An Approximate Dynamic Programming Algorithm for Multi-Stage Capacity Investment Problems

This paper studies a dynamic multi-facility capacity investment problem (MCIP) with discrete capacity. In this setting, capacity adjustment decisions are made sequentially based on observations of demand. First, we formulate this problem as a Markov decision process (MDP). Then, we design a customized fitted value iteration (FVI) algorithm. In particular, we approximate the value functions with a two-layer neural network with piecewise linear activation functions. However, the action selection procedure of FVI for MCIP can be time-consuming since the action space is discrete and high dimensional. To speed up the action selection, we recast the action selection problem as a two-stage stochastic programming problem. The resulting recourse function comes from the two-layer neural network, and it is solved with a specialized multi-cut decomposition algorithm. Numerical studies show that our algorithm provides high quality solutions to the MCIP, and also that the multi-cut algorithm can significantly speed up the action selection problem of FVI in comparison to the brute-force method and the integer L-shape algorithm. Finally, we show that the techniques developed here for MCIP are applicable to many other finite-time horizon MDPs with finite but high dimensional action spaces.

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