Explainable LP-MPC: Shadow Price Contributions Reveal MV-CV Pairings

Abstract

In the process industries, MPC (Model Predictive Control) is typically implemented as a two-stage controller with a Linear Program (LP) steady-state optimizer that generates economically optimal targets for the MPC algorithm. Abnormal behaviors in industrial LP optimizers are often difficult to rationalize, especially when a large number of manipulated variables (MVs) and controlled variables (CVs) are involved. We introduce a novel, post-hoc LP explainability method by recasting the role of shadow prices in the LP solution as an attribution mechanism for MV-CV relationships. The core idea is that the shadow price of a constrained CV is not just an intrinsic property of the LP solution, but can be split into contributions from individual unconstrained MVs and resolved into one-to-one MV-CV pairings using a linear sum assignment algorithm. The proposed MV-CV pairing framework serves as a practical explainability tool for online LP-MPC systems, enabling practitioners to diagnose suboptimal constraints and verify alignment of the controller's behavior with its original design.