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IEEE Transactions on Systems Man and Cybernetics Systems

Optimal Solving of a Scheduling Problem Using Quantum Annealing Metaheuristics on the D-Wave Quantum Solver

Wojciech Bożejko, Ryszard Klempous, Jarosław Pempera, Jerzy W. Rozenblit, Czesław Smutnicki, Mariusz Uchroński, Mieczysław Wodecki

Published Oct 2, 2024• 22 citations
Publisher

Abstract

Accurately solving discrete optimization problems is still a serious challenge for classical computing systems. Despite significant progress, it is still impossible to optimally solve examples of the size found in real-world applications. This article examines the problem of scheduling tasks on one machine. Each task has an associated execution time, desired completion date, and a late penalty feature weight. The order of executing tasks should be determined in a manner that minimizes the sum of delay costs. Despite the simplicity of the formulation, this problem is NP-hard and is representative of optimization challenges encountered in the practice of production control, management, and logistics.In this work, we propose using a quantum computing environment to solve it. The main disadvantage of calculations on quantum computers is their nondeterminism. Therefore, there is no guarantee of optimality of solutions to discrete optimization problems determined on a quantum computer.We propose a novel approach to constructing hybrid algorithms that guarantee the optimality of the solution. The algorithm for solving the task scheduling problem considered here is based on the branch and bound method. Calculations are performed alternately on a classical computer and a quantum computer. The lower and upper bounds of the criterion function are determined on a quantum computer. Computational experiments were performed on the D-Wave quantum computer. In small examples, the algorithm runs faster and generates significantly fewer solution tree vertices than its counterpart running on a classic computer. For larger examples, the proposed runtime-constrained method, acting as a heuristic, is an interesting alternative to using quantum computing hardware in relation to classical approaches based on silicon computers.

DOI
10.1109/tsmc.2024.3458873

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Accurately solving discrete optimization problems is still a serious challenge for classical computing systems. Despite significant progress, it is still impossible to optimally solve examples of the size found in real-world applications. This article examines the problem of scheduling tasks on one machine. Each task has an associated execution time, desired completion date, and a late penalty feature weight. The order of executing tasks should be determined in a manner that minimizes the sum of delay costs. Despite the simplicity of the formulation, this problem is NP-hard and is representative of optimization challenges encountered in the practice of production control, management, and logistics.In this work, we propose using a quantum computing environment to solve it. The main disadvantage of calculations on quantum computers is their nondeterminism. Therefore, there is no guarantee of optimality of solutions to discrete optimization problems determined on a quantum computer.We propose a novel approach to constructing hybrid algorithms that guarantee the optimality of the solution. The algorithm for solving the task scheduling problem considered here is based on the branch and bound method. Calculations are performed alternately on a classical computer and a quantum computer. The lower and upper bounds of the criterion function are determined on a quantum computer. Computational experiments were performed on the D-Wave quantum computer. In small examples, the algorithm runs faster and generates significantly fewer solution tree vertices than its counterpart running on a classic computer. For larger examples, the proposed runtime-constrained method, acting as a heuristic, is an interesting alternative to using quantum computing hardware in relation to classical approaches based on silicon computers.
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. The main disadvantage of calculations on quantum computers is their non-determinism. For optimization problems, of course, it is possible to get surprisingly good results but without a guarantee of the actual optimality of the result, i.e, an assurance that there exists no better solution which could potentially be found by the quantum machine. In this paper, we propose an novel approach that provides such a guarantee of optimality. We generate a solution that is optimal in the strict mathematical sense, without probabilistic considerations. To accomplish this, we use the D-Wave quantum machine working as a sampler implementing quantum annealing -- an approach considered a hardware metaheuristic -- to obtain upper and lower bounds on the value of the objective function of the problem under consideration. Then, we use the Branch and Bound scheme controlled by quantum annealing. This allows us to obtain very quickly -- in constant time -- the boundaries of the considered subproblems. Our approach is a combination of calculations realized on a quantum machine controlled by procedures implemented on a classical computer, allowing us to generate optimal solutions to the NP-hard problems of task scheduling on a single machine with a total weighted tardiness criterion. The proposed quantum algorithm shows a significant advantage over the classical CPU approach both in terms of calculation time and the number of generated solution tree nodes.
The work deals with the NP-hard problem of scheduling tasks on two machines with the sum-cost criterion. In this problem, each task must be performed sequentially on the first and then on the second machine. Each task is associated with: execution time, due date, and weight (revenue) related to completing the task on-time (i.e. before the due date). The goal is to determine the order of task execution, the same on both machines, that maximizes the total weighted number of on-time tasks (maximum profit). We propose a hybrid algorithm combining quantum and classical methods based on the branch and bound method. We propose to formulate the problem under consideration as a Constrained Quadratic Model (CQM), due to which we can compute upper bounds (using Lagrange relaxation) and lower bounds on the D-Wave quantum machine. Although the D-Wave quantum computer performs probabilistic calculations, the solution determined by the proposed algorithm is optimal.
Abstract The efficient management of complex production systems is a challenge in today’s logistics. In the field of intelligent and sustainable logistics, the optimization of production batches, especially in the context of a rapidly changing product range, requires fast and precise computational solutions. This paper explores the potential of quantum computers for solving these problems. Traditional computational methods are often limited when it comes to optimizing complex logistics systems. In response to these challenges, the paper proposes the use of a hybrid algorithm that combines quantum technologies with classical computational methods. Such integration allows the computational power of both types of technologies to be harnessed, leading to faster and more efficient identification of optimal solutions. In this work, we consider the knapsack problem, a classic NP-hard optimization problem that is commonly used to verify the effectiveness of new algorithm construction methods. The algorithm presented is based on the Branch and Bound method and aims to ensure solution optimality in the context of the non-determinism of quantum computers. Within the algorithm, computations are performed alternately on a classical processor and a quantum processor. In addition, the lower and upper bounds of the objective function are computed in constant time using the D-Wave quantum machine.
Abstract The transportation problem considered in this work consists of minimizing the sum of penalties for tardiness in the delivery of the product from the base to clients. We propose an innovative method for constructing hybrid algorithms using quantum annealing that guarantees the optimality of the solution. The hybrid algorithm alternately performs calculations on a classical computer equipped with a silicon CPU and a quantum annealer with a QPU processor. The entire method is based on the branch and bound scheme. The lower and upper bounds are calculated on the QPU, while the CPU is responsible for organizing the calculations, recalling the solution tree. The problem under consideration is of great practical importance because it is a component of many complex issues related to solving the traveling repairmen problem, which are very difficult or even impossible to solve today using classical methods. Undoubtedly, an important application of quantum computers in the future is NP‐hard optimization tasks related to transportation scheduling and logistics.
Quantum computing (QC) holds great promise to open up a new era of computing and has been receiving significant attention recently. To overcome the performance limitations of near-term QC, utilizing the current quantum computers to complement classical techniques for solving real-world problems is of utmost importance. In this paper, we develop QC-based solution strategies that exploit quantum annealing and classical optimization techniques for solving large-scale scheduling problems in manufacturing systems. The applications of the proposed algorithms are illustrated through two case studies in production scheduling. First, we present a hybrid QC-based solution approach for the job-shop scheduling problem. Second, we propose a hybrid QC-based parametric method for the multi-purpose batch scheduling problem with a fractional objective. The proposed hybrid algorithms can tackle optimization problems formulated as mixed-integer linear and mixed-integer fractional programs, respectively, and provide feasibility guarantees. Performance comparison between state-of-the-art exact and heuristic solvers and the proposed QC-based hybrid solution techniques is presented for both job-shop and batch scheduling problems. Unlike conventional classical solution techniques, the proposed hybrid frameworks harness quantum annealing to supplement established deterministic optimization algorithms and demonstrate performance efficiency over standard off-the-shelf optimization solvers.
Job Shop Scheduling is a combinatorial optimization problem of particular importance for production environments where the goal is to complete a production task in the shortest possible time given limitations in the resources available. Due to its computational complexity it quickly becomes intractable for problems of interesting size. The emerging technology of Quantum Annealing provides an alternative computational architecture that promises improved scalability and solution quality. However, several limitations as well as open research questions exist in this relatively new and rapidly developing technology. This paper studies the application of quantum annealing to solve the job shop scheduling problem, describing each step required from the problem formulation to the fine-tuning of the quantum annealer and compares the solution quality with various classical solvers. Particular attention is devoted to aspects that are often overlooked, such as the computational cost of representing the problem in the formulation required by the quantum annealer, the relative qubits requirements and how to mitigate chain breaks. Furthermore, the impact of advanced tools such as reverse annealing is presented and its effectiveness discussed. The results indicate several challenges emerging at various stages of the experimental pipeline which bring forward important research questions and directions of improvement.