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Surrogate-based feasibility analysis for black-box stochastic simulations with heteroscedastic noise

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Abstract

Feasibility analysis has been developed to evaluate and quantify the capability that a process can remain feasible under uncertainty of model inputs and parameters. It can be conducted during the design stage when the objective is to get a robust design which can tolerate a certain amount of variations in the process conditions. Also, it can be used after a design is fixed when the objective is to characterize its feasible region. In this work, we have extended the usage of feasibility analysis to the cases in which inherent stochasticity is existent in the model outputs. With a surrogate-based adaptive sampling framework, we have developed and compared three algorithms that are promising to make accurate predictions on the feasible regions with a limited sampling budget. Both the advantages and limitations are discussed based on the results from five benchmark problems. Finally, we apply such methods to a pharmaceutical manufacturing process and demonstrate its potential application in characterizing the design space of the process.

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Acknowledgements

The authors would like to acknowledge financial support from FDA (DHHS - FDA - 1 U01 FD005295-01) as well as National Science Foundation Engineering Research Center on Structured Organic Particulate Systems (NSF-ECC 0540855).

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Authors and Affiliations

  1. Department of Chemical and Biochemical Engineering, Rutgers – The State University of New Jersey, Piscataway, NJ, 08854, USA

    Zilong Wang & Marianthi Ierapetritou

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  1. Zilong Wang
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Correspondence to Marianthi Ierapetritou.

Appendix

Appendix

A. Derivation of \({\varvec{E}}{\varvec{I}}_{{\varvec{feas}}}\)

Rewrite the improvement of feasibility as follows:

$$\begin{aligned} I_{ feas}=\left\{ {\begin{array}{ll} {\hat{s}}(0-z),\, &{} \quad \mathrm {if}\, z\le {f}^{{t}^{'}}\le 0\\ 0,\, &{} \quad \mathrm {else} \\ {\hat{s}}(z-0),\, &{} \quad \mathrm {if}\, z\ge {f}^{{t}^{'}}>0, \\ \end{array}} \right. \end{aligned}$$

where \(z=\frac{y-{\hat{\mu }}}{{\hat{s}}},\, {f}^{{t}^{'}}=\frac{f^{t}-{\hat{\mu }}}{{\hat{s}}}\).

Due to the assumed normal distribution for y, in the case when \({f}^{{t}'}\le 0\), the expected value of \(I_{ feas}\) can be derived as follows:

$$\begin{aligned} E\left[ I_{ feas} \right]&={\hat{s}}\int _{-\infty }^{{f}^{{t'}}} {\left( 0-z \right) \phi \left( z \right) dz} \\&\qquad \qquad ={\hat{s}}\left[ \frac{e^{-\frac{z^{2}}{2}}}{\sqrt{2\pi } } \right] _{-\infty }^{{f}^{{t'}}} \\&\qquad \qquad ={\hat{s}}\phi \left( f^{t'} \right) \\&\qquad \qquad ={\hat{s}}\phi \left( \frac{f^{t}-{\hat{\mu }}}{{\hat{s}}} \right) . \\ \end{aligned}$$

In the case when \(f^{{t}^{'}}>0\), the same expression for \(E\left[ I_{ feas} \right] \) can be obtained by taking similar derivation steps as mentioned above.

Fig. 18
figure 18

“Branin” test function. a “easy” noise; b “hard” noise. Thick-dashed line: feasible region boundaries; filled contour: standard deviation of the noise term

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Fig. 19
figure 19

“Camelback” test function. a “easy” noise; b “hard” noise. Thick-dashed line: feasible region boundaries; filled contour: standard deviation of the noise term

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Fig. 20
figure 20

“Example3” test function. a “easy” noise; b “hard” noise. Thick-dashed line: feasible region boundaries; filled contour: standard deviation of the noise term

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Fig. 21
figure 21

“Sasena” test function. a “easy” noise; b “hard” noise. Thick-dashed line: feasible region boundaries; filled contour: standard deviation of the noise term

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B. Figures for the 2D test problems

Below, we show the figures depicting the feasible region boundaries (denoted with thick-dashed line) and noise standard deviation \(\xi \) (denoted with filled contour) for the four 2D test functions. “Branin” in Fig. 18; “Camelback” in Fig. 19; “Example3” in Fig. 20; “Sasena” in Fig. 21.

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Wang, Z., Ierapetritou, M. Surrogate-based feasibility analysis for black-box stochastic simulations with heteroscedastic noise. J Glob Optim 71, 957–985 (2018). https://doi.org/10.1007/s10898-018-0615-4

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