What is a mathematical model ?

According to the Oxford Dictionary, a mathematical model is “a physical, mathematical or logical system representing the essential structures of a reality and, at this level, capable of explaining or dynamically reproducing the operation".
A mathematical model, therefore, combines the rational methods and techniques of analysis and synthesis of the phenomena of organisation, in order to reproduce their behaviour.
In the economic sector, mathematical models serve as decision-making tools in order to, for example, maximise profits, minimise the costs of production, etc.
Some management problems contain so many parameters and variables that the model becomes complex to solve, making it necessary to reformulate it. At this point dedicated and specific techniques are called on which make it possible to express the constraints and variables of the problem in a different way. The aim of this reformulation is to focus on the decision-making levers and to define a mathematical model with achievable solutions, while giving a true representation of the problem. The full name of this expertise is modelling.

Example

We are going to illustrate this definition using a simple example. Namely a company that sells a quantity Q of chairs each month. The cost of manufacturing a chair (wood, fabric and labour) is 50 euros, and the selling price is 100 euros.
The company wants to maximise its profit. The mathematical model describing its performance will, therefore, be a maximisation of “Income – Costs”:



Let us now suppose that the factory does not have sufficient staff to produce more than 1,000 chairs per month. We must, therefore, add a constraint on the form



Let us also add another type of constraint: the production of a chair requires a “unit” of fabric and 4 “units” of wood. This means that for each chair produced, one unit of fabric is consumed and 4 units of wood. Let us take ‘t‘ as the quantity of fabric used per month and ‘b’ the quantity of wood used per month. Mathematically, this is expressed as follows:

Let us suppose now that the wood supplier of our company is limited to 3,000 units per month and that the fabric supplier is limited to 800 units per month. We must, therefore, add two more constraints ‘to our model’:

Finally, let us suppose that our company can buy a quantity of wood Beta from another supplier, but involving an additional cost of 10 euros per unit compared with its usual supplier.
How can this information be integrated into our model?
Firstly, the cost of production for the chairs manufactured with this new wood would be higher. Therefore, this must be taken into account in the maximisation of profit. Furthermore, the limitation on chair production changes, as henceforth there are two possible sources of supply. 

Ultimately, we are able to draw up a global model as follows:

Optimization

We now want to find the optimum production of chairs, so that the constraints are observed and the profit maximised.
For a small problem like this, it is easy to find the optimum solution “by hand”.
In total, 800 chairs will be produced per month. 800 units of fabric will be ordered and 3200 units of wood (3,000 from the usual supplier and 200 from the new supplier). The total profit will be 38,000 euros (80000 − 40000 − 2000).

Of course, for problems on a real scale, this may become extremely complicated. Let us imagine that the company also produces tables and cupboards, also requiring wood, fabric and metal, and also specialised labour. In such a hypothetical case, optimization through the use of algorithms becomes essential.
We will not enter into explanations now about how to resolve a problem, or into the description of the algorithms. However, we are briefly showing you how to resolve this problem with the help of an optimization language, such as AMPL for example.

Resolution by computer

Once the model is written mathematically, the main part of the work is complete. In fact this mathematical model just has to be written in mathematical language, as computers can process this language.
If we take the small model above, its equivalent in AMPL mathematical language would be as follows:

Then we just have to click on ”solve”, and we obtain the solution immediately: