## Create a linear program

1. You are the recently hired Chief Operations Officer at ABC Inc, a regional firm which produces specialized circuit boards used in the production of various makes and models of automobiles. The company currently owns three production plants, one much newer than the other two. Once the circuit boards are produced, the firm ships them to one of four warehouses throughout the state where they are placed in inventory until ordered by the end-user. Each circuit board sells for \$150.

Being the good COO that you are, you are always looking for ways to minimize costs and increase profitability. You suspect that there is much room for improvement in this regard. Specifically, you have reason to believe that production and stocking levels of each plant and warehouse could be modified to yield the desired results.

The details of this managerial challenge are as follows:

Production Costs – Plant A — \$100 per circuit board

Plant B — \$120 per circuit board

Plant C — \$90 per circuit board

Plant Capacity – Plant A ? 1,200 circuit boards

Plant B ? 1,200 circuit boards

Plant C ? 500 circuit boards

Demand Forecast – Warehouse #1 ? 700 circuit boards

Warehouse #2 ? 400 circuit boards

Warehouse #3 ? 600 circuit boards

Warehouse #4 ? 500 circuit boards

Shipping Costs

Warehouse #1

Warehouse #2

Warehouse #3

Warehouse #4

Plant A

\$10

\$15

\$40

\$30

Plant B

\$50

\$20

\$25

\$20

Plant C

\$25

\$45

\$30

\$22

Create a linear program that shows exactly how many circuit boards should be produced at each plant and then shipped to each warehouse in order to maximize resulting profits. Your completed model should show the optimal values of all 12 decision variables AS WELL AS the optimal value of the objective function.

2. (20 pts.) Given the following LP model,

Minimize (costs) Z = 4X1 + 8X2

Subject to Fiber 5X1 + 8X2 > 40

Protein 6X1 + 4X2 > 24

X1, X2 > 0

a.) What is the optimal value of the objective function?

b.) What are the optimal values of the two decision variables?

c.) Find the range of optimality for each objective function coefficient.

d.) How would a decrease of \$1 in the X1 coefficient of the objective function affect the optimal values of the decision variables?

e.) How would a decrease of \$1 in the X1 coefficient of the objective function affect the optimal value of the objective function?

f.) What is the dual value (AKA ??shadow price?) for the RHS of the protein constraint?

g.) What is the range of feasibility of the dual value for the RHS of the protein constraint?

h.) What impact on total cost would a decrease of 2 units in the RHS of the protein constraint have? Please explain the rationale for your answer.

3. (20 pts.) The manager of FYZ Incorporated has been presented with the following LP model:

Minimize (costs) Z = 30A + 45B

Subject to 5A + 2B > 100

4A + 8B > 240

B > 20

A and B > 0

She would like your assistance with several questions below.

a.) What is the proper name of the last constraint shown in the model?

b.) What is the optimal value of the objective function?

c.) What are the optimal values of the two decision variables?

d.) If the cost of B could be reduced to \$42 per unit, how many units of B would be optimal?

e.) If the cost of B could be reduced to \$42 per unit, what would the minimum total cost be?

f.) What is the dual value (AKA–?shadow price?) for the RHS value of the first constraint?

g.) What is the range of feasibility for the RHS value of the first constraint?

h.) By what amount would the total cost change, and in what direction, if the RHS value of the first constraint was changed to 110? Please explain the rationale for your answer.