$10\mathrm{.}29$ Six cells $({20}^{\text{'}}{20}^{\text{'}})$ are to be located in a $23($rectangular$)$ facility, as shown in the figure below. The material flow matrix is given in a separate table. The material handling cost is directly proportional to the distance traveled $($from centroid to centroid$).$ Assume rectilinear distance.

a$.$ Determine a lower bound for the material handling cost.

b$.$ Given an initial cell layout where product $1$ is located in cell $A,$ product $2$ is located in cell B$,$ etc., calculate the total material handling cost.

c$.$ Use the improvement procedure to determine a better layout and calculate its total material handling cost.

$\backslash$table$[[A,C,E$