44 CHAPTER 1. BACKGROUND IDEAS
for the model. On the other hand, if a predicted or modeled value varies
substantially in comparison to the parameter as it is slightly varied, then
the accuracy of measurement of the critical parameter assumes new impor-
tance. In sensitivity analysis, just as in all modeling, this comparison of
“varying substantially” should be measured with significant digits, relative
magnitudes, and rates of change. Here is another area where expressing pa-
rameters in dimensionless groups is important [34]. In some areas of applied
mathematics such as linear optimization and statistics, a side effect of the
solution method is that it produces sensitivity parameters. In linear opti-
mization, these are sometimes called the shadow prices and these additional
solution values should be used whenever possible.
Interpretation and Refinement
Finally the modelers must take the results from the previous steps and use
them to refine the interpretation and understanding of the real world situ-
ation. This interpretation step is represented in the diagram by connection
between the boxes labeled 4 and 1, completing the cycle of modeling. For
example, if the situation is modeling motion, then examining results may
show that the predicted motion is faster than measured, or that the object
does not travel as far as the model predicts. Then it may be that the model
does not include the effects of friction, and so friction should be incorporated
into a new model. At this step, the modeler has to be open and honest in
assessing the strengths and weaknesses of the model. It also requires an im-
proved understanding of the real world situation to include the correct new
elements and hypotheses to correct the discrepancies in the results.
The step between stages 4 and 1 may suggest new processes, or experi-
mental conditions to alter the model. If the problem suggests changes then
those changes should be implemented and tested in another cycle in the
modeling process.
A good summary of the modeling process is that it is an intense and
structured application of “critical thinking”. Sophistication of mathematical
techniques is not always necessary, the mathematics connecting steps 2 and
3 or potentially steps 3 and 4 may only be arithmetic. The key to good
modeling is the critical thinking that occurs between steps 1 and 2, steps 3
and 4, and 4 and 1. If a model does not fit into this paradigm, it probably
does not meet the criteria for a good model.
Good mathematical modeling, like good critical thinking, does not arise