1.5. MATHEMATICAL MODELING 43
optima or the rates of change of optima with respect to the constraints. If the
model is a differential equation or a difference equation, then the solution may
have some mathematical substance. For instance, for a ballistics problem, the
model may be a differential equation and the solution by calculus methods
yields the equation of motion. For a problem with randomness, the derivation
may find the mean or the variance. For a connectivity graph, one might be
interested in the number of cycles, components or the diameter of the graph.
If the model is a computer program, then this step usually involves running
the program to obtain the output.
It is easy for students to focus most attention on this stage of the process,
since the usual methods are the core of the typical mathematical curriculum.
This step usually requires no interpretation, the model dictates the methods
that must be used. This step is often the easiest in the sense that it is the
clearest on how to proceed, although the mathematical procedures may be
daunting.
Testing and Sensitivity
Once this step is done, the model is ready for testing and sensitivity analysis.
This is the step that connects the boxes labeled 3 and 4. At the least,
the modelers should try to verify, even with common sense, the results of
the solution. Typically for a mathematical model, the previous step allows
the modelers to produce some important or valuable quantity of the model.
Modelers compare the results of the model with standard or common inputs
with known quantities for the data or statement of the problem. This may
be as easy as substituting into the derived equation, regression expression, or
equation of motion. When running a computer model or program, this may
involve sequences of program runs and related analysis. With any model,
the results will probably not be exactly the same as the known data so
interpretation or error analysis will be necessary. The interpretation will
take judgment on the relative magnitudes of the quantities produced in light
of the confidence in the exactness or applicability of the hypotheses.
Another important activity at this stage in the modeling process is the
sensitivity analysis. The modelers should choose some critical feature of the
model and then vary the parameter value that quantifies that feature. The
results should be carefully compared to the real world and to the predicted
values. If the results do not vary substantially, then perhaps the feature or
parameter is not as critical as believed. This is important new information