The single machine scheduling problem subject to random machine breakdowns is addressed in this paper, where jobs arrive at system dynamically. A modified version of Optimized Surrogate Measure Heuristic (OSMH) developed by Mehta and Uzsoy (1999) is proposed, i.e., Modified Optimized Surrogate Measure Heuristic (MOSMH). The idea behind the heuristic MOSMH is to make full use of the idle times probably existing in initial schedule. The stability of the predictive schedule is measured by the sum of the weighted absolute deviation between the planned job completion times and the realized one. Extensive computational results show that the heuristic MOSMH proposed in this paper significantly improves the schedule stability without deteriorating the efficiency performance and the predictive schedule obtained by MOSMH is more insensitive to the various breakdown scenarios than the heuristic OSMH.