Geometric Facility Location Problems

More documents

Recommendations

Info

Here (x 1 ,y 1 ), (x 2 , y 2 ), …., (x n ,y n ) are the set of points for which we have to compute the minimum enclosing circle. Let (x,y) be the center of the minimum enclosing circle and let r be its radius. Then we have the minimum enclosing circle problem formulated as the following optimization problem: Minimize r 2 Subject to: (xx i ) 2 + (yy i ) 2 ≤ r 2 i = 1, 2, …, n If we substitute z = x 2 + y 2 – r 2 then we have the following convex programming formulation of the problem: Minimize x 2 + y 2 – z Subject to: 2xx i 2yy i + z + (x i 2 + y i2 ) ≤ 0 i = 1, 2, …, n Thus we have to minimize a quadratic function over a set of linear constraints involving 3 variables.
FermatWeber Problem Problem Definition: For a given set of m points x 1 , x 2 , …, x n with each x i ∈ R d find a point y from where the sum of all Euclidean distances to the x i ’s is the minimum. This problem is also known as <strong>Geometric</strong> 1Median problem : arg min n ∑ d y ∈ R i= 1 x i − y The special case when n = 3 and d = 2 arises in the construction of Minimal Steiner Trees and was originally posed as problem by Torricelli.
Page 1 and 2: Pinaki Mitra Dept. of CSE IIT Guwah
Page 3 and 4: Illustration of the Proof of Heron
Page 5 and 6: The minimum enclosing circle can be
Page 7: But the computation of Furthest Nei
Page 11 and 12: If we restrict d = 1 , i.e., for th
Page 13 and 14: In kmedian problem the elements f
Page 15 and 16: y i ∈{0,1} xij ∈{0,1} i = 1,2,.
Page 17 and 18: Line Facility In many practical app
Page 19 and 20: Thus we have d unknowns a 1 , a
Page 21: Similar type of problems for line f

Geometric Facility Location Problems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?