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 of the shops to the warehouse should be minimal. The frame consists of a board with n holes corresponding to the n shops at the locations x 1 , . . . x n {\displaystyle \mathbf {x} _{1},...\mathbf {x} _{n}} , n strings are tied together in a knot at one end, the loose ends are passed, one each, through the holes and are attached to weights below the board (see diagram). If the influence of friction and other odds of the real world are neglected, the strings are long enough to prevent weights being jammed into their holes, and no single weight is so heavy as to pull the knot through the hole and below the table, the knot will take a position of equilibrium v {\displaystyle \mathbf {v} } . It can be shown (see below), that point v {\displaystyle \mathbf {v} } is the optimal location which minimizes the weighted sum of distances
(1): D ( x ) = ∑ i = 1 n m i ‖ x i − x ‖ {\displaystyle \ D(\mathbf {x} )=\sum _{i=1}^{n}m_{i}\|\mathbf {x} _{i}-\mathbf {x} \|} . The optimization problem is called Weber problem.
Mechanical Problem - Optimization Problem
If the holes have locations x 1 , … , x n {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{n}} and the masses of the weights are m 1 , . . . , m n {\displaystyle m_{1},...,m_{n}} then the force acting at the i-th string has the magnitude m i ⋅ g {\displaystyle m_{i}\cdot g} ( g = 9.81 m/sec {\displaystyle g=9.81{\text{m/sec}}} : constant of gravity) and direction x i − v ‖ x i − v ‖ {\displaystyle {\tfrac {\mathbf {x} _{i}-\mathbf {v} }{\|\mathbf {x} _{i}-\mathbf {v} \|}}} (unitvector). Summing up all forces and cancelling the common term g {\displaystyle g} one gets the equation
(2): F ( v ) = ∑ i = 1 n m i x i − v ‖ x i − v ‖ = 0 {\displaystyle \ \mathbf {F} (\mathbf {v} )=\sum _{i=1}^{n}m_{i}{\frac {\mathbf {x} _{i}-\mathbf {v} }{\|\mathbf {x} _{i}-\mathbf {v} \|}}=\mathbf {0} } . (At the point of equilibrium the sum of all forces is zero !) This is a non-linear system for the coordinates of point v {\displaystyle \mathbf {v} } which can be solved iteratively by the Weiszfeld-algorithm (see below) The connection between equation (1) and equation (2) is:
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