A partial rescheduling (PR) heuristic is presented for single machine with unforeseen breakdowns. Unlike a full rescheduling strategy where all unfinished jobs are considered, a PR strategy reschedules partial unfinished jobs which form a PR problem, and shifts the rest jobs to the right according to the solution of the PR problem. The rescheduling problem considers a bi-criterion that optimizes both shop efficiency (i.e. makespan performance of the schedule) and stability (i.e. deviation from the original schedule). A genetic algorithm is developed to solve the PR problem. Extensive computational testing was conducted. The computational results show that the PR heuristic with bi-criterion can significantly improve schedule stability with little sacrifice in efficiency, and provide a reasonable trade-off between solution quality and computational efforts.