Fleet Size & Mix VRP: Theory & Algorithms

• FSMVRP is a comprehensive model that integrates fleet selection with routing constraints, including capacity, time windows, and uncertainty factors.
• The methodology leverages MILP formulations, robust heuristics like SCM, column generation, and deep RL to achieve near-optimal solutions efficiently.
• Domain-specific adaptations, such as for electric vehicles and multimodal fleets, offer practical insights for urban logistics and supply chain management.

The Fleet Size and Mix Vehicle Routing Problem (FSMVRP) is a foundational model in transportation science, integrating fleet selection (size and composition) with classical vehicle routing constraints such as capacity, time windows, multiple commodities, and other operational limitations. It encompasses both deterministic and robust (stochastic or adversarial) settings and forms the basis of numerous applied and theoretical research efforts across supply chain, logistics, urban delivery, and emerging applications such as electric vehicle fleet deployment, dynamic routing, and autonomous system planning.

1. Formal Problem Statement and Mathematical Foundations

At its core, FSMVRP extends the Capacitated Vehicle Routing Problem (CVRP) by allowing the optimization not only of vehicle routes but also of the number and types of vehicles in the fleet. Vehicle types can differ in capacity, cost, compatibility, propulsion technology, acquisition and usage costs, and operational feasibility.

The canonical deterministic FSMVRP is typically formulated on a directed graph $(V,E)$, with a depot $0$ and customer set $N=V\setminus\{0\}$. For a given set of vehicle types $T$ (each with capacity $Q_t$, fixed and/or variable costs $c^t$, compatibility sets, etc.), decision variables $x_{ij}^{tk}$ represent whether arc $(i,j)$ is traversed by vehicle $k$ of type $t$, and $y^{t,k}$ encodes fleet composition (purchase or deployment of vehicle $k$ of type $t$). Constraints enforce demand satisfaction, route feasibility (e.g., subtour elimination via MTZ or commodity-flow), capacity, type-customer compatibility, and, if present, temporal or geographical restrictions.

The objective is to minimize the combined fixed (fleet acquisition) and variable (operational/routing) costs, subject to these constraints. In robust or stochastic FSMVRP variants, the problem expands to include scenarios $K$ modeling, e.g., random demand, uncertain customer sets, or adversarial disruptions, leading to two-stage or min–max formulations with recourse and “here-and-now” variables (Makansi, 2024, Beatrici et al., 10 Dec 2025).

2. Robust and Stochastic FSMVRP Formulations

Robust FSMVRP introduces uncertainty in customer demand, service requirements, or constraint activation. The demand-robust FSMVRP, as in Makansi & Savla, represents the uncertainty in which subset of constraints (e.g., customer visits) will be active, and optimizes fleet size and vehicle mix so as to minimize the worst-case cost over all possible scenarios (Makansi, 2024).

Typical stochastic FSMVRP models (e.g., (Malladi et al., 2020, Beatrici et al., 10 Dec 2025)) posit a two-stage structure:

• First-stage decisions: select fleet composition and fixed routes, incurring acquisition costs.
• Second-stage (per scenario): adapt routes to realized demand/conditions, accounting for additional routing and penalty costs (recourse/outsourcing).

The mathematical structure couples MILP (mixed-integer linear programming) or path-based formulations with scenario-dependent continuous or integer variables, enforcing scenario-wise feasibility and incorporating scenario probabilities in the objective (to minimize expected total cost). For realistic instance sizes, direct MILP optimizations become intractable due to the explosion of binary variables.

3. Algorithmic and Heuristic Solutions

Given the combinatorial complexity, state-of-the-art FSMVRP research advances are largely algorithmic:

• Set Cover Mapping Heuristic (SCM): Reduces robust FSMVRP to a polynomial-size demand-robust weighted set cover problem by pre-generating and aggregating scenario-wise routes, dramatically shrinking the solution space. SCM achieves sub-2× empirical optimality gaps with exponential speedup relative to branch-and-bound MILPs (Makansi, 2024).
• Column Generation and Matheuristics: Long-horizon FSMVRP is addressed via column generation, treating daily routing as a subproblem and fleet assignment as a master problem. The Restricted-Master Heuristic (RMH) and Branch-and-Price (BAP) frameworks iteratively refine a master LP through dual-driven selection of “critical” days and solutions to embedded single-day FSMVRP subproblems (Bertoli et al., 2017).
• Sample Average Approximation (SAA): Monte Carlo SAA is paired with adaptive large neighborhood search (ALNS) to tackle stochastic FSMVRP by sampling demand and operational scenarios, efficiently estimating expected costs and robust fleet compositions (Malladi et al., 2020).
• Metaheuristics (MCTS, Kernel Search, VNS-TS): Monte Carlo Tree Search guides B&B algorithms, offering rapid candidate solutions for fleet size/composition and seeding exact optimization; Kernel Search iteratively builds a promising “kernel” of routes in path-based models, and Variable Neighborhood Search–Tabu Search hybrids efficiently solve massive-scale EV fleet mix, routing, and charger allocation under urban logistics constraints (Baltussen et al., 2023, Beatrici et al., 10 Dec 2025, Uhm et al., 2024).

Author/Method	Optimization Layer	Key Features/Scale
Makansi & Savla (SCM) (Makansi, 2024)	Set cover heuristic	Demand-robust, handles time windows/compatibility
Bertoli et al. (RMH/BAP) (Bertoli et al., 2017)	Column generation, matheur.	Long horizon, complex constraints
Goeke et al. (SAA+ALNS) (Malladi et al., 2020)	SAA + recourse search	Electromobility, realistic energy modeling
Cuda et al. (KS) (Beatrici et al., 10 Dec 2025)	Path-based MILP + KS	Consistent/stochastic last mile, scenario reduction
Duan et al. (FRIPN) (Wan et al., 30 Dec 2025)	Deep RL policy network	Fast, end-to-end fleet+route decisions, large-scale

4. FSMVRP Variations and Domain-Specific Constraints

FSMVRP exhibits substantial flexibility to encode domain-specific constraints:

• Split Deliveries and Commodities: FSMVRP can incorporate split-delivery routing, multiple commodities, and vehicle–commodity incompatibility. Commodity-flow MIP formulations and stable-fleet models achieve tight relaxations and strong empirical results (gap < 1.1% in practice); valid inequalities and CP warm-starts further strengthen performance (Mahéo et al., 2016).
• Electric and Multimodal Fleets: Recent models embed battery, fuel, and energy constraints to co-optimize fleet electrification, charger allocation, shift assignment, and routing under physical constraints (e.g., range, recharging, urban off-peak rules) (Araghi et al., 2 Dec 2025, Uhm et al., 2024). Multi-shift and off-hour delivery, charger co-design, and heterogeneity in vehicle types are explicitly modeled.
• Consistency and Recourse: Consistent FSMVRP addresses the need to maintain repeated assignment patterns over time (e.g., for customer satisfaction or regulatory requirements). Two-stage stochastic models with local recourse or path-based consistent routing, such as in the Italian postal case study, demonstrate practical approaches to balancing efficiency, cost, and reliability under uncertainty (Beatrici et al., 10 Dec 2025).

5. Computational Experience and Empirical Scalability

Comprehensive benchmarking across academic and industry datasets highlights several recurring empirical phenomena:

• Scalability Constraints: Complete MILPs become intractable beyond moderate instance sizes (e.g., >20–30 customers, >3–5 vehicle types, >10 scenarios) due to the combinatorial increase in binary variables required for each additional scenario or type (Makansi, 2024).
• Heuristic Effectiveness: SCM, column generation, and hybrid metaheuristics (e.g., MCTS+B&B or Kernel Search) attain nearly optimal solutions (empirical gaps often below 2%) on realistic instance sizes—orders of magnitude faster than brute-force or even advanced MILPs.
• Domain Adaptation: Metaheuristics and RL-based policies generalize to large-scale, real-world instances (hundreds–thousands of customers, multi-modal fleets, dynamic or uncertain settings). For example, deep RL approaches (FRIPN) generate solutions for $n=1000$ within tens of seconds, outperforming ALNS and traditional heuristics by significant margins (Wan et al., 30 Dec 2025).
• Long-Horizon and Real Data: Across long-term fuel distribution and last-mile delivery, matheuristics exploit demand variability to improve average fleet utilization (reducing total vehicle count and idle time by ~50% vs. traditional “union fleet” baselines) (Bertoli et al., 2017, Beatrici et al., 10 Dec 2025). Scenario reduction, path-based reformulation, and local recourse facilitate tractability and managerial interpretability.

6. Future Directions and Theoretical Gaps

Ongoing research addresses several open challenges:

• Approximation Guarantees: Few FSMVRP heuristics provide provable worst-case approximation ratios; empirical ratios (e.g., SCM’s sub-2×) may not translate to robust theoretical guarantees (Makansi, 2024). This remains an important direction for theoretical development.
• Learning-Augmented and Online FSMVRP: Deep RL and hybrid methodologies (e.g., FRIPN) offer rapid and scalable FSMVRP solutions; extending these to handle richer constraints (multi-depot, time windows, energy, online arrivals) is a key research frontier (Wan et al., 30 Dec 2025).
• Robustness and Recourse Modeling: Demand-robust and stochastic FSMVRP formulations increasingly reflect realistic operational uncertainty. There remains significant interest in quantifying the value of information (e.g., EVPI, VSS), managing recourse, and deriving practical policies for recourse routing and outsourcing (Beatrici et al., 10 Dec 2025, Malladi et al., 2020).
• Integration with Infrastructure Planning: Simultaneous optimization of fleet, chargers, and facility locations—especially under electrification and decarbonization constraints—defines a rich cross-disciplinary research agenda (Uhm et al., 2024, Araghi et al., 2 Dec 2025).

FSMVRP thus encapsulates both foundational modeling advances in vehicle routing and a rapidly expanding space of computational, algorithmic, and domain-driven innovations, reflecting the increasing complexity of modern transportation and logistics systems.