Capacitated Green Vehicle Routing Problem

• CGVRP is a vehicle routing problem that integrates vehicle capacity with energy limitations and environmental emission reduction requirements.
• It employs models like MILP, VNS, and memetic algorithms to optimize routes while managing refueling station visits and energy constraints.
• The framework drives innovations in sustainable logistics by balancing operational efficiency with low-emission objectives and regulatory compliance.

The Capacitated Green Vehicle Routing Problem (CGVRP) is an extension of the classical Vehicle Routing Problem that integrates environmental constraints and objectives—such as fuel or energy limitations, greenhouse gas (GHG) emission minimization, and the operational constraints introduced by alternative-fuel or electric vehicles. It combines vehicle capacity constraints with requirements for alternative-fuel station (AFS) visits, energy management, and—in advanced variants—models the emissions generated by routing choices. The CGVRP thus forms a critical testbed for next-generation logistics optimization under sustainability constraints, with growing industrial and academic relevance (Woller et al., 11 Nov 2025, Sabet et al., 2022, Xu et al., 6 Apr 2025, Kazemian et al., 2015).

1. Mathematical Formulations and Model Structures

The CGVRP is typically formalized on a directed graph $G=(V,A)$, with node set partitioned into depot(s), customer set, and alternative-fuel/refueling stations. Vehicles are identical, each having a maximum load $Q$ and, depending on the subvariant, a maximum energy or fuel capacity (battery for EVs, tank for AFVs). Modern CGVRP formulations extend the classical CVRP by introducing variables and constraints for energy consumption and station visits, alongside load and routing decisions.

Example: Minimalistic CGVRP (CEC-12 Benchmark) (Woller et al., 11 Nov 2025)

• Sets:
  • $D = \{0\}$: depot, $I = \{1,...,n\}$: customers, $F$: AFS, $F'$: multiset of $\beta = 2|I|$ copies of each AFS, $V = D \cup I \cup F'$
• Parameters:
  • $w_{ij}$: Euclidean distances; $b_i$: customer demand; $C$: load capacity; $Q$: battery capacity; $h$: consumption rate per unit distance
• Decision variables:
  • $x_{ij} \in \{0,1\}$: routing arc usage
  • $u_i$: load on arrival at node $i$, $y_i$: battery on arrival at node $i$

Objective: Minimize $\sum_{i,j \in V, i \neq j} w_{ij} x_{ij}$

Critical Constraints:

• Each customer is served once
• Each AFS copy is visited at most once
• Flow conservation (in-degree/out-degree)
• Load propagation: $u_j \leq u_i - b_i x_{ij} + C(1-x_{ij})$
• Battery management:
  • At customer: battery only decreases
  • At depot/AFS: full recharge

Generalized CGVRP Models further incorporate:

• Multi-objective cost/emission trade-offs (Sabet et al., 2022)
• Time windows, speed dependencies, nonlinear refueling (Kazemian et al., 2015, Sabet et al., 2022)
• Private AFS with limited simultaneous refueling spots and waiting times (Xu et al., 6 Apr 2025)

2. Core Variants and Constraint Families

The CGVRP literature spans diverse operational settings. Key variants include:

Variant Family	Key Distinctions
Fossil-fuel GVRP	Fuel tank, emission models (PRP, eco-routing)
Alternative-fuel/EVRP	Battery constraints, AFS/charger visits
Heterogeneous fleet	Vehicle-specific capacities/emissions
Time-dependent GVRP	Time-varying speeds, emission/fuel cost functions
Capacity/Compartmentalized	Multi-compartment, transshipment, pickups/deliveries
AFS with capacity limits	AFS with finite refueling spots, queuing/waiting
Multi-objective CGVRP	Joint minimization of distance, fuel, emissions

Specialized constraints incorporate nonlinear charging, stochastic waiting at stations, upstream lifecycle emission accounting, and limited charger/slot capacity at AFS (Sabet et al., 2022, Xu et al., 6 Apr 2025).

3. Algorithmic Approaches and Computational Benchmarks

Solution techniques for the CGVRP reflect the NP-hardness and operational complexity of combining capacity and energy management:

Exact Methods

• MILP, Branch-and-Cut, Dynamic Programming—effective for instances up to |C| ≈ 30–40 (Sabet et al., 2022)
• Limited scalability for nonlinear energy/charging or large networks

Classical and Advanced Metaheuristics

Variable Neighborhood Search (VNS) for CGVRP (Woller et al., 11 Nov 2025):

• Initialization: Density-based clustering, modified Clarke-Wright, one-pass AFS insertion
• Shaking: Generalized double-bridge (“p=2”)
• Local Search: Randomized Variable Neighborhood Descent (RVND); 2-opt, string exchanges, AFS-relocation
• Acceptance: Only improving moves, shake/diversify after plateau

Memetic Algorithm (METS) with SCTS/ELS (Xu et al., 6 Apr 2025):

• Global: Split giant tours by duration or by max driving-range (SCTS)
• Local: Constant-time evaluation, conditional AFS-insert move, nine customized neighborhood moves
• Population diversity and feasibility are adaptively balanced via biased fitness

Simulated Annealing and Other Metaheuristics (Kazemian et al., 2015, Sabet et al., 2022):

• Segment reversal, relocate, intra/inter-route swaps, temperature-based acceptance, time-dependent speed levels

Algorithmic Summary Table

Method	Strengths	Limitations
MILP	Optimal for small instances	Poor scalability
VNS	Robust, parameterized, scalable	Tuning, no opt. guarantee
METS (memetic)	Fast, scalable, constant-time moves	Needs diverse population
Sim Ann/Tabu/ALNS	Good quality, flexible	Parameter sensitivity

4. Benchmark Studies and Numerical Insights

CEC-12 benchmark (Woller et al., 11 Nov 2025) and M/GPS-derived sets (Xu et al., 6 Apr 2025) represent standard testbeds. Representative metrics are:

• VNS achieved mean gaps of 2.14% (16/17 at best-known solution), outperforming Simulated Annealing and Genetic Algorithm baselines (SA: 4.46%, GA: 7.11%)
• On large instances (|I|≥350), VNS provided 2–6% gap improvement over post-competition BACO
• METS memetic approach improved or matched 57/60 best-knowns for n≤100 and all 20 large-scale (n≤1000) real-world instances, with mean gaps 1.6%–3.98% vs. prior GRASP methods (Xu et al., 6 Apr 2025)
• Simulated Annealing found optimal or near-optimal solutions for medium (n≤50) and large-scale (n up to 100) problems within 2.8% gap of best-known exact solutions, at a fraction of the computation time (Kazemian et al., 2015)

All studies routinely report mean and standard deviation over multiple seeds, confidence intervals for convergence, and statistical tests confirming robust performance differentials (Woller et al., 11 Nov 2025, Xu et al., 6 Apr 2025).

5. Structural and Operational Insights

• Capacity vs. Energy Constraints: Capacity-based routing alone yields long sequences requiring post hoc AFS insertions; energy/battery constraints fragment routes, cluster boundaries often present as natural AFS insertion points.
• AFS Placement/Assignment: Dedicated operators (AFS-realloc in VNS, CAI in METS) significantly improve fine-tuning beyond standard permutation or swap neighborhoods, reducing final gap by up to 1% (Woller et al., 11 Nov 2025, Xu et al., 6 Apr 2025).
• Efficiency Trade-offs: Incorporating environmental costs (GHG, fuel) alters optimal speed, route length, and fleet size trade-offs, inducing non-trivial interaction between emission objectives and load balancing (Kazemian et al., 2015).
• Temporal Dynamics: Time-dependent travel speeds and station waiting times add substantial realism and complexity, with metaheuristics retaining scalability where exact relaxation fails (Kazemian et al., 2015, Xu et al., 6 Apr 2025).

6. Research Gaps and Future Directions

Recent surveys highlight the following open areas (Sabet et al., 2022, Woller et al., 11 Nov 2025, Xu et al., 6 Apr 2025):

• Nonlinear charging/refueling modeling in MILP/exact algorithms
• Mixed-fleet CGVRP and facility location-routing integration (multi-echelon, supply chain design)
• Realistic stochasticity: demand uncertainty, waiting times, station breakdowns
• Multi-objective formulations: jointly optimize distance, waiting, GHG
• Integration of deep reinforcement learning, IoV data, and federated/meta-learning for adaptive dynamic routing
• Lifecycle emission modeling (accounting for H₂ fuel, battery production, grid carbon intensity)

Benchmark code and data for new variants (e.g., METS's Beijing set) are increasingly available for reproducibility and scaling (Xu et al., 6 Apr 2025).

7. Significance and Practical Implications

The CGVRP framework is essential for operationalizing low-emission logistics, supporting regulatory compliance (carbon taxes, cap-and-trade), and optimizing new-energy vehicle fleets. Metaheuristic algorithms—especially VNS and memetic frameworks—provide robust, scalable solutions robust to the integration of new constraints, empirical performance demands, and future advancements in vehicle and fuel technology (Woller et al., 11 Nov 2025, Xu et al., 6 Apr 2025). The ongoing evolution of solution methods enables practical deployment in GIS-based logistics platforms, especially in urban and regulated environments, and motivates future work on dynamic, multi-agent, and data-driven vehicle routing under environmental constraints (Sabet et al., 2022, Kazemian et al., 2015).