NP-hard scheduling problems in planning process automation in discrete systems of certain classes

Abstract

In this paper, we consider an intractable problem of total tardiness of tasks minimization on single machine. The problem has a broad applications solutions during planning process automation in systems in various spheres of human activity. We investigate the solutions obtained by the exact algorithm for this problem earlier developed by M.Z. Zgurovsky and A.A. Pavlov. We propose an efficient approximation algorithm with O(n2) complexity with estimate of the maximum possible deviation from optimum. We calculate the estimate separately for each problem instance. Based on this estimate, we construct an efficient estimate of the deviation from the optimum for solutions obtained by any heuristic algorithms. Our statistical studies have revealed the conditions under which our approximation algorithm statistically significantly yields a solution within 1–2% deviation from the optimum, presumably for any problem size. This makes possible obtaining efficient approximate solutions for real practical size problems that cannot be solved with known exact methods.