(173d) Designing Optimal Mixtures Using Generalized Disjunctive Programming: Convex Hull Reformulation
The design of mixtures is a topic of great importance across many industrial sectors, as part of process development and of product design. Computer-Aided Mixture/blend Design (CAMbD)  has emerged as an attractive methodology which has been successfully applied to designing various types of mixtures. Within the CAMbD framework, these design problems have often been posed as Mixed-Integer Nonlinear Programming (MINLP) problems, where the optimal molecular structure is determined with respect to a set of property constraints. In most existing methodologies a reduced version of the general CAMbD problem has been addressed: the number of mixture ingredients is fixed (usually a binary mixture is considered) and the identity of a compound (or of all compounds) that can participate in the mixture is chosen from a given set of candidate molecules. Solving the resulting optimization problems can be quite challenging, as these are combinatorial in nature due to the presence of binary variables and to the highly nonlinear expressions that relate composition, structure and physical properties.
In working towards a generalised CAMbD problem, a novel methodology where the number, identity and compositions of mixture constituents are optimised simultaneously, was recently proposed [2,3]. In order to address the difficulties arising from the complexity of expressing the problem within a mathematical framework, a logic-based methodology, Generalized Disjunctive Programming (GDP) , was used to formulate the discrete choices of a mixture design problem. The design methodology was first developed for a standard formulation with a fixed number of components (restricted problem) and then generalised to allow the number of mixture ingredients to vary up to a maximum number (general problem). In this approach, the designed components in the mixture are selected from a list of candidate pure compounds. The GDP problems are formulated as MINLP problems in order to exploit the available algorithms for solving optimisation problems. The formulations are converted into MINLP problems by replacing the Boolean variables with binary ones and converting the logic propositions into linear inequalities. In this initial work [2,3], conditional constraints inside the disjunctions were formulated using the Big-M (BM) approach . The BM approach is the simplest representation of a GDP problem in a mixed-integer form. However, it introduces a large parameter (Big-M) which in most cases is specified intuitively and can yield to poor relaxations. Here, we extend this work by converting the GDP models into MINLP problems using the Convex Hull (CH) reformulation . The CH reformulation introduces a new set of disaggregated variables and new constraints, increasing the size of the problem. It can however be shown  that when the discrete domain of the CH formulation is relaxed, it gives at least as tight or tighter relaxations than BM.
The design methodology is applied to two case studies: the first one aimed to design an optimal solvent mixture to dissolve ibuprofen and the second one to design the most effective solvent mixture to extract acetic acid from water. The results of the problems studied show that the proposed formulations offer a promising approach to the problem of mixture design, as the simultaneous design of the optimal number, identity and compositions of the components that participate in a mixture, is achieved. The CH reformulation is compared to the Big-M approach and both methods appear suitable for formulating mixture problems effectively. Finally, better performance is obtained with a solvent mixture in both examples, showing the benefit of using mixtures instead of pure solvents to attain enhanced properties.
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