(417c) Design Centering through Derivative-Free Optimization

In the pharmaceutical industry, design space is defined as “the multidimensional combination and interaction of input variables and process parameters that have been demonstrated to provide assurance of quality” [1]. The product quality can meet specifications as long as the process parameters vary within the proposed design space. Process parameters [2] correspond to degrees of freedom or variables that can be manipulated in the operation of a manufacturing process, and which can be measured and set within the controller tolerance for a desired value. Design spaces are defined using predictions from a model (either empirical or deterministic). Thus, the uncertainty in the model parameters plays an important role. The values of the model parameters are estimated from data, and commonly assumed to follow a Gaussian distribution from which confidence intervals can be defined.

Design space definition is one of the key components in pharmaceutical research and development. Flexibility index [3] and design centering [4] are two ways to estimate a candidate design space. Flexibility index is used to describe an operational range, which represents a maximum scaled departure of all process parameters from the given nominal conditions, such as a largest rectangle inscribed in the feasible space of steady-state operation in which control or recourse variables are adjusted. Since the design space is limited by the qualification ranges for process parameters, the result of flexibility index can be used to approximate the design space as an inscribed largest feasible region, which may be a rectangle, ellipse or other appropriate and acceptable shapes. If the nominal conditions of the process parameters are unknown, the flexibility index problem can be extended to a design centering problem, which focuses on determining the optimal nominal conditions, while maximizing the feasible operating region.

The goal of a design centering problem is to find an optimal nominal point, which corresponds to the largest flexibility index; thus, the design centering problem can be solved based on the flexibility index model. From a mathematical viewpoint, the design centering problem corresponds to a bi-level optimization model [5]. In the upper-level problem, the nominal point is searched within the feasible region; in the lower-level problem, an exact flexibility index should be calculated for each nominal point. Flexibility index problems are commonly formulated as a mixed-integer linear or nonlinear programming (MILP/MINLP) model [6-8]. For a general bi-level optimization problem, the most important issue is how to guarantee finding the global optimal solution of the lower-level model at each iteration. Similarly, for the design centering problem, the key issue is to guarantee locating the global optimal flexibility index for each nominal point.

In this work, since the design centering is a bi-level optimization problem, we first propose a novel formulation of the flexibility index for the lower-level optimization. A search direction is used to define the candidate flexibility index along a direction from the given nominal point for a specified shape of the feasible region. By applying the Karush-Kuhn-Tucker (KKT) conditions and complementarity conditions, the flexibility index model can be transformed into a single-level MINLP model. Thus, a global optimization solver (e.g., BARON) can be used to solve this model, and the exact flexibility index for a given nominal point can be obtained. This new formulation of flexibility index mainly has three features: (1) Compared with the traditional tri-level model, the proposed formulation is a bi-level model, and the corresponding MINLP model is simpler because it only requires once the KKT reformulation; (2) The traditional Haar condition [6] is not required, because the proposed model has ability to find the direction corresponding to the active constraint directly; (3) The proposed method can be extended to any shape of feasible region, as long as the representation of the shape can be provided.

For the upper-level optimization of the design centering model, a derivative-free optimization (DFO) method [9] is applied to search the nominal point in the entire feasible region. The main steps of the proposed design centering method are: (1) Perform the Latin Hypercube Sampling (LHS) strategy over the space of process parameters, where upper and lower bounds are required; (2) Check the feasibility of each sampling point. (3) Solve the DFO model of design centering problem for each feasible LHS point. The proposed flexibility index model is used to calculate the exact flexibility index at each iteration, and the obtained flexibility index for each candidate nominal point is stored. (4) The optimal nominal point is the one that has the largest value of flexibility index. In the case studies, Pyomo [10] is applied to define the models. BARON is called to solve the MINLP model through the interface of Pyomo and GAMS. Rectangular and elliptical sets for the process parameters are considered in each case. Several cases with convex and nonconvex feasible regions demonstrate the computational efficiency of the proposed design centering method.

Keywords: design space; flexibility index; design centering; derivative-free optimization.

Reference:

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