In this work, an integrated scheduling and Economic Nonlinear Model Predictive Control (ENMPC) framework is designed for the optimal operation of a fermentation process comprising multiple fed-batch fermenters operating in parallel, using as a case study a lignocellulosic biomass biorefinery that produces bioethanol, a promising but limited alternative to fossil fuels. The integrated scheduler and controller aim to find optimal decisions among staggered reactors operating simultaneously, being able to imitate continuous operation. Overall, the proposed operation strategy is able to economically distribute feed flows and yeast used in each reactor, by considering coupled scheduling and control interactions, and user defined constraints, e.g., avoiding excessively large feed flow changes. The yeast, the non-constant substrate feeding policies, and the optimal fed-batch times that maximize profit and reject feedstock composition disturbances are obtained. The results show that, in contrast to traditional scheduling and constant feeding rate policies, variable feeding rates integrated with scheduling decisions may lead to reductions in operating costs, while yielding a similar ethanol productivity, which could be a step forward to achieving large-scale sustainable bioethanol production in global decarbonization efforts.

Optimization of chemical processes is challenging due to nonlinearities arising from chemical principles and discrete design decisions. The optimal synthesis and design of chemical processes can be posed as a Generalized Disjunctive Programming (GDP) problem. While reformulating GDP problems as Mixed-Integer Nonlinear Programming (MINLP) problems is common, specialized algorithms for GDP remain scarce. This study introduces the Logic-Based Discrete-Steepest Descent Algorithm (LD-SDA) as a solution method for GDP problems involving ordered Boolean variables. LD-SDA transforms these variables into external integer decisions and uses a two-level decomposition: the upper-level sets external configurations, and the lower-level solves the remaining variables, efficiently exploiting the GDP structure. In the case studies presented in this work, including batch processing, reactor superstructures, and distillation columns, LD-SDA consistently outperforms conventional GDP and MINLP solvers, especially as the problem size grows. LD-SDA also proves superior when solving challenging problems where other solvers encounter difficulties finding optimal solutions.

Mixed integer nonlinear programming (MINLP) in chemical engineering originated as a tool for solving optimal process synthesis and design problems. Since then, the application of MINLP has expanded to encompass control and operational decisions that are in line with the arising challenges faced by the industry, e.g., sustainability, competitive markets, and volatile supply chain environments. Nowadays, process plants are transitioning from traditional manufacturing practices to automated solutions able to integrate decision-making within manufacturing enterprises. This paradigm shift aims to increase profits, optimize resource utilization efficiency, promote long-term sustainability, minimize waste, and enhance responsiveness under uncertainties and perturbations. Accordingly, the development of reliable, computationally efficient, and robust MINLP algorithms capable of simultaneously handling process design, planning, scheduling, or control decisions are crucial to achieving Industry 4.0 integration goals. This work explores potential research opportunities and recent advances toward the development of integrated decision-making frameworks, focusing on their underlying state-of-the-art optimization tools. We provide an overview of emerging deterministic MINLP optimization algorithms for simultaneous decision-making problems. Furthermore, we constructively discuss the versatility and limitations of these optimization tools. We also highlight how novel optimization theories, both within and outside the chemical engineering domain, can be incorporated into advanced MINLP frameworks suitable for process integration.

This work presents a hybrid stochastic-deterministic algorithm for optimal design of process flowsheets, i.e., finding the optimal design variables and operating conditions of multiple interconnected units using rigorous phenomenological chemical engineering models. Unlike previous studies that propose hybrid deterministic and stochastic algorithms in sequential and nested arrangements, the present work proposes a parallel configuration to perform the hybridization. The proposed hybrid algorithm combines a stochastic method (SM) with the deterministic Discrete-Steepest Descent Algorithm with Variable Bounding (DSDA-VB). The SM and DSDA-VB strategies interact in parallel by exchanging new feasible solutions identified by the SM and improved search bounds determined by the DSDA-VB. The proposed method is illustrated using a thermally coupled system and a sequence of reactive, extractive, and traditional distillation columns. The results indicate that the proposed algorithm outperforms the traditional Differential Evolution with Tabu List (DETL) algorithm, showing faster and improved convergence.

This work proposes a general discrete-time simultaneous scheduling and dynamic optimization (SSDO) formulation based on the state-task network (STN) representation. This formulation explicitly considers variable processing times, which is a key aspect in the integration of scheduling and control decisions. The resulting Mixed-Integer Nonlinear Programming (MINLP) problem is solved using a custom Discrete-Steepest Descent Algorithm (D-SDA), which is designed to efficiently explore the ordered discrete decisions in the formulation, i.e., processing times and batching variables. The performance of the proposed solution framework is illustrated using two case studies adapted from the literature. The results show that the D-SDA explores the feasible region of ordered discrete decisions more efficiently than a general-purpose MINLP solver, leading to more profitable solutions in shorter computational times.

This study presents the first application of a logic-based Benders decomposition (LBBD) technique in the field of simultaneous scheduling and dynamic optimization (SSDO), applied to network batch processes with a discrete-time scheduling formulation. The proposed algorithm employs neighborhood information of ordered discrete decisions (e.g., batching variables) to generate cuts, rather than relying on traditional cut generation techniques based on dual information that are implemented in generalized Benders decomposition (GBD) algorithms. The proposed algorithm relies on solving multiple subproblems per iteration, which is a feature that allows the generation of multiple cuts per iteration thus producing accurate approximations of the objective function in shorter computational times. This results in the herein proposed multicut logic-based discrete Benders decomposition (MLD-BD) algorithm, which enables features such as a pruning strategy, and a cut-off technique. Two case studies are used to demonstrate the computational advantages of the MLD-BD framework against GBD and heuristic methodologies.

This work addresses the discrete-time simultaneous scheduling and open-loop control (SSOC) of network batch processes with variable processing times through a tailored Generalized Benders Decomposition (GBD) framework. This SSOC problem is a challenging mixed-integer nonlinear programming (MINLP) problem because variable processing times introduce more binary variables to a discrete-time scheduling formulation and may generate new infeasibilities if those variables are poorly selected. Variable processing times are key in SSOC since they affect both the flexibility of the schedule, and the dynamic performance of batch systems. The key novelty of the proposed GBD approach is the addition of initial and auxiliary feasibility cuts to facilitate the handling of infeasibilities generated by variable processing times. The performance of the proposed GBD framework is tested using a case study adapted from the literature. A GBD methodology that implements traditional feasibility cuts is used as a benchmark. While the conventional GBD method was unable to converge to a feasible solution, the proposed GBD framework found a feasible solution within the first two interactions and then converged by closing the absolute MINLP gap. Therefore, the proposed GBD framework is a promising strategy to solve SSOC problems involving batch processes often found in the pharmaceutical, energy, and food industries.

This study introduces the logic-based discrete-Benders decomposition (LD-BD) for Generalized Disjunctive Programming (GDP) superstructure problems with ordered Boolean variables. The key idea is to obtain Benders cuts that use neighborhood information of a reformulated version of Boolean variables. These Benders cuts are iteratively refined, which guarantees convergence to a local optimum. A mathematical case study, the optimization of a network with Continuous Stirred-Tank Reactors (CSTRs) in series, and a large-scale problem involving the design of a distillation column are considered to demonstrate the features of LD-BD. The results from these case studies have shown that the LD-BD method exhibited good performance by finding attractive locally optimal solutions relative to existing logic-based solvers for GDP problems. Based on these tests, the LD-BD method is a promising strategy to solve optimal synthesis problems with ordered discrete decisions emerging in chemical engineering applications.

The simultaneous optimization of continuous and discrete design variables, operating conditions, and controller's tuning parameters of reactive distillation (RD) columns is investigated in this work. For this purpose, the capabilities of a recently proposed modular economic optimization strategy based on a Discrete-Steepest Descent (D-SDA) framework are investigated. The D-SDA is a decomposition method that aims to improve an initial design by systematically modifying its discrete decisions, e.g., number of stages, until a design that optimizes the process economics while meeting the desired specifications is found. A case study involving the production of ethyl tert-butyl-ether (ETBE) in a RD unit was considered. The simultaneous design and control of the RD column was solved under two scenarios, i.e., product changeovers between four different grades and the production of a single grade of ETBE under a step disturbance in the feed composition. The results show that the modular strategy can specify economic design and control schemes in reasonable computational times.

This work presents the optimal design and operation of catalytic distillation (CD) units considering discrete and continuous design and operation variables combined with rigorous non-linear dynamic process models. The key novelty in this work is that optimal process design and dynamic transitions between different product grades in CD units are simultaneously intensified using a deterministic optimization framework. The proposed optimization method is based on a Discrete-Steepest Descent Algorithm (D-SDA), which enables the designer to improve an initial feasible design, while providing local optimality guarantees. The D-SDA has been effective to handle the optimal steady-state design of CD columns. The present study brings dynamic transitions into consideration thus providing theoretical foundations regarding the optimal design of CD columns in the transient domain. The production of ethyl tert-butyl-ether (ETBE) is considered as a case study, where dynamic transitions between different grades of ETBE coupled with design considerations are investigated. This work shows that a single CD unit can be optimally designed to produce multiple ETBE grades, while simultaneously optimizing its dynamic performance to attain different steady states. Accordingly, this study provides new insights regarding the economic and operational benefits of performing the simultaneous optimal design of multitask CD columns with dynamic transitions.