GPS-X Technical Reference

375 Optimizer
In the objective function expressions given above the following nomenclature is used:
zi,j = the measured value of response j in experiment i.
fi,j
j
m
nj
= the value of response variable j predicted by the process model in experiment i
= the heteroscedasticity parameter for response j
= the number of measured response variables
= the number of experiments (i.e. observations) for response j
These objective function types are accessed by clicking on the inverted triangle beside the
Optimize icon and then selecting Type from the drop-down menu.
Using the Optimizer for Parameter Estimation
In general, the maximum likelihood objective function should be used when doing
parameter estimation. This objective function calculates statistically optimal parameter
estimates based on assumptions on the nature of the measurement errors. The sum of
squares objective function is a special case of the maximum likelihood objective function
derived using further simplifying assumptions and can also be used. It is equivalent to the
maximum likelihood objective function when there is only one response or target
variable. Further details on parameter estimation and the maximum likelihood and sum
of squares objective functions can be found in Appendix A of this chapter.
The other objective functions can be used for curve fitting when calculating statistically
optimal parameter estimates is not a concern.
Using the Optimizer for Process Optimization
In addition to data fitting applications, GPS-X can be used for process optimization. For
example, GPS-X can calculate the operating conditions for your process model that
optimize some measure of process performance, such as operating cost or effluent
quality.
To solve this type of problem in GPS-X you need to treat the problem as a data
fitting exercise. For example, if you want to minimize the value of a certain
model variable you need to specify an arbitrarily small target value for the model
variable in a .dat file and have the optimizer minimize the difference between
the calculated variable value and the target value using the absolute difference
objective function. This is equivalent to minimizing the model variable directly.
The target value should be made small enough so that the optimizer cannot reach
it.
You are not limited to using one performance measure. You can select a number of
different performance measures and have GPS-X optimize these variables simultaneously
by fitting them to user-supplied targets.
GPS-X Technical Reference