Balancing adaptability and predictability: K-revision multistage stochastic programming
Chengwenjian Wang, Alexander S. Estes, Jean-Philippe P. Richard
TL;DR
The paper addresses balancing adaptability and commitment in multistage stochastic programming by introducing the $K$-revision MSP, which fixes an initial plan for all future stages and allows at most $K$ revisions per scenario. It provides a rigorous theoretical foundation, proving NP-hardness for the $K$-revision hypercube and offering a combinatorial ELBE-subtree characterization that enables two MIP formulations: a Complete Plan (CP) formulation and a Subtree (ST) formulation, along with tightening and reformulation techniques. The authors develop a constraint-generation algorithm and an extended formulation for the ST model, and a size-reduction and a novel facet-defining inequality approach for CP, enabling scalable solving across diverse MSP applications including SAGHP, lot-sizing, and capacity planning. Computational experiments show that the $K$-revision approach yields near-optimal performance with improved solution predictability, with CP and ST variants excelling in complementary tree structures and the combined CP++ generally offering the strongest relaxations. Overall, the work provides both theoretical insights and practical algorithms for sequential decision-making under uncertainty with controlled plan revisions. All mathematical notation is denoted with $...$ as required, for example $K$, $T$, $oldsymbol{x}$, and $oldsymbol{y}$.
Abstract
A standard assumption in multistage stochastic programming is that decisions are made after observing the uncertainty from the prior stage. The resulting solutions can be difficult to implement in practice, as they leave practitioners ill-prepared for future stages. To provide better foresight, we introduce the K-revision approach. This new framework requires plans to be specified in advance. To maintain flexibility, we allow plans to be revised a maximum of K times as new information becomes available. We analyze the complexity of K-revision problems, showing NP-hardness even in a simple setting. We examine, both theoretically and computationally, the impact of the K-revision approach on the objective compared with classical multistage stochastic programming models and the partially adaptive approach introduced in [1, 2]. We develop two MIP formulations, one directly from our definition and the other based on a combinatorial characterization. We analyze the tightness of these formulations and propose several methods to strengthen them. Computational experiments on synthetic problems and practical applications demonstrate that our approach is both computationally tractable and effective in reaching near-optimal performance while increasing the predictability of the solutions produced.