This paper addresses the scheduling problem in assembly job shop systems with machine breakdowns. The present study intends to rank the dispatching rules that are applied in an assembly job shop production system. A simulation model of an assembly job shop is developed for the purpose of this ranking and eight dispatching rules from the literature are incorporated in the simulation model. The product structures considered in this study are single-level assembly structure, two-level assembly structures and three level assembly structures. The machines are subjected to non-availability due to of breakdowns. Five performance measures are considered for analysis. The performance of each dispatching rule for each performance measure is calculated individually. To identify which of the dispatching rule provides the optimum result when all the performance measures are equally important, grey relational analysis is adopted to rank the dispatching rules. The results indicate that earliest completion time rule performs better in comparison with the other rules investigated in this study for single-level structure with multiple objectives.