Klein Chemicals, Inc., produces a special oil$-$based material that is currently in short supply. Four of Klein's customers have already placed orders that together exceed the combined capacity of Klein's two plants. Klein's management faces the problem of deciding how many units it should supply to each customer. Because the four customers are in different industries, different prices can be charged because of the various industry pricing structures. However, slightly different production costs at the two plants and varying transportation costs between the plants and customers make a "sell to the highest bidder" strategy unacceptable. After considering price, production costs, and transportation costs, Klein established the following profit per unit for each plant$-$customer alternative.

Plant Customer

D$11$ D$22$ D$33$ D$44$

Clifton Springs $1$ $$32$ $$34$ $$32$ $$40$

Danville $2$ $$34$ $$30$ $$28$ $$38$

The plant capacities and customer orders are as follows.

Plant Capacity $($units$)$

Clifton Springs $5,000$

Danville $3,000$

Distributor Orders $($units$)$

D$1$

$2,000$

D$2$

$5,000$

D$3$

$3,000$

D$4$

$2,000$

$($a$)$

Develop a network model and a linear programming formulation of this problem.

$($i$)$

network model $($Submit a file with a maximum size of $1$ MB$.)$

$($ii$)$

linear programming formulation

Let xij $=$ number of units i shipped to client j$,$ using the indices from the given table. $($It may be necessary to combine plants or distributors in a single node in order to solve this problem. Use index number $5$ for this type of node. Enter "DNE" in any unused answer blanks.$)$

Max

s$.$t$.$

Orders from Clifton Springs

Orders from Danville

Orders from$/$for Dummy Node

Orders for D$1$

Orders for D$2$

Orders for D$3$

Orders for D$4$

xij $>=0$ for all i$,$ j$.$

$($b$)$

How many units should each plant produce for each customer to maximize profits?

Optimal Solution

Units Cost

Clifton Springs$$D$1$

$

Clifton Springs$$D$2$

$

Clifton Springs$$D$3$

$

Clifton Springs$$D$4$

$

Danville$$D$1$

$

Danville$$D$2$

$

Danville$$D$3$

$

Danville$$D$4$

$

Total Cost $

$($c$)$

Which customer demands will not be met?

Distributor $1$ will have a shortfall of units.

Distributor $2$ will have a shortfall of units.

Distributor $3$ will have a shortfall of units.

Distributor $4$ will have a shortfall of units.