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This work presents a model of single-machine scheduling problem. The machine is failure-prone and subject to random breakdowns. The processing time is a deterministic sequence that is randomly compressible, which may be from the introduction of new technology or addition of new equipment. Taking into account the cost for the random breakdowns and the random compressible processing time, our objective is to find the optimal scheduling policy to minimize an objective function. Under simple conditions, it is shown that the optimal sequence possesses a V-shape property.

This paper investigates semi-online hierarchical scheduling problems on two identical machines, with the purpose of minimizing the [Formula: see text]-norm of the machines’ loads. We consider two semi-online versions with knowing the total processing time [Formula: see text] of all jobs, or knowing the total processing time [Formula: see text] of the jobs of hierarchy [Formula: see text] for [Formula: see text] in advance. For the two semi-online versions, the best possible online algorithms are designed with competitive ratios of [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the maximum value of the function [Formula: see text] in [Formula: see text]. When [Formula: see text], our results cover the known results for minimizing the makespan.

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