1. EMGT 501
Mid-Term Exam
Due Day: Oct Midnight

2. PERT/CPM: You and several friends are about to prepare a lasagna dinner. The tasks to be performed, their immediate predecessors, and their estimated durations are as follows:
Tasks that
Task Task Description Must Precede Time
A Buy the mozzarella cheese* minutes
B Slice the mozzarella A, C minutes
C Beat 2 eggs minutes
D Mix eggs and ricotta cheese C minutes
E Cut up onions and mushrooms minutes
F Cook the tomato sauce E, G minutes
G Boil large quantity of water minutes
H Boil the lasagna noodles G , K minutes
I Drain the lasagna noodles D, F,H minutes
J Assemble all the ingredients I, F, D, B minutes
K Preheat the oven minutes
L Bake the lasagna J, I, K minutes

3. (a) Construct the project network for preparing this dinner.
(b) Which of these paths is a critical path?
(c) Find the earliest start time and earliest finish time for each
activity.
(d) Find the latest start time and latest finish time for each activity.
(e) Find the slack for each activity. Which of the paths is a
critical path?
(f) Because of a phone call, you were interrupted for 10 minutes when you should have been cutting the onions and mushrooms. By how much will the dinner be delayed? If you use your food processor, which reduces the cutting time from 10 to 7 minutes, will the dinner still be delayed?

4. 2. Develop a case study regarding how to use PERT/CPM for your work.
Describe your case, focusing upon your managerial issue.
(b) Solve your case study.
(c) Make your recommendation based upon your solution.
Requirement: (a) Optimistic, Most Likely, Pessimistic Times should be used in your problem solving. (b) Include the probability estimation of the project completion time in your case study.

5. 3. Consider a decisional case of a Mutual Funds, Inc, located
in Socorro (NM). The firm has just obtained $100,000 by
converting industrial bonds to cash. Peter Anselmo, a chief
financial executive, is now looking for other investment
opportunities for funds. Peter has identified five investment
opportunities and projected these annual rates of return.
The investments and rates of returns are as follows:
Investment Projected Rate of Return
Atlantic Oil (%)
Pacific Oil (%)
Midwest Steel (%)
Huber Steel (%)
Government Bond (%)

6. The management has imposed the following investment
guidelines:
Both oil and steel industries should not receive more
than $50,000.
(b) Government bond should be at least 25% of the steel industry investment.
(c) The investment in Pacific Oil cannot be more than 60% of the total oil industry investment.
What portfolio recommendations-investments and
amounts-should be made for the available $100,000?

7. 4. Apply DEA to the following data set in order to measure their efficiency scores of the following eight firms:
Firm Inputs (x1, x2) Outputs (y1,y2)
A (15, 23) (20, 53)
B (45, 10) (39, 23)
C (60, 55) (27, 45)
D (10, 32) (43, 43)
E (10, 19) (90, 75)
F (27, 43) (34, 38)
G (43, 24) (50, 33)
H (27, 33) (22, 13)
I (21, 34) (56, 21)

8. 5. Develop a case in which you apply DEA to your performance evaluation related to your job.
Specify your case study
(b) Formulate a DEA model
(c) Solve the DEA formulation by LP
(d) Discuss its managerial implications

9. 6.Consider the game having the following payoff table.
(a) Formulate the problem to find an optimal mixed strategy according to the minimax criterion as a linear programming problem.
(b) Use the simplex method to find these optimal mixed strategies.

10. 7.
Show that the maximin strategy of Player I is equivalent to the mimimax strategy of Player II.
(b) Using the complementary slackness condition, discuss the relationship between the two strategies from a practical perspective.

11. EMGT 501
MID-TERM EXAM
ANSWER

12. 1. (a)(b)(c)(d) Start 25 15 15 A C 10 E 10 G K 15 13 25 10 B D F H I12
S=(40, 40)
F=(52, 52) J 10
S=(52, 52)
F=(62, 62)
Earliest
start time
Latest
start time
S=(62, 62)
F=(82, 82)
Duration L 20 I 12
S=(40, 40)
F=(52, 52)
The critical path
Finish
S=(82, 82)
F=(82, 82)
Earliest
finish time
Latest
finish time
G - F - I - J - L

13. (e)

14. (f)
10 delay – 5 slack = 5 min delay.
The dinner will still be delayed for 2 minutes. (2 = 7 - 5)

15. Option: this is also acceptable.3.
Option: this is also acceptable.

16. The portfolio recommendations-investments
The expected return is $8000.

17. 4.
The efficiency scores

18. 6. (a) Minimax Criterion: Minimize y5
Subject to 3y1 + 0y2 + 2y3 + y y5
2y y2 + 3y3 + 2y4 y5
3y1 + 5y2 + 0y3 + 3y4 y5
y1 + y2 + y3 + y4 = 1
y1 , y2 , y3 , y

19. 6. (b) Solving the problem gives: y1 = 0 y4 = 0
Objective Function Z =

20. 7. (a) Maximin Criterion for Player 1: Maximize x4
Subject to 3x1 + 2x2 + 3x3 x4
0x x2 + 5x3 x4
2x1 + 3x2 + 0x3 x4
x1 + 2x2 + 3x3 x4
x1 + x2 + x3 = 1
x1 , x2 , x3 , x4 0

21. We can rewrite the equations as:
Minimize x4
Subject to 3x1 + 2x2 + 3x3 + x4 0
0x x2 + 5x3 + x4 0
2x1 + 3x2 + 0x3 + x4 0
x1 + 2x2 + 3x3 + x
x1 + x2 + x3 = 1
x1 , x2 , x3 , x4 0

22. If we take the dual of this problem:
Maximize y5
Subject to 3y1 + 0y2 + 2y3 + y4 + y5 0
2y y2 + 3y3 + 2y4 + y5 0
3y1 + 5y2 + 0y3 + 3y4 + y5 0
y1 + y2 + y3 + y = 1
y1 , y2 , y3 , y4 , y

23. We can rewrite the dual problem as:
Minimize y5
Subject to y1 + 0y2 + 2y3 + y4 - y
2y y2 + 3y3 + 2y4 - y5 0
3y1 + 5y2 + 0y3 + 3y4 - y5 0
y1 + y2 + y3 + y = 1
y1 , y2 , y3 , y4 , y
This is equal to minimax strategy of Player II. If we solve it, we get the same objective function (Z = ). This proves that maximin strategy of Player I is equal to minimax strategy of Player II.

24. 7. (b)
Using CSC and the solution of Minimax Strategy of Player II, we can comment on Maximin Strategy of Player I:
Maximize x4
Subject to 3x1 + 2x2 + 3x3 x Redundant Constraint
0x x2 + 5x3 = x Binding Constraint
2x1 + 3x2 + 0x3 = x Binding Constraint
x1 + 2x2 + 3x3 x Redundant Constraint
x1 + x2 + x3 = Binding Constraint
x1 , x2 , x3 , x
Since solution of minimax strategy of Player II gives:
y1 = 0 y4 = 0
y2 = y5 =
y3 =
This allows us to ignore the redundant strategies and only concentrate on the binding ones (no slack).

25. Similarly, using CSC and the solution of Maximin Strategy of Player I, we can comment on Minimax Strategy of Player II:
Minimize y5
Subject to 3y1 + 0y2 + 2y3 + y4 y Redundant Constraint
2y y2 + 3y3 + 2y4 = y Binding Constraint
3y1 + 5y2 + 0y3 + 3y4 = y Binding Constraint
y1 + y2 + y3 + y4 = Binding Constraint
y1 , y2 , y3 , y
Since solution of maximin strategy of Player I gives:
x1 = 0 x3 =
x2 = x4 =
This allows us to ignore the redundant strategies and only concentrate on the binding ones (no slack).