Computational Approaches to Stochastic Vehicle Routing Problems

Further results on the probabilistic traveling salesman problem

The probabilistic traveling salesman problem (PTSP) is a variation of the classic TSP and one of the most significant stochastic network and routing problems. Designing effective and efficient algorithms for solving PTSP is a really challenging task, since in PTSP, the computational complexity associated with the combinatorial explosion of potential solution is exacerbated by the stochastic element in the data. In general, researchers use two types of techniques in their search algorithms for PTSP: analytical computation and empirical estimation. The analytical computation approach computes the cost f(\(\pi \)) of an a priori tour \(\pi \) using a closed-form expression. Empirical estimation simply estimates the cost through Monte Carlo simulation. This paper describes a simulation-based algorithm that constructs the solution attractor of local search for the PTSP and then finds the best a priori tour within the solution attractor. More specifically, our algorithm first uses a simple multi-start local search process to find a set of locally optimal a priori tours through Monte Carlo simulation and stores these tours in a matrix. Then, the algorithm uses an exhausted search process to find all tours contained in the solution attractor and identifies a globally optimal a priori tour \(\pi \) through Monte Carlo simulation.

The Probabilistic Traveling Salesman Problem with Deadlines (PTSPD) is a Stochastic Vehicle Routing Problem with a computationally demanding objective function. In this work we propose an approximation for that objective function based on Monte Carlo Sampling and using the novel approach of quasi-parallel evaluation of samples. We perform comprehensive computational studies that reveal the efficiency of this approximation. Additionally, we examine different Local Search Algorithms and present a Random Restart Local Search Algorithm for solving the PTSPD together with an extensive computational study on a large set of benchmark instances.

It is well-known that the cost of parcel delivery can be reduced by designing routes that take into account the uncertainty surrounding customers’ presences. Thus far, routing problems with stochastic customer presences have relied on the assumption that all customer presences are independent from each other. However, the notion that demographic factors retain predictive power for parcel-delivery efficiency suggests that shared characteristics can be exploited to map dependencies between customer presences. This paper introduces the correlated probabilistic traveling salesman problem (CPTSP). The CPTSP generalizes the traveling salesman problem with stochastic customer presences, also known as the probabilistic traveling salesman problem (PTSP), to account for potential correlations between customer presences. I propose a generic and flexible model formulation for the CPTSP using copulas that maintains computational and mathematical tractability in high-dimensional settings. I also present several adaptations of existing exact and heuristic frameworks to solve the CPTSP effectively. Computational experiments on real-world parcel-delivery data reveal that correlations between stochastic customer presences do not always affect route decisions, but could have a considerable impact on route cost estimates. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0005 .

We show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem. The probabilistic "traveling salesman" must decide on an a priori order in which to visit n cities (randomly distributed over a unit square) before learning that some cities can be omitted. We find the optimized average length of the pruned tour follows E(L(pruned))=sqrt[np](0.872-0.105p)f(np), where p is the probability of a city needing to be visited, and f(np)-->1 as np--> infinity. The average length of the a priori tour (before omitting any cities) is found to follow E(L(a priori))=sqrt[n/p]beta(p), where beta(p)=1/[1.25-0.82 ln(p)] is measured for 0.05< or =p< or =0.6. Scaling arguments and indirect measurements suggest that beta(p) tends towards a constant for p<0.03. Our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths, exploiting the sampling error to provide the analog of thermal fluctuations in simulated (thermal) annealing. The method has general application to the optimization of functions whose cost to evaluate rises with the precision required.

Probabilistic combinatorial optimization problems are generalized versions of deterministic combinatorial optimization problems with explicit inclusion of probabilistic elements in the problem definitions. Based on the probabilistic traveling salesman problem (PTSP) and on the probabilistic minimum spanning tree problem (PMSTP), the objective of this paper is to give a rigorous treatment of the probabilistic analysis of these problems in the plane. More specifically we present general finite-size bounds and limit theorems for the objective functions of the PTSP and PMSTP. We also discuss the practical implications of these results and indicate some open problems.

One of the main issues in the application of a particle swarm optimization (PSO) algorithm and of every evolutionary optimization algorithm is the finding of the suitable parameters of the algorithm. Usually, a trial and error procedure is used but, also, a number of different procedures have been applied in the past. In this chapter, we use a new adaptive version of a PSO algorithm where random values are assigned in the initialization of the algorithm and, then, during the iterations the parameters are optimized together and simultaneously with the optimization of the objective function of the problem. This idea is used for the solution of the probabilistic traveling salesman problem (PTSP). The algorithm is tested on a number of benchmark instances and it is compared with a number of algorithms from the literature.KeywordsParticle swarm optimizationVariable neighborhood searchProbabilistic traveling salesman problem

A general-purpose, simulation-based algorithm S-ACO for solving stochastic combinatorial optimization problems by means of the ant colony optimization (ACO) paradigm is investigated. Whereas in a prior publication, theoretical convergence of S-ACO to the globally optimal solution has been demonstrated, the present article is concerned with an experimental study of S-ACO on two stochastic problems of fixed-routes type: First, a pre-test is carried out on the probabilistic traveling salesman problem. Then, more comprehensive tests are performed for a traveling salesman problem with time windows (TSPTW) in the case of stochastic service times. As a yardstick, a stochastic simulated annealing (SSA) algorithm has been implemented for comparison. Both approaches are tested at randomly generated problem instances of different size. It turns out that S-ACO outperforms the SSA approach on the considered test instances. Some conclusions for fine-tuning S-ACO are drawn.

PTSP Solution Strategy for Motion Trajectory of UAV in Ubiquitous Sensor Network

The vehicle routing problem (VRP) attempts to find optimal (minimum length) routes for a set of vehicles visiting a set of locations. Solving a VRP calls for a cost matrix between locations. The size of the matrix grows quadratically with an increasing number of locations, restricting large-sized VRPs from being solved in a reasonable amount of time. The time needed to obtain a cost matrix is expensive when routing engines are used, which solve shortest path problems in the back end. In fact, details on the shortest path are redundant; only distance or time values are necessary for VRPs. In this study, an artificial neural network (ANN) that receives two geo-coordinates as input and provides estimated cost (distance and time) as output is trained. The trained ANN model was able to estimate with a mean absolute percentage error of 7.68%, surpassing the quality of 13.2% with a simple regression model on Euclidean distance. The possibility of using a trained model in VRPs is examined with different implementation scenarios. The experimental results with VRPs confirm that using ANN estimation instead of Euclidean distance produces a better solution, which is verified to be statistically significant. The results also suggest that an ANN can be a better choice than routing engines when the trade-off between response time and solution quality is considered.

The probabilistic travelling salesman problem (PTSP) is an extension of the well-known travelling salesman problem (TSP), which has been extensively studied in the field of combinatorial optimization. The goal of the TSP is to find the minimum length of a tour to all customers, given the distances between all pairs of customers whereas the objective of the PTSP is to minimize the expected length of the a priori tour where each customer requires a visit only with a given probability (Bertsimas, 1988; Bertsimas et al., 1990; Jaillet, 1985). The main difference between the PTSP and the TSP is that in the PTSP the probability of each node being visited is between 0.0 and 1.0 while in TSP the probability of each node being visited is 1.0. Due to the fact that the element of uncertainty not only exists, but also significantly affects the system performance in many real-world transportation and logistics applications, the results from the PTSP can provide insights into research in other probabilistic combinatorial optimization problems. Moreover, the PTSP can also be used to model many real-world applications in logistical and transportation planning, such as daily pickup-delivery services with stochastic demand, job sequencing involving changeover cost, design of retrieval sequences in a warehouse or in a cargo terminal operations, meals on wheels in senior citizen services, trip-chaining activities, vehicle routing problem with stochastic demand, and home delivery service under e-commerce (Bartholdi et al., 1983; Bertsimas et al., 1995; Campbell, 2006; Jaillet, 1988; Tang & Miller-Hooks, 2004). Early PTSP computational studies, dating from 1985, adopted heuristic approaches that were modified from the TSP (e.g., nearest neighbor, savings approach, spacefilling curve, radial sorting, 1-shift, and 2-opt exchanges) (Bartholdi & Platzman, 1988; Bertsimas, 1988; Bertsimas & Howell, 1993; Jaillet, 1985, 1987; Rossi & Gavioli, 1987). With its less than satisfactory performance in yielding solution quality, researchers in the recent years switch to metaheuristic methods, such as ant colony optimization (Bianchi, 2006; Branke & Guntsch, 2004), evolutionary algorithm (Liu et al., 2007), simulated annealing (Bowler et al., 2003), threshold accepting (Tang & Miller-Hooks, 2004) and scatter search (Liu, 2006, 2007, 2008). Because the genetic algorithm (GA), a conceptual framework of the population-based metaheuristic method, has been shown to yield promising outcomes for solving various complicated optimization problems in the past three decades (Back et al., 1997; Davis, 1991; O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

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.

In This Issue

The Vehicle Routing Problem (VRP) is one of the most frequently encountered optimization problems in logistics, which aims to minimize the cost of transportation operations by a fleet of vehicles operating out of a base. This paper introduces VRP Spreadsheet Solver, an open source Excel based tool for solving many variants of the Vehicle Routing Problem (VRP). Case studies of two real-world applications of the solver from the healthcare and tourism sectors that demonstrate its use are presented. The solution algorithm for the solver, and computational results on benchmark instances from the literature are provided. The solver is found to be capable of solving Capacitated VRP and Distance-Constrained VRP instances with up to 200 customers within 1 h of CPU time.

The Vehicle Routing Problem (VRP) is a cornerstone of operations research with broad real-world relevance. As variants increase in complexity, researchers increasingly adopt metaheuristics and matheuristics to find high-quality solutions. We focus on set partitioning (SP) and set covering (SC) formulations as enhancement mechanisms in matheuristics for VRPs. These methods exploit previously generated routes by decomposing and recombining solutions, either as a post-optimization step or iteratively. We conducted a meta-analysis of 30 implementations, selected from 54 eligible studies identified via systematic review of Web of Science and Scopus, complemented by backward snowballing. Using a random-effects model, we quantified the effect of SP/SC enhancements on solution quality. On average, SP/SC improved solutions by 0.51% (95% CI: 0.41%–0.61%). Although the numerical gains are modest, they are consistent and significant, highlighting the practical value of these classical formulations in hybrid heuristic frameworks. • Set Partitioning/Set covering enhances Vehicle Routing Problems heuristics. • Meta-analysis of 30 implementations quantifies impact on solutions. • Methods effective across diverse Vehicle Routing Problems variants. • Improves solutions by average 0.51% (95% interval: 0.41%–0.61%).