The multiple traveling salesman problem: an overview of formulations and solution procedures

Recent works using deep learning to solve routing problems such as the traveling salesman problem (TSP) have focused on learning construction heuristics. Such approaches find good quality solutions but require additional procedures such as beam search and sampling to improve solutions and achieve state-of-the-art performance. However, few studies have focused on improvement heuristics, where a given solution is improved until reaching a near-optimal one. In this work, we propose to learn a local search heuristic based on 2-opt operators via deep reinforcement learning. We propose a policy gradient algorithm to learn a stochastic policy that selects 2-opt operations given a current solution. Moreover, we introduce a policy neural network that leverages a pointing attention mechanism, which can be easily extended to more general k-opt moves. Our results show that the learned policies can improve even over random initial solutions and approach near-optimal solutions faster than previous state-of-the-art deep learning methods for the TSP. We also show we can adapt the proposed method to two extensions of the TSP: the multiple TSP and the Vehicle Routing Problem, achieving results on par with classical heuristics and learned methods.

K-DSA for the Multiple Traveling Salesman Problem

Combinatorial optimization problems are broadly studied in the literature. On the one hand, their challenging characteristics, such as the constraints and number of potential solutions, inspires their use to test new solution techniques. On the other hand, the practical application of these problems provides support of daily tasks of people and companies. Vehicle routing problems constitute a well-known class of combinatorial optimization problems, from which the Traveling Salesman Problem (TSP) is one of the most elementary ones. TSP corresponds to finding the shortest route that visits all cities within a path returning to the start city. Despite its simplicity, the difficulty in finding its exact solution and its direct application in practical problems in multiple areas make it one of the most studied problems in the literature. Algorithms inspired by biological phenomena are being successfully applied to solve optimization tasks, mainly combinatorial optimization problems. Those inspired by the collective behavior of insects produce good results for solving such problems. This article proposes the VRoptBees, a framework inspired by honeybee behavior to tackle vehicle routing problems. The framework provides a flexible and modular tool to easily build solutions to vehicle routing problems. Together with the framework, two examples of implementation are described, one to solve the TSP and the other to solve the Capacitated Vehicle Routing Problem (CVRP). Tests were conducted with benchmark instances from the literature, showing competitive results.

In this paper, a multiple depot, multiple traveling salesman problem is transformed into a single, asymmetric traveling salesman problem if the cost of the edges satisfy the triangle inequality. This improves on the previously known transformation for a 2-depot, multiple traveling salesman problem in the literature. To test the effectiveness of the transformation, some computational results are presented by applying the well known LKH heuristic on the transformed problem for instances involving Dubins vehicles. Results show that the transformation is effective and high quality solutions can be found for large instances in a relatively short time.

Unmanned aerial vehicles (UAVs) are being increasingly used for surveillance missions in civil and military applications. These vehicles can be heterogeneous in the sense that they can differ either in their motion constraints or sensing/attack capabilities. Given a surveillance mission that require a group of heterogeneous UAVs to visit a set of targets, this paper addresses a resource allocation problem of finding the optimal sequence of targets for each vehicle such that 1) each target is visited at least once by some vehicle, and 2) the total cost travelled by all the vehicles is minimized. This problem can be posed as a Heterogeneous, Multiple Depot, Multiple Traveling Salesman Problem (HMDMTSP). This paper presents a transformation of a Heterogeneous, Multiple Depot, Multiple Traveling Salesman Problem (HMDMTSP) into a single, Asymmetric, Traveling Salesman Problem (ATSP). As a result, algorithms available for the single salesman problem can be used to solve the HMDMTSP. To show the effectiveness of the transformation, the well known Lin-Kernighan-Helsgaun heuristic was applied to the transformed ATSP. Computational results show that good quality solutions can be obtained for the HMDMTSP relatively fast.

Location‐routing problem (LRP) thoroughly considers location allocation problem (LAP) and vehicle routing problem (VRP) which has been an integral part applied in modern logistics. A number of researchers at home and aboard have put forward their views by establishing fine models. On the basis of studying the previous research results by classification, summary, and comparative analysis, this study hence proposes a new solution‐fuzzy clustering model and algorithm to resolve two‐layer location‐routing problem based on a heuristic hybrid algorithm: Designing a hybrid genetic and simulated annealing algorithm (GASA) to optimize the initial value of the fuzzy C‐means clustering algorithm (FCM); considering the roving visit characteristics of vehicles to design the path by employing a special VRP problem—the multiple traveling salesman problem (MTSP). Theoretical analysis and experimental results show that the algorithm used in this study has the advantages of fast convergence speed and less iterations, which significantly improve the quality of the initial solution of FCM in LAP, shorten the vehicle patrol cycle in VRP to a great extent, improve the vehicle utilization, and save the vehicle patrol costs. A specific example is programmed by MATLAB to verify the feasibility of this method.

The Generalized Traveling Salesman Problem (GTSP) is a useful model for problems involving decisions of selection and sequence. The problem is defined on a directed graph in which the nodes have been pregrouped into m mutually exclusive and exhaustive nodesets. Arcs are defined only between nodes belonging to different nodesets with each arc having an associated cost. The GTSP is the problem of finding a minimum cost m-arc directed cycle which includes exactly one node from each nodeset. In this paper, we show how to efficiently transform a GTSP into a standard asymmetric Traveling Salesman Problem (TSP) over the same number of nodes. The transformation allows certain routing problems which involve discrete alternatives to be modeled using the TSP framework. One such problem is the heterogenous Multiple Traveling Salesman Problem (MTSP) for which we provide a GTSP formulation.

Vehicle routing problems (VRPs) are a fundamental class of combinatorial optimization problems (COPs) in computer science and operations research, with diverse real-world applications in logistics, transportation, and manufacturing. The intrinsic NP-hard nature makes VRPs exponentially expensive to be solved by exact solvers. As an alternative, heuristic solvers deliver suboptimal solutions within reasonable time but require extensive hand-crafted rules and domain expertise tailored to each specific problem. Recently, neural combinatorial optimization (NCO), which leverages machine learning to learn heuristics in a data-driven manner, has gained significant attention. These neural solvers demonstrate strong performance while reducing computational overhead and reliance on domain expertise compared to traditional solvers. However, they face significant generalization challenges. For example, their performance may degrade substantially when applied to instances with different data distributions, problem scales, or constraints than those encountered during training. This generalization issue severely limits their practical applicability. This thesis aims to systematically address these challenges and advance the development of generalizable neural solvers for VRPs. The thesis begins by enhancing the cross-distribution generalization of neural VRP solvers through an adversarial training approach. We introduce an ensemble-based Collaborative Neural Framework (CNF) that adversarially trains multiple models in a collaborative manner, promoting robustness against adversarial attacks while also boosting performance on clean instances. In this framework, an attacker generates hard instances by perturbing the data distribution, challenging the models to adapt and thereby strengthening cross-distribution generalization. Additionally, a neural router is designed to efficiently distribute training instances among the models, thereby improving load balancing and collaborative efficacy. Extensive experiments on two classic routing problems -- the travelling salesman problem (TSP) and the capacitated vehicle routing problem (CVRP) -- demonstrate the effectiveness and versatility of CNF in enhancing the cross-distribution generalization. Next, the thesis further expands neural solvers to simultaneously address both cross-size and cross-distribution generalization challenges. We propose Omni-VRP, a generic meta-learning framework that enables effective training of an initial model with the capacity for rapid adaptation to new tasks during inference, where each task corresponds to a set of instances with the same problem scale and data distribution. Additionally, we introduce a simple yet efficient approximation method to reduce the training overhead associated with second-order derivative calculations. Extensive experiments on both synthetic and benchmark instances of TSP and CVRP validate the effectiveness of our approach. Finally, the thesis pioneers the study of cross-problem generalization, pushing the boundaries of neural VRP solvers in learning generalizable representations across diverse constraints. We propose MVMoE, a unified neural solver capable of solving 16 VRP variants simultaneously in a zero-shot manner through attribute composition. To efficiently enhance model capacity, we employ a mixture-of-experts architecture with a hierarchical gating mechanism, striking a balance between empirical performance and computational complexity. Experimentally, our method significantly promotes zero-shot generalization on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. Overall, this thesis represents a pioneering exploration and substantial advancement in developing generalizable neural solvers for VRPs. It investigates innovative designs across various components, including model architectures, training algorithms, and problem settings, to enhance the generalization of neural solvers across diverse contexts such as varying data distributions, problem scales, and constraints. These contributions encourage neural solvers to learn more robust and generalizable representations, paving the way for the first generation of foundation models capable of addressing a wide range of COPs. Ultimately, the insights gained here not only propel the field of NCO forward but also enrich the broader landscape of learning-based optimization methods.

Routing problems are a class of combinatorial problems with many practical applications. Recently, end-to-end deep learning methods have been proposed to learn approximate solution heuristics for such problems. In contrast, classical dynamic programming (DP) algorithms guarantee optimal solutions, but scale badly with the problem size. We propose Deep Policy Dynamic Programming (DPDP), which aims to combine the strengths of learned neural heuristics with those of DP algorithms. DPDP prioritizes and restricts the DP state space using a policy derived from a deep neural network, which is trained to predict edges from example solutions. We evaluate our framework on the travelling salesman problem (TSP), the vehicle routing problem (VRP) and TSP with time windows (TSPTW) and show that the neural policy improves the performance of (restricted) DP algorithms, making them competitive to strong alternatives such as LKH, while also outperforming most other ‘neural approaches’ for solving TSPs, VRPs and TSPTWs with 100 nodes.KeywordsDynamic ProgrammingDeep LearningVehicle Routing

We present a solution to the Multiple Depots, Multiple Traveling Salesmen Problem (MTSP) with constant prots, which is a generalization of the single traveling salesman problem (TSP) in the following ways: there are several salesmen, the salesmen originate at various initial locations called depots, and the cities are visited only if it is protable to visit them, i.e., the benet of visit is higher than the cost of travel to that city. The solution is based on a binary programming formulation of the problem and solving this formulation using a generic branch-and-bound algorithm. We present results that show the ability of the proposed solution to solve problems with various levels of city benets and compare these results to other MTSP variants.

The Travelling Salesman Problem (TSP) is a well-known combinatorial optimization problem that belongs to a class of problems known as NP-hard, which is an exceptional case of travelling salesman problem (TSP), which determines a set of routes enabling multiple salesmen to start at and return to home cities (depots). The penguins search optimization algorithm (PeSOA) is a new metaheuristic optimization algorithm. In this paper, we present a discrete penguins search optimization algorithm (PeSOA) for solving the multiple travelling salesman problem (MTSP). The PeSOA evaluated by a set of benchmarks of TSP instance from TSPLIB library. The experimental results show that PeSOA is very efficient in finding the right solutions in a reasonable time

The Multiple Travelling Salesman Problem (MTSP) is a basic deformation of the Travelling Salesman Problem (TSP), which is a typical NP-Hard problem. For combinatorial optimization problems shaped like MTSP, researchers often use intelligent optimization algorithms such as genetic algorithms, differential evolutionary algorithms, simulated annealing algorithms and other intelligent optimization algorithms to approximate the solution. However, typical intelligent optimization algorithms have the disadvantage of being prone to local convergence and failing to produce theoretically optimal solutions. Based on this, we propose an HCA&DE algorithm that improves the DE algorithm by using the hill-climbing algorithm, and carry out data experiments using the solution set of TSPLIB, which proves the practicality and effectiveness of the algorithm.

In the aftermath of a large disaster, the routing of vehicles carrying critical supplies can greatly impact the arrival times to those in need. Because it is critical that the deliveries are both fast and fair to those being served, it is not clear that the classic cost-minimizing routing problems properly reflect the relevant priorities in disaster relief. In this paper, we take the first steps toward developing new methodologies for these problems. We focus specifically on two alternative objective functions for the traveling salesman problem (TSP) and the vehicle routing problem (VRP): one that minimizes the maximum arrival time (minmax) and one that minimizes the average arrival time (minavg). To demonstrate the potential impact of using these new objective functions, we bound the worst-case performance of optimal TSP solutions with respect to these new variants and extend these bounds to include multiple vehicles and vehicle capacity. Similarly, we examine the potential increase in routing costs that results from using these alternate objectives. We present solution approaches for these two variants of the TSP and VRP, which are based on well-known insertion and local search techniques. These are used in a series of computational experiments to help identify the types of instances in which TSP and VRP solutions can be significantly different from optimal minmax and minavg solutions.

The Multiple Traveling Salesman Problem (multiple-TSP) is a straightforward extension of the well-known Traveling Salesman Problem (TSP), in which more salesmen must visit a set of interconnected cities. Ant Colony Optimization (ACO) algorithms are designed to build sequentially the solutions, aspect which on multiple-TSP imposes new challenges. Compared to TSP which deals with one sample space - the set of cities (locations), multiple-TSP involves two sample spaces: the set of salesmen (agents) and the set of cities. Existing ACO algorithms addressing multiple-TSP are two-phase sampling procedures which, firstly, independently sample from the first set (the set of salesmen), and then conditionally sample from the second set. Our claim is that a joint sampling mechanism, which will exploit a joint probability space, is likely to lead to superior results. We validate our hypothesis by implementing five ACO-based algorithms to solve multiple-TSP: three of them are two-phase algorithms exploring various methods to sample from the salesmen space, while two of them implement the joint sampling scheme. The results are analyzed both in a single-objective manner that considers the minimization of the longest tour, and also from a bi-objective perspective that considers two conflicting objectives: 1) minimization of the total traveled distance and 2) work balancing – which amounts to minimizing the amplitude of the costs of individual tours.

Integrating forecasting in metaheuristic methods to solve dynamic routing problems: Evidence from the logistic processes of tuna vessels