I would like to explain the underlying basic mathematics and engineering of many apparently diverse problems I have solved in the last few years and how we have further developed some of the basic methodologies while doing research in a diversity of application areas.
Many engineering (including financial engineering and supply chain management) problems can be modeled, for the purpose of dynamic analysis in continuous time, by the ordinary differential equation:
where, t is an indexing variable, x(t) is the state variable of the system, u(t) is the control variable, and A and B are given matrices. In the most general case, the right-hand side of (I) could be a set of nonlinear functions. This is a well-known equation, however, when there is uncertainty, due to either stochastic or fuzzy, the model becomes more realistic but can present some challenging practical difficulties in its solution such as, non-linearity, non- smoothness, non-differentiability, and large-scale. The variables x(t) and u(t) may be bounded, in which case, for stochastic problems, little is known. We have made some progress in discrete time domain for such problems, as explained later. For steady-state analysis, and for analysis in the frequency domain, the problem may become an algebraic problem. In addition to (I), for the purposes of design, we usually have a cost function that either minimizes the total cost of operations or maximizes the total net benefits from operating the system, for example, in the form of
In the case of uncertainty (depends on the problem, for example, in water reservoirs they may come from net inflows in equation (I) and prices in equation (II) ), the cost function must be appropriately changed and additional probabilistic constraints may need to be added as explained later, case by case. In some of our problems, we might be only interested in knowing the effect of certain design decisions (for example, in pollution modeling, the difference between having a treatment facility or not) and this cost function may not be explicitly present.