We focus on the lot-sizing and scheduling problem with the additional considerations of machine eligibility, sequence-dependent setups, and uncertain demands. Multi-stage stochastic programming is proposed. We analyze the problem structure and suggest ways for modeling and solving large-scale stochastic integer programs. The analysis compares deterministic and stochastic model solutions to assess demand variance effects under the circumstances of increasing, fluctuating, and decreasing demands. The result shows that the expected cost performance of the stochastic programming model outperforms that of the deterministic model, in particular, when the demand is highly uncertain in the circumstance of an upward market trend. Our study can apply to the wafer fab manufacturing and other industries that heavily restricted by machine eligibility and demand uncertainties.