377460 Dynamic Feasibility Analysis of Black Box Processes
The concept of process feasibility describes the ability of a process to remain operable and satisfy all product specifications in the presence of uncertainty. This uncertainty can be introduced by variation in process inputs, error in estimated model parameters, or variability in process design and operating conditions. A rigorous mathematical formulation describing process feasibility was developed in the 1980s, with the goal of solving optimal design under uncertainty problems.1,2 This feasibility test problem is an optimization problem, which seeks to minimize the violation of process constraints over a set of uncertain parameters, state variables and control variables for a given process design. To date, approaches to solving the feasibility test problem have generally involved deterministic optimization solvers. These require explicitly known constraints and also make certain assumptions regarding the convexity of the feasible region.3 However, recent developments in derivative-free optimization have enabled the determination of feasible regions for processes with black box constraints, without making inferences regarding the convexity of the feasible region.4,5 This is useful in addressing the feasibility analysis of integrated process models, which often contain a large number of potentially complex equations. In these applications, closed-form expressions for the constraints may not be available. In addition, the feasible region for these processes can have complex shapes, being nonconvex or even disjoint in nature.
This work is concerned with surrogate-based optimization methods, which involve the identification of a surrogate function to represent a complex process model. The surrogate function is less complex and faster to evaluate that the original model.6 The process is then optimized using the surrogate representation rather than the complex process model. In previous work, the ability of surrogate-based methods to identify complex feasible regions has been demonstrated.4 In addition, surrogate-based feasibility analysis has been applied to pharmaceutical processes to identify their design space during steady state operation.7 In practice, many pharmaceutical manufacturing processes are inherently time-variant. As such, the feasible operating region may not be constant with respect to time. In the current work, a method is proposed for the surrogate-based feasibility analysis of dynamic processes, where the shape and size of the feasible region may vary over time. This methodology involves the development of a response surface representation of the feasibility function followed by the refinement of the response surface using an adaptive sampling strategy. The feasible region is identified using an iterative optimization process in which new points are added to the response surface based on their anticipated ability to improve upon the current model. The response surface is developed using Kriging, an interpolating black box modeling methodology, which allows determination of prediction variance at each sampled point. The prediction variance informs an expected improvement function, which is used to direct sampling towards points on the boundary of the feasible region. The adaptive sampling process continues until the expected improvement is below a threshold value, at which time the feasibility surrogate function is identified and used to determine the feasible region for the process. The proposed method is demonstrated on a pharmaceutical manufacturing process described by a complex process model with black box constraints.
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