(635b) Design Centering Problem through Flexibility Analysis Concepts
A similar problem has been tackled in the area of electrical engineering, where Director and Hachtel (1977) addressed the problem of choosing a nominal design point that maximizes the number of VLSI circuits subject to given performance specification. The authors proposed the simplicial approximation approach, based on the explicit approximation of the boundary of the feasible region of an n-parameter design space by a polyhedron made up of n-dimensional simplexes. Based on this work, Goyal and Ierapetritou (2002) proposed an approach to identify the operating envelopes where process operation is feasible, safe and profitable. Only recently, Hardwood and Barton (2017) formulated the design-centering problem as a generalized semi-infinite program, and discussed reformulations of the design-centering problem to simpler problems such as finite nonlinear programs (NLPs) or standard semi-infinite programs (SIP).
In this work, we propose three approaches based on flexibility index concepts (Swaney and Grossmann, 1985) to solve the design-centering problem in order to find the safest nominal operating point of the design space. The approaches are presented with increasing degree of complexity. It is important to note that the difference between the flexibility index calculation and the design-centering problem is that for the former problem the nominal conditions are given, whereas in the latter we are not only maximizing the flexibility index but also simultaneously choosing the nominal conditions. Therefore, the design-centering problem can be considered an extension of the flexibility index problem.
First, assuming convexity in the feasible region, we propose the search over all directions method. The basic idea is to maximize the flexibility index ð¿ and to determine the nominal condition of the process parameters, , by simultaneously evaluating feasibility of all vertex directions, allowing only to find vertex solutions. The (N)LP model type is reatained, but the dimensionality of the problem grows by 2k with the number of process parameters k. However, the problem has a decomposable structure that can be exploited by a decomposition scheme.
In order to overcome the limitation of vertex solution and the assumption of convexity, the nominal value grid evaluation method is proposed. The space of the nominal condition of process parameters is discretized and the flexibility index is calculated at each discrete nominal point of the grid. The flexibility index is reformulated following the active set strategy (Grossmann & Floudas, 1987). This implies solving to global optimality an MI(N)LP at each grid point. The solution corresponds to the nominal conditions that have associated the largest value of flexibility index. It is important to note that since the flexibility index for each grid point is independently calculated, there is room for parallelization.
Finally, to avoid exhaustive enumeration of all grid nominal value points, we propose the nested reformulation, which is an extension to the original flexibility index (FI) formulation developed by Swaney and Grossmann (1985). This formulation involves an additional level of optimization where the flexibility index is maximized and the nominal point is considered as an optimization variable. To solve the FI problem, the innermost optimization problems are replaced by their optimality conditions and then the complementarity conditions are expressed with mixed-integer constraints, leading to an MINLP problem that must be solved to global optimality. As opposed to the original formulation, it is also required to include more optimization variables, such as lower and upper bound of the process parameters. In addition, the model feasibility is evaluated at the nominal point and at every combination of the process parameter bounds.
The proposed methods are illustrated through an application example of thermal deprotection reaction in a CSTR followed by a separation in an evaporator.
Keywords: Design centering, Flexibility Analysis, Design Space, Quality by Design.
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