Question 10.1: Solving a single-facility, rectilinear minisum location prob...

To illustrate the solution procedure, consider the problem of locating a new general purpose machine tool in a maintenance department. Five machines currently located in the department have the following coordinate locations: p_{1} =(1, 1), p_{2} = (6, 2), p_{3} = (2, 8), p_{4} = (3, 6), and p_{5} = (8, 4). The cost per unit distance traveled between the new machine and each existing machine is the same. The number of trips per day between the new machine and existing machines 1, . . ., 5 are 10, 20, 25, 20, and 25, respectively.

Table 10.1 Solving for the Optimum x-Coordinate

Table 10.2 Solving for the Optimum y-Coordinate

Ordering the x-coordinates of the existing facilities gives the sequence 1, 2, 3, 6, and 8. The corresponding weights are 10, 25, 20, 20, and 25. The sum of the weights is 100. As shown in Table 10.1, the partial sum of the ordered sequence of weights first equals or exceeds one-half the total (50) for i = 4; hence, x* = a_{4} = 3.
Ordering the y-coordinates of the existing facilities gives the sequence 1, 2, 4, 6, and 8.
The corresponding weights are 10, 20, 25, 20, and 25. The sum of the weights is 100. As shown in Table 10.2, the partial sum of the ordered sequence of weights first equals or exceeds one-half the total (50) for i = 4; hence, y* = b_{5} =4.
As shown in Figure 10.1, the optimum location for the new machine tool is X *
(3,4)^{2} . Rectilinear paths between the new facility and each existing facility are indicated by dashed lines in Figure 10.1.

The total weighted distance resulting from the location X (3, 4) is

=50+100+125+40+125=440

If the partial sum equals one-half the sum of all weights, then the optimum solution includes all points between the coordinate where the equality occurred and the next greater coordinate.