On a pair of job-machine assignment problems with two stages

We consider two assignment problems in which a number of jobs are assigned to the same number of machines that operate in parallel, but in two stages. They are known as the ‘2-stage time minimizing assignment problem’ and the ‘bi-level time minimizing assignment problem’. These problems have the following characteristics in common: • Each of the machines processes one job (non-preemptively, without help of other machines). • The job-machine assignments are partitioned into two successive stages of parallel processing. • The objective is to minimize the makespan, the sum of the primary and the secondary completion time. Both problems can be solved by (a series of) assignment problems. We improve the related computational complexities by applying reoptimization. Under some conditions a quadratic complexity is derived. We introduce a parameter weighing the relative importance of the primary and the secondary cost per time unit. The parametric problems can be solved, for all parameter values simultaneously, within the same reduced time complexity bounds. Scope and purpose: As it is often important to solve problems quickly, it is essential to reduce the computational complexity of available algorithms, as far as possible. We consider two problems which arise when parallel scheduling is done in two successive stages; they can be tackled by solving a series of linear assignment problems. We show that they can be solved more efficiently, using properties of the classical linear assignment problem. In practice, the cost per time unit in the two stages need not be equal. A parameter controlling the ratio between these costs defines a parametric version of each problem. The algorithms of reduced time complexity can be extended to these parametric problem versions