Wednesday, June 23, 2010

Minimizing the Length of Connecting Lines

There is a problem in Operations Research in which one is given three points and one has to find the connecting point, X, which makes the total length of the lines is a minimum.

The objective function, f, is the sum of the lengths of the lines and at the optimal point its variation is zero for all possible variations in position and as a consequence the variational coefficients are also zero.

These two conditions can be simplified to show that the sum of the unit vectors along the connecting lines is also zero. This means that the angles between these directions are all 120 degrees.

One can solve for the unknown lengths of two of the lines if the angle of one of them is assumed and in this way one can obtain two curves which intersect at the optimal point. The epsilons, ε, denote the unit vectors along the lines connecting at the optimal point.

Using a solve block one can easily find the optimal point. The function, r, solves for the unknown length of the lines for a given angle, θ, and the find function solves for the optimum angle yielding the solution, X.

The solution appears to be the intersection of two circles with one point of intersection at the optimal point and another point of intersection for which the distance from the common point, A, is zero. Circles are curves of constant curvature which may be due the angle between the two links being a constant.

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