🤖 machine learning

FiLMMeD: Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing

The paper proposes FiLMMeD, a novel unified neural-based model that leverages Feature-wise Linear Modulation, Preference Optimization, and curriculum learning to effectively solve 24 diverse Multi-Depot Vehicle Routing Problem variants, outperforming existing state-of-the-art baselines.

Original authors: Arthur Corrêa, Paulo Nascimento, Samuel Moniz

Published 2026-05-01

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CC BY 4.0

Original authors: Arthur Corrêa, Paulo Nascimento, Samuel Moniz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the manager of a massive delivery company. You have trucks, drivers, and thousands of customers. Your job is to figure out the most efficient way to get packages from your warehouses to the customers' doors. This is the Vehicle Routing Problem (VRP).

Now, imagine your company grows. You don't just have one warehouse; you have multiple depots scattered across the city. This makes the puzzle much harder. This is the Multi-Depot VRP (MDVRP).

To make things even more chaotic, real life throws curveballs every day:

Sometimes a customer wants a package picked up (a Backhaul).
Sometimes a truck has a strict time limit on how long it can drive (Route Length Limit).
Sometimes a customer only accepts deliveries between 9 AM and 11 AM (Time Window).
Sometimes a truck is allowed to stop at a different warehouse in the middle of its route to reload (Inter-depot Route).

Traditionally, solving these puzzles required hiring a different expert for every single combination of rules. If you added a new rule, you had to start from scratch.

This paper introduces FiLMMeD, a new "super-solver" designed to handle all these different rule combinations at once, without needing to be retrained every time. Here is how it works, using simple analogies:

1. The "Smart Glasses" (Feature-wise Linear Modulation)

Imagine you are wearing a pair of smart glasses. When you look at a map, the glasses change how you see the world based on the rules of the day.

If the rule is "Time Windows," the glasses highlight the clocks on the map and dim the rest.
If the rule is "Backhauls," the glasses highlight the pickup points and change the color of the delivery points.

In the paper, this is called FiLM (Feature-wise Linear Modulation). Instead of building a new brain for every rule, the AI uses a single brain but puts on these "smart glasses" (mathematical adjustments) to instantly adapt its internal thinking to the specific constraints of the current problem. This allows it to understand that a "Time Window" rule changes the importance of certain data points, while a "Backhaul" rule changes others.

2. The "Training Camp" (Curriculum Learning)

If you tried to teach a student to solve a complex math problem by throwing every possible variable at them on day one, they would likely fail. You need a Curriculum.

The authors realized that for Multi-Depot problems, you can't just mix all the rules randomly. You have to build up the difficulty:

Phase 1: Teach the AI simple problems with just one extra rule (e.g., just Time Windows).
Phase 2: Once it masters that, introduce problems with two rules.
Phase 3: Finally, let it tackle the "boss battles" with four or five rules combined.

This Curriculum Learning strategy acts like a training camp, ensuring the AI learns the basics of how rules interact before it gets overwhelmed by the most complex scenarios.

3. The "Coach's Feedback" (Preference Optimization)

Usually, AI learns by trying to guess the answer and getting a score (Reward) at the end. If the score is low, it tries again. This can be messy and slow, especially when juggling many different types of problems.

The authors tried a different approach called Preference Optimization (PO). Instead of asking, "How good is this route?" (giving a number), they ask the AI to compare two routes: "Is Route A better than Route B?"

Think of it like a coach watching two athletes run. The coach doesn't need to know the exact speed; they just need to know who finished first.
The paper claims this method is more stable and helps the AI learn faster and better than the traditional "score-based" method, especially when dealing with many different types of problems at once.

What Did They Achieve?

The team tested their new model, FiLMMeD, on 24 different versions of the Multi-Depot problem (including 8 brand-new types they invented) and 16 single-depot problems.

The Result: FiLMMeD consistently beat the previous best AI models. It found better routes, faster, and could handle complex combinations of rules that other models struggled with.
The "Zero-Shot" Win: Even when they tested the model on rule combinations it had never seen before, it still performed very well, proving that the "smart glasses" and "training camp" approach helped it truly understand the logic of the problem, not just memorize answers.

Summary

The paper presents a new AI system that acts like a universal delivery planner. By using smart glasses to adapt to specific rules, a step-by-step training camp to learn complexity, and a comparative coaching style to learn efficiently, it solves complex logistics puzzles better than any previous method, all without needing a new model for every new scenario.

1. Problem Definition

The paper addresses the Multi-Depot Vehicle Routing Problem (MDVRP), a complex combinatorial optimization (CO) problem central to modern logistics, particularly in e-commerce. Unlike the classic single-depot VRP, MDVRP involves dispatching vehicles from multiple depots to serve customers.

The specific challenge tackled is cross-problem generalization. Real-world logistics involve heterogeneous constraints that vary dynamically (e.g., time windows, backhauls, route length limits, open routes, and inter-depot transfers). Traditional neural-based solvers are typically trained on specific problem variants, making them brittle when faced with new constraint combinations. While Multi-Task Learning (MTL) has been applied to single-depot VRPs, it has largely failed to address the complexity of MDVRP variants, often suffering from:

Gradient Conflicts: Conflicting updates when training on diverse constraint sets simultaneously.
Poor Generalization: Inability to handle complex interactions between constraints (e.g., backhauls + time windows) without retraining.
Static Conditioning: Existing models treat constraints as high-level context rather than dynamically modulating internal representations.

2. Methodology: FiLMMeD

The authors propose FiLMMeD, a unified neural-based model designed to solve 24 distinct MDVRP variants (including 8 novel formulations with inter-depot routes) and 16 single-depot VRP variants. The architecture is built upon Transformer-based encoder-decoder frameworks (specifically extending MTPOMO, MVMoE, and CaDA).

Core Components:

Feature-wise Linear Modulation (FiLM):
- Mechanism: Instead of simple concatenation of constraint features, FiLMMeD introduces a conditioning vector $z$ (a Boolean vector indicating active constraints) to dynamically modulate node embeddings.
- Implementation: The model computes scaling ( $\gamma$ ) and shifting ( $\beta$ ) parameters based on $z$ via learnable linear projections. These parameters are applied to customer embeddings as an affine transformation: $FiLM(h|\gamma, \beta) = \gamma \cdot h + \beta$ .
- Effect: This allows the model to adapt its internal representations specifically to the active constraints of an instance, effectively disentangling the latent space and mitigating gradient conflicts between tasks.
Curriculum Learning (CL) Strategy:
- Motivation: Uniform sampling of all variants (common in single-depot MTL) leads to poor convergence in MDVRP due to complex constraint interactions.
- Strategy: The training curriculum is structured to progressively increase complexity:
  - Phase 1 (0-30%): Single-constraint variants + specific two-constraint variants known to be difficult (e.g., backhaul + time window).
  - Phase 2 (30-60%): Introduction of all two-constraint variants.
  - Phase 3 (60-90%): Introduction of three-constraint variants.
  - Phase 4 (90-100%): Full set including four-constraint variants.
- Goal: To guide the model from learning isolated constraint effects to mastering their complex interactions.
Preference Optimization (PO):
- Innovation: The paper introduces PO as a training paradigm for MTL, replacing the standard Reinforcement Learning (RL) loss (REINFORCE).
- Mechanism: Instead of optimizing scalar rewards, the model learns from pairwise preferences between trajectories generated for the same instance. If trajectory $A$ has a better cost than $B$ , the model is optimized to increase the probability of $A$ relative to $B$ .
- Advantage: PO reduces gradient variance and is scale-invariant (unaffected by the absolute magnitude of costs), addressing the instability and gradient interference common in MTL settings.

3. Key Contributions

First Unified MTL Model for MDVRP: FiLMMeD is the first model explicitly designed to handle a wide range of MDVRP variants (24 total), including novel inter-depot route formulations.
FiLM Mechanism for Constraint Conditioning: Demonstrates that dynamically modulating embeddings based on constraint attributes significantly improves generalization and disentangles the latent space compared to static feature concatenation.
Preference Optimization in MTL: Provides the first demonstration of PO in an MTL setting, showing it outperforms traditional RL in terms of convergence stability and solution quality.
Targeted Curriculum Learning: Establishes that MDVRP requires a guided curriculum (gradual exposure to constraint complexity) rather than uniform sampling, which is sufficient for single-depot problems.

4. Experimental Results

The authors conducted extensive experiments on 24 MDVRP variants and 16 single-depot VRPs.

Performance on MDVRP:
- FiLMMeD consistently outperformed state-of-the-art baselines (MTPOMO, MVMoE) and traditional heuristics (PyVRP) in terms of solution quality (Gap).
- On 50-node instances, FiLMMeD-MTPOMO reduced the average gap by 2.888% compared to its baseline.
- On 100-node instances, improvements were even more significant, with FiLMMeD-MVMoE outperforming its baseline by 5.288%.
- The model achieved superior zero-shot generalization on unseen constraint combinations.
Performance on Single-Depot VRP:
- When fine-tuned on single-depot variants, FiLMMeD (using PO) achieved competitive or superior results compared to baselines trained on massive datasets (e.g., 100M instances), despite being trained on fewer instances (30M).
- FiLMMeD-CaDA† (combining FiLM, PO, and ReLD) achieved the best overall performance, with an average gap of 1.46% on 50-node instances.
Ablation Studies:
- FiLM: Removing FiLM resulted in significantly worse performance on unseen variants (e.g., gap increased from 5.297% to 5.716% for MVMoE).
- Curriculum: Uniform sampling or "unified" training strategies led to poor convergence on MDVRP, validating the necessity of the proposed CL strategy.
- PO vs. RL: Models trained with PO showed faster convergence and lower average gaps compared to those trained with REINFORCE across all architectures.
Complexity: The FiLM mechanism adds negligible parameters (<0.01M, <1% increase) and minimal GPU memory overhead. Training times per epoch remain nearly identical to baselines.

5. Significance

This paper represents a significant step forward in neural combinatorial optimization for logistics:

Practical Applicability: By solving 24 variants in a single unified model, FiLMMeD addresses the "fragmentation" problem in logistics, where companies currently need separate solvers for different constraint sets.
Theoretical Advancement: It challenges the dominance of Reinforcement Learning in CO by demonstrating that Preference Optimization offers a more stable and effective training paradigm for Multi-Task Learning.
Architectural Insight: It proves that dynamic conditioning (FiLM) is superior to static context embedding for handling heterogeneous constraints, offering a blueprint for future cross-problem learning models.
Generalization: The success of the curriculum learning strategy highlights that the complexity of constraint interactions in multi-depot settings requires a structured learning path, a finding that may apply to other complex CO problems.

In summary, FiLMMeD establishes a new state-of-the-art for unified neural solvers in vehicle routing, offering a scalable, flexible, and highly generalizable solution for modern, dynamic logistics challenges.