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Data Analysis Australia

# 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.
• 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.

Examples of project experience where Data Analysis Australia has provided solutions for clients by applying mathematical modelling, simulation and/or optimisation techniques are listed below.

### Simulation Model for a Major Port Operator

Data Analysis Australia was able to assist a major port operator in understanding the potential for improvements when upgrading a bulk cargo jetty.  Upgrading the loaders and unloaders would be a waste if other constraints such as the schedule of ship arrivals would still limit the use.  Data Analysis Australia set up a simulation model for the operation of the jetty.  Critical to this was the understanding of ship arrival times - most ports operate on a first come first served basis so the use of a facility is dominated by the distribution of times between ship arrivals.  In addition, queuing of ships at a port drives up costs for shippers.  A statistical analysis of historical ship arrivals established their random nature and calibrated suitable probability models.  A similar analysis calibrated models for loading and unloading times.  An overall simulation model was then built in the package Extend.  Using this model, a number of scenarios were explored for upgrades and the most cost effective options chosen.

### Modelling Data for Workforce Planning for the Police Department

Workforce planning was modelled by Data Analysis Australia, a challenge for a specialist agency such as the Police, where years of training and experience were critical.  When the Western Australian Police were faced with the planning of training facilities, the critical question of "how big" could only be answered with a thorough model of the workforce structure.  Data Analysis Australia identified that any model would need to incorporate the promotion and attrition structure in the Police and the way that this was affected by factors such as age.  A Markov model that incorporated probabilities of transitions between ranks was built, initially in a simple form and then with age and gender groups.  The model allowed the impact of various recruitment strategies to be understood and quantitative estimates of training needs derived.

### Optimal Location Modelling of Metropolitan Magistrates Courts

Western Australia is geographically the largest State in Australia, and building and development is spreading both North and South along the coastal side of the Perth metropolitan area.  Given these developments, and the increasing population, it is important to plan appropriately for the placement of new Court buildings.  Data Analysis Australia created a Locational Model as part of the Metropolitan Courts Strategic Planning Project in December 2005.  This work covered various scenarios aiming to optimise the placement of justice facilities by minimising travel from offence locations to court locations projected out to 2031.  In 2006 the Locational Model was expanded to incorporate public transport.  In 2007/08 the model was updated further, incorporating recent offence data and scenarios with location constraints reflecting the Government's strategic development planning for certain areas.

For more information about any of our Areas of Expertise please contact one of our consultants, email us at daa(at)daa.com.au or call 08 9468 2533.