Consider the problem of locating a new machine to an existin

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.

Solution

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)