Consider a static IoT sensor network modeled as an undirected graph:G=(V,E)where:

V={v1,v2,…,vN} represents sensor nodes

In a large-scale smart city implementation, several edge gateways or sinks can be installed to minimize communication latency and reduce backbone overload. The suggested framework facilitates the selection of the multi-sink, allowing the heads of the clusters to be linked to the closest best sink for routing optimization.

Node attributes:(1)vi=(xi,yi,Eires,Degi2hop,Ci,wi) (2)dij=(xi−xj)2+(yi−yj)2

In smart city IoT deployments, sensing traffic may have different priority levels depending on application criticality (e.g., emergency monitoring, traffic control, environmental sensing). Therefore, each node vi is additionally associated with a priority weight wi, representing data urgency, where Si represents the set of one-hop neighboring nodes of the sensor node vi, vi, i.e., Si={vj∈V∣dij≤R}

The first-order radio energy model is adopted in the framework to capture communication energy expenditures precisely. Transmission energy is a function of transmission distance and packet size, with short-range transmission following the free-space model and long-range transmission following the multipath fading model. The energy received is represented as a linear function of packet size. Once transmission and reception are in operation, the residual energy is continuously updated. This energy formulation ensures that decisions on clustering and routing are made to capture realistic battery-depletion dynamics in IoT nodes.

First-order radio model:(3)Etx(k,d)={kEelec+kϵfsd2,d<d0kEelec+kϵmpd4,d≥d0 (4)Erx(k)=kEelec

Residual energy update:(5)Eires(t+1)=Eires(t)−Etx−Erx

The proposed framework is developed based on several key assumptions while addressing key design challenges inherent to IoT-enabled wireless sensor networks. Sensor nodes are randomly deployed in a two-dimensional sensing area and remain static after deployment. Each node is initialized with equal energy to ensure uniform initial conditions. The base station is assumed to be fixed and may be located either inside or outside the sensing region. Furthermore, nodes can estimate inter-node distances from received signal strength, and the communication links between nodes are assumed to be symmetric. Despite these assumptions, several challenges must be addressed to ensure efficient network operation. One of the primary issues is uneven energy dissipation across nodes, which leads to premature node failures and reduced network lifetime. Frequent re-clustering further increases control overhead and affects overall efficiency.

Additionally, maintaining stable and reliable multi-hop routing paths in dynamic network conditions remains a complex task. The framework must also address scalability concerns as node density increases in large-scale deployments. Moreover, there is an inherent trade-off among energy efficiency, latency, and load balancing, making it difficult to optimize all performance metrics simultaneously. These considerations motivate the need for a multi-objective optimization-based clustering and routing framework, which is formally described in the subsequent sections.

The proposed method uses both the two-hop neighborhood degree and spatial centrality measures to enhance the quality of cluster-head selection. The two-hop degree has a broader influence on connectivity and can select nodes that are highly topologically important to the algorithm, even if they are not immediate neighbors. Centrality is used to show the extent to which a node is spatially balanced in its neighborhood, thereby minimizing intra-cluster transmission distance when a node is chosen as a cluster head. All these measures improve load allocation and minimize communication cost within clusters. (6)Degi2hop=∣{vj∈V|dij≤2R} (7)Ci=1∣Si∣∑j∈Sidij2

The cluster-head selection is formulated as a Pareto-based multi-objective optimization problem with binary decision variables. This goal is achieved by reducing the inverse residual energy, spatial centrality cost, distance to the sink, cluster load variance, and energy unfairness, as measured by the Jain fairness index. Rather than integrating goals into a single weighted objective, non-dominated solutions are determined to maintain trade-offs between conflicting goals. Several constraints are implemented, including minimum cluster-head count, energy threshold eligibility, and unique cluster membership, which ensure viable and balanced clustering configurations.

Define binary CH variable:(8)xi∈{0,1}

We formulate CH selection as a Pareto multi-objective optimization problem:(9)minF(X)={f1(X),f2(X),f3(X),f4(X),f5(X)}where:

Energy objective:(10)f1=∑I=1NxiEires

Centrality objective:(11)f2=∑I=1NxiCi

Sink distance objective:(12)f3=∑I=1Nximinsj∈Vsdis

Load variance:(13)f4=Var(L)=1K∑k=1K(Lk−L¯)2

Energy fairness (Jain index): The Jain fairness index ensures uniform energy consumption across nodes and penalizes imbalance. (14)J=(∑i=1NEires)2N∑i=1N(Eires)2

Fairness objective:(15)F5=1J

Constraints Coverage:(16)∑i=1Nxi≥KMinwhere KMin denotes the minimum required number of cluster heads to guarantee network coverage ∀regionrk,∃CHi∈rk

This constraint ensures that every sensing region in the smart city deployment maintains at least one active cluster head, guaranteeing continuous urban monitoring coverage even as energy depletion progresses.

Energy threshold:(17)Eires≥Eth,∀i, where xi=1

Unique membership:(18)∑i=1Naij=1,∀j

To solve the multi-objective cluster-head selection problem, an NS-AAA is employed. Each algae individual represents a potential cluster-head configuration. The population evolves through growth and adaptation phases, where candidate solutions move toward better-performing individuals while maintaining diversity. Continuous position updates are converted to binary decisions using a sigmoid function. Pareto dominance ranking and elitist preservation ensure that high-quality non-dominated solutions are retained across generations. This approach enables efficient exploration of the solution space without premature convergence.

Each algae individual:(19)Xp=[x1(p),x2(p),…,xNp]

Pareto dominance:Xa≺Xb⟺∀k,fk(Xa)≤fk(Xb)and ∃k such that (20)fk(Xa)<fk(Xb)

The population is ranked using non-dominated sorting. The elitist preservation mechanism ensures that non-dominated solutions are retained across generations, guaranteeing monotonic improvement of the Pareto front. The stochastic growth and adaptation operators maintain diversity, reducing premature convergence and enabling efficient exploration of the solution space.

AAA Phases (21)Growth:Xinew=Xi+αr(Xbest−Xi) (22)Adaptation:Xinew=Xi+βr(Xj−Xk)

Binary mapping:(23)xi=1ifσ(Xi)>0.5 0otherwise (24)∑i=1Naij=1,∀j (25)σ(Xi)=11+e−Xi

Elitist selection retains Pareto front solutions.

After Pareto-optimal cluster-head candidates are generated, a fuzzy inference system is applied for final refinement. Residual energy, two-hop degree, and centrality are normalized and provided as inputs to a Mamdani-type fuzzy system. The system outputs a suitability score indicating the likelihood that a node will serve as a stable cluster head. Within each two-hop neighborhood, the node with the highest fuzzy score is selected. This refinement stage enhances robustness against minor energy fluctuations and improves cluster stability. Triangular membership functions are used for the input values (Low, Medium, High). A rule base of 9 rules is defined that combines residual energy, centrality, and degree. Defuzzification is performed using the centroid method.

Normalized inputs:(26)Ei′=EiresEmax,Di′=Degi2hopN,Ci′=CiCmax

Mamdani inference:(27)Chancei=F(Ei′,Di′,Ci′)

Final CH:(28)CHi=1ifChancei=max(Chancej),j∈2−hop

The framework uses the Lyapunov drift analysis to avoid the long-term energy imbalance. Each node has an energy deficit queue, which is the difference between the initial and residual energy. A Lyapunov function is defined in such a way as to quantify the deviation of system energy with time. A drift penalty term is also added to the routing objective, which discourages pathways in routing that increase the energy imbalance. This mechanism mathematically guarantees limited energy variation and enables sustainable network operation across multiple rounds. The energy deficit queue is defined as Qi(t)=Einitial−Ei(t), representing the cumulative energy consumption of node i, enabling monitoring of long-term imbalance. (29)Qi(t)=Einit−Eires(t)

Lyapunov function: The drift control parameter η is empirically set to 0.1 in simulations to balance routing optimality and energy stability. (30)L(t)=12∑i=1NQi2(t)

Drift:(31)ΔL=L(t+1)−L(t)

Routing objective augmented:(32)R′(P)=R(P)+ηΔL

This ensures energy stability over time.

Multi-hop routing paths between cluster heads and the sink are established after cluster formation. The routing goal minimizes a weighted sum of transmission energy, the number of hops, the inverse residual energy, and the path load. To ensure long-term energy sustainability, the Lyapunov drift component is factored into the final routing cost. The AAA optimization process selects the optimal path that uses energy resources in the most balanced way, without compromising communication reliability.

Path:(33)P={vi1,vi2,…,vs}

Routing objective:(34)R(P)=λ1∑(i,j)∈PEtx(dij)+λ2∣P∣+λ3∑i∈P1Eires+λ4Load(P)+λ5∑wiEires

The weights are set as: λ₁ = 0.25 (energy), λ₂ = 0.20 (hop count), λ₃ = 0.20 (residual energy), λ₄ = 0.20 (load), λ₅ = 0.15 (priority)

The priority term ensures that delay-sensitive smart city data is routed through energy-stable paths, improving Quality of Service for critical applications.

Optimal route:(35)P∗=argminR′(P)

Priority term guarantees that delay-sensitive smart city traffic is routed along energy-stable paths and enhances the quality of service for essential applications. The routing optimization problem is solved using the AAA framework; the set of candidate solutions is encoded as viable multi-hop paths between the cluster head and the sink. At each iteration, the candidate paths are updated using the growth and adaptation operators, and infeasible paths are repaired via connectivity validation. The route with the lowest augmented routing cost is chosen as the best path.

Re-clustering is not performed in all rounds to minimize unnecessary control overhead. The protocol, on the other hand, checks the ratio of minimum residual energy over the average network energy. A drop in this ratio to a predetermined level will result in re-clustering. This control strategy helps prevent overcomputing and ensures the network imbalance is corrected before critical nodes fail.

Re-clustering is activated if:(36)EminEavg<θ

This strategy enhances scalability and extends network lifetime by avoiding redundant optimization cycles while preserving global energy stability.

The comparison of the mean end-to-end delay across all simulated routing protocols over the course of the simulation rounds is shown in Fig. 3. iPAFAR has the lowest end-to-end delay across all simulation rounds. iPAFAR achieves around 39 ms at 1000 rounds and is much lower than LEACH-M (66 ms on average), FQA (53 ms on average), and MKNDPC (55 ms on average). This can be credited to the effective cluster-chain routing strategy and the zed next-hop selection, which minimize transmission distance and reduce buffering delays at intermediates. LEACH-M has the highest delay during the simulation. Its deterioration is more evident beyond 500 rounds, where the delay rises rapidly from 45 ms to 66 ms at 1000 rounds. BKA-TOA exhibits lower delay than RANP-PSO or ME-LEACH but it still cannot achieve the same delay efficiency as iPAFAR.

The Packet Delivery Ratio (PDR) represents the reliability of data transmission. It is defined as the ratio of successfully received packets at the sink to the total packets transmitted by sensor nodes. Fig. 4 illustrates the PDR performance of all protocols over 1000 simulation rounds.

It is observed that PDR gradually decreases with increasing rounds due to node failures and energy depletion. However, the proposed iPAFAR framework consistently maintains a higher PDR than existing protocols. At 1000 rounds, iPAFAR achieves approximately 63%–68% packet delivery, whereas LEACH-M drops below 40%, and FQA and MKNDPC show moderate performance around 50%–55%. The incorporation of energy fairness and Lyapunov-based stability reduces packet drops and improves route reliability, particularly in later stages of network operation.

Residual energy is the energy left in the network and directly affects network lifetime and stability.

Energy-saving measures ensure a long network lifetime. Based on Fig. 5, all protocols exhibit a decrease in residual energy with increasing rounds. LEACH-M has a high energy consumption rate and almost reaches zero J after 1000 rounds, indicating premature network failure. DV-HOP-DNN and RANP-PSO have around 5–10 J of energy, and BKA-TOA has around 17 J at the conclusion of the simulation. Conversely, the proposed iPAFAR conserves about 43 J of residual energy at 1000 rounds, which is much higher than that of any other protocol.

Fig. 6 shows the alive nodes at the 1000th simulation round for all the routing protocols examined.

As anticipated, the active nodes decrease gradually over time due to energy loss and the load on the cluster head. Nevertheless, the node death rates across different protocols differ significantly. The iPAFAR method suggested shows the greatest network survivability throughout the simulation. At 1000 rounds, iPAFAR still has about 47 live nodes, whereas LEACH-M has none, and FQA has fewer than 10. LEACH-M has the greatest node depletion. At 500 rounds, the number of alive nodes reduces drastically, and by the conclusion of the simulation, the number of alive nodes is almost 0. The survivability of RANP-PSO and ME-LEACH among intermediate protocols is relatively high compared with FQA and MKNDPC. Nevertheless, their node survival rate after 700 rounds drops by several folds, indicating minimal long-term load balancing. In general, the findings indicate that iPAFAR substantially increases network lifetime and stability, reducing premature node death and sustaining the cluster balance.

Fig. 7 provides the results of the Load Balance Index (LBI). As the number of network rounds increases, LBI values increase progressively across all protocols due to uneven residual energy use.

The Load Balance Index (LBI) measures the variation in residual energy across sensor nodes. It is computed as the average squared deviation of individual node energy from the network’s mean energy. A lower LBI value indicates better load balancing, as it reflects a more uniform energy distribution among nodes. However, iPAFAR consistently has the lowest LBI values throughout the simulation, indicating better workload distribution among the cluster heads. LEACH-M exhibits the greatest imbalance, especially at 300 rounds, which is why its nodes were quickly depleted, and, as shown in the earlier analysis, the network lifetime was shorter. Similarly, MKNDPC and FQA energy-adaptation strategies are inefficient.

The clustering time of the considered protocols varies across simulation rounds, as shown in Fig. 8. The offered iPAFAR algorithm maintains a relatively low, constant average clustering time of 5–6 ms, indicating that it is very efficient and consistent in creating clusters. Conversely, MKNDPC cluster heads and node assignment. LEACH-M has the shortest absolute clustering time (approximately 45 ms); however, this is due to the random selection of cluster-heads, which can affect the quality of the clusters and the energy balance. FQA and RANP-PSO have moderate clustering times, with minor variations across rounds, whereas BKA-TOA has a relatively stable, slightly higher value than iPAFAR. On balance, the findings show that iPAFAR can achieve a good trade-off between computational complexity and clustering efficiency. It is not the minimum, but its clustering time is always low, providing better network stability and energy balancing, making it more appropriate for long-term WSN deployments than traditional methods.

The jitter performance, as shown in Fig. 9, increases gradually with the number of rounds across all routing protocols as nodes lose energy and experience greater traffic congestion in the latter phases of the simulation. Nevertheless, the iPAFAR scheme always has the minimum jitter in all rounds. iPAFAR has a jitter of approximately 3.5 ms at 1000 rounds, and LEACH-M has the largest delay variation of more than 6 ms. The sharp increase in LEACH-M after 500 rounds indicates irregular routing paths and an uneven workload distribution among cluster heads. Conversely, iPAFAR exhibits smoother growth, better controls fluctuations, and indicates improved congestion management and adaptive routing.

The relatively stable protocols, like FQA and RANP-PSO, perform moderately and only become evidently unstable after 700 rounds. The findings verify that iPAFAR provides more stable timing and consistent transmission in dense WSN environments. The overall computational complexity of the proposed framework is approximately O(P×G×N), where P is population size, G is generations, and N is number of nodes. Additional overhead from fuzzy refinement and routing optimization remains linear, making the approach computationally feasible.