Distributionally Robust Multi-Stage Decision Problem and Satisficing

We study the classical multi-stage stochastic decision problem, wherein the exact distribution of uncertainty is unknown. We first employ an ambiguity set to characterize the potential distributions and model the problem within the conventional framework of distributionally robust optimization. We demonstrate that solving such a problem is computationally demanding. Then, we introduce the robust satisficing framework, which adopts a target-driven approach to account for all possible probability distributions. Under this framework, we use Kullback-Leibler divergence to measure the distance from potential distributions to the reference distribution. We show that the multi-stage stochastic decision problem can be reformulated into a series of dynamic programming problems, and the optimal policy can be determined through a Bellman-type backward induction. Finally, we apply this method to an inventory management problem and illustrate the optimality of base-stock policy under the robust satisficing framework. Through our numerical results, we show the promising performance of our approach in effectively addressing uncertain scenarios and achieving predefined target