$[2$:$49$ PM$,3/25/2024]$ Aarohi: A certain military deployment requires that supplies be delivered to $4$ locations. These deliveries come from $1$ of $3$ ports. Logistics planners wish to deliver the supplies in an efficient manner, in this case by minimizing total ton$-$miles. The port$-$destination

data, along with destination demand, are provided in the following table:

Port

Destination

D$1/$

D$2$

D$3$

D$4$

A

$751$

$88$

$103$

$56$

$105$

$76$

$101$

$85$

$43$

$80$

$95$

$62$

Demand

$500$

$600$

$450$

$700$ The ports are supplied by $1$ of $2$ supply ships. These ships travel to a particular port where their supplies are off$-$loaded and shipped to the requesting destinations. Ship S$1$ carries $1,200$ tons of supplies, while Ship $$2$ carries $1,120$ tons of supplies. These

ships can only go to a single port, and each port can only accommodate $1$ ship. Assume the costs for a ship to travel to a port are not part of the objective function.

The following is the ILP formulation and a spreadsheet model for the problem.

Let

Yij be $1$ if ship i travels to port j$,$ for i $=$ S$1,$ S$2$ and j $=$ A$,$ B$,$ C

Xik be the tons shipped from port j $=$ A$,$ B$,$ C to Destination k $=$ D$1,$ D$2,$

D$3,$ D$4$

$[2$:$52$ PM$,3/25/2024]$ Aarohi: MIN:

Subject to:

$75$ X$11+88$ X$12+103$ X$13+56$ X$14+105$ X$21+76$ X$22+101$ X$23+$

$85$ X$24$

$+43$ X$31+80$ X$32+95$ X$33+62$ X$34$

Y$11+$ Y$21<=1$

Y$12+$ Y$22<=1|$

Y$13+$ Y$23<=1|$

Y$11+$ Y$12+$ Y$13=1|$

Y$21+$ Y$22+$ Y$23=1|$ X$14+$ X$15+$ X$16+$ X$17<=1,200$ Y$11+1,120$ Y$21|$ X$24+$ X$25+$ X$26+$ X$27<=1,200$ Y$12+1,120$ Y$22$ X$34+$ X$35+$ X$36+$ X$37<=1,200$ Y$13+1,120$ Y$23|$

X$14+$ X$24+$ X$34>=500$

X$15+$ X$25+$ X$35>=600$

X$16+$ X$26+$ X$36>=450$

X$17+$ X$27+$ X$37>=700$

Yj$-$Xk$>=0$

Destinations

Ports

$01$

D$2$

D$3$

D$4$

$75$

$188$

$103$

$56$

$105$

$76$

$101|$

$85$

$43$

$180$

$195$

$62$

Ship Loads

$1,200$

$1,120$

D$1$

D$2$

$03$

D$4$

Shipped

Supply

S$1$

S$2$

$0$

$0$

$1,120$

$19$

$0$

$1,200$

$11$

Received

$12$

Demand

$500$

$600$

$450$

$700$

$13$

$14$

Cost