## Finding the Optimal Solution

In any environment there is constant pressure to do everything better, faster and cheaper. This often means finding the best way of using the available resources to maximise profit or outputs. Optimisation is the art and science of finding this best solution.

Optimisation involves setting up a mathematical model for the problem, specifying in quantitative terms the cost structures, the constraints and a clear idea of what is meant by an "optimal" solution. The model is an approximation of reality but one that is close enough to be useful. Most importantly, the model allows numerical algorithms to find a solution.

Mathematical methods are at the centre of modern optimisation. Of these, linear programming is by far the most successful method. First developed in World War II to solve problems such as convoy planning, it has been shown to have a very wide range of applications and, just as importantly, efficient algorithms have been found so that it can be used on problems of any size, even those with tens of thousands of parameters.

The first step in the optimisation process is to convert operating or system rules and procedures into a precise mathematical form. This step should never be underestimated, both in terms of the effort involved and the benefits that come from that precision. This is best handled through a partnership of someone familiar with the system and an experienced mathematician. It is often a process of discovery since frequently constraints have only been understood intuitively and rarely documented.

The question of *what* is being optimised is also critical. Sometimes it is obvious (it might be simply profit) but for some systems it can be subtle with several criteria needing to be optimised at the same time. This can be done through applying weights to each of the criteria reflecting their relative importance.

In some situations the optimisation process may give solutions that are quite unexpected. This may highlight a failure in identifying some feature of the system such as a constraint. Alternatively, since the mathematical methods have no preconceptions as to where the solution should lie, they might have suggested viable solutions that people too close to the system cannot see.

Data Analysis Australia uses optimisation modelling to assist clients in finding the best way to run their operations. An example is the siting of new services to ensure cost effective access over the metropolitan area, both now and in the future. This requires forecasts of where future demand will come from (a statistical problem in itself) and a way of measuring access. Typically the aim will be to minimise the distance or time that clients for the service will have to travel on average. The effect of moving the location of a service point is usually not trivial, since it will make it more accessible to some and less accessible to others, sometimes changing which service point is closest to a client. This complexity means that *ad hoc* solutions can be misleading and a mathematical optimisation can be substantially better.

Another advantage is that the optimisation model can be set up as a program that the clients can run themselves, examining various scenarios and evaluating the results. This can be a way of learning what affects their system, how it affects their system and is a cost effective way of testing proposed solutions.

Optimisation modelling can be used to find a solution to any problem where demand needs to be met using limited capacity.

For more information on optimisation modelling and its applications, please Contact Us.

*December 2004*