Problem 1 There are four facilities at the points 00 33 11 2

Problem 1

There are four facilities at the points (0,0), (3,3), (1,1), (2,2).

(a) A new facility is going to be located in such a way as to minimize the longest distance to any existing facility. Find the optimal location..

(b) Write the equations of the boundary lines of a contour corresponding to a minimal longest distance equal to 5.

(c) Draw the contour for part (b) and show the optimal solution on your graph.

(d) What is the optimal location of the new facility if the total distance from each existing facility to the new facility is minimized?

Solution

4 optimal facilities are at (0,0) ,  (1,1) ,  (2,2) ,  (3,3)
it is noteworthy that all 4 of these points lie in the single straight line with points  (0,0) and  (3,3) at teh extremum.
So to minimise the longest distance to any existing facility, a new facility has to be setup at the mid point of the two extremum points which is (0+3 /2 , 0+3 /2) = (3/2,3/2)
Thus the optimal location is (3/2, 3/2)

Since the existing faclities fall in the single line, the total distance from each existing facility to the new facility will be minimum at the average points of all 4 of these facilities which is (0+1+2+3 /4 , 0+1+2+3 / 4) = (6/4, 6/4) = (3/2 , 3/2)
Thus the optimal location of the new facility if the total distance from each existing facility to the new facility is to be minimised is (3/2 , 3/2)