Genetic Algorithms with Neural Cost Predictor for Solving Hierarchical Vehicle Routing Problems

When vehicle routing decisions are intertwined with higher-level decisions, the resulting optimization problems pose significant challenges for computation. Examples are the multi-depot vehicle routing problem (MDVRP), where customers are assigned to depots before delivery, and the capacitated location routing problem (CLRP), where the locations of depots should be determined first. A simple and straightforward approach for such hierarchical problems would be to separate the higher-level decisions from the complicated vehicle routing decisions. For each higher-level decision candidate, we may evaluate the underlying vehicle routing problems to assess the candidate. As this approach requires solving vehicle routing problems multiple times, it has been regarded as impractical in most cases. We propose a novel deep-learning-based approach called Genetic Algorithm with Neural Cost Predictor (GANCP) to tackle the challenge and simplify algorithm developments. For each higher-level decision candidate, we predict the objective function values of the underlying vehicle routing problems using a pre-trained graph neural network without actually solving the routing problems. In particular, our proposed neural network learns the objective values of the HGS-CVRP open-source package that solves capacitated vehicle routing problems. Our numerical experiments show that this simplified approach is effective and efficient in generating high-quality solutions for both MDVRP and CLRP and has the potential to expedite algorithm developments for complicated hierarchical problems. We provide computational results evaluated in the standard benchmark instances used in the literature.

A Hybrid Genetic Algorithm for the min-max Multiple Traveling Salesman Problem

This paper proposes a hybrid genetic algorithm for solving the Multiple Traveling Salesman Problem (mTSP) to minimize the length of the longest tour. The genetic algorithm utilizes a TSP sequence as the representation of each individual, and a dynamic programming algorithm is employed to evaluate the individual and find the optimal mTSP solution for the given sequence of cities. A novel crossover operator is designed to combine similar tours from two parents and offers great diversity for the population. For some of the generated offspring, we detect and remove intersections between tours to obtain a solution with no intersections. This is particularly useful for the min-max mTSP. The generated offspring are also improved by a self-adaptive random local search and a thorough neighborhood search. Our algorithm outperforms all existing algorithms on average, with similar cutoff time thresholds, when tested against multiple benchmark sets found in the literature. Additionally, we improve the best-known solutions for 21 out of 89 instances on four benchmark sets.

A Neural Separation Algorithm for the Rounded Capacity Inequalities

The cutting plane method is a key technique for successful branch-and-cut and branch-price-and-cut algorithms that find the exact optimal solutions for various vehicle routing problems (VRPs). Among various cuts, the rounded capacity inequalities (RCIs) are the most fundamental. To generate RCIs, we need to solve the separation problem, whose exact solution takes a long time to obtain; therefore, heuristic methods are widely used. We design a learning-based separation heuristic algorithm with graph coarsening that learns the solutions of the exact separation problem with a graph neural network (GNN), which is trained with small instances of 50 to 100 customers. We embed our separation algorithm within the cutting plane method to find a lower bound for the capacitated VRP (CVRP) with up to 1,000 customers. We compare the performance of our approach with CVRPSEP, a popular separation software package for various cuts used in solving VRPs. Our computational results show that our approach finds better lower bounds than CVRPSEP for large-scale problems with 400 or more customers, while CVRPSEP shows strong competency for problems with less than 400 customers.

A Hybrid Genetic Algorithm with Type-Aware Chromosomes for Traveling Salesman Problems with Drone

There are emerging transportation problems known as the Traveling Salesman Problem with Drone (TSPD) and the Flying Sidekick Traveling Salesman Problem (FSTSP) that involve the use of a drone in conjunction with a truck for package delivery. This study represents a hybrid genetic algorithm for solving TSPD and FSTSP by combining local search methods and dynamic programming. Similar algorithms exist in the literature. Our algorithm, however, considers more sophisticated chromosomes and simpler dynamic programming to enable broader exploration by the genetic algorithm and efficient exploitation through dynamic programming and local searches. The key contribution of this paper is the discovery of how decision-making processes should be divided among the layers of genetic algorithm, dynamic programming, and local search. In particular, our genetic algorithm generates the truck and the drone sequences separately and encodes them in a type-aware chromosome, wherein each customer is assigned to either the truck or the drone. We apply local searches to each chromosome, which is decoded by dynamic programming for fitness evaluation. Our dynamic programming algorithm merges the two sequences by determining optimal launch and landing locations for the drone to construct a TSPD solution represented by the chromosome. We propose novel type-aware order crossover operations and effective local search methods. A strategy to escape from local optima is proposed. Our new algorithm is shown to outperform existing algorithms on most benchmark instances in both quality and time. Our algorithms found the new best solutions for 538 TSPD instances out of 920 and 93 FSTSP instances out of 132.

A Deep Reinforcement Learning Approach for Solving the Traveling Salesman Problem with Drone

Reinforcement learning has recently shown promise in learning quality solutions in many combinatorial optimization problems. In particular, the attention-based encoder-decoder models show high effectiveness on various routing problems, including the Traveling Salesman Problem (TSP). Unfortunately, they perform poorly for the TSP with Drone (TSP-D), requiring routing a heterogeneous fleet of vehicles in coordination -- a truck and a drone. In TSP-D, the two vehicles are moving in tandem and may need to wait at a node for the other vehicle to join. State-less attention-based decoder fails to make such coordination between vehicles. We propose an attention encoder-LSTM decoder hybrid model, in which the decoder's hidden state can represent the sequence of actions made. We empirically demonstrate that such a hybrid model improves upon a purely attention-based model for both solution quality and computational efficiency. Our experiments on the min-max Capacitated Vehicle Routing Problem (mmCVRP) also confirm that the hybrid model is more suitable for coordinated routing of multiple vehicles than the attention-based model.

A Reinforcement Learning Approach for Rebalancing Electric Vehicle Sharing Systems

This paper proposes a reinforcement learning approach for nightly offline rebalancing operations in free-floating electric vehicle sharing systems (FFEVSS). Due to sparse demand in a network, FFEVSS require relocation of electrical vehicles (EVs) to charging stations and demander nodes, which is typically done by a group of drivers. A shuttle is used to pick up and drop off drivers throughout the network. The objective of this study is to solve the shuttle routing problem to finish the rebalancing work in the minimal time. We consider a reinforcement learning framework for the problem, in which a central controller determines the routing policies of a fleet of multiple shuttles. We deploy a policy gradient method for training recurrent neural networks and compare the obtained policy results with heuristic solutions. Our numerical studies show that unlike the existing solutions in the literature, the proposed methods allow to solve the general version of the problem with no restrictions on the urban EV network structure and charging requirements of EVs. Moreover, the learned policies offer a wide range of flexibility resulting in a significant reduction in the time needed to rebalance the network.