$(20$ pts$)$ Within an assembly plant, the movement of materials between different departments

is facilitated by a closed$-$loop conveyor system that operates in a single direction. The

provided diagram illustrates the layout configuration for the modular.

The facility comprises three assembly modules $($A$,$ B$,$ and C$)$ of equal size, an administrative

module $($D$),$ and a warehouse module $($E$).$ The diagram also indicates the designated points for

pick$-$up and delivery $($P$/$D$)$ within each module. It should be noted that the administrative and

warehouse departments are fixed and cannot be relocated, while the assembly areas A$,$ B$,$ and

$C$ have the flexibility to be moved. The given information includes the distances between $\frac{P}{D}$

points and the number of pallet loads transferred between departments.

By employing the pairwise exchange method, calculate alternative positions for assembly

modules $A,B,$ and $C$ in order to minimize the total cost resulting from multiplying pallet flows

by conveyor travel distances.