Computational Geometry

Facility Location Problems

Hooman Mohajeri MoghaddamWinter 2011

Facility Location ProblemA.K.A. Location AnalysisA branch of Operations Research Mathematical modeling and solution of

problems concerning optimal placement of facilities

In order to minimize transportation costs, avoid placing hazardous materials near housing, outperform competitors' facilities, etc.

ApplicationWarehouses for branches of a supermarket

a b c d

p-line Center ProblemLocating p > 0 linear facilities in the plane so

that the maximum weighted distance from the demand points to the nearest facilities is minimized.

Application?The general problem is known to be NP-Hard

(Megiddo and Tamir 1982 ).However for the case when p=1 the problem

has polynomial solution (Morris and Norback 1983).

Problem Statement demand points , in the plane, each with a

positive weight

where As you can see is the Euclidean distance of

point to the line .

How Hard is it?

Unweighted 1-Line-CenterWe obtain the following lemma for the

unweighted (all =1) case:Lemma 1(Lee and Wu 1986):

For the unweighted 1-line-center problem, the set of demand points which determine the optimal line , i.e., those points which are at maximum distance from , is a subset of vertices on the convex hull CH ( S) of S.

Furthermore, there are two points in D which determine an edge of the convex hull CH( S).

Proof. is indeed inside the Convex Hull

At least two points(on either side of ) have optimal values.

is fixed by at least a Convex Hull edge.

Algorithm(Unweighted) 1.Find the convex hull points. 2. Pick an arbitrary hull edge to start with and find a point on the

convex hull which is the farthest away from the line containing the edge. Set the optimum value to infinity initially.

3. Compare the distance determined by the (edge, point) pair with the current optimum value. If the new distance is smaller, use this distance as the new optimum value, and the (edge, point) pair is recorded as the optimum pair.

4. Choose the counterclockwise succeeding edge on the convex hull as the next candidate edge. If the starting edge in step 2 is reached, go to step 6.

5. Determine the point on the convex hull which is the farthest away from the new candidate edge.(scan counterclockwise along the hull points from the previous candidate point on until its succeeding point is nearer to the new candidate edge) With this new (edge, point) pair, go to step 3.

6. the (edge, point) pair recorded so far determine the optimal line. Thus can be easily calculated.

Maximum Gap On a Circle ProblemProblem Statement: Given points on the first quadrant

of the unit circle, find the maximum straight line distance between neighboring points on the unit circle.

Lemma 2: The Maximum Gap On A Circle problem requires time. (We don’t prove it here)

Theorem 1: Maximum Gap On A Circle can be reduced to the unweighted I-line-center problem.

Corollary: The unweighted 1-line-center in the plane can be found in optimal time.

Proof.Find the minimum and the maximum of the

x-coordinates of the points.For each point , add an additional point and

and .

Solve the 1-line-center for the points formed in step 2 and compute the maximum distance from those points to the optimum line. Answer that MaximumGap =

Weighted 1-Line-CenterIn order to solve this problem we transform

the points into weighted composite points.State the problem differently:

Then for two indices we have:

Weighted 1-Line-CenterUsing triangle inequality we have:

Which is the distance of the weighted composite

point to the line .