Six housing subdivisions within a city area are targeted for emergency service b

Question

Six housing subdivisions within a city area are targeted for emergency service by a centralized fire station. The centroid locations (in miles) and total value of the houses in the subdivisions are as follows:

a.Where should the new fire station be located such that the maximum rectilinear travel distance is minimized?

b.Where should the new fire station be located such that the sum of the rectilinear travel distance is minimized?

c.Construct a contour line passing through the point having coordinates (25, 20).

st 245346 055500 2 505550 52313 A B C D E F

Explanation / Answer

Rectilinear Facility Location:

Various objectives can be used

Minisum location problem

Minimax location problem

Following steps are followed to the maximum rectilinear travel distance is minimized:

Procedure: To obtain a minimax solution, let

c1 = minimum (ai + bi )

c2 = maximum (ai + bi )

c3 = minimum (-ai + bi )

c4 = maximum (-ai + bi )

c5 = max (c2 -c1 , c4 -c3 )

Optimum solution for the new facility location is on the line segment connecting the points

X1 * (x1 * , y1 * ) and Y2 * (x2 * , y2 * )

X1 * (x1 * , y1 * ) = 0.5(c1 -c3 , c1+c3+c5 )

Y2 * (x2 * , y2 * ) = 0.5(c2 -c4 , c2+c4 - c5 )

Max distance equals c5 /2

Following steps are followed to the sum of the rectilinear travel distance is minimized:

Procedure

1. Find x-coordinate:

Order the facilities based on the ascending order of their x-coordinates

Calculate partial sum of weights

Find the facility for which the partial sum first equals or exceeds one-half the total weight

The x-coordinate of the new facility will be the same as the x-coordinate of this facility

2. Find y-coordinate : Repeat the same for y-coordinate

The problem is solved below :

The total weighted distance between the new facility and its suppliers can be found as:

=Sum of Weights|X-A1|+Sum of Weights|Y-B1|

=200*|30-5|+200*|20-10|+400*|30-50|+400*|20-15|+500*|30-25|+500*|20-25|+300*|30-35|+300*|20-5|+400*|30-15|+400*|20-20|+600*|30-30|+600*|20-300|

Rank X Coordinate in Assending Order Weight Partial Sum of Weights 5 200 200 15 400 600 25 500 1100 30 300 1400 35 400 1800 50 600 2400 sum 2400 1200 Thus X Coordinate 30 half of sum of weights

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