(612a) The Inherent Robustness of Closed-Loop (Online) Scheduling: Definitions, Algorithms, and Theoretical Guarantees

For any manufacturing facility, generating high-quality production schedules is an important task. However, generating a single, fixed schedule for a facility is not sufficient in practice due to disturbances such as delays, breakdowns, and yield losses. In closed-loop (online) scheduling, these disturbances are addressed in real-time through feedback and frequent reoptimization of the schedule (Gupta and Maravelias, 2016). The solution to each optimization problem is termed an open-loop schedule while the implemented set of actions is termed a closed-loop schedule. Unless formulated properly, however, this frequent reoptimization may produce nonintuitive and arbitrarily poor closed-loop schedules even in the nominal case, i.e., there are no disturbances and the model is perfect. By representing closed-loop scheduling as an economic model predictive control (MPC) problem, we can leverage the results developed in the MPC community to provide insights and theoretical guarantees for closed-loop systems (Subramanian, Maravelias, and Rawlings, 2012). Specifically, we can enforce terminal equality constraints in the economic MPC problem to establish nominal performance guarantees. By constraining the optimization problems to terminate along a reference trajectory (determined a priori for the nominal system), we can guarantee that the nominal closed-loop cost is better (no worse) than the reference trajectory (Risbeck, Maravelias, and Rawlings, 2019). We assume for this work that a reasonable, but not necessarily periodic, reference trajectory is available for the nominal system.

Although, these nominal performance guarantees do preclude particularly poor nominal closed-loop performance, the main purpose of closed-loop scheduling, or any feedback method, is to address disturbances. Thus, the impact of uncertainty on the closed-loop performance, i.e., robustness, is a particularly important topic. Although some simulation studies have explored the robustness of closed-loop scheduling through a variety of case-studies and metrics, there is currently no precise definition of robustness for closed-loop scheduling (or any online scheduling method). As a recent example, Gupta and Maravelias (2020) provide an excellent framework to study the impacts of uncertainty on production scheduling for individual case studies. These studies, however, are restricted to empirical results for specific systems.

By contrast, this presentation focuses on defining the term robustness for a general closed-loop scheduling problem. The goal is then to formulate a closed-loop scheduling algorithm that is inherently robust to some nonzero margin of uncertainty. A system is deemed robust if an arbitrarily 'small' disturbance cannot cause significant and permanent degradation in performance. We use the term 'inherent' to indicate that the robustness achieved by this algorithm is afforded by feedback and does not require robust or stochastic optimization techniques.

We begin by characterizing the types of disturbances relevant to closed-loop scheduling, specifically delays and breakdowns, as large and infrequent. We then mathematically define the concept of robustness for closed-loop scheduling, i.e., economic robustness to large, infrequent disturbances. Through a motivating example, we demonstrate the implications of this property. Notably, we demonstrate that closed-loop scheduling algorithms without properly formulated terminal costs and constraints are not necessarily robust in this context. Furthermore, closed-loop systems that are not economically robust to large, infrequent disturbances can incur large and permanent loss in economic performance due to a single disturbance.

In addition to defining the term robustness, we also present a novel terminal cost and constraint that guarantee the inherent robustness of closed-loop scheduling and retain the nominal performance guarantee as well. The terminal equality constraint, originally used to guarantee nominal performance, is overly restrictive and may produce infeasible optimization problems. Thus, we expand the terminal equality constraint to a terminal region that permits excess inventory and/or backlog relative to the reference trajectory. With this expanded terminal region, we can establish that the optimization problem remains feasible for all potential realizations of the disturbance. We then systematically construct a corresponding terminal cost and establish that these terminal ingredients satisfy the same underlying condition as the terminal equality constraint. Through a few reasonable and verifiable assumptions, we establish that closed-loop scheduling with these terminal conditions is economically robust to large, infrequent disturbances, i.e., the proposed closed-loop scheduling algorithm is inherently robust. We conclude with another example to further demonstrate the implications of these results and compare the proposed algorithm to other online scheduling algorithms.

Works Cited:

D. Gupta and C. T. Maravelias, 2016. On deterministic online scheduling: Major considerations, paradoxes and remedies. Comput. Chem. Eng., 94, 312-330.

K. Subramanian, C. T. Maravelias, J. B. and Rawlings. A state-space model for chemical production scheduling. Comput. Chem. Eng., 47, 97-160, 2012.

M. J. Risbeck, C. T. Maravelias, and J. B. Rawlings. Unification of closed-loop scheduling and control: State-space formulations, terminal constraints, and nominal theoretical properties. Comput. Chem. Eng., 129:106496, 2019.

D. Gupta and C. T. Maravelias. Framework for studying online production scheduling under endogenous uncertainty. Comput. Chem. Eng., 135:106670, 2020.

R. D. McAllister, J. B. Rawlings, and C. T. Maravelias. Inherent robustness of closed-loop scheduling. In preperation for Comput. Chem. Eng., 2021.