GBC has $3$ plants and $5$ warehouses.

Baked goods shipped from plants to warehouses in containers that

consists of a standard assortment of GBC$$s baked goods.

The # of containers available for shipment this week are:

$180$ Pittsburgh; $280$ Memphis; $150$ Omaha

This week$$s demand $($# of containers$)$:

$120$ Newark; $100$ Chicago; $160$ Atlanta; $80$ Dallas; $150-$ LA

Shipping costs per container are given in the table below

Shipping costs

$($hundred $$)$

Warehouses

Newark Chicago Atlanta Dallas LA

Plants

Pittsburgh $4651219$

Memphis $1048514$

Omaha $1393610$

The following model has been developed to determine the minimum cost shipping plan

Min $4$ X$11+6$ X$12+5$ X$13+12$ X$14+19$ X$15+$

$10$ X$21+4$ X$22+8$ X$23+5$ X$24+14$ X$25+$

$13$ X$31+9$ X$32+3$ X$33+6$ X$34+10$ X$35$

Subject to:

Supply

Node S$1$: X$11+$ X$12+$ X$13+$ X$14+$ X$15=180$

Node S$2$: X$21+$ X$22+$ X$23+$ X$24+$ X$25=280$

Node S$3$: X$31+$ X$32+$ X$33+$ X$34+$ X$35=150$

Demand

Node D$1$: X$11+$ X$21+$ X$31=120$

Node D$2$: X$12+$ X$22+$ X$32=100$

Node D$3$: X$13+$ X$23+$ X$33=160$

Node D$4$: X$14+$ X$24+$ X$34=80$

Node D$5$: X$15+$ X$25+$ X$35=150$

Non$-$Negativity: Xij $>0$ for i$-1,2,3$ and j$=1,2,3,4,5$

Consider the following $3$ scenarios and specify how the LP model changes $($refer both to the network

structure$/$graph and the LP$)$

OIM $310$: GBC Page $2$

A$.$ Suppose that the supply at plant $1($Pittsburgh$)$ increased from $180$ to $200$ creating excess of total

supply over total demand. The following table provides the cost incurred at each plant for storing a

unit that is not shipped and needs to be stored in the warehouse.

Plant Storage cost per unit

$1$: Pittsburgh $$0\mathrm{.}15$

$2$: Memphis $$0\mathrm{.}20$

$3$: Omaha $$0\mathrm{.}10$

Modify the network and the associated LP model $($from page $1)$ to determine the shipping plan that will

have the minimum total costs for scenario A $($total costs include shipping costs & storage costs$)$