Abstract
A general modeling framework for mixture design problems, which integrates Generalized Disjunctive Programming (GDP) into the Computer-Aided Mixture/blend Design (CAMbD) framework, was recently proposed (S. Jonuzaj, P.T. Akula, P.-M. Kleniati, C.S. Adjiman, 2016. The formulation of optimal mixtures with Generalized Disjunctive Programming: A solvent design case study. AIChE Journal 62, 1616–1633). In this paper we derive Hull Relaxations (HRs) of GDP mixture design problems as an alternative to the big-M (BM) approach presented in this earlier work. We show that in restricted mixture design problems, where the number of components is fixed and their identities and compositions are optimised, BM and HR formulations are identical. For general mixture design problems, where the optimal number of mixture components is also determined, a generic approach is employed to enable the derivation and solution of the HR formulation for problems involving functions that are not defined at zero (e.g., logarithms). The design methodology is applied successfully to two solvent design case studies: the maximization of the solubility of a drug and the separation of acetic acid from water in a liquid–liquid extraction process. Promising solvent mixtures are identified in both case studies. The HR and BM approaches are found to be effective for the formulation and solution of mixture design problems, especially via the general design problem.
Highlights
The design of mixtures is an important and challenging problem with numerous industrial applications
Generalized Disjunctive Programming (GDP) is a logic-based approach for formulating discrete/continuous optimization problems that extends the disjunctive programming proposed by Balas (1985) and involves Boolean and continuous variables that are related via disjunctions, algebraic equations and logic propositions (Beaumont, 1991; Turkay and Grossmann, 1996)
The mole fraction of the third component is at the user-specified lower bound of xcL3 = 0.001, which means that only a small amount of ethanol is added to the mixture, and it does not have any significant impact on the solubility of the drug
Summary
The design of mixtures is an important and challenging problem with numerous industrial applications. GDP is a logic-based approach for formulating discrete/continuous optimization problems that extends the disjunctive programming proposed by Balas (1985) and involves Boolean and continuous variables that are related via disjunctions, algebraic equations and logic propositions (Beaumont, 1991; Turkay and Grossmann, 1996) It has been employed by Grossmann and co-authors in several applications in the area of process systems engineering, such as the design of process network systems (Raman and Grossmann, 1994; Vecchietti et al, 2003; Ruiz and Grossmann, 2013; Trespalacios and Grossmann, 2015), the design of distillation columns (Grossmann and Trespalacios, 2013), strippacking (Sawaya and Grossmann, 2005) and scheduling problems (Raman and Grossmann, 1994; Sawaya and Grossmann, 2005; Méndez et al, 2006; Castro and Grossmann, 2012). We compare the performance of the BM and HR relaxations for solution of GDP formulations of mixture design problems, but we do not apply cutting planes or a basic step approach
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