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International Conference on Computation System and Information Technology for Sustainable Solutions (CSITSS)

Parallelized Clustering-Based Optimization for CVRP: Leveraging Quantum Computing and GPU Acceleration

Geetha V, Arfaat Tamboli, Salaj Saxena, Dhrithi Kadam

Published Nov 7, 2024• 0 citations
Publisher

Abstract

The Capacitated Vehicle Routing Problem is one of the most well-known issues in operations research and transportation logistics. It comprises optimizing a set of vehicle routes to serve a set of clients, subject to vehicle capacity restrictions. Reducing the total distance that the cars go is the aim of the challenge. The Capacitated Vehicle Routing Problem has been studied by academics for a very long time. This topic is receiving a lot of interest because it can be applied to almost every industry, including marketing, logistics, healthcare, and others. One of the main obstacles, though, is that when the problem is attempted to be solved for a greater number of clients and vehicles, the problem becomes so complex that standard computers are either unable to solve it or require several months or even years to generate an output. We proposed an approach for this restriction that enables the problem to be resolved with quantum computers. We use two algorithms in our methodology, Constrained Clustering and Fuzzy C-Means, to attain our results. Clustering and routing are the two phases each of these algorithms go through. This method involves grouping all connected consumers into different clusters and then conducting routing within each cluster. They work with notable efficiency, producing optimal solutions in a noticeably smaller span of time. The results show that the CC approach outperforms FCM in most instances, achieving an average optimality gap of 13.04%, compared to 16.5% for FCM. Furthermore, both approaches perform significantly better than non-clustered methods.

DOI
10.1109/csitss64042.2024.10817035

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The Capacitated Vehicle Routing Problem is one of the most well-known issues in operations research and transportation logistics. It comprises optimizing a set of vehicle routes to serve a set of clients, subject to vehicle capacity restrictions. Reducing the total distance that the cars go is the aim of the challenge. The Capacitated Vehicle Routing Problem has been studied by academics for a very long time. This topic is receiving a lot of interest because it can be applied to almost every industry, including marketing, logistics, healthcare, and others. One of the main obstacles, though, is that when the problem is attempted to be solved for a greater number of clients and vehicles, the problem becomes so complex that standard computers are either unable to solve it or require several months or even years to generate an output. We proposed an approach for this restriction that enables the problem to be resolved with quantum computers. We use two algorithms in our methodology, Constrained Clustering and Fuzzy C-Means, to attain our results. Clustering and routing are the two phases each of these algorithms go through. This method involves grouping all connected consumers into different clusters and then conducting routing within each cluster. They work with notable efficiency, producing optimal solutions in a noticeably smaller span of time. The results show that the CC approach outperforms FCM in most instances, achieving an average optimality gap of 13.04%, compared to 16.5% for FCM. Furthermore, both approaches perform significantly better than non-clustered methods.
Quantum optimization has recently drawn considerable attention as one of the possible applications of noisy intermediate-scale quantum computation, yet the problem of qubit requirement remains a major bottleneck when combinatorial optimization problems are converted into quantum circuits. This issue becomes especially critical in solving the capacitated vehicle routing problem (CVRP) with the quantum approximate optimization algorithm (QAOA), since the number of required qubits increases polynomially with respect to the number of nodes. This study investigates whether a heuristic divide-and-conquer strategy can be adapted to the quantum setting so as to improve qubit efficiency while preserving the optimization capability to a reasonable extent. The proposed method decomposes a single CVRP into multiple traveling salesman problems (TSPs) by the sweeping-based clustering method, searches for the sector configuration with the smallest angle sum by Grover’s search algorithm, and then solves each sector-wise TSP with the QAOA aided by the gravitational search algorithm. Experiments on five benchmark datasets show that the proposed approach attains feasible solutions within 3.4 to 12.7% of the reinforcement-learning baseline on the main test set. These results suggest that the proposed approach serves as a plausible quantum heuristic framework for constrained routing optimization, with the advantage of reducing the qubit burden by decomposing the original problem into smaller subproblems.
Abstract The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.
The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that has been of great interest for decades for both, science and industry. The CVRP is a variant of the vehicle routing problem characterized by capacity constrained vehicles. The aim is to plan tours for vehicles to supply a given number of customers as efficiently as possible. The problem is the combinatorial explosion of possible solutions, which increases superexponentially with the number of customers. Classical solutions provide good approximations to the globally optimal solution. D-Wave's quantum annealer is a machine designed to solve optimization problems. This machine uses quantum effects to speed up computation time compared to classic computers. The problem on solving the CVRP on the quantum annealer is the particular formulation of the optimization problem. For this, it has to be mapped onto a quadratic unconstrained binary optimization (QUBO) problem. Complex optimization problems such as the CVRP can be translated to smaller subproblems and thus enable a sequential solution of the partitioned problem. This work presents a quantum-classic hybrid solution method for the CVRP. It clarifies whether the implemenation of such a method pays off in comparison to existing classical solution methods regarding computation time and solution quality. Several approaches to solving the CVRP are elaborated, the arising problems are discussed, and the results are evaluated in terms of solution quality and computation time.
This study introduces a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problems (CVRP), combining the Column Generation method and the Quantum Alternating Operator Ansatz (QAOAnsatz). The column generation method divides the problem into a master problem and subproblems to effectively solve the problem. We solved the subproblem by using QAOAnsatz to enhance the optimal solution search. Moreover, we generated the problem Hamiltonian using an Augmented Lagrangian-inspired Method to reduce the number of required qubits to handle the inequality. As a result, our experiments obtained an optimal solution for a CVRP problem with 6 customers. This study demonstrates a practical quantum application in logistics and sets a new technique for hybrid quantum-classical optimization frameworks.
Many relevant problems in industrial settings result in NP-hard optimization problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced variant, the Travelling Salesperson Problem (TSP). Even with today's most powerful classical algorithms, the CVRP is challenging to solve classically. Quantum computing may offer a way to improve the time to solution, although the question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ) devices can achieve a practical advantage compared to classical heuristics. The most prominent algorithms proposed to solve combinatorial optimization problems in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and the more general Variational Quantum Eigensolver (VQE). However, implementing them in a way that reliably provides high-quality solutions is challenging, even for toy examples. In this work, we discuss decomposition and formulation aspects of the CVRP and propose an application-driven way to measure solution quality. Considering current hardware constraints, we reduce the CVRP to a clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA and VQE and investigate the influence of various hyperparameters, such as the classical optimizer choice and strength of constraint penalization. Results of QAOA are generally of limited quality because the algorithm does not reach the energy threshold for feasible TSP solutions, even when considering various extensions such as recursive and constraint-preserving mixer QAOA. On the other hand, the VQE reaches the energy threshold and shows a better performance. Our work outlines the obstacles to quantum-assisted solutions for real-world optimization problems and proposes perspectives on how to overcome them.
This study proposes a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problem (CVRP) by integrating the Column Generation (CG) method with the Quantum Alternating Operator Ansatz (QAOAnsatz). The CG method divides the CVRP into the reduced master problem, which finds the best combination of the routes under the route set, and one or more subproblems, which generate the routes that would be beneficial to add to the route set. This method is iteratively refined by adding new routes identified via subproblems and continues until no improving route can be found. We leverage the QAOAnsatz to solve the subproblems. Our algorithm restricts the search space by designing the QAOAnsatz mixer Hamiltonian to enforce one-hot constraints. Moreover, to handle capacity constraints in QAOAnsatz, we employ an Augmented Lagrangian-inspired method that obviates the need for additional slack variables, reducing the required number of qubits. Experimental results on small-scale CVRP instances (up to 6 customers) show that QAOAnsatz converges more quickly to optimal routes than the standard QAOA approach, demonstrating the potential of this hybrid framework in tackling realworld logistical optimization problems on near-term quantum hardware.