Multi-objective optimization for smart cities: a systematic review of algorithms, challenges, and future directions

Fundamental model of MOO problems

Modern smart cities, as large-scale complex systems, inherently pose scientific challenges related to multi-objective coordinated optimization. MOO offers a rigorous computational approach to address key urban challenges, such as resource allocation, service scheduling, and operational efficiency by using mathematical models and algorithmic strategies. A typical smart city optimization problem can be formalized as a vectorized multi-objective model:

(1) $$\begin{array}{cc}\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}:& F(x)=[f_1(x),f_2(x),\dots ,f_m(x){]}^T\hfill \\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}:& g_i(x)\le 0,i=1,2,\dots ,p\hfill \\ & h_j(x)=0,j=1,2,\dots ,q\hfill \\ & x\in \mathrm{\Omega }\hfill \end{array}$$where the decision variable vector $x=[x_1,x_2,\dots ,x_n{]}^T$ represents controllable urban parameters, such as traffic signal timing, energy network topology, and infrastructure layout. The objective function vector is defined as $F(x)=[f_1(x),f_2(x),\dots ,f_m(x){]}^T$, where each component $f_i(x)$ denotes a specific urban performance indicator, including traffic efficiency, carbon emissions intensity, and public service quality. In most urban systems, optimizing one performance metric often adversely affects others. Figure 2 presents two sample objective functions. As shown, improving one function often leads to degradation in the other, thus motivating the need for multi-objective trade-off analysis.

The optimization problem is further subject to a set of constraints. The inequality constraints $g_i(x)\le 0(i=1,\dots ,p)$ and equality constraints $h_j(x)=0(j=1,\dots ,q)$ represent physical limitations and operational rules embedded in urban systems.

For example, in urban logistics optimization, a common constraint is the vehicle capacity limit, which can be mathematically expressed as:

(2) $$\sum _{i=1}^n{d}_i{x}_{ik}\le Q_k,\mathrm{\forall }k\in \{1,\dots ,K\}$$where:

• $x_{ik}\in \{0,1\}$ is a binary variable indicating whether customer $i$ is served by vehicle $k$;

• $d_i$ is the demand of customer $i$ (e.g., in $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$);

• $Q_k$ is the maximum load capacity of vehicle $k$;

• $K$ is the total number of available vehicles.

Due to the conflicting nature of multiple objectives, conventional single-objective optimization methods are insufficient. Instead, the concept of a Pareto-optimal solution set is used, comprising solutions in which no single objective can be improved without compromising at least one other.

Pareto optimality and the Pareto front

In MOO, a solution is considered Pareto-optimal if it is not dominated by any other feasible solution in the search space. Formally, given two solutions $x^a$ and $x^b$, $x^a$ dominates $x^b$ (denoted $x^a\prec x^b$) if and only if:

(3) $$\mathrm{\forall }j\in \{1,2,\dots ,m\},f_j(x^a)\le f_j(x^b)\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\exists }k,f_k(x^a)<f_k(x^b).$$

This condition ensures that $x^a$ is no worse than $x^b$ across all objectives and strictly better in at least one. The set of all such non-dominated solutions forms what is known as the Pareto-optimal solution set.

To concretely illustrate this concept, consider the following bi-objective minimization problem:

(4) $$f_1(x)=x^2+2,f_2(x)=(x-2{)}^2+2.$$

Within the interval $x\in [0,2]$, no solution minimizes both objectives simultaneously. Instead, each solution represents a distinct trade-off between $f_1$ and $f_2$. The set of non-dominated solutions in this domain constitutes the Pareto-optimal set in the decision space.

Figure 3 illustrates the corresponding Pareto front in the objective space. Notably, at $x=1$, both objectives achieve balanced values, i.e., $f_1(1)=f_2(1)=3$, demonstrating a characteristic equilibrium point.

Mathematically, the Pareto front is defined as the image of the Pareto-optimal decision set ${\mathrm{\Omega }}^{\ast }$ mapped into the objective space by the vector-valued function $F(x)$:

(5) $${\mathcal{F}}^{\ast }=\left\{\left(f_1(x),f_2(x)\right)\mid x\in {\mathrm{\Omega }}^{\ast }\right\}.$$

This front encapsulates all trade-off solutions that cannot be improved in one objective without compromising another. Consequently, identifying the Pareto front is a central goal of most MOO algorithms, forming the foundation for informed decision-making in real-world applications. However, in practice, the shape of the Pareto front can exhibit complex geometries, such as discontinuities, multimodality, or loss of dominance pressure, especially in high-dimensional urban contexts. These complex Pareto front geometries, such as discontinuities or weak dominance pressure, pose serious challenges to algorithm convergence and diversity preservation. Consequently, understanding these structures is essential for the design and evaluation of MOO algorithms, as further discussed in later section.

Classification of multi-objective optimization algorithms

MOO algorithms have evolved through interdisciplinary inspiration, drawing from biological evolution, mathematical theory, physical sciences, and artificial intelligence. Bio-inspired algorithms, such as genetic algorithms (GA), particle swarm optimization (PSO), and ant colony optimization (ACO), mimic natural selection, swarm intelligence, and pheromone-based decision-making for global search. Mathematical theory-driven approaches, including the weighted sum method and Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D), leverage mathematical optimization techniques to structure and decompose multi-objective problems. Physics-inspired methods, such as simulated annealing (SA) and gravitational search algorithm (GSA), model optimization as energy minimization or force-driven convergence. With the advancement of artificial intelligence, machine learning-enhanced approaches, like neural network surrogate models and reinforcement learning-based optimization, integrate deep learning and dynamic decision-making to improve computational efficiency and adaptability. These algorithmic categories, along with their foundational principles and historical development, are summarized in Table 4.

Mapping objectives, domains, and constraints for smart city MOO

To enhance scope clarity and align MOO with urban priorities, Fig. 7 presents a structured conceptual framework. This visualization organizes the review around three key components: Fig. 7A core optimization objectives and their expected benefits, Fig. 7B major urban sectors where MOO techniques are applied, and Fig. 7C societal and governance constraints that influence model design and outcomes. Rather than listing algorithms by domain, the framework emphasizes both the technical applications of MOO and their strategic relevance in smart city contexts. It provides a high-level entry point that integrates environmental goals, operational challenges, and social considerations, offering a holistic perspective on how optimization interacts with real-world urban systems.

Algorithmic classification and scenario suitability

The classification of MOO algorithms for smart city applications depends on both their theoretical foundations and empirical performance across various urban scenarios. A clear classification framework helps reveal how each algorithm addresses complex urban challenges involving multi-dimensional trade-offs. For instance, NSGA-II has been applied to edge server placement in smart cities, improving infrastructure deployment and energy efficiency (Zhao et al., 2021). Similarly, a multi-objective genetic algorithm optimized vehicle routing in green city initiatives, enhancing urban transportation systems (Zhao, Bian & Mei, 2024).

Nature-inspired algorithms (NIA). This category represents the largest class of MOO algorithms, including evolutionary algorithms (EA) (e.g., NSGA-II, NSGA-III), swarm intelligence (SI) methods (e.g., PSO, ACO), and other techniques like gray wolf optimizer (GWO), firefly algorithm (FA), bat algorithm (BA). These algorithms mimic natural or physical processes through stochastic search, enabling flexible exploration of complex, nonlinear solution spaces. EA techniques demonstrate robust convergence in infrastructure planning and energy optimization problems (Maleki et al., 2022; Reitberger et al., 2024). However, their high computational demands necessitate distributed computing for scalability. SI algorithms excel in dynamic, real-time scenarios such as traffic flow optimization and IoT network management due to their adaptability (Sajid et al., 2021; Panda, Muthuraman & Elsts, 2024). Yet, they risk premature convergence and may require adaptive parameter tuning or hybridization. Other NIA approaches like GWO and FA have proven effective in resource allocation and traffic management but depend on robust feedback mechanisms to handle dynamic conditions (Othman, Darwish & Abd El-Moghith, 2023; Kazmi et al., 2025).

Physics-based and model-driven algorithms. These methods use deterministic modeling based on physical laws, contrasting with the stochastic nature of NIA. For example, energy flexibility was improved in a building–vehicle sharing network using optimization modeling (Zhou et al., 2020), and urban microclimates were co-designed with buildings for thermal optimization (Zhao, Li & Wang, 2025). These methods are suitable for energy transfer analysis, infrastructure design, and thermal regulation tasks (Wu et al., 2024; Shafiq et al., 2024). Their main strength lies in consistent, high-fidelity results crucial for smart city planning, though real-time scalability may be hindered by computational intensity.

Machine learning-assisted optimization. Integrating machine learning with MOO adds adaptability and predictive capabilities. For instance, spatial projection pursuit using geospatial data was employed to assess urban vitality (Zhang et al., 2024b). Deep learning enables modeling of complex, nonlinear patterns, supporting adaptive planning in logistics, energy systems, and land use (Preuveneers, Tsingenopoulos & Joosen, 2020; Zhang et al., 2024a). While effective in pattern recognition and predictive modeling, these approaches require computational resources and benefit from lightweight neural architectures and transfer learning.

Hybrid frameworks. These combine NIA’s exploratory abilities with machine learning’s predictive accuracy. For example, hybrid frameworks have been applied in fog–cloud computing environments requiring real-time responsiveness (Masoumi & van Genderen, 2024; Mokni & Yassa, 2024). By combining evolutionary search and data-driven models, they deliver both adaptability and efficiency. Yet, integration complexity remains a challenge, calling for modular and scalable system designs.

Cross-domain optimization strategies. These strategies synchronize multiple urban systems such as transportation, energy, and communication to work in concert. Coordinating across domains is pivotal for improving overall system efficiency and resilience (Stoyanova & Monti, 2019). For instance, optimizing network topology improved infrastructure layout (Zhou et al., 2019), enabling better service coordination. Such strategies are crucial in addressing the intertwined nature of urban problems and ensuring system synergy.

In summary, nature-inspired algorithms offer adaptability for dynamic problems, while physics-based and model-driven methods provide reliable results for critical infrastructure. Machine learning enhances adaptability and scalability in predictive tasks. Hybrid models combine strengths from both domains, and cross-domain strategies ensure holistic system integration. For instance, Zhang et al. (2024a) developed an adaptive MOO traffic signal model balancing efficiency and energy, exemplifying integrated algorithmic design for real-time urban systems.

Objective conflicts and resolution approaches

Resolving conflicting objectives is central to MOO in smart cities, where trade-offs between sustainability, efficiency, cost, and adaptability frequently arise. Based on our review of 117 studies, Fig. 8 visualizes the distribution of objective counts, showing that most MOO tasks fall into three categories of dimensional complexity: low, mid, and high. Each category presents unique modeling and computational challenges (Ishibuchi, Tsukamoto & Nojima, 2008). Detailed evaluation criteria and benchmark functions relevant to these conflict levels are discussed in subsequent sections to avoid redundancy.

Low-dimensional conflicts: rapid convergence and high adaptability

Low-dimensional conflicts (2–3 objectives) are frequently observed in energy systems, transportation networks, and smart building optimization. In scenarios balancing energy efficiency, thermal comfort, and operational cost, NSGA-II demonstrated robust convergence, achieving an 18% reduction in energy use while preserving indoor comfort and economic feasibility (Lotfi & Hassan, 2024).

In traffic optimization, PSO with adaptive inertia weight strategies simultaneously minimized travel time, emissions, and fuel consumption. Results reported include a 20% reduction in energy use and an 8% decrease in average travel time (Liu et al., 2023a). Despite its effectiveness, PSO’s performance deteriorates in non-convex landscapes, often requiring hybridization.

Mid-dimensional conflicts: balancing complexity, scalability, and convergence

With 4–10 objectives, mid-dimensional problems often arise in integrated land-use, transportation, and ecosystem planning. A hybrid NSGA-III + MOEA/D framework reduced ecosystem service loss by 15%, enhanced transportation accessibility by 10%, and maintained economic viability (Maleki et al., 2022). These algorithms effectively handle the increasing diversity and decision complexity inherent in urban interdependencies.

Further, combining NSGA-III with GWO and deep neural networks (DNNs) has proven effective in fog/IoT settings, improving resource utilization by 30% and reducing latency by 25% (Mokni & Yassa, 2024).

High-dimensional conflicts: underexplored but computationally demanding

High-dimensional conflicts (more than 10 objectives) remain underexplored due to the complexity of dominance preservation and solution diversity. New methods such as many-objective grasshopper optimization algorithm (MaOGOA) (Kalita et al., 2024) and MaOEA/TS (Zhao et al., 2024) address these limitations by integrating adaptive reference points and staged optimization. These approaches demonstrate promising results on test suites like Many-objective Test Function suite (MaF) and large-scale multi objective optimization problems (LSMOP) but are rarely applied in real-world smart city scenarios.

To operationalize high-dimensional optimization in practice, future work must balance scalability, interpretability, and convergence robustness, especially when trade-offs span multiple subsystems.

Evaluation criteria and benchmark resources in smart city MOO

To ensure fair comparison and reproducibility in MOO research, particularly in the diverse and dynamic context of smart cities—standardized evaluation criteria and benchmark resources are essential. This section integrates geometric considerations of Pareto fronts, performance metrics, benchmark functions, and real-world datasets to form a cohesive foundation for rigorous algorithm assessment.

1. Geometric characteristics and optimization implications. The structure of the Pareto front, whether known (e.g., convex or linear) or unknown (e.g., multimodal or disconnected), influences the effectiveness of optimization strategies. When the front is analytically known, mathematical techniques like the Karush–Kuhn–Tucker (KKT) conditions or $\epsilon$-constraint methods are employed. However, in complex urban systems, the front is often unknown, requiring evolutionary strategies that use non-dominated sorting, diversity maintenance, and adaptive parameter control. Table 5 summarizes the core evaluation metrics commonly used to assess the quality of Pareto-optimal solution sets in such scenarios.

2. Benchmark function suites for algorithm testing. Benchmark functions provide standardized environments to validate algorithm scalability, robustness, and adaptability. While ZDT, DTLZ, and WFG are often associated with low-, mid-, and high-dimensional test cases respectively, these suites are in fact highly configurable. Depending on the design of experimental studies, each can be adapted to various objective counts and Pareto front complexities (Deb et al., 2002b; Huband et al., 2006). Table 6 summarizes the typical characteristics of these dimensional categories along with representative benchmark functions commonly used in the literature.

3. Real-world datasets in smart city domains. To support the development and validation of MOO algorithms under realistic conditions, a range of domain-specific datasets have been employed across smart city research. These datasets enable algorithm testing under real-world constraints, capturing the complexity, noise, and multi-stakeholder objectives inherent to urban systems.

• Transportation: The METR-LA and PeMS datasets, collected from California highway sensors, are widely used in traffic forecasting and dynamic routing optimization (He, 2025; Caltrans Division of Traffic Operations, 2024). Additionally, the NYC Taxi Trip dataset supports large-scale mobility modeling and demand-responsive scheduling (New York City Taxi and Limousine Commission, 2024).

• Energy systems: The Pecan Street dataset provides high-resolution electricity consumption data at the household level, supporting demand response and smart grid optimization (Pecan Street Inc., 2024). The UMass Smart* dataset offers microgrid and appliance-level energy traces for distributed energy management research (University of Massachusetts Amherst, 2024).

• Water management: Hydrological modeling tools such as SWAT and EPA SWMM offer synthetic and empirical datasets for simulating stormwater drainage, flood risk, and infrastructure resilience (U.S. Department of Agriculture, 2024; Rossman & Simon, 2022). The WaterTAP dataset, curated by the U.S. Department of Energy, facilitates the optimization of water treatment and distribution systems (Atia et al., 2024).

• Cross-domain applications: Several benchmark initiatives, such as CityLearn and IEEG Smart City Challenge, integrate multiple urban subsystems (e.g., energy–mobility–comfort) to evaluate algorithm adaptability in coupled environments, bridging the gap between siloed research and real-world complexity.

These datasets serve as crucial resources for validating algorithm robustness and transferability, enabling MOO studies to move beyond theoretical constructs and address pressing urban challenges with real-time and multi-objective constraints. Together, these standardized resources form a methodological bridge between algorithm development and smart city deployment, fostering reproducible, scalable, and domain-adaptive MOO research.

Algorithmic strengths, weaknesses, and selection criteria

Bio-inspired algorithms. Bio-inspired algorithms have emerged as a dominant paradigm in MOO for smart city applications, driven by their population-based exploration capabilities and robust performance in complex, nonlinear, and dynamic environments. These algorithms have demonstrated high adaptability in diverse urban tasks such as land-use allocation, energy system control, transportation scheduling, and computational resource distribution. However, their performance can deteriorate in high-dimensional settings due to convergence slowdowns, increased computational overhead, and limited ability to preserve the diversity of solutions.

To interpret Table 7, we classify the performance of bio-inspired MOO algorithms across three core dimensions: convergence, diversity, and computational complexity, based on both structural characteristics and benchmark results reported in the literature.

In terms of convergence, NSGA-III (Seada, Abouhawwash & Deb, 2016; Gupta & Nanda, 2019) exhibits superior performance, primarily due to its reference direction—based niche preservation mechanism. Gupta & Nanda (2019) reported inverted generational distance (IGD) values as low as 0.005 across 3- to 8-objective benchmark problems, underscoring its high convergence accuracy. Similarly, Emotion Driven Monocular Face Capture and Animation (EMOCA) (Othman, Darwish & Abd El-Moghith, 2023) achieved strong convergence results in IoT routing scenarios, with fault tolerance reaching 0.98 under large-scale deployment conditions. In contrast, NSGA-II (Sato, Sato & Miyakawa, 2017; Li et al., 2022a) and SPEA2 (Liu & Zhang, 2019) demonstrate moderate convergence. For instance, Li et al. (2022a) employed SVR-TOPSIS analysis and found that NSGA-II achieved acceptable but suboptimal Pareto front accuracy. The standard Strength Pareto Evolutionary Algorithm 2 (SPEA2) implementation delivered consistent results but was outperformed by its improved variants on ZDT3 and DTLZ2 test suites (Liu & Zhang, 2019). ACO (Chitty, 2018; Sabery, Danishyar & Mubarez, 2023), while competitive in simpler contexts, reported up to 9.4% deviation from optimal solutions on TSPLIB instances, reflecting limited convergence capability in high-dimensional settings.

With respect to diversity, MOEA/D (Fu et al., 2021; Alagumathi & Thangavelu, 2024), NSGA-III (Seada, Abouhawwash & Deb, 2016), and EMOCA (Othman, Darwish & Abd El-Moghith, 2023) demonstrate a consistently strong ability to maintain well-distributed solution sets. Fu et al. (2021) introduced adaptive weight vector strategies in MOEA/D-AW, yielding significantly improved distribution on WFG benchmarks. Seada, Abouhawwash & Deb (2016) further showed that local search integration into NSGA-III enhances diversity preservation in complex Pareto landscapes. EMOCA integrates crowding distance evaluation throughout all evolutionary phases, effectively maintaining near-uniform spacing among non-dominated solutions. In contrast, NSGA-II, PSO, and ACO typically offer only moderate diversity maintenance. For example, Liu, Li & Zhu (2021) observed that standard PSO is susceptible to swarm stagnation, although diversity can be improved using biologically inspired variants such as SSI-PSO. Similarly, Chitty (2018) noted that ACO suffers from local optima entrapment without explicit diversity enhancement mechanisms.

Regarding computational complexity, PSO and MOEA/D are widely recognized for their computational efficiency. PSO operates with linear time complexity relative to particle count and dimensionality (Liu, Li & Zhu, 2021), making it suitable for real-time applications and large-scale problem settings. MOEA/D (Alagumathi & Thangavelu, 2024), leveraging subproblem decomposition and neighborhood-based update schemes, avoids the computational burden associated with nondominated sorting ( $O(M{N}^2)$) that is characteristic of NSGA-II and NSGA-III. On the other hand, NSGA-III (Gupta & Nanda, 2019), SPEA2 (Jiang & Yang, 2015), and EMOCA (Othman, Darwish & Abd El-Moghith, 2023) are associated with high computational demands. Jiang & Yang (2015) estimated that SPEA2’s k-nearest neighbor–based density estimation results in a worst-case complexity of $O(M{N}^3)$. NSGA-III’s reference-point association and EMOCA’s multi-phase crowding strategies further increase runtime overhead, which may limit scalability in time-sensitive urban applications.

This comparative synthesis enables a more nuanced understanding of the trade-offs among convergence reliability, diversity maintenance, and computational feasibility. It offers practitioners a decision-support reference for selecting appropriate bio-inspired MOO algorithms tailored to domain-specific constraints and resource availability.

To systematically evaluate the domain-specific applicability of bio-inspired MOO algorithms, a structured scoring framework was developed to integrate both empirical usage patterns and algorithmic performance characteristics. The results are summarized in Table 8, which presents a composite star-rating matrix across major smart city domains. Each algorithm—domain pair was assigned a composite score ranging from 0 to 6, based on two components. This 0–6 range ensures sufficient granularity while maintaining interpretability when translated into a five-level star system.

• •

  (i) Literature frequency (0–3 points): Quantified by the number of reviewed studies (out of 117) that applied the algorithm in the corresponding smart city domain:

  • –

    10 or more studies: 3 points

  • –

    5 to 9 studies: 2 points

  • –

    1 to 4 studies: 1 point

  • –

    0 studies: 0 points

• •

  (ii) Performance suitability (1–3 points): Assigned based on the algorithm’s convergence capability, solution diversity, and computational complexity, as summarized in Table 7. Specifically:

  • –

    3 points: “High” in two or more dimensions, or “High” in one and “Medium” in the others

  • –

    2 points: combination of “Medium” and one “High” or one “Low”

  • –

    1 point: two or more “Low” scores, or dominant “High” complexity without offsetting strengths

The combined score (0–6) was then translated into a five-level star rating to visualize algorithm suitability across domains.

These comparative insights are further supported by empirical applications across smart city domains. For instance, NSGA-II remains a dominant choice in land-use and environmental planning due to its reliable convergence and manageable complexity in low-dimensional tasks (Deb et al., 2002a; Liu et al., 2023b). Its successful deployment extends to blockchain-based smart contract testing (Alkhazi & Alipour, 2023), flood mitigation planning (Lin, Sun & Nijhuis, 2024), and stormwater infrastructure design, where it improved runoff control by 20% (Liu et al., 2023b). However, as the number of objectives increases, NSGA-II suffers from declining diversity and slower convergence, as noted by Feng et al. (2021). In contrast, NSGA-III and EMOCA demonstrate superior diversity maintenance and scalability in high-dimensional contexts (Maleki et al., 2022; Othman, Darwish & Abd El-Moghith, 2023), aligning with their high performance scores in Table 7.

MOEA/D, by decomposing the problem into scalar subproblems, achieves low computational overhead and has proven efficient for embedded applications such as IoT node placement and real-time transportation scheduling (Zhang & Li, 2007; Wang, Jin & Doherty, 2015). MOGA has shown superior performance in resource allocation, achieving optimal trade-offs in logistics and land use (Li et al., 2024), while NSGA-II remains effective in route optimization under emission constraints (Sajid et al., 2021).

In network-centric environments, EMOCA excels due to its crowding-based diversity control, enabling a 23% reduction in communication latency and enhanced fault resilience in IoT sensor networks (Othman, Darwish & Abd El-Moghith, 2023). PSO and ACO are favored for their low-latency convergence in dynamic routing and scheduling tasks, such as waste collection and service placement (Kazmi et al., 2025; Ahmad et al., 2020), though both are prone to premature convergence in complex landscapes.

Energy-focused applications highlight further trade-offs: NSGA-II has reduced CO₂ emissions by 7.5% in building-vehicle energy-sharing networks (Zhou et al., 2020) and improved fire safety in ventilation optimization (Xu et al., 2024), yet remains constrained by high computational demands, as seen in hydroelectric optimization (Uen et al., 2018).

To address such limitations, adaptive variants have emerged. Adaptive MOGA introduces self-adjusting mutation control to improve convergence in vehicle routing (Zhao et al., 2024); Hybrid PSO enhances farmland conservation with improved search capabilities but increased runtime (Wang et al., 2020); and Dynamic PSO incorporates inertia weight adjustment for improved stability in real-time optimization (Wang, 2020).

To synthesize the comparative evidence across algorithm families, Table 9 consolidates key algorithm classes with their typical domains, primary strengths and weaknesses, and representative quantitative outcomes. This table complements the prior two by offering more fine-grained, case-level guidance for real-world MOO problem settings.

In summary, while no single algorithm demonstrates universal superiority, this integrated evaluation highlights how convergence-diversity tradeoffs, computational constraints, and real-world objectives guide the practical selection of bio-inspired algorithms in smart cities. Future research should aim to enhance hybridization strategies and improve adaptability in dynamic and high-dimensional scenarios.

Mathematical optimization methods. Mathematical optimization methods are widely used in task scheduling, energy management, logistics, land use, and water resource management in smart cities to address multi-objective conflicts. Traditional techniques, such as mixed-integer programming (MIP), MOEA/D, and the weighted sum method (WSM), provide rigorous constraint handling and high computational precision. However, their computational complexity and limited adaptability in dynamic environments have led researchers to explore hybrid approaches and machine learning (ML)-enhanced optimization for improved efficiency and flexibility.

In task scheduling and energy management, optimization must balance execution efficiency, energy consumption, and carbon emissions while adapting to changing conditions. Abdel-Basset et al. (2021) proposed a Pareto-optimal strategy to reduce computational load and improve throughput. However, as a static optimization method, it struggles with real-time demand fluctuations. Zhao et al. (2023) applied WSM to balance cost, load shedding, and renewable energy integration, improving cost efficiency but highlighting challenges in economic feasibility and system resilience. A similar limitation is observed in energy management, where Algieri, Morrone & Bova (2020) employed MIP for CHP system optimization, achieving 7.5% higher efficiency and 15% increased energy self-consumption. Yet, MIP lacks adaptability to fluctuating energy demand, necessitating more flexible strategies.

To address this, machine learning-enhanced optimization has been introduced. Li et al. (2022b) proposed a data-driven model for energy management, reducing dependency on precise mathematical modeling while improving adaptability. Similarly, Pinki Kumar et al. (2025) combined fuzzy MOO and multi-criteria decision-making to develop an adaptive scheduling strategy. Unlike (Abdel-Basset et al., 2021), which relies on fixed optimization weights, this approach dynamically adjusts parameters, improving flexibility at the cost of higher computational demands.

Urban logistics and energy optimization also show an evolution in optimization strategies. Jiao & Zhang (2025) applied a gradient-based optimization approach to optimize logistics distribution and land use, significantly reducing transportation costs and carbon emissions. However, as a local optimization method, it is prone to converging to suboptimal solutions in complex multi-objective conflicts. However, as a local optimization method, it is prone to converging to suboptimal solutions in complex multi-objective conflicts.

Table 10 summarizes the representative mathematical optimization algorithms applied in smart city scenarios, highlighting their respective domains, advantages, limitations, and quantitative performance outcomes.

Physics-based optimization. Physics-based optimization plays a critical role in MOO, particularly in the domains of urban energy management and heat transfer. Computational fluid dynamics (CFD) is widely adopted to optimize urban airflow and thermal energy distribution (Cao et al., 2024). While CFD offers high-fidelity simulation and accuracy, its computational demands severely limit its use in real-time applications. As an alternative, data-driven thermal modeling has been explored to enhance computational efficiency. For instance, Shafiq et al. (2024) developed a CNN–LSTM model for HVAC system optimization, achieving 15.7–22.3% gains in energy efficiency and 21.8–28.5% improvements in occupant comfort. However, while such machine learning models improve feasibility, they often sacrifice interpretability and precision, thereby highlighting the trade-off between accuracy and computational practicality.

Beyond heat transfer, cross-domain optimization frameworks that integrate district heating, electricity, and water distribution networks are increasingly applied to improve overall urban energy efficiency. Nevertheless, these frameworks impose significant computational burdens, which has motivated the adoption of metaheuristic physics-inspired algorithms such as simulated annealing (SA) and gravitational search algorithm (GSA).

SA and GSA, both rooted in physical laws, have found extensive application in smart city optimization. SA mimics thermodynamic annealing processes and has demonstrated effectiveness in logistics and routing tasks. For example, Salehi-Amiri et al. (2022) employed SA for waste management optimization, yielding notable improvements in route efficiency. On the other hand, GSA, inspired by Newtonian gravitation, has shown promise in communication and network-related problems. Okafor et al. (2024) demonstrated that an Agile GSA (AGSA) variant improved 5G vehicular network efficiency by 57.43% over GA and PSO. Although GSA exhibits strong performance in network and mobility optimization, its application in urban energy and thermal management remains underexplored.

As smart cities continue to demand real-time, high-efficiency optimization, hybrid GSA–AI models represent a promising direction. The transition from pure physics-based techniques to hybrid metaheuristic–physics approaches reflects an emerging trend in urban optimization, where algorithms like SA and GSA are becoming essential for balancing computational efficiency with solution precision.

Table 11 summarizes representative physics-based optimization algorithms applied in smart city contexts, highlighting their domains, strengths, limitations, and real-world performance outcomes.

Machine learning-enhanced optimization. Machine learning-enhanced multi-objective optimization (ML-MOO) has demonstrated significant potential in a wide range of smart city applications. Compared to traditional optimization methods, ML-MOO combines machine learning’s predictive capabilities with the decision-making power of MOO, resulting in substantial gains in computational efficiency and reduced reliance on costly simulation models. Applications span energy management, transportation optimization, environmental monitoring, and infrastructure planning. However, ML-MOO also faces persistent challenges, including limited interpretability, constrained generalization, and high computational demands.

ML-MOO has notably improved computational efficiency. For example, Liu et al. (2024a) employed an XGBoost + NSGA-II framework for urban energy consumption optimization, achieving a 420 $\times$ acceleration compared to traditional CFD simulations. Similarly, Shafiq et al. (2024) utilized a CNN–LSTM model for thermal regulation in smart buildings, reducing energy usage by 22.3% while improving thermal comfort by 28.5%. In transportation, Dong et al. (2024) adopted deep reinforcement learning (DRL) for adaptive traffic signal control, which substantially reduced vehicle wait times and enhanced flow efficiency.

Despite these advancements, interpretability remains a core limitation. Yuan et al. (2024) applied SHAP analysis to enhance the interpretability of XGBoost in flood risk optimization, yet the method still lacked complete transparency in decision-making. Similarly, Jalali et al. (2024) achieved a 47% reduction in prediction error for air pollutant forecasting using an HNN + COOT model, but the opaque contribution of individual predictors hindered its usability in policy settings.

Generalization also remains a critical bottleneck. Wu et al. (2024) applied a Pix2Pix GAN + Genetic Algorithm approach for Beijing Hutong renovation optimization, but the model’s dependence on high-quality training data limited its adaptability to shifting data distributions. In the domain of water resource management, Tan & Yao (2023) found that ML-based optimization models failed to generalize effectively across different geographic regions, highlighting the challenge of transferability.

In addition to interpretability and generalization concerns, high computational cost remains a practical barrier. Preuveneers, Tsingenopoulos & Joosen (2020) reported that deep learning-based optimization increased computational overhead by over 30% compared to conventional algorithms in IoT and edge computing settings, thereby restricting deployment feasibility. Moreover, the majority of ML-MOO applications are currently confined to data-rich and infrastructure-abundant environments. Table 12 summarizes representative ML-enhanced optimization algorithms, highlighting their strengths, limitations, and empirical performance across diverse smart city domains.

To address these limitations, future research must prioritize (1) enhancing model transparency through symbolic regression and causal inference, (2) improving generalization using transfer learning and self-supervised learning to reduce reliance on labeled data, and (3) optimizing computational efficiency through lightweight architectures, model quantization, and edge AI deployment. By tackling these challenges, ML-MOO can evolve into a more interpretable, adaptable, and practical framework for managing complex, data-intensive smart city systems.

Based on the characteristics of different optimization problems, Table 13 provides general guidelines for selecting the most suitable algorithm depending on the problem structure and application scenario.