Statistics Exam
Question 1 20 points
1. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints:
a. 15X + 7Y ? 75
b. -5X + 15Y ? 150
c. 3Y ? 21
d. -4X ? 60
e. 5X + 3Y ? 0
Question 2 20 points
2. Consider the following linear program:
Maximize 32X + 15Y
s.t.
Constraint 1 5X + 20Y ? 120
Constraint 2 8X + 15Y ? 150
Constraint 3 3X – 6Y ? 30
X,Y ? 0
a. Show the graph and the feasible region
b. Shade the feasible region
c. Identify the optimal solution point on your graph
d. What are the values of X and Y at the optimal solution?
X Y
e. What is the optimal value of the objective function?
Question 3 20 points
3. Consider the following linear program. The cost per Dinner plate is $0.5 and the cost per Soup plate is $0.25:
Miniminze 18D + 12S
s.t.
Constraint 1 D ? 60
Constraint 2 S ? 36
Constraint 3 0.25D + 0.75S ? 70
Constraint 4 1.5D – 0.5S ? 120
D, S ? 0
Let D = Dinner plate
Let S = Soup plate
a. Insert a graph to show each constraint and the feasible region.
b. Shade the feasible region.
c. Identify the optimal solution point on the graph.
d. What is the optimal value of the objective function?
e. What is the optimal value of the objective function (dollar value)?
f. What is the optimal value for Dinner plates?
D
g. What is the optimal values for Soup plates?
S
h. How much would cost to make the optimal values of Dinner plates (dollar value)?
i. How much would cost to make the optimal values of Soup plates (dollar value)?
j. Which constraints are binding?
Question 4 20 points
4. By using solver, answer the following questions based on the the following LP model:
Maximize 70A + 50B
s.t.
Constraint 1 12A + 13B ? 1000
Constraint 2 15B ? 900
Constraint 3 12A – 20B ? 800
Constraint 4 15A ? 850
A, B ? 0
a. Insert the sensitivity and limit reports.
b. What is the optimal objective value of the objective function?
c. What are the optimal values of the two decision variables?
A B
d. What are the ranges optimality?
e. What are the range of feasibility for each constraint?
f. What are the dual values for each constraint?
g. Would it be beneficial to increase the maximum value of constraint 1 to 1200? Explain.
h. What would happen to constraint 2 if we decrease the final value to 400?
i. What would happen if we would change the RHS of constraint 3 to 650? Explain.
j. What would happen if we would change the RHS of constratint 4 to 860? Explain.
Question 5 20 points
5. The solver solution of the problem is shown on the side:
a. What is the optimal solution?
b. What is the value of objective function?
c. Which constraints are binding? Interpret each.
d. Specifiy the objective function ranges.
e. Identify each of the right-hand-side ranges.
f. Suppose that the variable Pasta decreases from its current value to 3. How does the optimal solution change, if at all?
g. Suppose that variable Salad increases from its current value to 10. How does the optimal solution change, if at all?
h. Would be beneficial to increase the maximum amount shown in constraint 2 to 20? Why or why not, and what would happen to the objective function?
*Shadow Price= Dual Value
i. Would be beneficial to increase the amount shown in constraint 1 to 30? Why or why not, and what would happen to optimal solution?
j. If the available value of constraint 3 is increased by 50, will the dual value of the constraint change? Explain.
Extra Credit – 10 points
Car Weight (in lbs) Car MPG
3250 28
3675 23
3840 19
3935 20
2140 43
4010 22
2565 34
3450 22
2900 28
3345 25
3545 24
3050 31
2540 34
2410 36
2865 30
3810 22
The values above on column A list the weight in pounds of 16 different cars made in 2020. The values in Column B are the corresponding gas mileages for each car.
Insert a linear programming graph with the regression model and R-squared value.
a. What is the regression model/equation?
b. What is the R-squared value?
c. What does the graph tell us about the general relationship between these two variables?
d. I am planning to purchase a relatively new (2020) car but I am concerned about getting good gas mileage.
Using this regression equation, tell me friend the expected gas mileage at 2750 pounds.
e. The value found on the question above sounds high. What would you tell me based on all the information found to justify your response?