Single-Processor Scheduling Problem With Dynamic Models of Task Release Dates

The problem of scheduling a set of tasks on a single (critical) processor under given technological precedence constraints is considered. Before a task is released for processing on the critical processor, it must undergo some preprocessing treatment. This treatment is a dynamic process in which the speed of change of the preprocess state depends on the amount of resource. We assume that the speed can be described as a concave, continuous, positive, and strictly increasing function of resource amount. The total consumption of resource at each moment is limited. The time which is needed to pass from the initial state to the terminal one is called a task release date. The objective is to minimize the maximum task completion time. Such a problem appears e.g., in steel-mill systems, where ingots (before hot rolling on the blooming mill) have to achieve the required temperature in the preheating process in soaking pits. The computational complexity of the problem is analyzed. Due to the problem properties, the difficult dynamic problem of the optimal resource allocation (for a fixed task schedule) is reduced to a task of convex programming. A general approximation algorithm of constructing task schedule along with its experimental and worst case analysis is also presented.