Example: pipeline for constant flow rate.
Capital cost pipe diameter, D - maybe should be D^{0.6} or D^{0.9}.
Operating cost
D^{-4.75} - assumes friction factor
Re^{-0.25}.
Typically, capital cost versus operating cost.
f(x) | - the objective function |
x | - the decision variable |
constraints: | |
Constraints define feasible region.
Minimise f(x) with respect to (w.r.t.) x, subject to (s.t.) ,
Example: (use D instead of x) minimise f(D) = 2.0 D + 0.4 + 0.5 D^{-4.75} (s.t.) , .
If expression available for derivative..
Solve
Nature of each SP at x_{sp} is given by sign of second derivative f^{''}(x):
+ | implies local minimum |
- | implies local maximum |
0 | don't know |
Example:
implies
m.
and
Hence local minimum.
Beware of discontinuities
Simple theory assumes that both f(x) and f^{'}(x) are continuous. If not, global minimum may exist at or near a discontinuity in f or f^{'}.
Profit, throughput, reactor conversion,...
`Maximise f(x)' is equivalent to `Minimise -f(x)'.
Without loss of generality, a computer code can treat all problems as minimisations.
Next - Section 5.3.2: Non-gradient Methods
Course Organiser Last modified: Tue Aug 25 10:29:53 BST