|
|
Gerry Myerson wrote:
> In article <1158110928.746640.261200@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> "Chip Eastham" <hardmath@xxxxxxxxx> wrote:
>
> > Gerry Myerson wrote:
> > > In article <1158061997.291948.225520@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > "bem" <cwei83@xxxxxxxxx> wrote:
> > >
> > > > A company has 5 factories A, B, C, D, and E, located at the points
> > > > (10,10), (30,50), (16.667,29),(0.555,29.888) and (22.2221,49.988),
> > > > respectively, in the xy-plane. Assume that the distance between 2
> > > > points represents the driving distance, in miles, between the
> > > > factories. The company plans to build a warehouse at some point in the
> > > > plane. It is anticipated that during an average week there will be 10,
> > > > 18,20, 14, and 25 deliveries made to factories A, B, C, D, and E,
> > > > respectively. Ideally, to minimize the weekly mileage of delivery
> > > > vehicles, where should the warehouse be located?
> > >
> > > Put it at (x, y). Then the distance to, say, factory B is
> > > square root of ( (x - 30) squared plus (y - 50) squared ).
> > > Call that d_B(x, y). The mileage is then
> > > M(x, y) = 10 d_A(x, y) + ... + 25 d_E(x, y),
> > > so you're trying to minimize a function of 2 variables.
> > > Do you know how to do that?
> >
> >
> > Gerry Myerson wrote;
> >
> > > I never heard of powell's method, nor of Nelder Mead.
> >
> > Powell's method refers to a quasi-Newton method with updates
> > to the approximate Hessian:
> >
> > http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/unconstrained/quasi.html
> >
> > while Nelder-Mead is a "robust" method for "pivoting" across
> > faces of a simplex to obtain improved function values:
> >
> > mathworld.wolfram.com/Nelder-MeadMethod.html">http://mathworld.wolfram.com/Nelder-MeadMethod.html
>
> OK, thanks. Are these better than setting the partial derivatives
> to zero and solving for x and y? or are these numerical methods
> for solving for x and y because you can't, or won't, solve analytically?
Yes, these are "real world" techniques, although the
Davidon-Fletcher-Powell update method has been largely supplanted by
BFGS (Broyden-Fletcher-Goldfarb-Shannon) update.
The fact that the exercise is worded clearly and for a two-dimensional
problem suggests classwork, of course.
Naturally a small 2D weighted sum of squares problem can be settled
exactly by solving a 2x2 system, so one would have the advantage of
knowing an exact solution with which to compare the rates of
convergence of Powell's method and Nelder-Mead.
regards, chip
|
|