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Computers & Operations Research

Large-scale vehicle routing problems: Quantum Annealing, tunings and results

Alex Syrichas, Alan J. Crispin

Published May 25, 2017• 47 citations
Publisher

Abstract

Quantum Annealing was previously applied to the vehicle routing problem and the results were promising. For all benchmark instances in the study, optimal results were obtained. However, 100% success rate was not achieved in every case, and tuning the control parameters for larger instances proved cumbersome. This work addresses these remaining difficulties. An empirical approach is taken wherein measurements of run-time behaviour are exploited to transform existing good values of control parameters so that they can be used successfully for other problem instances. The course of this work shows a method which simplifies hand-tuning so that the heuristic performs successfully when applied to larger instances, and also demonstrates a tuning method which establishes control parameter values for instances which belong in broadly defined groupings. In addition, new best known solutions for large-scale instances, and initial results for the distance-constrained variant of the vehicle routing problem are presented.

DOI
10.1016/j.cor.2017.05.014

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Quantum Annealing was previously applied to the vehicle routing problem and the results were promising. For all benchmark instances in the study, optimal results were obtained. However, 100% success rate was not achieved in every case, and tuning the control parameters for larger instances proved cumbersome. This work addresses these remaining difficulties. An empirical approach is taken wherein measurements of run-time behaviour are exploited to transform existing good values of control parameters so that they can be used successfully for other problem instances. The course of this work shows a method which simplifies hand-tuning so that the heuristic performs successfully when applied to larger instances, and also demonstrates a tuning method which establishes control parameter values for instances which belong in broadly defined groupings. In addition, new best known solutions for large-scale instances, and initial results for the distance-constrained variant of the vehicle routing problem are presented.
Metaheuristic approaches to solving combinatorial optimization problems have many attractions. They sidestep the issue of combinatorial explosion; they return good results; they are often conceptually simple and straight forward to implement. There are also shortcomings. Optimal solutions are not guaranteed; choosing the metaheuristic which best fits a problem is a matter of experimentation; and conceptual differences between metaheuristics make absolute comparisons of performance difficult. There is also the difficulty of configuration of the algorithm - the process of identifying precise values for the parameters which control the optimization process. Quantum annealing is a metaheuristic which is the quantum counterpart of the well known classical Simulated Annealing algorithm for combinatorial optimization problems. This research investigates the application of quantum annealing to the Vehicle Routing Problem, a difficult problem of practical significance within industries such as logistics and workforce scheduling. The work devises spin encoding schemes for routing and scheduling problem domains, enabling an effective quantum annealing algorithm which locates new solutions to widely used benchmarks. The performance of the metaheuristic is further improved by the development of an enhanced tuning approach using fitness clouds as behaviour models. The algorithm is shown to be further enhanced by taking advantage of multiprocessor environments, using threading techniques to parallelize the optimization workload. The work also shows quantum annealing applied successfully in an industrial setting to generate solutions to complex scheduling problems, results which created extra savings over an incumbent optimization technique. Components of the intellectual property rendered in this latter effort went on to secure a patent-protected status.
In this paper we propose a new strategy for solving the Capacitated Vehicle Routing Problem using a quantum annealing algorithm. The Capacitated Vehicle Routing Problem is a variant of the Vehicle Routing Problem being characterized by capacitated vehicles which contain goods up to a certain maximum capacity. Quantum annealing is a metaheuristic that uses quantum tunneling in the annealing process. We discuss devising a spin encoding scheme for solving the Capacitated Vehicle Routing Problem with a quantum annealing algorithm and an empirical approach for tuning parameters. We study the effectiveness of quantum annealing in comparison with best known solutions for a range of benchmark instances.
The Vehicle Routing Problem (VRP) is an NP-Hard optimization problem that has been widely studied over the last decades due to its wide practical applications. The goal of the VRP is to design efficient routes for vehicles performing service functions such as the distribution of goods from one or more central locations. There exist many variants to the VRP. These variations arise due to application-specific constraints such as vehicle capacities, delivery time windows, split deliveries, and many more. Due to the complexity of the problem, solvers are usually designed for specific variants and do not generally scale well to larger problem instances. As we approach the physical scaling limits of Moore's law, different novel special-purpose hardware are being designed for solving combinatorial optimization problems such as quantum and quantum-inspired devices. However, these devices are able to solve problems up to fixed variable sizes. In this work, we carry out experiments on the Fujitsu Digital Annealer (DA), quantum-inspired hardware, and show that we are able to get near-optimal results for instances of the Capacitated VRP (CVRP) that can be directly solved on the DA. We further develop a hybrid method for solving large-scale VRP instances. In order to achieve this, we develop a novel candidate route generation method, whose core solves a multi-objective clustering problem on the DA. Once a set of candidate routes has been selected, we then solve a Set Partitioning Problem on the DA to get a final solution. We apply our method to the CVRP and the CVRP with Time-Windows and show that we are able to get near-optimal solutions for problem instances significantly larger the base solvers could solve directly, within the same amount of computation time.
Vehicle Routing Problem is a well-known optimization issue, which has many extensions resulting from practical appliances. The problem is NP-hard, therefore it may be impossible to find an optimal solution for big instances, with traditional computation methods. Recent development of quantum technology, may be a response for this challenge, which however requires a different formulation of the model. The problem is then modeled using Quadratic Unconstrained Binary Optimization (QUBO) equations. Such model can be run using quantum annealers (QA), designed specifically for optimization purposes. This paper is a systematic survey of Vehicle Routing Problem models that are formulated using QUBO, to be solved on quantum computers. The main goal of the paper is to check research gaps in this subject, compare and explain different approaches of modeling various problem extensions with QUBO. The results obtained in reviewed papers will be presented and analyzed. Moreover, the theoretical basis for quantum computation will be presented, with focus on D-Wave company solutions.
In the current competitive industrial environment, the study of NP-hard issues is becoming more and more significant. In logistics, the Vehicle Routing Problem (VRP) is one of the popular challenges, especially in cold chain logistics, which requires the management of perishable goods. This problem involves identifying the optimal paths for vehicles responsible for transporting goods to different locations. Due to its combinatorial nature and the complexity of finding optimal solutions, it is identified as an NP-hard problem that requires an unreasonable time frame to solve. Recently, quantum computing has proven to be a promising approach for addressing some of the complex challenges. Among quantum methodologies, quantum annealing (QA) has demonstrated the potential to accelerate computation times in solving optimization problems. By leveraging the principles of quantum tunneling and superposition, QA can explore solution spaces more efficiently. Therefore, this study explores the application of QA to address VRP in cold chain logistics, aiming to develop solutions that minimize travel distances during deliveries. The case study demonstrates that QA can achieve, on average, at least 85% of optimal performance while providing approximately 95% faster run times compared to classical genetic algorithms (GA). This contributes to greater efficiency in cold chain logistics operations.
In this paper, we explore the potential for quantum annealing to solve realistic routing problems. We focus on two NP-Hard problems, including the Traveling Salesman Problem with Time Windows and the Capacitated Vehicle Routing Problem with Time Windows. We utilize D-Wave's Quantum Annealer and Constrained Quadratic Model (CQM) solver within a hybrid framework to solve these problems. We demonstrate that while the CQM solver effectively minimizes route costs, it struggles to maintain time window feasibility as the problem size increases. To address this limitation, we implement a heuristic method that fixes infeasible solutions through a series of swapping operations. Testing on benchmark instances shows our method achieves promising results with an average optimality gap of 3.86%.