Locomotive Assignment Problem (LAP)

• Locomotive Assignment Problem (LAP) is a combinatorial optimization challenge that assigns locomotives to services while satisfying complex operational constraints.
• Extended formulations use integer programming and polyhedral models to incorporate time-indexed decisions, transition costs, and connectivity for real-world applications.
• Hybrid approaches combining machine learning heuristics, graph neural networks, and conflict constraints improve solution speed and scalability for large-scale rail operations.

The Locomotive Assignment Problem (LAP) is a large-scale combinatorial optimization problem inherent to rail operations, in which available locomotives must be assigned to train services, routes, or time slots over a planning horizon. Solutions must satisfy complex operational constraints—including capacity, connectivity, conflicting requirements, and real-world considerations such as service timing, maintenance, and repositioning—while minimizing a cost function reflecting factors like fleet size, repositioning costs, and labor. There is a substantial body of literature that addresses LAP from integer programming, polyhedral, machine learning, and combinatorial modeling perspectives, each providing unique computational trade-offs and modeling capabilities.

1. Mathematical Formulations and Polyhedral Extended Models

LAP is traditionally modeled using integer or binary variables within the classic Assignment Problem (AP) structure: $$\min \sum_{l, r} c_{lr} x_{lr} \qquad \text{s.t.}\quad \sum_{r} x_{lr} = 1\;\forall l,\qquad \sum_{l} x_{lr} = 1\;\forall r,\qquad x_{lr}\in\{0,1\}$$ where $l$ indexes locomotives, $r$ indexes jobs/routes, and $c_{lr}$ encodes assignment costs. Extensions for LAP must capture time-indexed decisions, equipment movement, and various domain-specific requirements.

A general approach to handle the combinatorial complexity is to treat LAP within an extended network-flow polyhedral framework—as in the $O(m^9)$ network flow LP model of the assignment problem polytope (Diaby, 2016). In this formulation, decision variables $(w_{i,r})$ represent assignment of locomotive $i$ to time slot or duty $r$, and high-dimensional “linking” variables $$(x_{\langle i,r\rangle,\langle j,s\rangle,\langle k,p\rangle})$$ enforce consistency and operational connectivity over multiple stages. Core assignment constraints are: $$\sum_{r=1}^{m} w_{i,r} = 1\;\forall i,\qquad \sum_{i=1}^{m} w_{i,r} = 1\;\forall r$$ with linear extension constraints such as: $$w_{i,r} - \sum_{j\ne i}\sum_{k\ne i,j}\overline{x}(\langle i, r\rangle, \langle j, s\rangle, \langle k, p\rangle) = 0,\qquad \text{for suitable }r, s, p$$ The extended formulation concept allows the incorporation of transition costs, sequencing, and resource flow logic not expressible in the classical AP. These models, although polynomial in size, can have high exponents (e.g., $O(m^9)$), and therefore practical deployment requires careful exploitation of problem sparsity and network structure.

2. Algorithmic and Computational Perspectives

Integer programming solvers remain the primary tool for solving LAP instances, with model size and complexity scaling rapidly with the number of locomotives, trains, and planning periods. A salient innovation is the integration of reduction rules, cutting planes, or tailored decomposition strategies to maintain tractability at industrial scales.

Recent work presents a network-based integer programming framework over a cyclic space-time network, where nodes denote terminals at specific times and arcs denote movements, light travel, or work events (Kim et al., 29 Jul 2025). Special binary and integer decision variables capture pick-up, set-out, and repositioning events, and the objective function aggregates costs from locomotive ownership, deadheading, light travel, and event penalties. The resulting model is:

• Exact, guaranteeing global optimality
• Compatible with cyclic (weekly) assignment structures
• Rich enough to capture endogenous positioning of work events

To address the extreme growth in potential light travel arcs, the framework introduces formal reduction rules:

• Earliest reachability: only the first available repositioning options are considered at destination nodes
• Latest origin filtering: among multiple equivalent departures, only the latest is retained to avoid idle time These rules eliminate redundant arcs and dramatically reduce problem size, enabling, for the first time, exact solutions on real, large-scale LAP instances.

In contrast, Minimum Cost Flow (MCF) heuristics can quickly generate feasible near-optimal plans by restricting the arc set a priori, but cannot guarantee global optimality or adaptability to arbitrary cost structures.

3. Extensions: Conflict Constraints and Generalizations

Many operational environments require explicit modeling of conflicts—such as overlapping train paths, incompatible assignments, or safety separations. The Assignment Problem with Conflicts (APC) generalizes LAP by adding constraints of the form: $x_{lr} + x_{l'r'} \leq 1$ for every conflicting assignment pair $((l,r), (l',r'))$. Such models are naturally expressed as Mixed-Integer Linear Programs (MILPs) and efficiently solvable using modern constraint programming solvers, notably Google OR-Tools’ CP-SAT (Montemanni et al., 3 Jun 2025). CP-SAT excels in large-scale, highly-constrained settings through advanced propagation techniques, yielding near-optimal or optimal solutions with minimal implementation effort.

The inclusion of conflict constraints is directly relevant in practice for modeling schedule incompatibilities, regulatory constraints, or maintenance exclusions in LAP. Experimental findings indicate that CP-SAT achieves results comparable to, and often better than, other exact or heuristic solvers, especially as the volume and density of conflicts increase.

4. Machine Learning–Driven and Heuristic Approaches

The increasing regularity of LAP instances in real-world railways motivates leveraging historical solution data and online learning. One notable paradigm (Rocca et al., 14 Mar 2024) applies a one-shot learning (OSL) heuristic for mixed-integer programs with SOS1 constraints—a common feature in LAP where each train (“S1 constraint”) must be assigned exactly one consist. The OSL methodology involves:

• Running a short “probing” phase with branch-and-bound
• Recording the empirical distribution of variable assignments within each S1 constraint
• Freezing high-confidence decisions by adding cuts $x_v^{k'} = 1$ for those S1 constraints where the mode $k'$ dominates
• Solving the reduced problem to optimality

Experimental evidence on real LAP datasets demonstrates that OSL, especially via the Probe and Freeze (PNF) approach within CPLEX, achieves an average primal gap as low as 0.23%—substantially outperforming classical heuristics.

Complementary methods utilize entropy-based minimization together with online Fast Random Forest (FRF) learning (Rocca et al., 2022). Here, variable “stability” (low entropy in past decisions) is used as a surrogate for prediction confidence, and the most stable variables are fixed early in the branch-and-bound search. For regular, recurring LAP instances, this approach yields an order of magnitude speedup over standard branch-and-bound (CPLEX) and, in some large instances, delivers better solutions within equivalent compute budgets.

5. Graph Neural Networks and Differentiable Assignment Solvers

Solving large, structured LAPs is further advanced by formulating assignment as a problem on bipartite graphs and applying deep graph neural networks for scalable, learnable solvers (Liu et al., 2022). The GLAN approach:

• Converts the assignment cost matrix into a bipartite agent-job graph, representing locomotives and tasks/routes as nodes and cost-weighted assignment options as edges
• Applies several iterations of edge and node convolution, with message passing and attention-based aggregation
• Decodes assignment decisions through a learned, end-to-end process
• Outputs binary (or fractional) assignment labels directly GLAN demonstrates robust scalability and consistent assignment accuracy as problem sizes grow, outperforming classical algorithms (including Sinkhorn normalization and RNN-based assignment solvers) especially as $n$ reaches thousands. In transport applications, such as locomotive assignment, GLAN's model can easily accommodate context-dependent costs, constraints, and even learn implicit operational patterns from data. Its differentiable, end-to-end design is particularly valuable when embedding LAP solutions within more complex learning or tracking frameworks.

6. Duality, Relative-Interior Solutions, and Quadratic Generalizations

Linear Assignment Problems are polyhedrally integral and permit dual interpretations crucial for bounding and decomposition. Efficient computation of dual solutions in the relative interior of the optimizer set has strong practical implications, especially when LAP appears as a subproblem in larger quadratic assignment decompositions (Dlask et al., 2023). A linear-time algorithm can be used to shift any dual-optimal solution for LAP to the relative interior, by exploiting properties of the equality subgraph and alternating cycles, ensuring all matchable edges are strictly feasible. In block-coordinate ascent relaxations of generalizations such as the Quadratic Assignment Problem (QAP), using relative-interior duals in the LAP block results in tighter bounds and improves convergence. Reduction techniques also exist to handle incomplete LAPs (e.g., those permitting unassigned tasks) by mapping them onto equivalent, complete instances.

7. Practical Impact and Applications

The strategic assignment of locomotives using these models and algorithms has direct operational impact in large freight rail carriers—minimizing fleet requirements, repositioning and maintenance costs, and facilitating robust weekly plans (Kim et al., 29 Jul 2025). The scalability of exact integer programming via advanced reduction rules activates planning for real, industrial-size instances with tens of thousands of variables and constraints. Simultaneously, advancements in fast learning-based heuristics, graph network solvers, and constraint satisfaction methods generate actionable solutions in timeframes compatible with operational decision cycles.

The cross-fertilization of polyhedral, combinatorial, and learning-based methods underscores both the ongoing challenge and rich opportunity in the LAP domain. Robust, scalable assignment methodologies developed in this context have influenced not only railway scheduling but also more general large-scale assignment problems in transportation, logistics, and multi-object tracking.