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# Linear Programming

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1. Linear Programming Properties
Which of the following statements is not true?

a) An infeasible solution violates all constraints.
b) A feasible solution point does not have to lie on the boundary of the feasible solution.
c) A feasible solution satisfies all constraints.
d) An optimal solution satisfies all constraints.

2. Minimization Graphical Solution
Solve the following linear model graphically and select the set of extreme points that make up the possible feasible solutions.

a) (x1=12, x2=0, z=120), (x1=6, x2=5, x3=160), (x1=0, x2=8, z=160)

b) (x1=0, x2=12, z=240), (x1=6, x2=5, x3=160), (x1=20, x2=0, z=200)

c) (x1=0, x2=12, z=240), (x1=20/3, x2=16/3, x3=520/3), (x1=20, x2=0, z=200)

d) (x1=12, x2=0, z=120), (x1=20/3, x2=16/3, x3=520/3), (x1=0, x2=8, z=160)

3. Minimization Graph Surplus Variables
Based on the optimal solution from the previous problem, which of the following statements is true if s1 represents the slack from the first constraint and s2 represents the slack from the second constraint?

a) Constraint 1 has no slack
Constraint 2 does have slack

b) Constraint 1 does have slack
Constraint 2 has no slack

c) Both constraints 1 and 2 have no slack.

4. Maximization Feasible Solutions
Given the following maximization linear programming model, which of the possible solutions provided below is NOT feasible?

a) x1 = 0 and x2 = 120

b) x1 = 75 and x2 = 90

c) x1 = 90 and x2 = 75

d) x1 = 135 and x2 = 0

5. Maximization Graphical Solution
Graphically solve the linear programming model from the previous problem and determine the set of extreme points that make up the set of feasible solutions.

a) (x1=0, x2=120, z=240), (x1=90, x2=75, z=420), (x1=240, x2=0, z=720)

b) (x1=0, x2=120, z=240), (x1=90, x2=75, z=420), (x1=135, x2=0, z=405)

c) (x1=0, x2=225, z=450), (x1=90, x2=75, z=420), (x1=135, x2=0, z=405)

d) (x1=0, x2=225, z=450), (x1=90, x2=75, z=420), (x1=240, x2=0, z=720)

NOTE: The linear programming model and accompanying Excel sensitivity report in problem 6 are also to be used in problems 7, 8, and 9.

6. Excel Sensitivity Analysis 1
The following model was solved using Excel

Excel the produced the following sensitivity report.

What is the maximum profit that can be made on Product 1 WITHOUT affecting the optimal product mix?

a) \$36.71 b) \$50.00 c) \$61.53 d) \$65.31

7. Excel Sensitivity Analysis 2
Using the model and the sensitivity report from the previous problem, a manager is offered the opportunity to make a bulk purchase of 150 additional hours for Process 1 at a cost of \$1050. Which of the following should the manager do without impacting the current product mix?

a) Refuse because the purchase price per additional hour is greater than the shadow price.

b) Accept because the purchase price for each additional hour is less than the shadow price and this does not impact the current product mix?

c) Refuse because this would impact his current optimal product mix.

8. Excel Sensitivity Analysis 3
Using the model and sensitivity report from the previous two problems, how many pounds of Material A are left over from the optimal solution? NOTE: If you calculations are off by + or - 0.10 from any of these choices, then select the choice that is closest to your solution.

a) 0 b) 70.92 c) 111.45 d) 149.67

9. Excel Sensitivity Analysis 4
Use the model and sensitivity report from the previous three problems.

A manager elects to purchase 50 additional pounds of Material B at \$2.00 per pound, how much additional profit can be made from this purchase?

a) \$30 b) \$50 c) \$100 d) \$130

NOTE: The linear programming model and accompanying sensitivity report from QM for Windows are to be used for problems 10, 11, and 12.