and hesitation in urban flood-related characteristics. Similarly, each edge $\left( {v_{i} , v_{j} } \right)$ represents a connection or interaction between two regions such as shared drainage infrastructure or water-flow dependency. Each edge is assigned values $\mu_{2} \left( {v_{i} , v_{j} } \right), \gamma_{2} \left( {v_{i} , v_{j} } \right)$, where $\mu_{2}$ indicates the strength of drainage cooperation, and $\gamma_{2}$ denotes possible inefficiencies or conflicts between the regions. These values satisfy the condition $\mu_{2} + \gamma_{2} \le { }1$ and follow intuitionistic fuzzy graph constraints: $\mu_{2} \left( {v_{i} , v_{j} } \right) \le \mu_{1} \left( {v_{i} } \right) X \mu_{1} \left( {v_{j} } \right)$, and $\gamma_{2} \left( {v_{i} , v_{j} } \right) \le \gamma_{1} \left( {v_{i} } \right) X \gamma_{1} \left( {v_{j} } \right)$ × γ₁(vⱼ). For example, vertex $v_{1}$ has values (0.6, 0.4), and edge $v_{1} v_{3}$ has values (0.5, 0.2), which satisfy all the required conditions. This fuzzy representation helps modelling uncertainty in relationships, particularly for spatial urban environments such as drainage or flood-prone regions.

Definition 2.4¹

A bipolar intuitionistic fuzzy graph(BIFG) of a graph G* = (V, E) is a pair G(A, B) where $A=\left({\mu }_{A}^{P}, {\mu }_{A}^{N}, {\gamma }_{A}^{P}, {\gamma }_{A}^{N}\right)$ is a bipolar intuitionistic fuzzy set in V and $B=\left({\mu }_{B}^{P}, {\mu }_{B}^{N}, {\gamma }_{B}^{P}, {\gamma }_{B}^{N}\right)$ is a bipolar intuitionistic fuzzy set in V × V such that

$${\mu }_{B}^{P}\left(x,y\right)\le \left({\mu }_{A}^{P}\left(x\right), {\mu }_{A}^{P}\left(y\right)\right) , \forall (x,y)\in V\times V$$

$${\mu }_{B}^{N}\left(x,y\right)\ge \left({\mu }_{A}^{N}\left(x\right), {\mu }_{A}^{N}\left(y\right)\right) , \forall (x,y)\in V\times V$$

$${\gamma }_{B}^{P}\left(x,y\right)\ge \left({\gamma }_{A}^{P}\left(x\right), {\gamma }_{A}^{P}\left(y\right)\right) , \forall (x,y)\in V\times V$$

$${\gamma }_{B}^{P}\left(x,y\right)\le min\left({\gamma }_{A}^{P}\left(x\right), {\gamma }_{A}^{P}\left(y\right)\right) , \forall (x,y)\in V\times V$$

Figure 3 displays a BIFG depiction.

Fig. 3

Bipolar intuitionistic fuzzy graph.

Definition 2.5⁵

An intuitionistic fuzzy graph $G=(A,B)$ if ${\mu }_{B}\left(xy\right)=min({\mu }_{A}\left(x\right),$ ${\mu }_{A}\left(y\right))$ and ${v}_{B}\left(xy\right)=max({v}_{A}\left(x\right),$ ${v}_{A}\left(y\right))$ for all $xy\in E$ is referred to as a strong intuitionistic fuzzy graph.

Definition 2.6⁵

The fuzzy graph $G :(V,\sigma ,\mu)$ is given. If the membership function value of an arc in a fuzzy graph is at least as high as the strength of the hyperlink between its two ends after deletion, the arc is said to be strong.

Definition 2.7⁸

The Effective edges $e=\left(p,q\right)$ in G are defined as follows if

$${\mu }^{+}\left(p,q\right)={\mu }^{+}\left(p\right)\wedge {\mu }^{+}\left(q\right)$$

$${\mu }^{-}\left(p,q\right)={\mu }^{-}\left(p\right)\vee {\mu }^{-}\left(q\right)$$

$${\gamma }^{+}\left(p,q\right)={\gamma }^{+}\left(p\right)\vee {\gamma }^{+}\left(q\right)$$

${\gamma }^{-}\left(p,q\right)={\gamma }^{-}\left(p\right)\wedge {\gamma }^{-}\left(q\right)$ $for all \left(p,q\right)in G.$ The example for E-edges is shown Fig. 4.

Fig. 4

Effective edges of BIFG.

Definition 2.8⁸

When p is in G, its strong neighbourhood is $N\left(p\right)\left\{\left(p,q\right)is an E-edge\right\}.$ For p in a graph G, a closed strong neighbourhood is $N\left[p\right]=N\left(p\right)\cup p$. In a bipolar intuitionistic fuzzy graph G, the vertex cardinality or order O(G) can be expressed as $\left|V\left(G\right)\right|= {\sum }_{p\in V\left(G\right)}\left(\delta \left(p\right)\right).$ As stated by $\left|E\left(G\right)\right|= {\sum }_{p,q\in V\left(G\right)}\left(\delta \left(p,q\right)\right)$, the edge cardinality is obtained.

Theorem 2.9⁶

Consider the fuzzy graph G. Every vertex u in a dominating set must meet at least one of the following requirements. A technique to describe S of G is as a minimal dominant set. (i) The subgraph created by S in G does not contain u as a strong neighbour for each vertex in S. (ii) A node called $v$ exists that is not a member of S such that ${N}_{S }\left(v\right)\cap S=\{u\}$.

Proof

Suppose S is a fuzzy graph G’s minimal dominant set. Then, the set ${S}{\prime}=S-\left\{u\right\}$ is not a dominant set of G for each vertex $u\in S$. A vertex $v\in V-S{\prime}$ that is not dominated by any vertex in $S-\{u\}$ so exists $v=u or v\in V-S$. If $v=u$, then u is separated from every vertex in S. Vertex v is close to vertex u only in S if $v\in V-S$ is dominated by S rather than S–{u}. This indicates that ${N}_{S }\left(v\right)\cap S=\{u\}$, thus the condition (ii) is satisfying.

On the other hand, assume that S is a dominant set and that one of the aforementioned requirements is met for each vertex u. It is now necessary to demonstrate that S is a dominating set. There is a vertex u that belongs to S such that $S-\{u\}$ is a dominating set, assuming that S is not a dominating set. Consequently, at least one vertex $S-\{u\}$ near to u. It follows that the first requirement is not met. Additionally, each vertex in V-S is a strong neighbour to at least one vertex in $S-\{u\}$ and if $S-\{u\}$ is a dominant set. As a result, the second need is also not achieved. Hence, conditions (i) and (ii) conflict with one another.

Theorem 2.10⁸

The dominating set S associated with every connected fuzzy graph G is a dominating set, and so is the complementary set $V-S$.

Proof

Let T be any spanning tree of fuzzy graph G, and let u be one of its vertices. Next, the vertices in T are separated into two distinct sets, $S and {S}{\prime}=S-\{u\}$, which are made up of the vertices in T that are odd and even distances from u. In that case, $S and {S}{\prime}$ are the dominant sets of fuzzy graph G.

For if $v\in V,$ then either $d(u,v)$ is odd or even in T. If $d(u,v)$ is odd, it will be a neighborhood to a vertex of even distance from u. Alternatively, a vertex that is oddly distant from u will be closest to it. In this case, $v\notin V-S$ is a dominating set in G; if $v\notin S$, it is a neighbourhood to a vertex S in G, indicating that S is a dominating set.

Theorem 2.11⁸

If there are no isolated vertices in a fuzzy graph G, then the complement of each minimal dominant set is V-S.

Proof

Let S be a fuzzy graph G’s minimal dominant set. Make the assumption that no vertex V in S dominates $v$. In other words, no vertex in V-S is closest to ${N}_{S}\left(v\right)\subset S$ on $v.$ V must be near to one vertex in $S-\{v\}$. Since G excludes isolated vertices. Consequently, S-{v} dominates $v,$ and $S-\{v\}$. likewise dominates V-S.

The fact that it is contradictory that $S-\{v\}$ is a dominant set. Consequently, every vertex in S is dominated by at least one vertex in V-S. This suggests that V-S is a dominant set.

Definition 2.12²

If the set S − {v} is not a dominating set for each vertex v ∈ S, then the fuzzy dominating set S of a fuzzy graph G is referred to as a minimal dominating set of G. Figure 5 shows an example of a minimal dominating set. The minimal dominance number of a graph G is the number of vertices in its minimum dominating set (MDS). The symbol for it is γ(G). Figure 5 displays a fuzzy graph’s MDS.

$$S_{1} = \left\{ {a,{ }c,{ }e} \right\};\;V - S_{1} = \left\{ {b,{ }d,{ }f,{ }g} \right\}$$

$$S_{2} = \left\{ {c,d,e} \right\};\;V - S_{2} = \left\{ {a,{ }b,{ }f,{ }g} \right\}$$

$$S_{3} = \left\{ {f,{ }g} \right\};\;V - S_{3} = \left\{ {a,{ }b,{ }c,{ }d,{ }e} \right\}$$

Fig. 5

Fuzzy graph’s minimal vertex dominating set = {f, g}.

Therefore, the minimal dominating set of a fuzzy graph is γ(G) = 2.

Definition 2.13²

Consider the fuzzy graph $G=(\sigma ,\mu )$. Assume that G has two edges, ${e}_{i}$ and ${e}_{j}$. If ${e}_{i}$ is an effective edge and ${e}_{i}$ is closest to ${e}_{j}$, then define that ${e}_{i}$ dominates ${e}_{j}$. If they include ${e}_{i}\in {S}{\prime}$ that ensures that ${e}_{i}$ dominates ${e}_{j}$ for every ${e}_{j}\in E\left(G\right)-{S}{\prime}$, then a subset ${S}{\prime}$ of E(G) can be considered to be an effective edge dominating set.

Theorem 2.14⁸

If a fuzzy graph $G=\left(\sigma ,\mu \right)$ contains an effective edge dominating set, then for each $\nu$ in $G$, ${d}_{E}>0$.

Proof

Consider ${S}{\prime}$ being the collection of all of G’s effective edges. One edge $\left(x,y\right)\in$ $E\left(G\right)- {S}{\prime}$ in which $\left(x,y\right)$ is not close to any edge in ${S}{\prime}$ belongs if ${S}{\prime}$ is not an edge dominating set. The edge $\left(x,x{\prime} \right)\in {S}{\prime}$ since the vertex $x$ has an effective neighbour, state that $x{\prime}$, and $\left(x,y\right)$ is ineffective, and ${d}_{E}(x)>0$. As a result, the edges $\left(x,y\right)$ and $\left(x,x{\prime} \right)\in {S}{\prime}$ neighbour to each other. It is contradictory that the set ${S}{\prime}$ dominates the edge $\left(x,y\right)$.

Note

If ${S}{\prime}$ is a dominating set of minimal effective edge, then $E\left(G\right)= {S}{\prime}$ need not be an edge dominating set.

An effective edge dominant set made up of the set of all effective edges in a fuzzy graph $G=\left(\sigma ,\mu \right)$ is not always present.

Definition 2.15⁶

When ${S}{\prime}$ of G is a minimal edge dominating set, then no suitable subset of ${S}{\prime}$ is an edge dominating set. When |${S}{\prime}$ |= ${\gamma }{\prime}(G)$, then an edge dominating set ${S}{\prime}$ of G is called a minimum edge dominating set. In Fig. 6 below, the edge dominating set example is shown.

Fig. 6

Fuzzy graph’s minimal edge dominating set = {b, e}.

The arcs in Fig. 6 above are strong for the edges ($\left( {v_{1} , v_{2} } \right), \left( {v_{2} ,v_{3} } \right), \left( {v_{3} ,v_{4} } \right) and \left( {v_{1} ,v_{5} } \right)$ but not for the edges $\left( {v_{4} ,v_{5} } \right).$ Thus When b and e are the respective edge values, $S^{\prime} = \left\{ { \left( {v_{2} ,v_{3} } \right), \left( {v_{1} ,v_{5} } \right)} \right\}$ is a minimal edge dominating set.

Definition 2.16⁸

Consider the BIFS of vertices $V = \{ p, \mu^{ + } \left( p \right), \mu^{ - } \left( p \right), \gamma^{ + } \left( p \right),\gamma^{ - } \left( p \right))|p\varepsilon V)$. The formula for a scoring function δ(p) of V is $\delta \left( p \right) = \left( {\frac{{2 + \mu^{ + } \left( p \right) - \mu^{ - } \left( p \right) - \gamma^{ + } \left( p \right) + \gamma^{ - } \left( p \right)}}{4}} \right) , \forall p \in V$, where p is equal to V.

Definition 2.17⁸

A BIFS of vertices is represented as $\underline {E} = \{ \mu^{ + } \left( {p,q} \right), \mu^{ - } \left( {p,q} \right), \gamma^{ + } \left( {p,q} \right),\gamma^{ - } \left( {p,q} \right))| p,q \varepsilon V)$. The definition of a scoring function $\delta \left( {p,q} \right)$ of $\underline {E}$ is defined as $\delta \left( {p,q} \right) = \left( {\frac{{2 + \mu^{ + } (\left( {p,q} \right) - \mu^{ - } \left( {p,q} \right) - \gamma^{ + } \left( {p,q} \right) + \gamma^{ - } \left( {p,q} \right)}}{4}} \right),\forall p,q \in V.$

Methodology

This study offers a hybrid optimization framework that combines Bipolar Intuitionistic Fuzzy Graphs with Machine Learning Optimization (BIFG-MLO) to improve water logging prediction and enable the efficient installation of permeable pavement systems in Thoothukudi, India’s smart cities. The methodology employs a multi-phase, systematic approach:

Data acquisition and pre-processing

Usually, administrative planning is necessary when an unforeseen catastrophe threatens the city. Two southern Indian states that experience inundation and swamping during the biennial monsoon season are Kerala and Tamil Nadu. On December 2023, Thoothukudi and four southern Tamil Nadu districts, especially Tirunelveli and Chennai, were completely hammered by unprecedentedly torrential rains for approximately 24 h, leaving a number of communities deserted and millions of people stuck inside their homes. Thoothukudi received 96 cm of rainfall in just 24 h, well exceeding the little town’s yearly rainfall in a regular year, although the district’s average rainfall was 38 cm. Therefore, in the course of an enormous weather occurrence, rainfall might gather an immense quantity of water, causing it to get blocked or overflow, which leads to flooding. To mitigate these issues, commercial, industrial, and residential locations can adopt permeable roadway approaches, which allow water to move easily through the surfaces of pavement, just as it does in the earth’s structure beneath them. Through lowering rainwater stagnation, enhancing sanitation, and safeguarding livelihoods—particularly for daily wage earners who are most impacted by protracted flooding—these improvements are anticipated to benefit about 24,283 families, or 97,132 individuals. The stormwater mitigation plan for Thoothukudi combines social planning, engineering, and smart city integration in a thorough manner. It tackles the long-term objective of creating a resilient, sustainable city as well as the pressing necessity to lessen urban floods.

The application incorporates the preceding catastrophe management problem, which depicts Thoothukudi’s districts devastated by inundation, whereby edges bind the vertices to the cities in the vicinity. The most adjacent cities in the BIFG have an enduring association regarding the effective edges. The porous concrete can be constructed by implementing fuzzy graph domination problems in order to regulate swamping zones caused by hurricanes, torrential downpour etc. The data sets for the current study were gathered from the India Meteorological Department, as indicated in Table 1, with the positive degrees representing average rainfall and low-level locations and the negative degrees representing their adverse consequences. As a result, the national and state governments can prioritize low-level regions for the implementation of the permeable surface infrastructure based on dominance concepts. Instead of allowing water to accumulate, flow into drains, and end up in streams as waste, permeable pavers collect and gradually release runoff into the soil. Reduced runoff volumes and rates from permeable pavement can help reduce the cost of water treatment, erosion, and floods. This leads to choosing these affected regions to set up the advanced pavement system, which is more crucial because this helps to manage stormwater more sustainably with the greatest coverage of neighbouring cities.

Table 1 Average Rainfall in Smart Cities of Thoothukudi.

The India Meteorological Department (IMD) and pertinent civic agencies provided meteorological and hydrological data, such as average rainfall, drainage capacity, and urban topography. Ottapidaram, Sawyerpuram, Kovilpatti, Kandasamypuram, Tuticorin Melur, P&T Colony, National Highway 45B, and Nazareth are the eight smart cities in Thoothukudi to which the data was mapped. Fuzzy membership values were assigned to each city, which represented:

• Positive impact (e.g., rainfall intensity, urban exposure)

• Negative impact (e.g., water logging, drainage inefficiency)

Bipolar intuitionistic fuzzy graph (BIFG) construction

Consider the eight smart cities as the vertices, and nearby smart cities are connected through edges, which can be framed using bipolar intuitionistic fuzzy networks. There are more neighbouring areas in the city, according to the E-edges. The water level of the region during a downpour is shown by the positive membership and non-membership ratings, while their negative grades indicate the inadequate drainage system. An indicator of the degree of both positive and negative impacts across neighbouring city regions is the edge between the two smart cities. The vertex values are represented as {Rainfall intensity, Water stagnation, Urban hazard exposure, Inadequate drainage system}, and the edge values are represented as the distance between the smart cities.

Algorithm for data conversion

Transforming Average Rainfall into a Bipolar Intuitionistic Fuzzy Graph (BIFG) offers an advanced representation of rainfall analysis in smart cities that addresses uncertainty and dual perspectives (benefit/risk). Here is a generic computational method for creating a BIFG appropriate for urban analysis from average rainfall data.

Step 1: Data Integration.

Compile average rainfall information for several smart city node installations or urban zones.

The city should be arranged as a graph G = (V, E), where:

Vertices (V) represent smart cities.

Edges (E) represent inter-city connectivity based on geographic proximity and flood impact correlation.

Step 2: Define Fuzzification.

Each vertex and edge are assigned four fuzzy components:

(μ⁺, μ⁻, γ⁺, γ⁻) denoting positive and negative degrees of membership and non-membership. The effective edge formulation ensures that the graph accurately captures uncertain and incomplete spatial relationships.

Step 3: Construction of BIFG,

In the BIFG, each smart city is represented as a vertex, with linked roadways serving as edges. Environmental features, including waterlogging, drainage inefficiency, and rainfall intensity, are.

represented by the fuzzy values. Each vertex $p\in V$ in the BIFG is designated for four-tuple membership:

$$(p, {\mu }^{+}\left(p\right), {\mu }^{-}\left(p\right),{\gamma }^{+}{\left(p\right), \gamma }^{-}\left(p\right))$$

where: ${\mu }^{+}\left(p\right)$: Positive membership (e.g., Rainfall intensity).

${\mu }^{-}\left(p\right):$ negative membership (e.g., Water stagnation).

${\gamma }^{+}(p)$: Positive non-membership (e.g., Urban hazard exposure).

${\gamma }^{-}\left(p\right):$ Negative non-membership (e.g., Inadequate drainage system).

Step 4: Novel property of Bipolar Intuitionistic Fuzzification:

Every node emphasizes, $0\le {\mu }^{+}+ {\gamma }^{+} \le 1; 0\le {\mu }^{-}+ {\gamma }^{-} \le 1$

The degree of reluctance is the remainder to 1.

Step 5: Define the Edge BIFG parameters.

Each edge $(p,q)\in E$ is also defined by a tuple:

$({\mu }^{+}\left(p,q\right), {\mu }^{-}\left(p,q\right),{\gamma }^{+}{\left(p,q\right), \gamma }^{-}\left(p,q\right)$).

Step 6: Compute the Effective Edges:

These edge values are computed using the effective edge criteria:

$${\mu }^{+}\left(p,q\right)=\left( {\mu }^{+}\left(p\right),{\mu }^{+}\left(q\right)\right), {{\mu }^{-}\left(p,q\right)=max(\mu }^{-}\left(p\right),{\mu }^{-}\left(q\right))$$

$${\gamma }^{+}\left(p,q\right)=\left( {\gamma }^{+}\left(p\right),{\gamma }^{+}\left(q\right)\right), {{\gamma }^{-}\left(p,q\right)=min(\gamma }^{-}\left(p\right),{\gamma }^{-}\left(q\right))$$

Step 7: Representation of a graph:

Vertices: $V=\{p({\mu }^{+}, {\gamma }^{+}, {\mu }^{-},{\gamma }^{-})\}$

Edges: $E=\{(p,q)({\mu }^{+}, {\gamma }^{+}, {\mu }^{-},{\gamma }^{-})\}$

The BIFG that is produced can subsequently be utilised for susceptibility assessment, choice-making, or efficient resource management throughout the city. Moreover, an edge connects them both, indicating that they have a greater number of nearby places and are correspondingly affected by floods. BIFG could be portrayed as an infrastructure plan that unites the various regions of the community, with vertices indicating areas and edges indicating pathways, to benefit the law enforcement officers. Table 2 provides the edge membership values, while Fig. 7 illustrates the vertex membership values.

Table 2 BIFG values of the smart cities.

Fig. 7

Smart cities of Thoothukudi linking roads.

Dominance theory and graph metrics

Using domination theory, cities and connections were evaluated to identify:

• Dominating Sets (cities covering most water-logging impacts)

• Minimal Dominating Sets (MDS) ensuring optimized resource deployment

• Edge Domination for route prioritization in infrastructure planning

Model input and feature construction

Each smart city (vertex) is characterized using the following features:

Positive Membership (μ⁺): Average rainfall intensity

Negative Membership (μ⁻): Degree of water stagnation

Positive Non-membership (γ+): Effective drainage indicators

Negative Non-membership (γ⁻): Inadequate sewage and topographical depression.

These features are used to derive the vertex score $\delta \left(p\right)$ and edge score $\delta \left(p,q\right)$, which serve as input to ML classifiers. Vertex and edge domination numbers were calculated using scoring functions derived from fuzzy values, facilitating the ranking of smart cities by flood vulnerability. To quantify the city’s vulnerability and prioritise pavement implementation, we use the vertex scoring function:

$$\delta \left(p\right)= \left(\frac{2+ {\mu }^{+}\left(p\right) - {\mu }^{-}\left(p\right)- {\gamma }^{+}\left(p\right)+ {\gamma }^{-}\left(p\right)}{4}\right)$$

For edges:

$$\delta \left(p,q\right)= \left(\frac{2+ {\mu }^{+}(\left(p,q\right) - {\mu }^{-}\left(p,q\right)- {\gamma }^{+}\left(p,q\right)+ {\gamma }^{-}\left(p,q\right)}{4}\right)$$

Table 3 presents an analysis of 8 cities using four key indicators:

Table 3 City data analysis.

To compute the Vertex Scoring Function: $\delta \left(p\right)= \left(\frac{2+ {\mu }^{+}\left(p\right) - {\mu }^{-}\left(p\right)- {\gamma }^{+}\left(p\right)+ {\gamma }^{-}\left(p\right)}{4}\right)$

The δ(p) scores are accurately computed using the vertex scoring formula as shown in Table 4.

Table 4 City-wise metrics.

Determine the dominating sets and dominance numbers

Three dominating sets (D₁, D₂, D₃) were tested:

D₁ = {C₁, C₂, C₇, C₈}, Dominance Number = 4.365

D₂ = {C₂, C₇, C₈}, Dominance Number = 3.94

D₃ = {C₂, C₈}, Dominance Number = 3.002 (Lowest)

D₃ = {C₂, C₈} is the optimal minimal dominating set, covering all other cities with the fewest dominant vertices. These cities showed the highest dominance scores and water-logging vulnerability based on the algorithmic analysis. This method assures the reliable integration of rainfall data into sophisticated, comprehensible smart city decision models.

Dominating edge evaluation

In relation to intuitionistic fuzzy graphs or even classical fuzzy graphs, BIFGs can able to simultaneously depict the positive and negative features of relationships as well as their presence or absence. In real-world situations where information is frequently unclear, contradictory, or inadequate, this duality is essential. An illustration of the notion can be found below. Numerous current problems, such as social networks, college bus routing, radio stations, and communication networks, can be figured out by fuzzy graph dominances.

Permeable pavements are intended to be installed across smart cities, but the federal government finds it challenging to open them everywhere due to their high cost. As a result, it is feasible to suggest that they choose sparse locations that are near other locations. The most convenient pathway between the cities has been demonstrated by the selection of effective edges. Figure 8 depicts effective edges employing the dominance property.

Fig. 8

BIFG using effective edges.

Bipolar fuzzy graphs with effective edges have been utilized to construct networks with undetermined parameters. Effective edges serve as a vital to the long-term sustainability of a bipolar fuzzy graph (BFG) because they maintain links between two surrounding locations. In terms of accuracy and efficiency, the edge reduction method can be used to identify the most vulnerable routes between the two cities. Smart cities intend to install permeable asphalt pavement across the entire city; however, their elevated prices make it difficult for authorities to permit them widely. Therefore, it is feasible for them to suggest that they select sparse places that are proximate to other areas. Effective edges, as seen in Fig. 8, have been selected to illustrate the most viable approach between the cities with the aid of the dominance property. Using effective edge scoring $\delta \left(p,q\right)$, key connections between cities were ranked, as seen in Table 5. The edges with highest influence for stormwater routing and pavement planning include:

Table 5 Edge-wise metrics between cities.

Optimization and algorithmic implementation

Using Bipolar Intuitionistic Fuzzy Graphs (BIFGs) and a fuzzy graph domination algorithm improved with Swarm Intelligence concepts, an optimization framework is created to systematically prioritize flood-prone areas and ascertain the best location for permeable pavement systems. This hybrid method makes it possible to calculate dominance in metropolitan areas with ambiguous and insufficient environmental data in an effective manner. For real-time, data-driven inundation management specific to Thoothukudi’s particular disaster threats and infrastructure necessities, this optimised flowchart makes use of machine learning and the Internet of Things. The flowchart below will considerably benefit attempts to manage flooding issues in smart cities across Thoothukudi.

Graph-based optimization framework

In the BIFG, every smart city is represented as a vertex. The edges of the graph show the hydrological link and closeness between cities. In order to determine the best city subsets for intervention, the system creates minimal dominating sets (MDS) and assesses each city’s dominance using fuzzy membership values. The dominance score of each city(vertex) $\delta \left(p\right)$ and city-pair (edges) $\delta \left(p,q\right)$ are calculated using

$$\delta \left(p\right)= \left(\frac{2+ {\mu }^{+}\left(p\right) - {\mu }^{-}\left(p\right)- {\gamma }^{+}\left(p\right)+ {\gamma }^{-}\left(p\right)}{4}\right)$$

$$\delta \left(p,q\right)= \left(\frac{2+ {\mu }^{+}(\left(p,q\right) - {\mu }^{-}\left(p,q\right)- {\gamma }^{+}\left(p,q\right)+ {\gamma }^{-}\left(p,q\right)}{4}\right)$$

By taking into account both positive and negative indications (such as drainage failure and rainfall intensity), these scores evaluate the susceptibility of cities and their connections.

Algorithm for identifying dominant urban zones

Input: V: Set of smart cities

E: Edges representing city interconnections

F(v): Flooding severity function based on environmental parameters

Using the vertex scoring function:

$$\delta \left(p\right)= \left(\frac{2+ {\mu }^{+}\left(p\right) - {\mu }^{-}\left(p\right)- {\gamma }^{+}\left(p\right)+ {\gamma }^{-}\left(p\right)}{4}\right)$$

The concept of a dominating set is intended to help prioritize flood-affected areas for the development of permeable pavement systems. These systems play an important role in stormwater management, runoff reduction, and urban resilience. By focusing resources on the most vulnerable locations, the developers may maximize the impact of our programs and contribute to long-term sustainable urban growth that can be integrated into our present strategies. The minimal dominating set (MDS) with strong edges will cover the maximum cities. Depending on the vertex cardinality, the permeable pavement system can be implemented in the smart cities on a priority basis. The below Fig. 9 shows the dominating edges and their adjacent vertices.

$${\text{D}}_{1} = \left\{ {{\text{C}}_{1} ,\;{\text{C}}_{2} ,\;{\text{C}}_{7} ,\;{\text{C}}_{8} ,} \right\}$$

$${\text{V}} - {\text{D}}_{1} = \left\{ {{\text{C}}_{3} ,\;{\text{C}}_{4} ,\;{\text{C}}_{5} ,\;{\text{C}}_{6} ,} \right\}$$

Fig. 9

Dominating graph of BIFG D₁.

The adjacent edges are shown below, which is used to identify the most affected cities with low-level water logging areas in order to implement the permeable pavement system. D₁ has the following minimal dominating edges: {(C₁, C₃), (C₁, C₄), (C₂, C₃), (C₂, C₄), (C₃, C₇), (C₄, C₇), (C₅, C₇), (C₆, C₈)}. It is noted that C₁, C₂, C₇ and C₈ dominate the other vertices, and the dominance number of D₁ = 4.365.

It is evident from Fig. 10 above that C₂, C₇ and C₈ dominate the other vertices.

$${\text{D}}_{2} = \left\{ {{\text{C}}_{2} ,\;{\text{C}}_{7} ,\;{\text{C}}_{8} } \right\}$$

$${\text{V}} - {\text{D}}_{2} = \left\{ {{\text{C}}_{1} ,\;{\text{C}}_{3} ,\;{\text{C}}_{4} ,\;{\text{C}}_{5} ,\;{\text{C}}_{6} } \right\}$$

Fig. 10

Dominating graph of BIFG D₂.

For D₂, the minimal dominating edges are {(C₁, C₂), (C₂, C₃), (C₂, C₄), (C_3, C₇), (C₄, C₇), (C₅, C₇), (C₅ ,C₈), (C₆, C₇). (C₆, C₈)} & D₂'s dominance number = 3.94.

$${\text{D}}_{3} = \left\{ {{\text{C}}_{2} ,\;{\text{C}}_{8} } \right\}$$

$${\text{V}} - {\text{D}}_{3} = \left\{ {{\text{C}}_{1} ,\;{\text{C}}_{3} ,\;{\text{C}}_{4} ,\;{\text{C}}_{5} ,\;{\text{C}}_{6} ,\;{\text{C}}_{7} } \right\}$$

Figure 11 shows that D₂ has a dominance number of 3.002, and its minimal dominant edges are {(C₁, C₂), (C₂, C₃), (C₂, C₄), (C₂, C₇), (C₅, C₈), (C₆, C₈), (C₇, C₈)}. From Figs. 9, 10, 11, it has been observed that D₃ covers all the cities with a minimal dominating city, therefore, the vertex cardinality is γ(G) = 2. The most impacted smart city with stagnant water regions is identified using the vertex domination number. The dominance scores for the cities are calculated as follows in Table 6.

Fig. 11

Dominating graph of BIFG D₃.

Table 6 Vertex domination number.

The city Kovilpatti (C₃) has the highest vertex dominance score, indicating maximum vulnerability to flooding. According to the aforementioned statistics, the city C_3, followed by the C₆ area has been seriously damaged with multiple areas, experiencing waterlogging. Therefore, when establishing the system into setting, the cities that are closest to such areas must be given precedence. The set of edges utilised to deal with the network of connections routing issues have been identified as the edge domination. The permeability system connecting the cities has to be constructed for sustainability with the help of edge domination number in Table 7.

Table 7 Edge domination number.

Minimal dominating sets

Three candidate dominating sets were identified and evaluated based on dominance number:

$${\text{D}}_{1} = \left\{ {{\text{C}}_{1} ,\;{\text{C}}_{2} ,\;{\text{C}}_{7} ,\;{\text{C}}_{8} } \right\}$$

Covers all cities with a dominance number of 4.365.

$${\text{D}}_{2} = \left\{ {{\text{C}}_{2} ,\;{\text{C}}_{7} ,\;{\text{C}}_{8} } \right\}$$

Reduced set, dominance number 3.94.

$${\text{D}}_{3} = \left\{ {{\text{C}}_{2} ,\;{\text{C}}_{8} } \right\}$$

Minimal Dominating Set with dominance number 3.002.

Thus, D₃ = {C₂, C₈} is selected as the optimal configuration, minimizing infrastructure while ensuring full regional coverage.

Rank cities by dominance and vulnerability

Permeable pavement solutions are essential for efficient stormwater management and urban resilience in the context of sustainable urban development, particularly in areas like Thoothukudi that are prone to flooding or water stress. In order to facilitate effective infrastructure planning and resource allocation, this study presents a graph-based ranking system that assesses cities according to two crucial dimensions: vulnerability and dominance. In order to implement permeable pavement, this dual-ranking method identifies high-priority zones where infrastructure deployment will minimise risk and maximise impact. While low-dominance or low-vulnerability cities are deprioritized or targeted in subsequent phases, high-dominance and high-vulnerability cities are advised to take urgent action. Based on δ(p) values and edge analysis, the following ranking of cities is proposed for permeable pavement deployment.

From vertex domination and edge scores, the final ranking of cities (most to least vulnerable) is:

• C₃—Kovilpatti (δ = 0.61)

• C₆—P&T Colony (δ = 0.54)

• C₅—Tuticorin Melur (δ = 0.52)

• C₁—Ottapidaram (δ = 0.51)

• C₂—Sawyerpuram (δ = 0.49)

• C₇—NH 45B (δ = 0.48)

• C₈—Nazareth (δ = 0.46)

• C₄—Kandasamypuram (δ = 0.45)

Recommended priority cities for permeable pavement deployment

• Kovilpatti (C₃)

• P&T Colony (C₆)

• Tuticorin Melur (C₅)

Predictive analytics using machine learning

In today’s data-driven financial system, data analytics tools and techniques are critical for making reliable choices that boost productivity and help the business to stay ahead of the curve. Predictive analytics explores and estimates waterlogging-prone regions using big data, neural networks (AI), and geographic information systems (GIS). Machine learning (ML)-based predictive analytics presents a proactive method for locating, tracking, and streamlining water-logging-prone regions. This approach investigates how ML models can improve emergency responders’ and urban planners’ decision-making. Bipolar Fuzzy Graph (BFG) is developed on the conventional fuzzy graph paradigm to represent both antagonistic and sustaining links in a network by integrating positive and negative membership functions. Authorities can anticipate possible flood-prone regions and take preventive action by using predictive analytics, which is a vital arsenal to utilize for controlling the waterlogged zones. Incorporating the bipolar fuzzy graph approach into their prediction models can help decision makers improve their accuracy and efficiency. Water logging spots can be properly regulated by combining bipolar fuzzy graph techniques with predictive analytics to establish the execution of decision mechanisms. Through the utilisation of sophisticated graph theory and real-time data analysis, politicians, engineers, and urban planners can enhance a drainage system’s functionality, minimise the danger of flooding, and construct more secure metropolises.

The suggested approach combines the Bipolar Intuitionistic Fuzzy Graph (BIFG) model with predictive analytics to improve decision-making in urban flood control. Machine learning (ML) algorithms are used to categorize and predict flood-prone areas in Thoothukudi by examining drainage characteristics, historical rainfall, and spatial connectedness among smart cities. Pavement system modelling is elevated from strict deterministic approaches to a complex, uncertainty-aware framework by this methodology. When managing permeable pavements in smart city contexts, it improves forecast accuracy, cost-effectiveness, and sustainability, which has a direct positive impact on city planning, urban water management, and infrastructure durability. With the use of fuzzy graph theory and machine learning, this work offers a computationally complex, regionally contextual, and uncertainty-aware method for optimising permeable pavement design and management in the unique climate and infrastructure of Thoothukudi Smart City. This approach would move the field closer to true smart infrastructure management, which integrates real, imperfect, and diverse data sources, making urban water management more reliable and sustainable.

Classification and learning approach

To determine which cities are most susceptible to flooding, a supervised classification model is used. Labelled examples of waterlogging severity from previous flood occurrences and sensor datasets are used to train the system. Using fuzzy features, the model seeks to differentiate between high-risk and low-risk areas.

Algorithms applied include

• Random Forest

• Support Vector Machine (SVM)

• K-Nearest Neighbours (KNN)

• Hybrid Fuzzy–ML Ensembles

Each model is trained and validated using a stratified dataset to ensure balanced representation of all risk levels. This optimization technique integrates fuzzy graph domination and swarm intelligence to figure out and alleviate water-logging hazards in a savvy, versatile, and profitable manner. By modelling the geographic area as a fuzzy graph and using optimisation techniques, fuzzy graph dominance regulates zones that are likely to accumulate precipitation. A structured algorithm is designed below to accurately locate water-logging locations using Swarm Intelligence and Fuzzy Graph Domination.

Input: Consider the smart cities to be the vertices, with neighbouring smart cities connected through edges by assigning fuzzy weights.

Output: The degrees in the adjacent edges assist in locating the most impacted cities with low-level drowning zones.

• 1. 01. Initialize parameters: $\:(w,\:u\in\:V,\:f\left(v\right),v\in\:S)\:$

• 2. 02. Assign fuzzy membership values $\:\mu\:\left(u\right)\:for\:all\:u\varepsilon V$.

• 3. 03. Generate water accumulation values as edge weights.

• 4. 04. For each iteration (i = 1, 2, …) do:

• 5. 05. For each vertex u ∈ V:

• 6. 06. Determine fuzzy membership function:

  $$\:\mu\:\left(u\right)=v_{1}f\left(u\right)\:+v_{2}f\left(u\right)+v_{3}f\left(u\right)$$

// where $v_{1} ,v_{2} ,v_{3}$ are impact coefficients for low-lying zones.

• 7. Else:

• 8. 08. Compute the Dominating Set

$S\subseteq V$:

$\forall u\in V, \exists v\in S$ such that $\left(v,u\right)\in D$

• 9. Update optimization metrics

• 10. Optimize fuzzy memberships:

  $$\sum_{u\varepsilon S}\mu (u)$$

• 11. Refine the Dominating Set S to reduce redundancy

• 12. End For

• 13. Output cities (vertices) most prone to water-logging based on $\mu \left(u\right)$

Performance metrics

The preceding section discusses the present methods and the outcome-based analysis of the suggested model. Precision rate, recall rate, F1 score, specificity, False Positive Rate (FPR), True Positive Rate (TPR), and total accuracy are the metrics used for analysis. The confusion matrix is used to calculate standard metrics that evaluate the model’s classification quality:

$$Accuracy = \:\frac{(TP+FP)}{(TP+TN+FN+FP)}.$$

$$Precision = \:\frac{TP}{(TP\:+\:FP)}.$$

$$Recall = \:\frac{TP}{(TP\:+\:FN)}.$$

$$F1 Score = \:\left(\frac{Precision*Recall}{Precision+Recall}\right).$$

$$Specificity or True Negative Rate (TNR) = \:\frac{TN}{TN+FP}.$$

$$False Positive Rate (FPR) =\:\frac{FP}{FP+\:TN}.$$

$$True Positive Rate (TPR) =\:\frac{TP}{TP+FN}.$$

.

A robust sensor network, multi-source data integration, phased drainage infrastructure, sophisticated machine learning-based prediction, and a strong emphasis on resilience, maintenance, and community impact are the hallmarks of Thoothukudi’s smart city inundation management concept. Together, these factors allow for prompt response and proactive flood risk management, making Thoothukudi a model for other smart cities in India dealing with comparable issues.

In keeping with a people-centric smart city vision, special attention is paid to protecting vital urban infrastructure and causing the least amount of inconvenience to daily wage earners. The parameters and attributes used in the machine learning-driven, optimised inundation management model for Thoothukudi are broken down in greater detail below, along with contextual and comparative information pertinent to smart city operations. The several parameters of the suggested model, which is utilized to track the smart cities of Thoothukudi in comparison, are shown in Table 8 below:

Table 8 Performance metrics of smart cities.

Figure 12 below exhibits the level of precision that exists between the proposed and existing models. The proposed framework accomplished an unprecedented level of precision when assessed against existing models.

Fig. 12

Comparative study of positive predictive values.

Considering its recall score and overall efficacy, the proposed fuzzy graph model is ranked highest on the list of priorities for the deployment of permeable pavement in the smart cities, which is shown in Fig. 13.

Fig. 13

Analysis of recall score.

As seen by its F1 score in Fig. 14, the proposed model fared better than the prevailing model in terms of striking a balance between precision and recall. The correlated average of precision and recall, or the smart city’s F1 score, is statistically significant for improved interpretability. The F1 score is commonly used for comparing machine learning models, especially when evaluating classification tasks.

Fig. 14

Benchmarking Analysis of F1 Score.

In the context of a smart city, the suggested model shows a noticeably higher specificity than the current model, indicating that it is more adept at accurately detecting non-issue cases, minimizing needless interventions, and optimizing resource consumption. This result’s significance suggests that the difference is not only quantifiable but also significant to stakeholders, which may have an impact on their choice to implement the suggested approach more widely, as shown in Fig. 15.

Fig. 15

Relative Comparison of Specificity.

In smart city machine learning based applications, the suggested model outperforms current AI-based models in terms of threat detection precision from Fig16, achieving a noticeably lower False Positive Rate (FPR). Low FPR is necessary to preserve operational effectiveness and confidence in IoT-driven infrastructure, and this development improves the dependability of automated security systems in smart cities.

Fig. 16

Statistical comparison of FPR.

In terms of TPR/Recall for smart city safety applications, the statistical comparison depicts that specialized algorithms and advanced deep learning perform better than conventional machine learning models, as shown in Fig 17. In complicated metropolitan areas, the suggested approach is particularly effective for real-time threat identification and mitigation due to its near-perfect recall.

Fig. 17

Statistical comparison of TPR.

When compared to smart city findings, the suggested model achieves the highest accuracy score by outperforming the competition in important criteria that are in line with smart city assessment frameworks which is shown in Fig. 18. In these integrated measures, a model that achieves maximal accuracy probably performs exceptionally well, demonstrating strong data fusion and decision-making skills.

Fig. 18

Performance Evaluation for Accuracy.

Machine learning algorithms based on fuzzy graphs are ideal for drainage system optimization. They facilitate resource efficiency over intricate irrigation networks, respond to environmental change, and successfully handle uncertainty. These models’ relevance for intelligent and sustainable city growth is further increased by integrating them with sensor and Internet of Things data. To assess and contrast the accuracy of the proposed model with the existing ones, a graph is a viable and visually appealing tool to show performance differences. With an accuracy rate of 73.4%, precision rate of 76.9%, recall of 76.7%, F1 score of 71.4%, specificity of 76.9%, FPR of 63.1%, and TPR of 76.7%, as shown in the graphs above, the model that was suggested accomplishes better than the current models. This shows that the model’s capacity to accurately categorize inputs was substantially enhanced, confirming the efficacy of the incorporated improvements.

Results and discussions

The study determines the most optimal places for installing permeable pavement by using the domination number in a Bipolar Intuitionistic Fuzzy Graph (BIFG) framework. It comes to the conclusion that cities C₃ and C₆ are the most appropriate for these efforts because they have the highest domination numbers. Both the dominant number analysis and the evaluation of effective edges, which aid in prioritising implementation, yield the following ranking of cities: C₃ > C₆ > C₅ > C₁ > C₂ > C₇ > C₈ > C₄. Additionally, the method explicitly addresses uncertainty by using Bipolar Intuitionistic Fuzzy Sets (BIFS), which is an advance over traditional techniques. The optimisation strategy’s relative performance clearly shows its advantage in capturing the complex, real-world uncertainties that arise during flood resilience and smart city development. Earlier approaches usually employed simple fuzzy or crisp models, which were frequently unreliable when handling multiple points for ambiguity and did not take into consideration complex inter-city connections. The optimization algorithm with BIFG-ML based dominance analysis used in this study improves on earlier methods by utilising domination statistics to rank potential cities specifically for the development of strategic infrastructure. Unlike earlier general site-selection frameworks, this one provides more meaningful help by directly connecting edge effectiveness and dominance notions to real-world implementation. As a result, this uncertainty-aware and optimised methodology outperforms previous models that lacked such advanced decision-support elements, yielding a clearer, more actionable rating for the location of permeable pavement systems.

Numerical comparison table

A thorough comparison analysis with numerical tables and references will help to clarify the benefits and possible disadvantages of the suggested model. Traditional Engineering Models (TEM) and Data-Driven Optimisation Models (DDOM) for permeable pavements are two examples of traditional or cutting-edge models for optimising urban stormwater systems that can be compared to the Fuzzy Graph-Based Machine Learning Optimisation (FG-MLO) model for permeable pavements.

The comparative analysis in the Table 9 Zhang et al. [²⁶], ²⁷, Jones et al. ²⁸, Kurniawan et al. ²⁹ unequivocally demonstrates the BipolarFuzzy Graph-Based Machine Learning Optimisation (BFG—MLO) model’s superiority in terms of flexibility, sustainability, and optimisation efficiency. As a result, the FG-MLO model is more suited for incorporation into smart cities that prioritise effective resource management and long-term sustainability.

Table 9 Hypothetical numerical comparison table.

Conclusion

The most susceptible area to flooding in Thoothukudi has been identified in this paper using a bipolar intuitionistic fuzzy network. The dominance theorems in fuzzy graphs have important implications for understanding the structure and behaviour of complex systems, and have been used in many pragmatic scenarios. A well-designed permeable pavement technology is used to manage areas that are inundated by rainfall and cyclones, reducing the impact of storm occurrences, disaster management, and location selection which is done by using fuzzy graph dominant problems. The Bipolar Intuitionistic Fuzzy Graph-Based Machine Learning Optimisation (BIFG-MLO) framework provides decision-makers with a methodical and flexible approach that helps them manage environmental data uncertainties while optimising the placement and efficacy of permeable pavements. The BIFG-ML model maximises permeability and minimises surface runoff in flood-prone locations, improving urban flood resilience through iterative learning and improvement. Thus, the decision to set up the advanced asphalt system in these impacted areas is even more consequential since it supports higher-quality drainage systems with the most comprehensive protection of nearby communities. Through the utilisation of performance indicators like accuracy and precision, this strategy ensures the monitoring systems are both effective and malleable enough to adjust to changing circumstances. This technique has several benefits for tracking, evaluating, and enhancing complicated, erratic, or unpredictable systems. The proposed algorithm and fuzzy graph model collaborate with each other in order to produce prominent results in determining the degree of flood damage. Furthermore, to mitigate the effects of disasters and save lives for residents of smart cities, the government should give priority to funding the setup of permeable pavement systems. The results show practical possibilities to reduce disaster risks and promote sustainable city growth, which encourages the government to prioritise financing for robust infrastructure. The quality and granularity of environmental and infrastructure data, which may be scarce or inconsistent in some areas, determine how effective the strategy is. To help with budgetary considerations, incorporate long-term life-cycle analyses and cost-benefit evaluations for permeable pavement projects. Using BIFG-ML optimization approach, the study significantly enhances urban flood management; yet, practical implementation and wider applicability present problems. To improve resilience and sustainability for upcoming smart cities, more research should concentrate on validation, real-time adaptability, and economic viability. This will help the nation significantly decrease the effects of tropical cyclones, which will promote environmentally friendly growth.