Early research on VRPTW primarily employed exact algorithms, such as branch-and-bound and dynamic programming, to find optimal solutions. However, these methods often faced scalability issues with larger problem instances. Recent studies have aimed to enhance the efficiency of exact methods. For instance, Baldacci et al. [11] provided a comprehensive review of exact algorithms for VRPTW, discussing their applicability and limitations in large-scale scenarios.

To address the limitations of exact methods, metaheuristic algorithms have been widely adopted. Techniques like GAs, Ant Colony Optimization (ACO), and Particle Swarm Optimization (PSO) have been applied to VRPTW with notable success. Marinakis et al. [12] introduced a multi-adaptive PSO for VRPTW, demonstrating improved performance over traditional PSO variants. Furthermore, Zahedi and Khalilzadeh [13] proposed a hybrid metaheuristic algorithm combining the Nondominated Sorting Genetic Algorithm II (NSGA-II) with Teaching–Learning-Based Optimization (TLBO) to solve a bi-objective capacitated electric vehicle routing problem with time windows and partial recharging. Their method efficiently balances routing cost and environmental impact, illustrating the growing interest in sustainable and multi-objective VRP solutions.

Real-world applications of VRPTW often involve uncertainties such as stochastic customer demands and variable travel times. Traditional deterministic models may not adequately capture these uncertainties, leading to suboptimal solutions. Recent research has focused on stochastic and robust optimization methods to better handle these challenges. Florio et al. [14] discussed Bayesian learning techniques for correlated demands in stochastic VRP, offering insights into handling demand uncertainty effectively. Additionally, Indrianti et al. [15] developed a Green Vehicle Routing Problem (GVRP) model that integrates complex real-world constraints—including emission reduction and delivery limits—using GAs.

With the growing emphasis on sustainability, recent studies have integrated environmental objectives into VRPTW. Zahedi and Khalilzadeh [13] offered a hybrid metaheuristic algorithm combining NSGA-II and teaching–learning-based optimization for a bi-objective capacitated electric vehicle routing problem with time windows and partial recharging, addressing both economic and environmental considerations. Additionally, Fan [16] developed a hybrid adaptive large neighborhood search method for the time-dependent open electric vehicle routing problem with hybrid energy replenishment strategies, contributing to the advancement of sustainable transportation solutions.

The integration of adaptive mechanisms into hybrid metaheuristics has shown promise in enhancing the flexibility and efficiency of VRPTW solutions. Chen et al. [7] proposed an Improved Genetic Ant Colony Optimization (IGA-ACO) algorithm to efficiently solve VRPTW, integrating Solomon’s insertion heuristic for population initialization and accelerating convergence while optimizing route planning to meet vehicle capacity and time window constraints. This approach highlights the potential of combining adaptive strategies with hybrid metaheuristics to tackle complex routing problems effectively.

The landscape of VRPTW research has evolved from classical exact methods to sophisticated hybrid metaheuristics that incorporate adaptive mechanisms and uncertainty modeling. Recent advancements have also integrated sustainability considerations, reflecting the multifaceted challenges of modern logistics. Despite these developments, opportunities remain to further enhance the adaptability and robustness of VRPTW solutions, particularly through the integration of real-time data and machine learning techniques.

To provide a concise and comparative overview of recent contributions to the VRPTW and its variants, Table 1 summarizes the key characteristics of relevant studies, including the year of publication, problem variants addressed, uncertainty and sustainability aspects, the metaheuristic techniques employed, and a brief evaluation of their advantages and disadvantages.

The proposed method is initialized with a population of feasible routing solutions constructed using a randomized version of Solomon’s insertion heuristic [2]. A standard GA is then employed to evolve the population through selection, crossover, and mutation operators. After each generation, a LS procedure is applied to selected individuals to improve solution quality through neighborhood exploration. The process is iterated for a fixed number of generations or until convergence criteria are met.

Uncertain travel times are incorporated using a scenario-based simulation approach. For each solution, its fitness is evaluated by sampling from a set of stochastic realizations and computing an expected (or worst-case) cost. Time window violations are softly penalized to allow trade-offs under uncertainty.

Each individual in the population represents a complete set of vehicle routes. The chromosome is encoded as a permutation of customer indices, segmented into routes based on vehicle capacity and time window feasibility. The initial population is generated using a randomized greedy heuristic based on Solomon’s insertion rule.

To evolve the population toward better solutions, we apply standard genetic operators adapted to the VRPTW context. These include selection, crossover, and mutation, each designed to preserve route feasibility and encourage solution diversity:

Selection: We employ tournament selection with replacement to ensure selection pressure while maintaining diversity.

To refine candidate solutions, a LS procedure is applied after crossover and mutation. The neighborhood includes:

2-opt: Reverses a segment within a route to reduce distance.

The LS is applied using a first-improvement or best-improvement strategy, and applied probabilistically to avoid excessive computation.

Each solution is evaluated over a scenario set Ω of r sampled travel time matrices. The expected total travel time and time window violations are calculated: (15)f(s)=1|Ω|∑ω∈Ω(∑k=1m∑i=0n∑j=0nt~ijω⋅xijk(ω)+λ∑i=1nδik(ω))

This provides a robust fitness evaluation incorporating both stochastic travel costs and penalized infeasibility.

An adaptive strategy is used to control the mutation rate μ and the LS probability pLS. The adaptation is governed by the relative improvement in best fitness over the last τ generations, denoted as Δf: (16)Δf=fbestg−τ−fbestgfbestg−τwhere fbestg is the best fitness at generation g. Two thresholds ε1 and ε2 are used to modulate the adaptive behavior:

If Δf<ε1, indicating stagnation, we increase both μ and pLS to promote exploration.

In our experiments, we set ε1=0.002 and ε2=0.01, values determined empirically via sensitivity analysis (Appendix A). These thresholds strike a balance between avoiding premature convergence and promoting convergence speed across various instance types.

Algorithm 1 outlines the main steps of the proposed AHM. It integrates initialization, evaluation under uncertainty, evolutionary operations, LS, and adaptive control into a single iterative framework.

We use the well-known Solomon benchmark suite [2], which provides 56 VRPTW test instances categorized into three groups:

C-type: clustered customer locations with narrow time windows (e.g., C101–C206)

Each instance includes 100 customers, fixed depot location, vehicle capacity constraints, and strict time windows.

To simulate real-world travel time variability, we perturb deterministic travel times between customers with stochastic noise modeled as a Gaussian distribution: (17)t~ij=tij+εij,εij∼𝒩(0,σij2)

The variance σij2 is set proportionally to the deterministic travel time tij:(18)σij=ρ⋅tijwhere ρ is the uncertainty level (e.g., ρ=0.1 for 10% variability). For each candidate solution, we generate r=10 stochastic scenarios to evaluate robustness under uncertainty.

The assumption of Gaussian-distributed noise is widely adopted in transportation modeling due to its analytical tractability and its ability to approximate various sources of small, random fluctuations observed in urban traffic systems. Studies such as Florio et al. [14] have demonstrated the applicability of this assumption in capturing the effects of stochastic travel times in logistics.

Nonetheless, we recognize that real-world travel time distributions may deviate from Gaussian behavior, particularly in the presence of skewed delays, incidents, or heavy-tailed congestion effects. Alternative distributions such as log-normal or uniform could offer more realistic modeling in certain contexts. While a comprehensive evaluation of different noise models is beyond the scope of this study, we discuss their implications in Section 6.1 and identify this as an important direction for future research.

The algorithm’s parameters are configured as follows:

Population size: 100

All parameters were calibrated through preliminary testing and grid search on a subset of instances.

The main parameters used in our algorithm are listed in Table 2. These values were selected based on preliminary experiments and empirical tuning.

Experiments were executed on a machine with the following specifications:

CPU: Intel Core i7-12700H, 2.7 GHz

Each algorithm was run 10 times per instance to account for stochastic behavior, and average results are reported.

To assess performance under uncertainty, we use the following metrics:

Expected Total Travel Time: average cost across r stochastic scenarios.

For comparison, we benchmark our approach against a classic GA and a non-adaptive hybrid GA+LS.

To evaluate the effectiveness of the proposed AHM, we compare it against both traditional baselines and two recent state-of-the-art approaches from the literature:

Standard GA: a canonical GA with fixed parameters.

Table 3 presents the average performance across three representative Solomon instances (C101, R101, RC101), with 10 runs and 10 uncertainty scenarios per run. The proposed method consistently achieves superior results in expected travel time, time window feasibility, and robustness, while maintaining competitive runtime.

As shown in Fig. 3, the proposed method converges more rapidly and attains a significantly lower final cost than the other methods. This suggests that the combined effect of adaptivity and local refinement not only improves solution quality but also enhances convergence speed.

Overall, the proposed approach strikes a compelling balance between runtime, quality, and robustness—making it suitable for deployment in real-time or uncertainty-aware logistics environments.

To better understand how spatial distribution impacts solution robustness and efficiency, we analyze the performance of the proposed method across the three canonical categories of Solomon benchmark instances: C-type (clustered), R-type (random), and RC-type (mixed). Fig. 4 visualizes sample customer layouts from each category, revealing the inherent structural differences that influence routing complexity.

To ensure statistical validity, we evaluated five benchmark instances per category: C-type: C101, C102, C104, C106, C108; R-type: R101, R103, R105, R108, R110; RC-type: RC101, RC103, RC104, RC105, RC108. Each instance was solved using 10 independent runs, with 10 stochastic scenarios per run.

Fig. 5 compares the average expected travel time and cumulative time window violations across categories. As expected, performance is strongest on C-type instances due to spatial compactness, which facilitates more efficient route planning. R-type instances yield higher travel times and constraint violations, reflecting the challenge of servicing spatially dispersed customers. RC-type results fall between the two extremes, highlighting the method’s adaptability in hybrid spatial configurations.

To further evaluate robustness, Fig. 6 shows the distribution of expected travel times across all scenarios and runs. The C-type category demonstrates not only the lowest average cost but also the tightest interquartile range, confirming superior robustness under uncertainty. In contrast, R-type instances show greater variability, indicating sensitivity to stochastic disruptions.

These results reinforce that the proposed method adapts well across spatial structures and maintains reliable performance even under uncertain travel conditions. Its robustness is particularly pronounced in scenarios where spatial clustering can be leveraged.

To assess the robustness of the proposed method under increasing uncertainty, we vary the uncertainty level ρ∈{0.05,0.10,0.20}, which controls the magnitude of Gaussian noise applied to travel times. Table 4 presents the impact on expected travel time and cumulative time window violations for Solomon instance R101.

As expected, both cost and constraint violations increase with higher uncertainty levels. However, the algorithm maintains feasible solutions and smooth degradation, confirming its robustness to stochastic disturbances.

To evaluate the adequacy of the scenario count |Ω|, we conduct a sensitivity analysis on the same instance, varying the number of sampled scenarios per evaluation: |Ω|∈{5,10,20,50}. Table 5 summarizes the results.

The results indicate that while increasing the number of scenarios slightly improves robustness (lower standard deviation), the marginal benefit diminishes beyond |Ω|=10. Meanwhile, runtime grows substantially. This suggests that using 10 scenarios offers a well-balanced trade-off between computational cost and evaluation fidelity.

In summary, the proposed method handles varying uncertainty levels effectively and achieves stable performance with a moderate number of evaluation scenarios.

To assess the contribution of the adaptive mechanism, we disable it and compare results with the full version. Table 6 shows that adaptivity improves solution quality and reduces time window violations.

The experimental results demonstrate that the proposed AHM consistently produces high-quality and robust solutions across a diverse range of VRPTW instances, particularly under conditions of stochastic travel times. By integrating dynamic parameter tuning and LS, the method significantly outperforms both traditional and contemporary hybrid approaches.

Compared to recent algorithms such as the Multi-Adaptive PSO by Marinakis et al. [12] and the Hybrid GA-ACO by Chen et al. [7], the proposed method achieves lower expected travel times, fewer time window violations, and reduced solution variability. The convergence analysis further shows that adaptivity accelerates performance gains over generations, especially in complex or noisy routing scenarios.

Our robustness analysis (Sections 6.2, 6.3) confirms that the method remains effective across a wide range of spatial distributions and uncertainty levels. In particular, it shows superior stability on clustered instances (C-type), which typically benefit from tighter route packing and less travel-time variance. Moreover, our evaluation of the scenario count indicates that using 10 scenarios (|Ω|=10) offers a balanced trade-off between evaluation fidelity and computational cost, making it suitable for practical applications.

While the method incurs moderately higher runtime compared to a standard GA, the additional computational effort is justified in settings where solution feasibility and consistency are mission-critical—such as last-mile delivery, emergency routing, or green logistics [13,15,16].

In summary, the proposed approach offers a flexible, scalable, and uncertainty-aware routing solution that combines adaptive control, probabilistic modeling, and effective LS. Its ability to generalize across different VRPTW structures makes it a strong candidate for deployment in real-world transportation systems.

Future work could explore the following directions:

Real-time adaptation based on live traffic or delivery updates.

To complement the empirical performance evaluation, we analyze the time complexity of the proposed and baseline methods. The standard GA exhibits a time complexity of O(G×P×E), where G is the number of generations, P the population size, and E the cost of evaluating a solution. The hybrid GA+LS adds a LS cost L, resulting in O(G×P×(E+L)). Our adaptive GA+LS incorporates dynamic parameter tuning and robustness evaluation over r stochastic scenarios, leading to a total cost of O(G×P×r×(E+L)). Although this adds computational overhead, it substantially improves robustness and solution quality under uncertainty.