Dynamic flux balance analysis (DFBA) provides a platform for detailed design, control and optimization of biochemical process technologies. It is a promising modeling framework that combines genome-scale metabolic network analysis with dynamic simulation of the extracellular environment. Dynamic flux balance analysis assumes that the intracellular species concentrations are in equilibrium with the extracellular environment. The resulting underdetermined stoichiometric model is solved under the assumption of a biochemical objective such as growth rate maximization. The model of the metabolism is coupled with the dynamic mass balance equations of the extracellular environment via expressions for the rates of substrate uptake and product excretion, which imposes additional constraints on the linear program (LP) defined by growth rate maximization of the metabolism. The linear program is embedded into the dynamic model of the bioreactor, and together with the additional constraints this provides an accurate model of the substrate consumption, product secretion, and biomass production during operation. A DFBA model consists of a system of ordinary differential equations for which the evaluation of the right-hand side requires not only function evaluations, but also the solution of one or more linear programs. The numerical tool presented here accurately and efficiently simulates large-scale dynamic flux balance models. The main advantages that this approach has over existing implementation are that the integration scheme has a variable step size, that the linear program only has to be solved when qualitative changes in the optimal flux distribution of the metabolic network occur, and that it can reliably simulate behavior near the boundary of the domain where the model is defined. This is illustrated through large-scale examples taken from the literature.

The determination of vehicle routes fulfilling connectivity, time, and operational constraints is a well-studied combinatorial optimization problem. The NP-hard complexity of vehicle routing problems has fostered the adoption of tailored exact approaches, matheuristics, and metaheuristics on classical computing devices. The ongoing evolution of quantum computing hardware and the recent advances of quantum algorithms (i.e., VQE, QAOA, ADMM) for mathematical programming make decision-making for routing problems an avenue of research worthwhile to be explored on quantum devices. In this paper, we propose several mathematical formulations for inventory routing cast as vehicle routing with time windows, and comment on their strengths and weaknesses. The optimization models are compared from a quantum computing perspective, specifically with metrics to evaluate the difficulty in solving the underlying quadratic unconstrained binary optimization problems. Finally, the solutions obtained on simulated quantum devices demonstrate the relative benefits of different algorithms, and their robustness when put into practice.

This work analyzes the initial value problem in ordinary differential equations with a parametric lexicographic linear program (LP) embedded. The LP is said to be embedded since the dynamics depend on the solution of the LP, which is in turn parameterized by the dynamic states. This problem formulation finds application in dynamic flux balance analysis, which serves as a modeling framework for industrial fermentation reactions. It is shown that the problem formulation can be intractable numerically, which arises from the fact that the LP induces an effective domain that may not be open. A numerical method is developed which reformulates the system so that it is defined on an open set. The result is a system of semi-explicit index-one differential algebraic equations, which can be solved with efficient and accurate methods. It is shown that this method addresses many of the issues stemming from the original problem's intractability. The application of the method to examples of industrial fermentation processes demonstrates its effectiveness and efficiency.

This work presents a general theory for the construction of a polyhedral outer approximation of the reachable set ("polyhedral bounds") of a dynamic system subject to time-varying inputs and uncertain initial conditions. This theory is inspired by the efficient methods for the construction of interval bounds based on comparison theorems. A numerically implementable instance of this theory leads to an auxiliary system of differential equations which can be solved with standard numerical integration methods. Meanwhile, the use of polyhedra provides greater flexibility in defining tight enclosures on the reachable set. These advantages are demonstrated with a few examples, which show that tight bounds can be efficiently computed for general, nonlinear systems. Further, it is demonstrated that the ability to use polyhedra provides a means to meaningfully distinguish between time-varying and constant, but uncertain, inputs.