Vehicle Routing Problem

Updated 10 July 2026

Vehicle Routing Problem (VRP) is a combinatorial optimization issue where multiple vehicles serve customers from a single depot to minimize travel costs or distances.
Its rich variant space incorporates constraints like capacities, time windows, risk, and fairness, often leveraging metaheuristics, machine learning, and quantum formulations.
Hybrid approaches combining exact methods, adaptive metaheuristics, and ML-enhanced strategies yield scalable, efficient, and explainable solutions for complex routing scenarios.

The Vehicle Routing Problem (VRP) is a fundamental combinatorial optimization problem in logistics and transportation, first introduced by Dantzig and Ramser in 1959. It generalizes the Traveling Salesman Problem from finding a minimum-length tour for a single salesman to the scenario of multiple vehicles starting from a depot, serving multiple customers, and returning to the depot, while optimizing criteria such as cost or total distance traveled. In the classical VRP, all vehicles depart from and return to a single depot, each customer is served exactly once by one vehicle, vehicle capacities are respected, and the aim is to minimize total travel cost/distances or other objectives. The problem is NP-hard, and its modern literature spans exact methods, metaheuristics, machine learning, quantum formulations, and a large family of rich variants (Abdoune et al., 28 Jun 2025, Vidal et al., 2019).

1. Classical definition and mathematical structure

A standard graph-theoretical formulation represents the problem on a directed graph $G = (V, A)$ , where $V = \{v_0, v_1, \dots, v_n\}$ contains a depot $v_0$ and $n$ customers. The main parameters are the number of identical vehicles $K$ , vehicle capacity $Q$ , travel cost $c_{ij}$ , and customer demand $q_i$ . A common binary decision variable is $x_{ijk}$ , equal to 1 if vehicle $k$ travels from $V = \{v_0, v_1, \dots, v_n\}$ 0 to $V = \{v_0, v_1, \dots, v_n\}$ 1, and 0 otherwise (Abdoune et al., 28 Jun 2025).

The classical objective is

$V = \{v_0, v_1, \dots, v_n\}$ 2

subject to capacity constraints, single-visit constraints, flow conservation, route start and end at the depot, and binary restrictions on the routing variables. In the formulation summarized in the literature, the capacity constraint is

$V = \{v_0, v_1, \dots, v_n\}$ 3

each customer is visited and left exactly once, every route starts and ends at the depot, and $V = \{v_0, v_1, \dots, v_n\}$ 4 (Abdoune et al., 28 Jun 2025).

This formulation is foundational but limited. The review literature emphasizes that homogeneous fleets, delivery-only service, static and deterministic demands, fixed travel times, a single depot, and a single-objective focus are often too simplistic for practical use (Abdoune et al., 28 Jun 2025). That observation is central to the evolution of VRP research from a canonical cost-minimization model to a broad class of routing problems with operational, temporal, stochastic, environmental, and organizational constraints.

Vidal, Matl, and Laporte organize VRP refinements along three lines: considering more relevant objectives and performance metrics, integrating vehicle routing evaluations with other tactical decisions, and capturing fine-grained yet essential aspects of modern supply chains (Vidal et al., 2019). A complementary overview classifies major constraint categories as customer demands, service types, vehicle-related features, time-related constraints, and depot constraints, and lists prominent variants including CVRP, VRPTW, VRPFFS, SVRP, DVRP, PDVRP, SDVRP, VRPB, HVRP, VRPD, EVRP, OVRP, MDVRP/2E-VRP, GVRP, and PVRP (Abdoune et al., 28 Jun 2025).

A recurring misconception is that VRP is exhausted by minimizing total distance. The survey literature instead identifies profitability and performance ratios, service quality, equity, consistency, simplicity, reliability, and externalities as major objective families (Vidal et al., 2019). This reorientation is visible in several specialized models.

In VRP with profits, not all customers need be visited. The team orienteering problem seeks to maximize total collected profit under a maximum distance per route, the capacitated profitable tour problem seeks to maximize profits minus travel costs under capacity constraints, and the VRP with private fleet and common carrier allows some customers to be delegated to an external carrier subject to a cost. These variants are unified in the two-resource VRP with profits, in which customer selection, assignment to vehicles, and sequencing of deliveries are combined decisions (Vidal et al., 2014).

Fairness appears explicitly in the Vehicle Routing Problem with Vector Profits, where each site has a vector of profits, one component per stakeholder, and the objective is to maximize the profit sum for the least satisfied stakeholder. Its max-min criterion is written as

$V = \{v_0, v_1, \dots, v_n\}$ 5

which is then reformulated as a mixed integer linear program with an auxiliary variable $V = \{v_0, v_1, \dots, v_n\}$ 6 (Lee et al., 2017). This moves VRP beyond scalar reward and toward stakeholder-balancing formulations.

Risk-aware routing provides another objective expansion. In a CVRP for road freight, logistics and accident risk costs are jointly minimized through a weighted objective

$V = \{v_0, v_1, \dots, v_n\}$ 7

where $V = \{v_0, v_1, \dots, v_n\}$ 8 is a safety level coefficient. Accident probabilities were estimated from official government data, risk costs were estimated via Monte Carlo simulation using cargo insurance loss data, and safer routes reduced risk cost by up to approximately 18% (Bilato et al., 2022).

Recent work also introduces structurally new routing classes. The Vehicle Routing Problem with Resource-Constrained Pickup and Delivery decouples dropoff and pickup: different vehicles may handle the two operations at the same location, resources process autonomously at the customer, and the objective is to minimize makespan. The model creates inter-route dependencies absent from standard pickup-and-delivery formulations and reflects autonomous robotics deployment, portable medical equipment distribution, disaster relief operations, construction tool rental, and agricultural sensor networks (Saseendran et al., 27 Feb 2026).

3. Exact methods, metaheuristics, and complexity

The algorithmic literature on VRP spans exact optimization, pseudo-polynomial dynamic programming, local search, large neighborhood search, clustering, adaptive memory, and hybrid pipelines. No single paradigm dominates across all variants.

On the exact side, the Vehicle Routing Problem with Vector Profits is solved through linear programming relaxation and column generation. The number of candidate routes is exponential, so the method solves a restricted LP, uses dual prices to generate improving columns, and then solves a reduced MILP. In the reported case studies, the optimality gap was very small, typically less than 3% (Lee et al., 2017). For general complexity, recent parameterized results show that uncapacitated VRP is fixed-parameter tractable when parameterized by treewidth, whereas for CVRP various parameterizations, including treewidth, are paraNP- and $V = \{v_0, v_1, \dots, v_n\}$ 9-hard; there is, however, an XP algorithm for CVRP when parameterized by both treewidth and the vehicle capacity (Döring et al., 12 Sep 2025).

Local and large-neighborhood methods remain central because many rich variants resist exact treatment. For VRP with profits, a large neighborhood search with implicit customer selection explores an exponential number of solutions in pseudo polynomial time. It combines standard VRP neighborhoods with a repeated Select algorithm based on resource constrained shortest paths. Computationally, this approach produced 52 new best known solutions, and even a local-improvement method to the first local optimum achieved an average gap of 0.09% on classic team orienteering benchmark instances (Vidal et al., 2014).

Cluster-first, route-second decompositions remain relevant in capacity-constrained settings. A binary tree k-means procedure recursively splits delivery points with $v_0$ 0 until each cluster satisfies a volume threshold of $v_0$ 1, producing compact sub-areas that can subsequently be routed. The method enforces feasibility but may generate small clusters and low space utilization, suggesting post-merge or re-clustering as a refinement (Hakim et al., 2020).

Rich VRP practice often relies on staged metaheuristics. One case study formulates a rich VRP with vehicle capacities, heterogeneous fleet, multiple depots, pickup and delivery pairs, time windows, fixed pause times, order-specific requirements, and other industrial constraints. It uses a two-stage strategy—VRP-stage for assignment and TSP-stage for sequencing—and a Timeline algorithm for time windows and pause times, together with Genetic Algorithm and Ant Colony Optimization. Across eight scenarios, the approach handled all given constraints in a reasonable time, and the Genetic Algorithm consistently produced feasible solutions (Lesch et al., 2021).

A more recent real-world study combines a randomized constrained Clarke-Wright heuristic, local search, and an adaptive memory procedure. The industrial case includes capacities, time windows, soft time windows, heterogeneous vehicles, dynamic fuel consumption, multi-trip delivery, crew skills, split delivery, and time-dependent routes. Relative to current state-of-the-art algorithms for vehicle routing problem with a large number of constraints, the reported average savings were 2.03% in delivery time and 20.98% in total delivery costs (Peric et al., 2024).

For the new VRP-RPD, exact methods are reported to be computationally intractable for instances beyond 16 customers. The proposed solution is a sequential two-stage metaheuristic pipeline in which a GPU-accelerated Adaptive Large Neighborhood Search produces a high-quality incumbent that is then used as a warm-start seed for a Biased Random-Key Genetic Algorithm. On TSPlib-derived benchmarks from 17 to 1000 nodes and multiple processing-time variants, the pipeline reduced makespan by up to 74% over baseline heuristics (Saseendran et al., 27 Feb 2026).

4. Machine learning and neural combinatorial optimization

Bai et al. divide hybrid analytics-and-ML research into two streams: ML-assisted VRP modelling and ML-assisted VRP optimisation (Bai et al., 2021). In the first stream, machine learning is used to estimate uncertain demands, time windows, and travel times, to support stochastic programming, robust optimization, chance-constrained programming, and data-driven forecasting. In the second, it is used to guide decomposition, neighborhood selection, hyper-heuristics, surrogate evaluation, and constructive policies such as pointer networks, graph neural networks, and actor-critic models (Bai et al., 2021).

Neural combinatorial optimization now targets increasingly rich VRPs. A recent approach for a time-constrained capacitated VRP with a finite, homogeneous vehicle fleet introduces ASAP, the Attention Score Amplifier for Pointer Network. The model uses an encoder-decoder architecture, is formulated in line with the Policy Optimization with Multiple Optima protocol, and is trained via Proximal Policy Optimization in an OpenAI Gym environment. Its objectives are minimizing total route distance and maximizing vehicle utilization. Benchmarked against PyVRP, the method outperformed heuristics in all tested cases, with the performance gap widening on larger and unseen instances and reaching up to 46% improvement in best cases (Deineko et al., 2024).

Another direction combines graph attention and quantum parameterization within deep reinforcement learning. The Quantum Graph Attention Network replaces conventional MLPs at critical readout stages with parameterized quantum circuits inside a DRL framework trained by PPO. In the reported comparison, classical GAT used 324,493 trainable parameters, whereas Q-GAT used 153,137 classical parameters and 1,350 quantum parameters, for a total of 154,487. The model reduced trainable parameters by more than 50%, converged after approximately 20 epochs versus approximately 40 for the classical GAT, and reduced routing cost by about 5% compared with classical GAT baselines (Giang et al., 19 Nov 2025).

These results do not eliminate the role of classical optimization. The review literature consistently presents ML as augmenting modelling realism and improving algorithmic performance in offline and online settings, rather than replacing analytical structure (Bai et al., 2021). This suggests that the most influential current systems are hybrid: they learn predictions, policies, or decompositions, but remain tightly coupled to routing constraints and combinatorial search.

5. Quantum formulations and hybrid quantum algorithms

Quantum VRP research typically begins by mapping routing decisions to QUBO or Ising form. In one formulation, a VRP instance $v_0$ 2 involves $v_0$ 3 vehicles and $v_0$ 4 locations plus a depot $v_0$ 5, binary variables $v_0$ 6, and squared Euclidean distances $v_0$ 7. The objective is

$v_0$ 8

with penalties enforcing that each location departs once, each location arrives once, the depot sends $v_0$ 9 edges, and the depot receives $n$ 0 edges. The penalized objective is then written as $n$ 1, converted into QUBO, and mapped to an Ising Hamiltonian through $n$ 2 (Azad et al., 2020).

Within that encoding, QAOA alternates a mixer Hamiltonian and a cost Hamiltonian,

$n$ 3

starting from $n$ 4 and optimizing the variational parameters classically (Azad et al., 2020). Simulations on IBM Qiskit considered the instances $n$ 5, $n$ 6, and $n$ 7, requiring $n$ 8 qubits, namely 12 qubits for $n$ 9 and 20 qubits for $K$ 0 and $K$ 1. The reported conclusion is explicitly heuristic: performance depends not only on the classical optimizer, the number of layers $K$ 2, and parameter initialization, but also on the problem instance itself, and there is no guarantee that finite- $K$ 3 QAOA yields the global optimum (Azad et al., 2020).

Noise sensitivity is a major limitation. A VQE-based VRP solver for 3- and 4-city instances, evaluated under amplitude damping, bit-flip, phase-flip, bit-phase-flip, and depolarizing channels, found that performance depends heavily on the noise model. Amplitude damping was the least damaging, whereas phase-flip, bit-phase-flip, and depolarizing noise were highly destructive; in noisy situations, deeper circuits were not beneficial (Mohanty et al., 2022).

Quantum machine learning has also been explored through quantum support vector machines. In 3-city and 4-city scenarios, corresponding to 6-qubit and 12-qubit circuits, amplitude encoding achieved the highest accuracy but was unsimulatable at 12 qubits due to circuit depth, angle encoding provided the best trade-off between accuracy and circuit complexity, higher-order encoding degraded with layer count, and IQP encoding had the lowest accuracy. Across the tested configurations, COBYLA performed best among the IBM Qiskit optimizers (Mohanty et al., 2023).

The present quantum evidence is therefore narrow but technically informative: small VRP instances can be cast into Ising or kernel-based quantum pipelines, yet optimizer dependence, circuit depth, qubit count, and noise remain dominant constraints (Azad et al., 2020, Mohanty et al., 2022, Mohanty et al., 2023).

6. Explainability, practice, and current directions

Practical deployment increasingly requires not only good routes but explanations. RouteExplainer is a solver-agnostic, post-hoc explanation framework that explains the influence of each edge in a generated route by extending counterfactual explanations based on the Action Influence Model to VRP. It adds an edge classifier to infer intentions, a dedicated loss function, and explanation-text generation by LLMs. Quantitatively, the edge classifier was evaluated on TSPTW, PCTSP, PCTSPTW, and CVRP; it processed 10,000 samples in less than 3 seconds, achieved Macro-F1 scores of 85–95% across most VRPs, achieved 69–78% on the 3-class PCTSPTW setting, and lost only 1–2% accuracy on counterfactual-route datasets (Kikuta et al., 2024).

The need for explainability is reinforced by the gap between textbook VRP and industrial routing. Rich VRP studies explicitly incorporate time windows, pause times, multiple depots, pickup and delivery pairs, heterogeneous fleets, hazardous-goods or certificate constraints, dynamic fuel consumption, multi-trip delivery, crew skills, split delivery, and time-dependent routes (Lesch et al., 2021, Peric et al., 2024). A postal-routing study further modeled a multi-objective problem on a roadmap with 25 vehicles and 30,000 deliveries per day, emphasizing route-length limits and fairness across postmen routes (Müller et al., 2018).

The review literature identifies several common shortcomings and active directions. One is scalability: current algorithms are often benchmarked on instances with a few hundred customers, whereas real applications require thousands (Vidal et al., 2019). Another is the need for dynamic, stochastic, and multi-depot models, wider adoption of heterogeneous fleets and green technologies, and greater multi-objective solution methods (Abdoune et al., 28 Jun 2025). In ML-assisted VRP, further challenges include interpretability, reliability, training data requirements, generalization, and the lack of integrated cross-disciplinary platforms (Bai et al., 2021).

Taken together, these strands define the contemporary VRP as a research field rather than a single problem. Its classical core remains the joint optimization of assignment and sequencing under routing constraints, but its current frontier includes fairness, profits, safety, explainability, learning-assisted decision making, quantum encodings, and structurally novel models such as resource-constrained pickup and delivery. The field’s continued evolution follows directly from the fact that practical routing is no longer a single-objective depot-to-customer exercise, but a multi-constraint, data-rich, and increasingly interactive optimization domain (Abdoune et al., 28 Jun 2025, Vidal et al., 2019).