1. Mixed-Integer Nonlinear Programming (MINLP) and Global Optimisation (GO) algorithms
Conventional Nonlinear Programming (NLP) considers the optimisation of general process models containing only continuous decision variables. MINLP methods allow the solution of problem formulations containing also discrete variables, i.e. variables taking only integer values of which binary variables are a special case. The latter can be used to model existence of units in the optimal solution and general parts of the model that can be represented by binary logic (True/False).
The power of MINLP models is in the fact that they can model both continuous and discrete decisions simultaneously, a requirement of many realistic modeling applications. Applications can be found in the area of Process Design, e.g. by construction of models containing all possible units and routes in the same model, comprising what is known as a superstructure, in the area of Operations Scheduling, such as in batch scheduling to find the optimal sequence of tasks and units to produce multiple products, and in the scheduling of maintenance/cleaning actions as for example in the case of Heat Exchanger Networks. Novel areas of application include the optimal modification prediction for biochemical reaction pathways subject to gene knock-outs or inclusion of new genes to achieve desired production objectives.
In term of algorithms there are a number of avenues to explore, drawing knowledge from an extensive bibliography. Typically solutions procedures and solvers exist for convex MINLP's, for which solution can be guaranteed to be the globally optimal one. However, engineering practice results more often in nonconvex MINLP's, and these pose a challenge to existing technology. New algorithms have appeared over the last decade or so, each with advantages and disadvantages.
The major issue is that to solve these problem models to guaranteed global optimality requires space partitioning, as even without discrete variables nonconvex NLP's are combinatorially hard. Based on the great importance that MINLP models have in both current and evolving engineering and scientific applications, this poses a challenge that we would like to explore in our group. Both rigorous and approximation methods are to be explored; the former to contribute to the theory of such methods and extend the model sizes that can be tackled, and the latter to offer practical solutions, albeit not globally guaranteed, that are valuable in current engineering applications.
2. Dynamic modelling and optimal treatment of chronic illnesses
Dynamic modelling of chronic illnesses, such as in the case of HIV infection (which was studied in our group in the past) and cancer progression, plays an important role in the understanding of the evolution of the disease. The challenge is twofold: to derive accurate mechanistic models based on population balances and related kinetics validated against published data, and to then link these models with appropriate representations of pharmacokinetics so as to provide an accurate model for the impact treatment has on the progression of these illnesses.
Once models as described above are proven to be valid, then the next phase in such research will involve dynamic optimisation studies (optimal control) which will be aiming to discover the optimal way of administering treatment as a function of time.
The cases of HIV and cancer have been mentioned, but this is not exclusive: other illnesses fall into the category of chronically managed diseases and we would be open to all such applications in our group.
3. Biochemical Reaction Pathways: metabolic control theory, sensitivity analysis, optimal control
In this broad area of research we are interested in the application of Process Systems Engineering methods in the modelling and solution of problems arising in biochemical pathways. In particular, we would like to explore general formulations involving metabolic control and genetic regulation in tandem, in a dynamic setting. The study of such systems will be both theoretical as well as computational. The general aim is to derive sensitivities of a biochemical pathway to regulating parameters of the system so as to predict the impact of genetic modifications using a model-based approach.
There is a vast number of published works in the general area to build new directions from, as well as novel ideas we have in our group in collaboration with international researchers in the field.
4. Scheduling the cleaning actions of heat Exchanger networks
Fouling is a major problem in many chemical industries. It is responsible for large energy and throughput losses, resulting in financial penalties and negative environmental impact. Fouling is tied together with ageing, which is the transformation of the initial soft deposit into a more cohesive form in time, due to exposure in process conditions. Thus, the growing of two layers occurs on a heat transfer surface.
One effective mitigation strategy is the regular cleaning of the fouled heat transfer units. In this ongoing project, two cleaning modes are considered, which differ in their effectiveness in removing aged material. The first, a fast and cheap in-situ technique can only remove the soft fouling layer, while the second, a time-consuming and expensive ex-situ one is able to remove both layers.
Optimisation tools can be used to schedule the timing and the selection of cleaning actions as well as the unit or units to be cleaned at each instant, in order to minimise the cost of fouling and impact on productivity. This is a scheduling problem of combinatorial nature, hence the developed models describing the cleaning scheduling yield a Mixed Integer Non Linear Programming (MINLP) optimisation problem. The primary objectives of this project can be divided in two categories:
Firstly, it aims to provide better understanding of how this coupled fouling-ageing phenomenon affects the heat transfer process and how it can be best mitigated by applying mixed cleaning campaigns. Most importantly it aims to demonstrate the potential benefit of performing experimental studies in order to obtain kinetic data about the this phenomenon.
Secondly, this work aims to contribute to the theory and solution techniques of large MINLPs, convex or nonconvex. This can be done by proposing specific model structures that are easier to solve or by investigating different problem formulations which will reduce the number of integer variables and constraints significantly. Also, it will be useful to construct approximation methods which will offer on one hand, good starting points for rigorous solvers and on the other hand, practical solutions to engineering applications.