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Lecture notes in networks and systems

An Exact Quantum Annealing-Driven Branch and Bound Algorithm for Maximizing the Total Weighted Number of on-Time Jobs on a Single Machine

Wojciech Bożejko, Jarosław Pempera, Mariusz Uchroński, Mieczysław Wodecki

Published Jan 1, 2023• 2 citations
Publisher
DOI
10.1007/978-3-031-35173-0_8

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Fixed interval scheduling problem with minimal idle time with an application to music arrangement problem

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The Operational Fixed Interval Scheduling Problem aims to find an assignment of jobs to machines that maximizes the total weight of the completed jobs. We introduce a new variant of the problem where we consider the additional goal of minimizing the idle time, the total duration during which the machines are idle. The problem is expressed using quadratic unconstrained binary optimization (QUBO) formulation, taking into account soft and hard constraints required to ensure that the number of jobs running at a time point is desirably equal to the number of machines. Our choice of QUBO representation is motivated by the increasing popularity of new computational architectures such as neuromorphic processors, coherent Ising machines, and quantum and quantum-inspired digital annealers for which QUBO is a natural input. An optimization problem that can be solved using the presented QUBO formulation is the music reduction problem, the process of reducing a given music piece for a smaller number of instruments. We use two music compositions to test the QUBO formulation and compare the performance of simulated, quantum, and hybrid annealing algorithms.

arXiv:2310.14825•Oct 23, 2023•0 citations
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.
The Operational Fixed Interval Scheduling Problem aims to find an assignment of jobs to machines that maximizes the total weight of the completed jobs. We introduce a new variant of the problem where we consider the additional goal of minimizing the idle time, the total duration during which the machines are idle. The problem is expressed using quadratic unconstrained binary optimization (QUBO) formulation, taking into account soft and hard constraints required to ensure that the number of jobs running at a time point is desirably equal to the number of machines. Our choice of QUBO representation is motivated by the increasing popularity of new computational architectures such as neuromorphic processors, coherent Ising machines, and quantum and quantum-inspired digital annealers for which QUBO is a natural input. An optimization problem that can be solved using the presented QUBO formulation is the music reduction problem, the process of reducing a given music piece for a smaller number of instruments. We use two music compositions to test the QUBO formulation and compare the performance of simulated, quantum, and hybrid annealing algorithms.