Capacitated Location-Routing Problems (CLRP)

• CLRPs are integrated combinatorial optimization problems combining depot selection and vehicle routing to minimize costs under capacity constraints.
• They involve complex interactions of location, allocation, and routing decisions, necessitating exact, approximation, and heuristic solution techniques.
• Practical applications span multi-depot logistics, sustainable delivery planning, and inventory routing, with recent advances leveraging machine learning and metaheuristics.

The Capacitated Location-Routing Problem (CLRP) is a foundational class of integrated combinatorial optimization problems at the intersection of facility location and vehicle routing. CLRPs model situations where one must select a subset of depots to open from a discrete set, assign customers to these depots, and design cost-minimizing vehicle tours to meet all demand, subject to depot and vehicle capacity constraints. CLRPs subsume capacitated facility location and capacitated vehicle routing as special cases, but present added combinatorial and computational complexities due to the interplay between location, allocation, and routing decisions. The scientific literature has established a variety of rigorous mathematical models, approximation frameworks, heuristic methods, and, more recently, machine- and learning-based solution strategies for CLRPs.

1. Mathematical Formulation and Structural Properties

A standard CLRP instance consists of a set of candidate depot locations $I$, with depot $i \in I$ offering opening cost $o_i$ and capacity $w_i$, a set of customers $J$, each $j \in J$ having demand $d_j$, and a metric cost structure $C = (c_{uv})$ for $u, v \in I \cup J$. Vehicles, typically homogeneous with capacity $Q$ and fixed per-use cost $F$, are routed from open depots to serve the assigned customers. Let $y_i \in \{0,1\}$ indicate whether depot $i$ is opened, $z_k \in \{0,1\}$ encode whether vehicle $k$ is used, and $e_{uv} \in \{0,1\}$ or flow variables indicate whether arc $(u, v)$ is traversed. The canonical mixed-integer programming model is (see (He et al., 14 Mar 2024)):

$$\min \sum_{i \in I} o_i y_i + F\sum_k z_k + \sum_{(u,v) \in A} c_{uv} e_{uv}$$

subject to:

• At least one depot is opened.
• Each customer is visited exactly once in and out (edge-based constraints).
• Flow conservation on customers.
• Vehicle-depot assignment: customers can only be served via open depots and active vehicles.
• Depot and vehicle capacity constraints: $\sum_{j \text{ served by } i} d_j \le w_i y_i$, and for each route, $\sum_{j \text{ on route } k} d_j \le Q z_k$.
• Standard subtour-elimination constraints (e.g., MTZ or connectivity cuts).

This formulation generalizes the capacitated vehicle routing problem (CVRP, one depot) and the capacitated facility location problem (no routing). CLRPs are NP-hard as their subproblems inherit this hardness.

2. Exact and Approximation Algorithms

2.1 Polyhedral and Integer Programming Approaches

Classical approaches employ branch-and-cut or branch-price-and-cut over strong mixed-integer formulations. The relationship to facility location and set partitioning enables design of lower bounds based on spanning tree and facility location relaxations (Heine et al., 2021). A key insight is the presence of exponentially many variables (routes), leading to the use of column generation in path-based models, or combinatorial relaxations with lifted cover inequalities for handling capacity coupling (Nordlund et al., 2023). State-of-the-art exact algorithms can solve moderate-size benchmark instances to proven optimality but scale poorly for large $n$.

2.2 Constant-Factor and Bifactor Approximation

Several constant-approximation algorithms are established:

• Reduction to k-Median-Forest: For the variant where exactly $k$ depots are opened, CLRP reduces to the $k$-median-forest problem, which involves minimizing connection costs plus the cost of a minimum spanning tree after depot contraction. Local-search algorithms with $t$-swap yield a $(3+2/t)$-approximation, resulting in a $(12+\varepsilon)$-approximation for CLRP for any constant $\varepsilon > 0$ (Goertz et al., 2011).
• Tree and Path-Split Approximations: A $4.169$-approximation is attained by combining solutions to a scaled Uncapacitated Facility Location (UFL) instance and a suitably constrained forest, followed by splitting to tours. If customer demand is splittable, a $4.091$-factor is proven via path-packing plus greedy splitting (Zhao et al., 16 Mar 2025). These give the best known guarantees for metric CLRPs with unsplittable and splittable demands.
• Bifactor Approximations: A bifactor $(4+2\alpha/\varepsilon)$-approximation is achieved for general CLRPs with both facility and vehicle capacities, where $\alpha \ge 1$ is an underlying facility location approximation guarantee and $\varepsilon$ is an allowed fractionally violated facility capacity; as $\varepsilon \to 0$, the guarantee becomes asymptotically constant. Rounding LP solutions ensures each depot's capacity is exceeded by at most $\varepsilon Q$ (Heine et al., 2021).

The following table summarizes major theoretical guarantees:

Algorithm Class	Approximation Factor (unsplittable)	Capacity Violation
$t$-swap local search ($t\to\infty$)	$3$ (k-median-forest) $\implies 12$	None
Tree-Split (UFL + forest)	$4.169$	None
Path-Split (splittable demands)	$4.091$	None
Bifactor (w/ CFL subroutine, $\varepsilon$)	$4+2\alpha/\varepsilon$	$\leq \varepsilon Q$
Randomized LP rounding (star IRPFL)	$12$–$48$ (variously)	None

3. Advanced Metaheuristics

CLRPs are highly amenable to trajectory-based metaheuristics. Notable approaches include:

• Multi-Population Memetic Algorithms: Multi-niche frameworks partition the population by depot configurations and deploy a specialized multi-depot edge assembly crossover (mdEAX), intensive local search restricted to nearest neighbors, and penalty-guided repair for capacity violations. Experimental results on over 281 benchmarks indicate that such frameworks (e.g., HGAMP_m) can improve a large fraction of best-known solutions and robustly outperform prior methods on both medium and large instances (He et al., 14 Mar 2024).
• Hybrid Constructive and Local Search: Initial depot sets are seeded using combinatorial heuristics such as coverage ratio heuristics with secondary filters based on MDVRP subproblem cost. Evolutionary improvement alternates crossover, repair, mutation, and strategic replacement based on both solution distance and fitness.
• Component Analysis: Empirical ablation confirms the necessity of population diversity over depot configurations, and mechanisms that preserve structural diversity (via niche subpopulations and diversified mutation) are associated with statistically significant performance gains.

4. Machine Learning and End-to-End Paradigms

Recent developments have embedded learned predictors into classic optimization pipelines with substantial empirical and computational advantages:

• Neural Surrogate Models: Permutation-invariant deep sets or graph neural networks are trained offline to estimate VRP routing costs for arbitrary depot–customer assignments (e.g., NEO-LRP, GANCP). These neural surrogates are integrated directly into MIP or genetic algorithms to evaluate candidate depot assignments efficiently without explicit VRP subproblem solutions (Kaleem et al., 7 Dec 2024, Sobhanan et al., 2023). Numerical experiments confirm speedups of orders of magnitude (decision times of seconds for $n\leq 200$) with only marginal (≤2%) optimality loss on large instances compared to state-of-the-art metaheuristics.
• End-to-End Reinforcement Learning: Transformer-style encoder–decoders with heterogeneous querying attention can jointly model depot selection and tour generation as a single Markov Decision Process (MDP). The DRLHQ approach employs tailored masks and query vectors to ensure feasibility of assignments and routes, achieving gaps of 7.6% (CLRP) and 6.1% (open CLRP) on classical benchmarks (Miao et al., 4 Nov 2025).
• Hierarchical GNN-Predicted Fitness: Population-based optimizers use a GNN trained on the CVRP subproblem to predict fitness within a genetic-algorithm search, with repair strategies to maintain feasibility. Empirical evaluation highlights competitive solution quality with minimal parameter-tuning overhead (Sobhanan et al., 2023).

5. Domain Extensions and Practical Impact

• Multi-Echelon and Sustainable Extensions: Two-echelon CLRPs, where decisions cascade from central depots to intermediate satellites, integrate customer delivery mode selection, vehicle fleet heterogeneity, and explicit emission modeling. The 2E-LRP-ECB introduces piecewise-linear emission penalties at stops and explicitly ties customer eco-behavior to system emissions; Pareto-frontier analysis with knee-point selection is employed for jointly minimizing distance and emissions (Bonomi et al., 26 Sep 2025).
• Budget and Split-Delivery Generalizations: Reframing CLRPs to include installation budget constraints for drop-off points, split deliveries, or non-closed walks in biomass or waste management applications, yields complex extended-graph formulations and new cut-generation techniques for enforcing connectivity and Eulerian trail constraints (Albareda-Sambola et al., 8 May 2025).
• Inventory and Temporal Extensions: Star Inventory Routing Problems with Facility Location (SIRPFL) generalize CLRPs across time, integrating holding costs and delivery schedules. Approximation guarantees are established for both splittable and unsplittable multi-trip capacitated variants (Jiao et al., 2019).

6. Computational Benchmarks and Empirical Insights

State-of-the-art frameworks are evaluated on extensive benchmark sets, including the Trunschke–Burke, Barreto, and Schneider–Löffler instances, with customer sizes $n$ up to 600 and depot options up to 30. Empirical highlights include:

• Multi-population schemes (e.g., HGAMP_m) achieved $–0.07\%$ mean gap to best-known solutions, improving or matching hundreds of BKS.
• Neural-embedded models solved $n=200$ CLRP instances within $1\%$ of BKS in $<5$ seconds on average (Kaleem et al., 7 Dec 2024).
• GANCP, leveraging GNN-based cost predictors, achieves $1.7\%$–$2.8\%$ average optimality gaps in seconds across classical CLRP benchmarks (Sobhanan et al., 2023).
• Deep RL (DRLHQ) achieves $6\%$–$8\%$ gaps, with superior generalization scaling compared to direct MIP or heuristic search (Miao et al., 4 Nov 2025).
• Approximation algorithms provide theoretical factors close to optimality (4.169, 4.091), but large-scale practical heuristics exhibit gaps as low as $<1\%$ at scale.

A plausible implication is that hybridizing combinatorial, metaheuristic, and neural learning components is now routine for both advancing computational boundaries and providing near-optimal solutions for industry-scale CLRPs.

7. Open Problems and Future Research

Key areas of current investigation include:

• Closing the theoretical gap toward pure constant-factor approximations for CLRPs with hard facility and vehicle capacities (i.e., with no allowed violation).
• Robust frameworks for multi-objective CLRPs (cost, emission, equity), leveraging exact, heuristic, and learning-augmented methods, as explored in (Bonomi et al., 26 Sep 2025).
• Extension of neural surrogate-based optimization to more general side constraints (time windows, precedence, stochastic demands).
• Integration and feedback between branch-and-bound MIP solvers and learned route-cost (GNN/DNN) surrogates for instance transfer and online improvement.
• Fully end-to-end approaches for hierarchical vehicle routing encompassing both location-routing and temporal/inventory decisions, with sustainable objectives.

The CLRP remains a central model and testbed for integrated, capacitated, and scalable optimization, as advances in both theory and large-scale empirical solution methodology continue to redefine tractable problem sizes and solution quality standards for integrated logistics planning.