$|$ Consider the following layout problem for a facility with $12$ departments. Each department has similar area so for modeling purposes assume they have the same area and the building is configured such that the departments fit in a $3$ by $4$ grid as show below. Assume that the initial layout is for the department order $1-12$ as shown.

The facility's seven highest volume parts are shown below with their part routings and daily transport volumes in units of pallets.

Write a computer code to utilize a pairwise interchange algorithm to find the best possible layout. Assume

$1.$ The objective is to minimize flow times cost times distance.

$2.$ The cost per unit distance is one.

$3.$ Distance is measured using the rectilinear distance metric.

$4.$ Distance is measured assuming material moves from the centroid of one department to the centroid of another department.

$5.$ Your algorithm operates in a greedy manner, that is it considers all possible exchanges and takes the best one.

$6.$ Use the initial layout shown in the diagram above with departments $1-4$ in row $1,$ etc.

Submit your final layout and briefly discuss why you believe your ordering of departments is logical$,$ your final objective function value, and a paper copy of your computer code.