# Mathematical Modelling, Simulation and Optimisation

Complex real-life structures and systems can often be represented in a mathematical or statistical way.  They can be conceptual models to better understand or simplify a process, or detailed data driven mathematical models to provide a theoretical understanding of relationships in a quantifiable way.  Importantly, they need to be built so that the main characteristics are captured while simplifying the superfluous.

Developing a suitable model is the key.  Once the model has been developed, simulations of the real-life system can be built in an artificial environment enabling a wide range of scenarios to be tested by changing the influential factors and observing the outputs under various conditions.  Some simulations are deterministic, while some involve random aspects. Data Analysis Australia can then use the models for:

• Understanding the inter-relationships of components in the system;
• Sensitivity analysis to determine factors that most influence outputs;
• Predicting the long-term behaviour of the system;
• Predicting how the system responds when changes are made to the factors influencing it; and
• Optimising the system without the risks of upsetting the real operations.

Models are usually a trade-off between getting the detail right and maintaining sufficient simplicity to permit analysis.  A combination of experience and experiment is often required.  Mathematical theory is usually necessary for making judgements as to what is important.  The experience of Data Analysis Australia in statistical data is often critical in building models.  Some optimisations require simulation while others use more formal methods, such as linear and integer programming, to find the optimal solutions without the need for simulation.  Data Analysis Australia's newsletter article titled Shoemakers' Lasts and Model Ships describes our approach to modelling in more detail.