# Deep Reinforcement Learning for Solving the Fleet Size and Mix Vehicle Routing Problem

arXiv: https://arxiv.org/abs/2512.24251 · HTML: https://mlsanity.com/paper/2512.24251/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 30 Dec 2025.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 10 of 10)

- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Exploratory Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1909.04063/) (2019)
- [RL4CO: an Extensive Reinforcement Learning for Combinatorial Optimization Benchmark](https://mlsanity.com/paper/2306.17100/) (2023)
- [Deep Reinforcement Learning for Solving the Heterogeneous Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/2110.02629/) (2021)
- [Deep Reinforcement Learning for Electric Vehicle Routing Problem with Time Windows](https://mlsanity.com/paper/2010.02068/) (2020)

## Abstract (arXiv)

The Fleet Size and Mix Vehicle Routing Problem (FSMVRP) is a prominent variant of the Vehicle Routing Problem (VRP), extensively studied in operations research and computational science. FSMVRP requires simultaneous decisions on fleet composition and routing, making it highly applicable to real-world scenarios such as short-term vehicle rental and on-demand logistics. However, these requirements also increase the complexity of FSMVRP, posing significant challenges, particularly in large-scale and time-constrained environments. In this paper, we propose a deep reinforcement learning (DRL)-based approach for solving FSMVRP, capable of generating near-optimal solutions within a few seconds. Specifically, we formulate the problem as a Markov Decision Process (MDP) and develop a novel policy network, termed FRIPN, that seamlessly integrates fleet composition and routing decisions. Our method incorporates specialized input embeddings designed for distinctdecision objectives, including a remaining graph embedding to facilitate effective vehicle employment decisions. Comprehensive experiments are conducted on both randomly generated instances and benchmark datasets. The experimental results demonstrate that our method exhibits notable advantages in terms of computational efficiency and scalability, particularly in large-scale and time-constrained scenarios. These strengths highlight the potential of our approach for practical applications and provide valuable inspiration for extending DRL-based techniques to other variants of VRP.