Société de Calcul Mathématique SA
Modeling: well posed problem, concrete results
Mathematics consists initially of a language, which makes it possible to transcribe problems of quantitative nature: this is modeling. Once this transcription is made, tools are available to solve these problems, partially or completely. Then, one brings back the solution in its context of origin.
For example, problems of stock management, transportation in an optimal way, optimization of a wallet, result in the maximization of a certain function, under certain constraints. These problems are solved, and one converts the solution into practical terms.
To carry out a model means above all to understand what occurs, not to be satisfied with an empirical solution. To model a process means to describe it in a scientific quantitative way, for example using equations (physical, chemical, etc). It makes it possible to study the evolution of the process, to simulate various solutions, by modifying certain parameters. Thus, an industrial will wonder: "What occurs if I move a site of production?". A transporter: "will I reduce my costs if I take this route?"
For most problems, there are empirical solutions, which you can use almost immediately. Why not be satisfied with them, and why should we care for precise modeling? There are four reasons, and they are essential:
A software borrowed from your neighbor, an empirical solution, do not represent a precise answer to your problem. Conversely, modeling gives you a solution which is at the same time effective and reliable, because all the parameters, all the possible cases, were considered. If for example an optimal route is sought, one will take account of all the obligations specific to each situation: travel time, places where one needs to stop, regulations, obligation to avoid certain zones or certain periods, etc.
The optimal solution thus obtained is obviously more effective than an approximate solution; it is also more reliable, because one knows the respective influence of the various parameters: some have a critical importance and some are secondary. On the opposite, a commercial software will give average values to many parameters, and will not allow them to be modified.
Once modeling has been carried out, quantitative tools are available, making it possible to know the significant parameters and to evaluate the cost of the programs. To make sure that the algorithms of resolution of the problem were well-designed is an essential element of the control of the costs.
Conversely, if one has used approximate solutions, one must use sophisticated devices in order to compensate and maintain global performances. But sophisticated devices are much more expensive than algorithms!
If you buy a commercial software, if you copy the approximate solution of your neighbor, you are dependent on this software, on this solution. You will use them without really understanding them. This may seem money-saving, at first sight, but soon you will regret, because:
- What occurs if initial data are modified (problems of sensitivity to the initial conditions) ?
- How do I have to choose the initial data so that, at the end, the solution has the required properties (inverse problems) ?
Modeling of the process allows you to secure full control in full independence; you know what has to be done, you know what is significant, and you do not depend upon commercial tools. You have a complete control of the situation, which is satisfying both at the intellectual level and from an economic point of view.