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Wikipedia / Varignon frame

The Varignon frame, named after Pierre Varignon, is a mechanical device which can be used to determine an optimal location of a warehouse for the distribution of goods to a set of shops. Optimal means that the sum of the weighted distances is the optimal location which minimizes the weighted sum of distances

  1. (1):.

The optimization problem is called Weber problem.

Mechanical Problem - Optimization Problem

one gets the equation

  1. (2):.

(At the point of equilibrium the sum of all forces is zero !)

This is a non-linear system which can be solved iteratively by the Weiszfeld-algorithm (see below)

The connection between equation (1) and equation (2) is:

  1. (3):

a local extremum and the Varignon frame provides the optimal location experimentally.

Example

For the following example the points are

and the weights

  1. .

. The second picture shows level curves which consists of points of equal but not optimal sums. Level curves can be used for assigning areas, where the weighted sums do not exceed a fixed level. Geometrically they are implicit curves with equations

  1. (see equation (1)).

Special cases n=1 and n=2

  • .

  • .

  • (see diagram). In this case the level curves (points with the same not-optimal sum) are confocal ellipses as common foci.

Weiszfeld-algorithm and a fixpoint problem

Replacing in formula (2) one gets:

  1. (4):

:

  1. .

This algorithm is called Weiszfeld-algorithm.

Formula (4) can be seen as the iteration formula for determining the fixed point of function

  1. (5)

with fixpoint equation

(see fixed point)

Remark on numerical problems:.

See also

References

  • Uwe Götze: Risikomanagement, Physica-Verlag HD, 2013, , S. 268

  • Andrew Wood, Susan Roberts : Economic Geography, Taylor & Francis, 2012, , p.22

  • H. A. Eiselt, Carl-Louis Sandblom :Operations Research, Springer Berlin Heidelberg, 2010, , p.239

  • Robert E. Kuenne: General Equilibrium Economics, Palgrave Macmillan UK, 1992, , p.226