(282f) Hierarchical Planning-Scheduling-Control - Optimality Surrogates upon Optimality Surrogates

Mathematical optimization is key to the hierarchical integration of process operations ranging from process and supply chain design down to planning, scheduling, and control [1]. Traditionally, upper-level decisions are taken while disregarding lower-level considerations, and then fed as setpoints to the lower levels. Ensuring lower-level feasibility and optimality however leads to large-scale, potentially multilevel, formulations which are not just computationally intractable, but also mathematically difficult [2]. Optimality surrogates and black-box optimization have already been leveraged to find good solutions to multilevel formulations in the integration of planning, scheduling, and control [3]. In this work, we explore if we can use optimality surrogates for both the integration of planning and scheduling and the integration of scheduling and control.

The use of surrogate models for the integration of planning-scheduling-control is well-established and is related to the concepts of meta-, scale-bridging, or reduced models [4-7]. Advances in data-driven techniques have been twofold for this community [8]: First, supervised learning has shifted the focus of surrogate type from system identification to supervised learning techniques such as Artificial Neural Networks, Gaussian Processes, and Decision Trees. Second, software developments have enabled the seamless merging of Machine Learning and optimization pipelines. OMLT for example automates the reformulation of trained neural networks and decision trees into mixed-integer linear constraints and their subsequent embedding into Pyomo blocks [9]. We draw another distinction in the use of approximation models that only map the upper-level relevant variables as a function of the variables that feed from the upper into the lower level [5, 10]: We call `feasibility surrogates' and `optimality surrogates' the use of any approximation model used to map lower-level feasibility or lower-level optimal response variables or objectives as a function of upper-level complicating variables or `setpoints'. ‘Optimality surrogates’ have analogues and similarities across many disciplines, from multi-parametric model predictive control [11], to learning value functions in Reinforcement Learning [12].

In this work, we start by training optimal control surrogates that map a set of sampled batch production targets to their corresponding optimal control responses. The optimal control surrogate can then be seamlessly integrated into the scheduling optimization formulation, collapsing a bi-level into a single-level integrated scheduling-control formulation. Similarly, we train an integrated scheduling-control surrogate on planning targets and their corresponding integrated scheduling-control optimization results. As such, we can find an approximate solution to the tri-level planning-scheduling-control problem by solving a single-level planning formulation with these embedded approximate optimal scheduling and control cost surrogates. Given our choice in surrogates, this is an MILP which can be tractably solved by state-of-the-art solvers as compared to monolithic MINLP formulations.

The use of surrogates inherently involves a trade-off between computational tractability and optimization solution quality. Employing composite optimality surrogates (approximations of approximations) increases the risk of small model inaccuracies being amplified to the point that they cannot be used for optimization. Consequently, we use multi-objective Bayesian optimization [13] for hyperparameter tuning of the optimality surrogate architecture to trade-off the optimization solution time and solution quality obtained after embedding the integrated scheduling-control surrogates into the planning optimization problem.

Sometimes, decision-making at the different hierarchical levels is intractable enough to prevent any attempt at integration. We highlight under which conditions optimality surrogates can be used towards a tractable planning-scheduling-control solution of a multi-site, multi-product case study [3]. We place special emphasis in our discussion on the role that understanding the interplay of Machine Learning and optimization pipelines has on navigating the inherent accuracy-tractability trade-off, and discuss other tools to hedge the risk that surrogate model inaccuracies pose. Finally, we discuss how we can adaptively leverage data and update models in practice to efficiently solve the integrated problem on a monthly basis.

References

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