Capacitated Pickup and Delivery Problem with Time Windows

• CPDPTW is a combinatorial optimization problem that schedules paired pickup and delivery tasks under strict capacity and time-window constraints.
• It incorporates complex constraints such as demand balancing, route continuity, and precedence, making it a benchmark for advanced mathematical programming and metaheuristic approaches.
• Research on CPDPTW drives innovations in exact solvers, decomposition techniques, metaheuristics, and emerging quantum and learning-based optimization methods.

The Capacitated Pickup and Delivery Problem with Time Windows (CPDPTW) is a combinatorial optimization problem central to logistics, transportation, and modern service platforms. It generalizes the classical vehicle routing problem by requiring that each vehicle must fulfill paired pickup–delivery requests, each constrained by vehicle capacities and strict time windows at both origin and destination locations. The interplay of route construction, temporal feasibility, and demand pairing makes CPDPTW a benchmark problem for advanced mathematical programming, metaheuristics, and increasingly, quantum and learning-based optimization methods.

1. Formal Definition and Constraint System

The CPDPTW requires planning a set of vehicle routes such that all transportation requests—each specified by a distinct pickup and delivery node, demand (positive or negative), and [early, late] time window—are serviced without violating operational constraints. The key elements and restrictive conditions are:

• Demand and Vehicle Capacity: Let $N$ be the set of nodes (depot, pickups, deliveries), each with demand $q_i$ where $q_i > 0$ for pickups, $q_i < 0$ for deliveries. Each vehicle $k$ has capacity $Q_k$. The sequence $(y_{ik})$ tracks the cumulative load to enforce $0 \leq y_{ik} \leq Q_k$.
• Time Windows: Each node $i$ has associated time window $[e_i, l_i]$. Service must begin within those bounds: $e_i \leq A_i \leq l_i$.
• Pairwise Precedence: For each request, the pickup must occur before the corresponding delivery: if $i$ is a pickup for delivery $j$, then $A_i + s_i \leq A_j$ (for service durations $s_i$).
• Route Assignment and Continuity: Each node is visited exactly once, each route starts and ends at the depot, and vehicle flows are continuous, governed by binary decision variables $x_{ijk}$ (vehicle $k$ travels from $i$ to $j$).
• Objective Functions: Typical objectives include minimizing total travel cost, fleet size, tardiness penalty, or weighted combinations thereof. For example, a generic linear objective is

$$\min f = \sum_{k} \sum_{i} \sum_{j} C_k d_{ij} x_{ijk}$$

with $C_k$ the cost coefficient and $d_{ij}$ the Euclidean travel distance.

• Dynamic and Generalized Variants: Dynamic CPDPTW models handle stochastic request arrivals and time-dependent travel times, and extensions may allow for mid-route load transfers, cross-docking, or multi-agent (robot/crew) assignment (Stoia et al., 14 Aug 2024, Gkiotsalitis et al., 2023, Avgerinos et al., 4 May 2025, Chen et al., 2021).

2. Exact and Decomposition-Based Solvers

Standard approaches for small to moderate instances employ exact mathematical programming or combinatorial decomposition techniques:

• Mixed-Integer Programming (MIP, MILP): Initial formulations employ binary variables $x_{ijk}$, service time variables $A_i$, and load variables $y_{ik}$. The primary challenge is scalability due to the combinatorial explosion in the number of variables and constraints when encoding time windows, precedence, capacity, and assignment (Gkiotsalitis et al., 2023).
• Logic-Based Benders Decomposition (LBBD): The problem is split into a master problem (vehicle/request assignment and arcs) and a subproblem (precise routing, timing, synchronisation at transfer locations). At each iteration, infeasible or suboptimal proposals are cut off via logic-based Benders cuts:

$$z \geq \bar{z} - \bar{z} \left(|E| - \sum_{(i,j)\in E} x_{ij}\right).$$

This reduces optimality gaps and improves lower bounds, especially when warm-started from high-quality heuristic solutions (Avgerinos et al., 4 May 2025).

• State-Space-Time and Multidimensional Networks: By constructing a network whose vertices are tuples $(i, t, s)$—node, time, cumulative served requests—the dynamic programming backbone can prune infeasible labelings and directly embed constraints for capacity, precedence, and time windows (Mahmoudi et al., 2016).
• Branch-and-Cut and Valid Inequalities: Nonlinear constraints (such as ride time for perishables) are linearized using additional variables and valid inequalities are injected to tighten the MILP relaxation, accelerating convergence to global optimality (Gkiotsalitis et al., 2023).

3. Metaheuristics and Hybrid Approaches

Metaheuristic methods dominate large-scale and real-time variants due to their flexibility and computational efficiency:

• Genetic Algorithms (GAs): Solutions are typically encoded as permutations of node sequences (chromosomes), with dual populations for node assignment (Pnode) and per-vehicle allocation (Pvehicle). Correction operators enforce precedence and capacity; crossover and mutation operators are designed to preserve feasibility (Dridi et al., 2010, Dridi et al., 2010, Dridi et al., 2010). Multi-criteria fitness functions (number of vehicles, tardiness, cost) are evaluated using Pareto dominance in multiobjective versions (Dridi et al., 2010, Dridi et al., 2011, Dridi et al., 2010).
• Memetic Algorithms: Advanced frameworks such as MATE combine evolutionary search with powerful local improvement phases. Key features include diversity-preserving initialization (e.g., grid-based RCRS heuristics), route-inheritance (RARI) crossover with regret-based repair, and constant-time move evaluation for efficient local search, enabling scalability to thousands of requests (Liu et al., 2020).
• Large Neighborhood Search (LNS): LNS iteratively destroys (removes a set of requests) and repairs (reinserts them, guided by insertion difficulty and similarity clustering) parts of the solution to explore the solution space. Advanced removal heuristics use feature-based dissimilarity measures and randomized insertion, making LNS robust and adaptable (Avgerinos et al., 4 May 2025).
• Parallel and Dynamic Heuristics: Parallel guided ejection search enables large-scale fleet minimization via insertion, squeeze, and out-exchange operations across multiple processors. Complexity is managed by careful design of move selection and cooperative solution sharing (Blocho et al., 2017).

4. Advanced and Emerging Methods

The CPDPTW continues to fuel algorithmic innovation, with recent approaches extending classical metaheuristics and exploring new computational paradigms:

• Deep Learning with Attention Encoders: Iterative constructive algorithms built on attention encoder–decoder models, extended to CPDPTW, represent nodes with rich feature embeddings and context-aware decoders. A notable addition is an insertion heuristic conducting quadratic-time feasibility checks for precedence and time window satisfaction—a limitation when scaling to large instances, but it provides rapid approximate solutions (Rabecq et al., 2022).
• Quantum Annealing and Hybrid Quantum–Classical Frameworks: Formulations as constrained quadratic models (CQM) allow the use of quantum annealers (e.g., D-Wave Leap CQM) for subroute optimization. Key constraints such as simultaneous pickups-deliveries, time windows, and vehicle mobility restrictions are encoded into objective and constraint quadratics, with classical orchestration for filtering and solution management, as in the Q4RPD technique (Osaba et al., 2 Apr 2025). Reinforcement learning using parametrized quantum circuits demonstrates improved parameter efficiency and solution scaling, with quantum entanglement layers directly representing pairing and spatial relationships in the route (Moosavi et al., 7 Aug 2025).
• Integrated Planning in Multi-Agent Systems: For robotic warehouse delivery and multi-driver scenarios, assignment and route planning are integrated using marginal-cost assignment heuristics solved via metaheuristic Large Neighborhood Search. This joint approach is superior to sequential assignment and path-finding in handling capacity, time windows, and collision avoidance (Chen et al., 2021, Lucci et al., 2021).

5. Extensions: Dynamic, Perishable, and Collaborative CPDPTW

Modern operational environments increasingly demand models that extend CPDPTW by integrating additional forms of dynamism or collaboration:

• Dynamic Route Planning and Capacity Expiration: When both vehicle availability and customer deliveries evolve stochastically, the CPDPTW is embedded in a Markov decision process. Destroy-and-repair heuristics informed by cost function approximation (CFA) dynamically reallocate outstanding and newly arrived requests, exploiting temporal capacity (remaining available service time) as a constraint (Stoia et al., 14 Aug 2024).
• Transfers, Crossdocking, and Perishables: Variants allow for mid-route transfers at crossdock points, which introduce new layers of vehicle synchronization, ride time (as a perishability constraint), and often nonlinear constraints. MILP linearization, valid inequalities, and branch-and-cut offer tractable solutions up to small/medium scales (Gkiotsalitis et al., 2023, Avgerinos et al., 4 May 2025).
• Collaborative and Crowdshipping Models: Routing on collaborative platforms may involve heterogeneous fleets, including crowdshippers appearing and disappearing with stochastic availability and capacity. Real-time destroy-and-repair heuristics (e.g., DRACE) jointly decide vehicle assignment, insertion, and bundling, explicitly modeling time-dependent travel and crowd-sourced labor (Stoia et al., 14 Aug 2024).

6. Benchmarking, Practical Impact, and Research Directions

The practical performance of CPDPTW algorithms is typically evaluated on established or novel benchmark sets:

Method/Feature	Instance Scale (requests)	Time Window Strictness	Special Constraints	Reported Impact
MILP/Branch-and-Cut (Gkiotsalitis et al., 2023)	up to 10	strict/perishable	Crossdocking, perishability	Global optimality, computationally heavy
LBBD (Avgerinos et al., 4 May 2025)	≤30 (exact), ≤100 (LNS)	loose/tight	Transfers, synchronisation	Tight lower bounds, warm start effective
MATE Memetic (Liu et al., 2020)	up to 1000	mixed	Simultaneous pickup-delivery	Outperforms SOTA, scalable
PQC-based RL (Moosavi et al., 7 Aug 2025)	up to realistic on-demand scales	varied	Quantum circuit constraints	Improved parameter efficiency
Deep Attention (Rabecq et al., 2022)	up to 100	standard	Feasibility via insertion	Fast, suboptimal for hard constraints
DRACE (Stoia et al., 14 Aug 2024)	dynamic/day-scale	stochastic, time-dependent	Dynamic crowdshipping	Improved real-time response, cost

Efforts to improve realism focus on: the introduction of instance generators reflecting geographic and temporal diversity (Avgerinos et al., 4 May 2025), new large-scale logistics benchmarks based on real-world data (Liu et al., 2020), and extended support for urban mobility restrictions, perishables, and hybrid resource assignment (Osaba et al., 2 Apr 2025, Gkiotsalitis et al., 2023).

7. Open Problems and Future Challenges

Despite extensive progress, CPDPTW remains computationally intractable for very large instances, especially when realistic extensions (transfers, perishability, dynamic arrivals) are incorporated. Significant open challenges include:

• Scalable, adaptive hybrid methods blending classical metaheuristics with learning- or quantum-based local solvers.
• Joint optimization of multi-actor assignments (robots, human drivers, crowdshippers) with evolving capacities and uncertain availabilities.
• Integrating more complex synchronisation, for example when combining crew scheduling with route optimization, or handling joint operations at crossdock or transfer points.
• Development of domain-agnostic, open-access large instance generators and standardized performance metrics.
• Algorithmic interpretability and robustness in real-time and uncertain environments, particularly for dynamic dispatching and route updates with minimal offline tuning.

The CPDPTW stands as both a theoretical touchstone for combinatorial optimization and a practical driver of algorithmic advances at the intersection of mathematical programming, metaheuristics, learning, and quantum computation.