(485c) A Multiparametric Mixed-Integer Quadratic Approximation Algorithm for the Solution of Process Engineering Problems Under Uncertainty

Many process engineering problems, and in particular those encountered in the field of chemical engineering, are complex and often non-linear. They involve two types of decisions: (i) discrete decisions ?those which define a specific (optimal) structure of the underlying system and (ii) continuous decisions?those which optimize its operation. Typical examples include synthesis problems in chemical process design, and production planning and scheduling of multiproduct batch plants among others. These types of problems can be approached very efficiently via mathematical programming techniques such as mixed-Integer nonlinear programming (MINLP) (Grossman and Biegler, 2004).

On the other hand, the inevitable presence of fluctuations in some process variables such as demand, prices or quality in raw materials, and the lack of exact knowledge of certain parameters such as kinetic data or physical properties require that uncertainty be taken into account in MINLP process models. As a result, various approaches which address the uncertainty in MINLP models have been proposed (Sahidinis, 2004): stochastic programming (Ierapetritou and Pistikopoulos, 1996), fuzzy programming (Balasubramanian and Grossmann, 2003), and robust optimization (Lin et al., 2004).

Recently, Pistikopoulos and co-workers (Pistikopoulos et al., 2007a; 2007b) have shown that process engineering problems under uncertainty can also be addressed via multiparametric mixed-integer nonlinear programming (mp-MINLP). Acevedo and Pistikopoulos (1996) proposed a mp-MINLP algorithm based on the outer-approximation/equality relaxation algorithm of Kocis and Grossmann (1986). Dua and Pistikopoulos (1999) proposed three mp-MINLP algorithms based on the reformulation of a master problem using (i) deterministic, (ii) outer-approximation (OA), and (iii) generalized benders decomposition (GBD) principles.

While these algorithms proved to be successful they remained computationally intensive. In this work, we present a novel mp- MINLP algorithm for the solution of process engineering problems under uncertainty. The algorithm is based on a decomposition strategy where a sequence of mp-NLP (primal) subproblems and deterministic MINLP (master) subproblems are solved. Because the solution of primal subproblems is the most expensive step of mp-MINLP algorithm, we propose a multi-parametric quadratic approximation (mp-QA) algorithm for their solution. We show that the application of the mp-QA algorithm in the primal step results in computational savings since only a few NLPs have to be solved. Finally, we present numerical examples which demonstrate the computational advantages of the proposed algorithm.

References:

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