(441d) New Developments in Flexibility Analysis in the Framework of Design Space Definition

The importance of the Design Space (DS) definition lies in the assurance of quality, a key goal of Quality by Design, and in the broadening from an acceptable operating set point, to a collection of tolerable operating regions. Working within the DS is considered acceptable and not additional actions are needed (e.g., notification). Operating outside the DS represents a deviation and would normally initiate an investigation. Defining properly the limits of the design space is vital to provide manufacturing flexibility and quality assurance. Even though process DS has been only recently formally introduced in the pharmaceutical industry, it is not a completely new concept. Identifying design and operating constraints as well as the necessary control scheme for a process are concepts and ideas that have been extensively studied in other industries and literature (Boukouvala et al., 2017). The use of concepts introduced by the process system literature, such as process feasibility and flexibility, have proven to be useful in the DS identification (García-Muñoz et al., 2015).

Flexibility addresses the guaranteed feasibility of operation of a plant over a range of conditions, with the ultimate goal being on how to design a process for guaranteed feasible operation over a specified parameter range (Grossmann et al., 2014). There are three main problems to address: the flexibility test problem, flexibility index calculation and the design under uncertainty with flexibility constraints. First, the flexibility test problem (Halemane and Grossmann, 1983) for a given design consists in determining whether by proper adjustment of the controls variables the process constraints hold for any realization of uncertain parameters. To indicate how much flexibility can actually be achieved in a given design (Swaney and Grossmann, 1985), the flexibility index is defined as the largest scaled factor that multiplies the deviation of the uncertain set for which the inequalities hold. As for design optimization problem, the selection of the design variables is involved so as to minimize cost and either satisfy the flexibility test, or maximize the flexibility index, where the latter problem gives rise to a multi-objective optimization problem.

However, these formulations are based implicitly on the assumption that control variables can compensate any variation in the uncertain parameter set and that during operation stage uncertain parameters can be determined with enough precision. Ostrovsky et al. (2004) and Rooney and Biegler (2004) extended the analysis by taking into account the level of parametric uncertainty in the mathematical models at the operation stage, by grouping the uncertain parameters, 𝜃 ∈ Θ, into two types: unmeasured and measured. The flexibility constraint was then extended to account for unmeasured process variability, 𝜃_𝑢. The flexibility constraint was re-defined as a multi-level optimization problem.

In this work, we propose new formulations for the extended flexibility analysis. The basic idea relies on recursively reformulating the inner optimization problems by the Karush-Kuhn-Tucker conditions, and with a mixed-integer representation of the complementarity conditions to solve the resulting multilevel optimization problem. Three types of models with uncertain parameters are tackled and solved with the proposed strategy: 1) linear programming model, 2) nonlinear programming model with monotonic variation of unmeasured uncertain, and finally, 3) nonlinear programming model. Similar mixed-integer formulations to the traditional flexibility analysis are obtained for the first two cases, whereas a novel nonconvex MINLP is obtained for the latter case. Specifically, for the nonlinear programming model, we study a thermal de-protection reaction performed in a CSTR followed by an evaporation step. Numerical results are compared to the traditional flexibility analysis (Grossmann and Floudas, 1987), leading to larger values of constraint violation and lower values of flexibility indices due to the fact that the recourse action can only compensate for measured uncertain parameter variations.

Keywords

Flexibility Analysis, Quality by Design, Design Space

References

Boukouvala, F., Muzzio, F. J. and Ierapetritou, M. G. (2017) Methods and Tools for Design Space Identification in Pharmaceutical Development, in Comprehensive Quality by Design for Pharmaceutical Product Development and Manufacture (eds G. V. Reklaitis, C. Seymour and S. García-Munoz), John Wiley & Sons, Inc., Hoboken, NJ, USA.

García-Muñoz, S., Luciani, C.V., Vaidyaraman, S., Seibert, K.D. (2015). Definition of Design Space Using Mechanistic Models and Geometric Projections of Probability Maps. Organic Process Research & Development, 19, 1012−1023.

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Swaney, R. E. & Grossmann, I. E. (1985). An Index for Operational Flexibility in Chemical Process Design. Part 1 and Part 2 Formulation and Theory. AIChE Journal, 31, 621.