Efficient Scheduling in Open Shops

Abstract

A new class of polynomially solvable problems within the class of open shop problems is explored. For the problems with m machines and n jobs, an optimal schedule is constructed by an algorithm with the running time polynomial in m and n when there exists a machine, call it i’, whose load M i’ exceeds the load of any other machine by a given amount of “dominance” M i’ — M i’ ≥ (2m - 4)K, i ≠ i’, where K is the maximum processing time of an operation. A weakened dominance condition M i’ — M i ≥ (m - 1)K, i ≠ i’, combined with M = max Mi ≥ (5.45m - 7)K, also provides efficient construction of an optimal schedule. In both cases the length of an optimal schedule is M. An approximation O(nm 2) algorithm is also described which, for any function η(m) and any input of the open shop problem satisfying M ≥ η(m)K, yields a schedule with an absolute performance guarantee dependent on the function η(m) and independent of the number of jobs n. In particular, for M > (7m - 6)K, the length of the schedule does not exceed M + (m - 1)K/3, which improves that of any greedy schedule.