(430a) Hierarchical Planning-Scheduling-Control - Is Derivative-Free Optimization All You Need?

Companies within the process industries rely on mathematical optimization for their operations to remain competitive in an environment of increasingly stringent safety, environmental, and economic requirements. Significant value can be captured by integrating units across all hierarchical levels of decision-making (from design, planning, scheduling, to control) [1]. Conventionally, decision-making happens sequentially: upper-level decisions are taken while disregarding lower-level considerations, and then fed as setpoints to the lower levels. In the hierarchical approach, upper-level decision-makers consider not only lower-level feasibility, but also optimality. This results in multi-level formulations, which are numerically intractable and mathematically difficult [2]. Data-driven techniques in the form of optimality surrogates and black-box optimization can be leveraged to find a tractable solution to tri-level planning-scheduling-control formulations [3]. In this work, we investigate if we can use black-box optimization for the end-to-end integration of planning-scheduling-control, avoiding the introduction of model inaccuracies associated with optimality surrogates.

There are two main options to decrease the computational burden: We can exploit 1) the mathematical structure of the optimization problem in decomposition or distributed optimization schemes [4], or 2) relaxation or aggregation techniques that relax the original formulation or replace parts of it using surrogate models that are easier to handle [5]. While research into the intersection of data-driven techniques and decomposition algorithms is less common [6], surrogate modelling is inherently 'data-driven’, and as such has been studied widely across many fields under various names [7-8]. The advances of Machine Learning and surrogate modelling have also spurred developments in the use of (model-based) derivative-free optimization (DFO). DFO algorithms are typically classified into either 'direct' methods that directly handle function evaluations, or 'model-based' methods that rely on the intermediate construction and optimization of surrogates [9]. We use the term DFO synonymously with data-driven, black-box, simulation-based, or gradient-free optimization [10-11]. While, DFO gains traction within the chemical engineering community, DFO has also been exploited to solve integration problems that traditionally are targeted via decomposition algorithms, such as in coordination [12] and multi-level [13] problems in process systems engineering and even in the hierarchical integration of process operations [14].

While scale-bridging optimality or feasibility surrogates present a popular way of alleviating the computational burden of integrated planning-scheduling-control formulations, these approaches inevitably incur the risk of losing solution accuracy however. For applications with little tolerance for approximation errors, we propose the end-to-end use of derivative-free optimization to solve a multi-site, multi-product, tri-level planning-scheduling-control problem.In our example, at each iteration, a derivative-free optimization solver suggests a new combination of 472 planning-specific variables that fix all degrees of freedom in the planning level. The planning targets are fixed and fed as setpoints into 12 scheduling problems (one for each planning horizon step). Then, each scheduling problem is solved in parallel to find the batch production targets that achieve the planning targets at minimal makespan. For each scheduling problem (12), for each event-equipment-production combination (7 events x 2 equipment items), we solve an optimal control problem to determine the optimal processing time and energy cost to produce said batch size. The optimal makespan decisions are fed back into the scheduling problem to check the feasibility of the batch sequence at the update processing times. Finally, the planning-level objective is augmented by the optimal scheduling and control cost and returned to the derivative-free optimization solver to determine the new iteration of planning-level variables.

We show that given the distributed nature of the information flow in this hierarchical problem we can leverage parallelization to solve each DFO evaluation in a fifth of the time required to solve the planning problems with embedded optimality surrogates. But most importantly, we do not have to forfeit model accuracy and solution quality in the scheduling-control integration. We discuss the trade-offs involved in the choice between the DFO and optimality surrogate approaches in terms of tolerance against model approximation errors, available online versus offline solution time, and warm-start approaches. We also discuss ways to exploit mathematical structure in the DFO variables to objective mappings to make the DFO approach even more competitive.

References

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