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IEEE International Symposium on Computational Intelligence and Informatics (CINTI)

Solving two-machine flow shop scheduling problem with total weighted number of on-time tasks maximization criterion on D-Wave quantum computer

Wojciech Bożejko, Ryszard Klempous, Mariusz Uchroński, Mieczysław Wodecki

Published Nov 20, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.1109/cinti59972.2023.10382117

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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.
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.
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.
Efficient production planning is essential in modern manufacturing to improve performance indicators such as lead time and to reduce reliance on human intuition. While mathematical optimization approaches, formulated as job shop scheduling problems, have been applied to automate this process, solving large-scale production planning problems remains computationally demanding. Moreover, many practical scenarios involve conflicting objectives, making traditional scalarization techniques ineffective in finding diverse and useful Pareto-optimal solutions. To address these challenges, we developed a quantum-classical hybrid algorithm that decomposes the problem into two subproblems: resource allocation and task scheduling. Resource allocation is formulated as a quadratic unconstrained binary optimization problem and solved using annealing-based methods that efficiently explore complex solutions. Task scheduling is modeled as a mixed-integer linear programming problem and solved using conventional solvers to satisfy detailed scheduling constraints. We validated the proposed method using benchmark instances based on foundry production scenarios. Experimental results demonstrate that our hybrid approach achieves superior solution quality and computational efficiency compared to traditional monolithic methods. This work offers a promising direction for high-speed, multi-objective scheduling in industrial applications.
In recent years, the Flow shop Scheduling Problem (FSP) has gained significant attention, prompting researchers to develop various methods, from conventional to hybrid optimization algorithms. Driven by the rapid growth of quantum computing capabilities, quantum approaches applied to optimization problems have attracted an increasing research effort. This work attempts to solve the permutation flow shop scheduling problem using quantum annealing by formulating the problem as a Quadratic Unconstrained Binary Optimization model (QUBO). Two QUBO formulations tailored to address the FSP are used. The first formulation is based on the position-based modelling while the second is based on five approximations of FSP as a Traveling Salesman Problem (TSP). The QUBO formulations are tested, using D-Wave's quantum annealers, on the well-known Taillard FSP benchmark and compared against each other. Results show that the proposed position-based QUBO reaches better solutions than the TSP-based QUBOs. This work is an attempt to highlight the increasing effectiveness of quantum annealers in addressing complex optimization problems and their limitations compared to conventional classical methods.