In this paper we consider the inventory-distribution planning under uncertainty for industrial gas supply chains by extending the continuous approximation solution strategy proposed in part I. A stochastic inventory approach is proposed and it is incorporated into a multi-period two-stage stochastic mixed-integer nonlinear programming (MINLP) model to handle uncertainty of demand and loss or addition of customers. This nonconvex MINLP formulation takes into account customer synergies and

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... neously predicts the optimal sizes of customers' storage tanks, the safety stock levels and the estimated delivery cost for replenishments. To globally optimize this stochastic MINLP problem with modest computational time, we develop a tailored branch-and-refine algorithm based on successive piece-wise linear approximation. The solution from the stochastic MINLP is fed into a detailed routing model with shorter planning horizon to determine the optimal deliveries, replenishments and inventory. A clustering-based heuristic is proposed for solving the routing model with reasonable computational effort. Three case studies including instances with up to 200 customers are presented to demonstrate the effectiveness of the proposed stochastic models and solution algorithms.