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Deterministic single machine scheduling to minimize total weighted absolute deviations of job completion times from a common due date (abbreviated by TWD) is a typical scheduling model in just-in-time manufacturing environment, and it is NP-hard. However, LPT (largest processing time) job schedule is optimal for the case where jobs' weights are proportional to their processing times. In this paper, we consider the stochastic counterpart of the TWD problem with proportional weights, where the processing times are arbitrary positive random variables, while the common due date is an exponentially distributed random variable. The optimal solution of the problem is derived. Moreover, the case where the machine is subject to stochastic breakdowns is also discussed. It is shown that the results can be extended to the situation where the machine is subject to stochastic breakdowns when the counting process describing machine breakdowns is characterized by a Poisson process and the down times are independent identically distributed.