Mathematical Programming or Mathematical Optimization is the art of formulating optimization models in a format that can be interpreted and solved by an optimization engine to maximize or minimize an objective function subject to some constraints.
The objective function is a real function that governs the relationship of the desired outcome to the manipulated Variables that can be changed to reach optimality. These Variables can be design capacities for equipment or systems or their operating throughputs or any other operating or design condition that may affect the objective function. The variables are usually limited to fall within certain range of acceptable limits which form the constraints to mathematical model. Such constraint can be the capacity limits of equipment or their operating conditions limits as an example.
There are several types of mathematical programing techniques, depending on the objective and constraints formulation. These types include:
- Linear Programing (LP), when the objective and all constraints are linearly related to the variables.
- Non-Linear Programing (NLP), when the objective or any of the constraints are related to any of the variables through a non-Linear function (e.g. quadratic, logarithmic)
- Integer Programming, when any of the decision variables is constrained to take integer values only (e.g. on/off, number of equipment installed, etc.). depending on the linearity of the problem, this leads into two subcategories, namely: Mixed Integer Linear Programing (MILP) and Mixed Integer Non-Linear Programing (MINLP)
- Stochastic Programming, when any of the input parameters of the model (e.g. raw material cost or availability, demand or product price) are uncertain and there is a need to evaluate the impact of such uncertainty on the decision variables and how it relates to the objective. In this case the objective function is to maximize the Expected gain (or minimize the expected loss) over all the possible scenarios of the uncertain parameters realizations.
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