Consider the problem of locating a new machine to an existing layout considering

Question

Consider the problem of locating a new machine to an existing layout considering the position of four machines. These machines are located at the following coordinates of a plane: (3,0), (0,-3), (2,1), and (1,4). Let the coordinates of the new machine be (x1,x2). Formulate the problem of finding an optimal location for the new machine as a linear program given that the objective is to minimize the sum of the distances from the new machine to the four existing machines. Use the "street" distance (rectilinear norm); for example, the distance from (x1,x2) to the first machine located at (3,0) is absolute (x1-3) + absolute (x2-0). In your formation, clearly define the variables, and state the objective function and constraints with proper justification.

Explanation / Answer

Objective function: z = Min (d1 + d2 + d3 + d4) = Min [ |x1-3|+|x2|+|x1|+|x2+3|+|x1-2|+|x2-1|+|x1-1|+|x2-4|]

subject to constraints:

|x1-1| + |x2-4| < 4 (because minimum distance of point (1,4) from all the other points is 4)

|x1-3| + |x2-0| < 2 (because minimum distance of point (3,0) from all other points is 2)

|x1-0| + |x2+3| < 6 (because minimum distance of point (0,-3) from all other points is 6)

|x1-2| + |x2-1| < 2 (because minimum distance of point (2,1) from all other points is 2)

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