Progressive hedging (PH) is a classical decomposition algorithm for solving multistage stochastic problems. However, due to the exponentially growing model size of real-world enterprise-wide optimization problems, critical issues arise when implementing PH in practice. In this work, we propose a novel PH-based algorithm to address integrated planning and scheduling problems under demand uncertainty in a general mathematical formulation. Strategies are proposed to accelerate and guarantee the convergence of the algorithm. Through application of the enhanced PH to solve variants of a typical state-task network example and a real-world ethylene plant case, computational results demonstrate that the proposed algorithm outperforms directly invoking commercial solvers and gets a better solution within nearly two-thirds of the direct solution time on a serial computer. The advantage of the multistage stochastic programming method is also demonstrated by comparing the model solution with the counterparts of an expected value-based deterministic model and a two-stage stochastic model.