$2.$ Logical logistics $(50$ pts$)$

As the Head of Logistics for CHOAM Ltd$.($a company that sells electric motors$),$ you want to or$-$ ganize the shipment of your semi$-$ready product from three factory locations $($A$,$ B$,$ C$)$ to three destination locations $($X$,$ Y$,$ Z$)$ where your customers reside. The final production step $($i$.$e$.,$ packag$-$ ing and sealing$)$ before shipment to customers must take place at either of two distribution centers $($U$,$ V$).$ In other words, $($i$)$ from A$,$ B$,$ or C$,$ you must first ship to U or V$,$ and $($ii$)$ only from U or V can you then ship to X$,$ Y$,$ or Z$.$ The following tables illustrate the per motor shipment costs:

Cost per motor shipped $($$$)$ A

B

C

Cost per motor shipped $($$$)$ U

V

X

U V

$455041543463$

Y Z

$787267357081$

Factories A$,$ B$,$ and C each have capacities of $300,200,$ and $500$ motors respectively. Moreover, the respective demands at locations X$,$ Y$,$ and Z are $550,200,$ and $250$ motors.

Formulate an optimization problem to minimize your total shipment cost using the following de$-$ cision variables:

$2$

NAU $=$ Number of motors shipped from A to U NAV $=$ Number of motors shipped from A to V NBU $=$ Number of motors shipped from B to U NBV $=$ Number of motors shipped from B to V NCU $=$ Number of motors shipped from C to U NCV $=$ Number of motors shipped from C to V

NUX $=$ Number of motors shipped from U to X NUY $=$ Number of motors shipped from U to Y NUZ $=$ Number of motors shipped from U to Z NV X $=$ Number of motors shipped from V to X NVY $=$ Number of motors shipped from V to Y NV Z $=$ Number of motors shipped from V to Z

$$ You do NOT need to build an Excel model to actually solve this optimization problem. Only the formulation would suffice in this case. However, your model must be clear i$.$e$.,$ you must clearly specify your objective function and constraints.

$$ HINT: In addition to the regular supply and demand constraints, you$$ll require a flow bal$-$ ance constraint for each distribution center. A flow balance constraint says that the total shipment into a distribution center must equal the total shipment out of that distribution center.