Decision Support by Multicriteria Optimization in Process Development: An Integrated Approach for Robust Planning and Design of Plant Experiments
Introduction
Models for computer-aided process design contain a multitude of physical, chemical and, when it comes to cost functions, also economical and sustainability related parameters. Often these parameters have either been estimated by model adjustment to experimental data, or their values are given by experience. In both cases, these values are not known exactly, but rather within specific uncertainty ranges (we prefer using the term uncertainty range to confidence interval. The term confidence interval has a statistical meaning that is not needed in our discussion). The impact of these ranges on the output functions can be estimated by sensitivity analysis (Saltelli et al. 2008). In sensitivity analysis, different sampling-techniques for scenarios in the space of the uncertain model parameters are available to obtain insight into the importance and the size of the resulting uncertainty ranges in the output functions of the model. The application of Monte-Carlo sampling (e.g. in Xin et al., 2000) demonstrated the importance of analysing the uncertainties in the calculated process designs.
On the other hand, the engineer aims at understanding the influence of the available process parameters on the output functions. This is essential in order to arrive at a process design in the space of process parameters that meets certain criteria in the space of the output functions. Generally, these optimization problems contain different, often conflicting objectives, like cost and quality measures. In this situation, Multicriteria optimization is a well-established tool to arrive at best compromises (Pareto solutions) between conflicting criteria (adaptive sampling strategies combined with deterministic optimization algorithms have been demonstrated by Bortz et al. (2014)). Navigating the Pareto solutions then visualizes the trade-offs interactively to the planner (Asprion et al. 2011).
In this contribution, we aim at including the sensitivities with respect to uncertain model parameters as further objectives into a multicriteria optimization setting. We call this procedure Robust Planning (RP). The result of RP will be the quantification of tradeoffs between increasing robustness to uncertain model parameters and the other objectives. This approach complements the definitions of robust optimization (for a review, see Ben-Tal et al. (2009)) and chance-constraint programming (Ostrovsky et al. 2009). In these approaches, the focus is on an absolute definition of robustness in the sense that solutions can be classified as either robust or non-robust. Our strategy however puts emphasis on the compromise one has to accept in order to arrive at a more robust solution compared to a more sensitive one.
It turns out that in practice, optimal experimental design (OED) is supported in a similar multicriteria setting: On the one hand, the model should be accurate for process designs which are practically relevant (e.g. at low costs, high product qualities). Thus, experiments should preferably be carried out at these designs. On the other hand, in order to reliably estimate model parameters, experiments are favored at which measureable output functions are sensitive to the respective model parameters. A common approach discussed in literature (Franceschini and Macchietto 2008) – focuses on the maximization of different sensitivity measures. Therefore, one defines scalar measures of the sensitivity matrix encoding the sensitivity of certain output functions to relevant model parameters. However, these scalarizations of the sensitivity matrix lead not only to contradictory results (Telen et al., 2012), but also constitute a weighting of matrix entries which has to be known a priori. In practice, however, this knowledge does not exist, leading eventually to a loss in optimization potential.
A possible solution suggested in this contribution consists in following a similar route as sketched above for robust process design: For each measurable output function of interest, an adequate sensitivity measure is proposed as one objective to be maximized, among others which also have to be met during the experiment. By doing so, the tradeoff between sensitivity, i.e. reliable estimates of model parameters, and good representation of the designs which are practically most relevant is quantified. On this basis, the planner can decide for a process design that realizes the compromise that best suits the current situation. Thus in our approach, RD and OED merely differ in the way how sensitivity measures are treated: As objectives to be minimized for RD, or as objectives to be maximized for OED.
The article is structured as follows. In Section 2 it is described how robustness can be incorporated into a conventional optimization problem. Section 3 contains the illustration of this approach by an example. The article ends with the conclusion in Section 4.
Section snippets
Sensitivity as an optimization objective
Sensitivity analysis is used here to quantify the uncertainty range of an output function with respect to given distributions over uncertainty ranges of thermodynamic model parameters around their reference values. During RP (OED), process parameters are to be adjusted such that, among others, adequate robustness measures derived from the uncertainty ranges of certain output functions are maximized (minimized). Our starting point is a conventional optimization problem that has been set up and
Example
In this section, we want to compare two types of distillation processes for separating a binary mixture of chloroform and acetone into pure components. As these two substances form a maximum boiling azeotrope, a separation in a simple single distillation column is not possible. Two variants for separating the azeotropic mixture are considered: the first is a pressure-swing distillation which takes advantage of the pressure dependency of the azeotrope. In the second one, an entrainer (benzene)
Conclusions
We presented a practical guideline how to include robustness as one objective among others in a multicriteria optimization framework. If the goal consists in finding robust designs, then robustness as an objective is to be maximized. However, in the framework of optimal experimental design, it is to be minimized. The guideline consists of three steps, starting from the original optimization problem, followed by a sensitivity analysis of the original Pareto points, possibly including an
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