Question

you can use rectilinear or Euclidean (direct linear) distance

We want to find a location for a new facility. There are existing machines 1, 2 and 3, which are located at the points (0,0), (4,6) and (10, 0), respectively. There are 1, 3 and 5 trips per day, respectively, between the machines and a new facility. Describe your logic to find the best location for a new facility.

Answer 1

This is a single facility minisum problem,

Inorder to solve these kind of problems, we have to consider the following rules

1. The x co-ordinate of the new facility will be that of an existing facility
2. the y- co-ordinate of the new facility will be that of an existing facility
3. The optimum x co-ordinate will be in a way that no more than half of the total weight will be to the left of x coordinate, and no more than half of the total weight will be to the right of x coordinate
4. The optimum y co-ordinate will be in a way that no more than half of the total weight will be to the left of y coordinate, and no more than half of the total weight will be to the right of y coordinate

Where "weight" means No. of trips with respect to this problem

Let us assume,

co ordinate of machine .

co. ordinate of machine

"weight"/ No. of trips taken per day

cumulative sum of weight

Considering rectilinear distance, we need to find the x and y values satisfying the objective function given by,

, where m is the no. of machines

To do this, following procedure can be followed

List all machines in ascending order of x co ordinate

Machine Sl No ()
1	0	1	1
2	4	3	4
3	10	5	9

Half of total sum of weight,

The closest x co.ordinate may be chosen given that

no more than half of cumulative weight can be to the left or right of x co ordinate. So, the x co ordinate of new facility is 4.

Applying the same process to the y co-ordinate,

List all machines in ascending order of y co ordinate

Machine Sl No ()
1	0	1	1
3	0	5	6
2	6	3	9

Half of total sum of weight,

The closest x co.ordinate may be chosen given that,

no more than half of cumulative weight can be to the left or right of y co ordinate. So the y co ordinate of new facility is 0.

Hence the ideal location for the new facility is (4,0)