Routing and scheduling problems with time windows have long been important optimization problems for logistics and planning. Many classical heuristics and exact methods exist for such problems. However, there are no satisfactory methods for generating routes using quantum computing (QC), for mainly two reasons: inequality constraints, and the trade-off of feasibility and solution quality. Inequality constraints are typically handled using slack variables; and feasible solutions are found by filtering samples. These challenges are amplified in the presence of noise inherent in QC. Here, we propose a greedy algorithm that generates routes by using information from all samples obtained from the quantum computer. By noticing the relationship between qubits in our formulation as a directed acyclic graph (DAG), we designed an algorithm that adaptively constructs a feasible solution. We prove its convergence to a feasible solution, and illustrate its efficacy by solving the Fleet Sizing Vehicle Routing Problem with Time Windows (FSVRPTW). Our computational results show that this method obtains a lower objective value than the current state-of-the-art annealing approaches, both classical and hybrid, for the same amount of time using D-Wave Hybrid Solvers. We also show its robustness to noise on D-Wave Advantage2 through computational results as compared to the filtering approach on DWaveSampler, even when the filtering approach is given a longer annealing time, and a larger sample size.
Routing and scheduling problems with time windows have long been important optimization problems for logistics and planning. Many classical heuristics and exact methods exist for such problems. However, there are no satisfactory methods for generating routes using quantum computing (QC), for mainly two reasons: inequality constraints, and the trade-off of feasibility and solution quality. Inequality constraints are typically handled using slack variables; and feasible solutions are found by filtering samples. These challenges are amplified in the presence of noise inherent in QC. Here, we propose a greedy algorithm that generates routes by using information from all samples obtained from the quantum computer. By noticing the relationship between qubits in our formulation as a directed acyclic graph (DAG), we designed an algorithm that adaptively constructs a feasible solution. We prove its convergence to a feasible solution, and illustrate its efficacy by solving the Fleet Sizing Vehicle Routing Problem with Time Windows (FSVRPTW). Our computational results show that this method obtains a lower objective value than the current state-of-the-art annealing approaches, both classical and hybrid, for the same amount of time using D-Wave Hybrid Solvers. We also show its robustness to noise on D-Wave Advantage2 through computational results as compared to the filtering approach on DWaveSampler, even when the filtering approach is given a longer annealing time, and a larger sample size.
Intelligent Transportation Systems (ITS) occupy a pivotal position in the contemporary societal landscape, with one of their widespread applications being the vehicle routing problem, renowned for its classification as an NP-hard challenge. Traditional algorithms often require considerable time to address such problems or can only provide approximate solutions. Recent progress in quantum computation, however, offer a promising alternative by utilizing the parallelism inherent in quantum mechanics. In this study, we propose to solve the vehicle routing problem using the Grover adaptive algorithm. Compared to classical algorithms and existing quantum methods, our innovative algorithm achieves a quadratic speedup, with a time complexity ofO(KMN). Additionally, it is applicable to a broader range of real-world scenarios, as it can directly handle linear inequality constraints without introducing slack variables. Notably, the algorithm operates independently of classical optimizers and gradient-based methods, thereby circumventing potential challenges such as the “barren plateaus” phenomenon. Experimental results from four instances demonstrate the feasibility and correctness of the proposed approach, highlighting its capability to handle inequality constraints easily.
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%.
Vehicle routing refers to the generic problem of finding the optimal routes for a given fleet of vehicles with respect to a given measure of optimality and subject to a given set of constraints. Finding the exact solution to such a problem is NPhard. Hence, we often use heuristics to find feasible solutions. We use the DWave Hybrid solver to find a solution to the vehicle routing problem, and compare the time complexity of finding the solution as well as the quality of the solution found against a classical alternative, namely Gurobi v9.5. We observe that the quantum annealer shows better scaling with respect to size when compared to Gurobi.
Quantum computing (QC) stands at the cusp of revolutionizing computation, yet its near-term potential, constrained by Noisy Intermediate-Scale Quantum (NISQ) devices, remains underexplored. This dissertation investigates how hybrid quantum-classical algorithms can address combinatorial optimization challenges in logistics, focusing on vehicle routing and drone delivery—NP-hard problems with exponential solution spaces that defy classical exhaustive methods. Amidst NISQ limitations like limited qubits and high noise, we confront key challenges: encoding complex constraints, e.g., time windows, battery capacity, into quantum models, balancing quantum and classical components for scalability, and accessing scarce quantum resources. By integrating quantum annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) with classical heuristics, this work demonstrates QC’s practical utility, advancing both optimization theory and real-world logistics applications. The dissertation unfolds across four core contributions. First, the Hybrid Quantum Tabu Search (HQTS) optimally solves the 50-location Capacitated Vehicle Routing Problem (CVRP), achieving a 4.72% optimality gap. Secondly, we refine HQTS to reach a 2.15% gap with enhanced QA utilization, outperforming prior hybrids on standard datasets. Third, adapting HQTS for the CVRPTWwith D-Wave’s Constrained Quadratic Model solver yields a 3.85% gap on 100-location instances, leveraging a novel heuristic to ensure time-window feasibility where quantum outputs falter. Lastly, QUADRO, a pioneering framework for drone delivery, employs QAOA-driven routing and hybrid scheduling, minimizing transit time and makespan for fleets of 2–11 drones needing fewer than 100 qubits, rivaling classical benchmarks on adapted Augerat datasets. These advances, tested via D-Wave’s Advantage system and Qiskit simulations, showcase hybrid strategies that mitigate NISQ constraints, delivering scalable, feasible solutions to logistics challenges. This work establishes near-term QC’s viability for combinatorial optimization, bridging theoretical promise with practical impact. Synthesizing quantum exploration with classical refinement achieves competitive performance, e.g., matching or surpassing Google ORTools in select CVRPTW cases, while illuminating NISQ limits, such as feasibility degradation beyond small scales. These hybrid strategies deliver scalable, feasible solutions, from a 2.15% optimality gap for CVRP to effective drone fleet management with fewer than 100 qubits, illuminating quantum computing’s potential despite its current constraints. Ultimately, this dissertation positions hybrid QC as a transformative tool for optimization challenges, offering a robust foundation for scaling quantum optimization into operational reality.
Quantum computing (QC) is expected to solve incredibly difficult problems, including finding optimal solutions to combinatorial optimization problems. However, to date, QC alone is still far to demonstrate this capability except on small-sized problems. Hybrid approaches where QC and classical computing work together have shown the most potential for solving real-world scale problems. This work aims to show that we can enhance a classical optimization algorithm with QC so that it can overcome this limitation. We present a new hybrid quantum-classical tabu search (HQTS) algorithm to solve the capacitated vehicle routing problem (CVRP). Based on our prior work, HQTS leverages QC for routing within a classical tabu search framework. The quantum component formulates the traveling salesman problem (TSP) for each route as a QUBO, solved using D-Wave's Advantage system. Experiments investigate the impact of quantum routing frequency and starting solution methods. While different starting solution methods, including quantum-based and classical heuristics methods, it shows minimal overall impact. HQTS achieved optimal or near-optimal solutions for several CVRP problems, outperforming other hybrid CVRP algorithms and significantly reducing the optimality gap compared to preliminary research. The experimental results demonstrate that more frequent quantum routing improves solution quality and runtime. The findings highlight the potential of integrating QC within meta-heuristic frameworks for complex optimization in vehicle routing problems.
This research addresses the Vehicle Routing Problem (VRP) with uncertain data and proposes a novel approach using quantum computing techniques. The problem involves optimizing vehicle routes considering uncertain customer demands, time windows, and vehicle capacities. We formulate the problem mathematically and develop an algorithmic framework to tackle it. The approach incorporates multiple scenarios based on the uncertainty distribution and selects the one with the minimum cost to optimize the vehicle routes. Through a numerical example, we demonstrate the effectiveness of the proposed approach in generating optimal routes that minimize the total distance traveled by the vehicles. The results highlight the solution quality, adaptability to uncertainty, and potential benefits in terms of cost reduction and resource utilization. While the computational efficiency of quantum computing approaches is a consideration, this research provides a promising direction for addressing uncertain optimization problems in logistics and transportation. Future research should focus on scalability and refinement of the algorithm to further enhance its applicability in real-world scenarios.