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

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.