Question

Find the optimal solution for the mini-sum location model assuming: (a) squared Euclidean distances; (b) Euclidean distances. For each case find the optimal location of a new machine, assuming existing machines located at (4, 3), (7, 5), (11, 8), and (13, 4) with weights equal to 1/3, 1/6, 1/3, and 1/6, respectively?.

Answer 1

Solution

Given

Machine Number Location Weights

Machine 1 (4,3) 1/3

Machine 2 (7,5) 1/6

Machine 3 (11,8) 1/3

Machine 4 (13,4) 1/6

Now arrange the machines in the increasing order of their x-coordinates

Machine Number ai (x-coordinates) wi Sigma wi Machine Number bi wi sigma wi

M1 4 1/3 1/3 M1 3 1/3 1/3

M2 7 1/6 1/2 M4 4 1/6 1/2

M3 11 1/3 5/6 M2 5 1/6 2/3 M4 13 1/6 1 M3 8 1/3 1

x-coordinate for the new location = sigma wi/2 = 1/2 y-coordinate for the new location = sigma wi/2 =1/2

min

Now using the above formula

a) Squared Euclidean distances

1/3 x (4- 1/2)^2 + 1/2 x (7-1/2)^2 + 1/3 x (11-1/2)^2 + 1/6 x (13-1/2)^2

=2112/24 = 88

88 is the new x-coordinate

for the y-coordinate

1/3 x (3-1/2)^2 + 1/2 x ( 4-1/2)^2 + 2/3 x (5-1/2)^2 + 1/3 x (8-1/2) ^2

= 971/24 = 40.46

So, the new location is (88, 40.46)

b) Euclidean distance

1/3 x sqrt((4- 1/2)^2) + 1/2 x sqrt((7-1/2)^2 )+ 1/3 x sqrt ((11-1/2)^2) + 1/6 x sqrt((13-1/2)^2)

=10

1/3 x sqrt((3-1/2)^2) + 1/2 x Sqrt(( 4-1/2)^2) + 2/3 x sqrt((5-1/2)^2) + 1/3 x sqrt((8-1/2) ^2)

=8.08

So, the new location coordinates are (10, 8.08)