# solution

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
D1 1 D2 2 D3 3 D4 4
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)

D1

2,000

D2

5,000

D3

3,000

D4

2,000

B) 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.

C)

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 Danvill:

Orders from/for Dummy Node:

Orders for D1:

Orders for D2:

Orders for D3:

Orders for D4:

xij = 0 for all i, j.