This paper addresses the relaxations in alternative models for disjunctions, big-M and convex hull model, in order to develop guidelines and insights when formulating Mixed-Integer Non-Linear Programming (MINLP), Generalized Disjunctive Programming (GDP), or hybrid models. Characterization and properties are presented for various types of disjunctions. An interesting result is presented for improper disjunctions where results in the continuous space differ from the ones in the mixed-integer space. A cutting plane method is also proposed that avoids the explicit generation of equations and variables of the convex hull. Several examples are presented throughout the paper, as well as a small process synthesis problem, which is solved with the proposed cutting plane method. #

In this work we propose an optimization model for the design of a biotechnological multiproduct batch plant. A first level of detail posynomial model is constructed for each unit, as well as decisions regarding the structural optimization of the plant. A particular feature of this model is that it contains composite units in which semicontinuous items operate on the material contained by batch items. This occurs in the purification steps, in particular with the microfilters operating between retentate and permeate vessels, and with the homogenizer and ultrafilters operating on the material contained in a batch holding vessel. Also, the unit models rely on batch operating time expressions that depend on both the batch size and the size of semicontinuous items. The model takes into account all of the available options to increase the efficiency of the batch plant design: unit duplication in-phase and out-of-phase and intermediate storage tanks. The resulting mathematical model for the minimization of the plant capital cost is a mixed integer non-linear program (MINLP), which is solved to global optimality with an implementation of the outer approximation/ equality relaxation/ augmented penalty (OA/ER/AP) method. A plant that produces four recombinant proteins in eight processing stages is used to illustrate the proposed approach. An interesting feature of this example is that it represents an attempt to standardize a plant for the production of both therapeutic and nontherapeutic proteins; the model applied is generic and can thus be applied to any such modular plant. Results indicate that the best solution in terms of minimal capital cost contains no units in parallel and with intermediate storage tank allocation.

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In this work we propose a model that simultaneously optimizes the process variables and the structure of a multiproduct batch plant for the production of recombinant proteins. The complete model includes process performance models for the unit stages and a posynomial representation for the multiproduct batch plant. Although the constant time and size factor models are the most commonly used to model multiproduct batch processes, process performance models describe these time and size factors as functions of the process variables selected for optimization. These process performance models are expressed as algebraic equations obtained from the analytical integration of simplified mass balances and kinetic expressions that describe each unit operation. They are kept as simple as possible while retaining the influence of the process variables selected to optimize the plant. The resulting mixed-integer nonlinear program simultaneously calculates the plant structure (parallel units in or out of phase, and allocation of intermediate storage tanks), the batch plant decision variables (equipment sizes, batch sizes, and operating times of semicontinuous items), and the process decision variables (e.g., final concentration at selected stages, volumetric ratio of phases in the liquid-liquid extraction). A noteworthy feature of the proposed approach is that the mathematical model for the plant is the same as that used in the constant factor model. The process performance models are handled as extra constraints. A plant consisting of eight stages operating in the single product campaign mode (one fermentation, two microfiltrations, two ultrafiltrations, one homogenization, one liquid-liquid extraction, and one chromatography) for producing four different recombinant proteins by the genetically engineered yeast Saccharomyces cerevisiae was modeled and optimized. Using this example, it is shown that the presence of additional degrees of freedom introduced by the process performance models, with respect to a fixed size and time factor model, represents an important development in improving plant design.