EMGT 501

Mid-Term Exam

Due Day: Oct 23- Midnight

1. 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 thatTask Task Description Must Precede Time A Buy the mozzarella cheese* 25 minutes B Slice the mozzarella A, C 15 minutes C Beat 2 eggs 10 minutes D Mix eggs and ricotta cheese C 13 minutes E Cut up onions and mushrooms 10 minutes F Cook the tomato sauce E, G 25 minutes G Boil large quantity of water 15 minutes H Boil the lasagna noodles G , K 10 minutes I Drain the lasagna noodles D, F,H 12 minutes J Assemble all the ingredients I, F, D, B 10 minutes K Preheat the oven 15 minutes L Bake the lasagna J, I, K 20 minutes

(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?

2. Develop a case study regarding how to use PERT/CPM for your work.

(a) 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.

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 ReturnAtlantic Oil 7.3 (%)Pacific Oil 10.3 (%)Midwest Steel 6.4 (%) Huber Steel 7.5 (%)Government Bond 4.5 (%)

The management has imposed the following investment guidelines:

(a) 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?

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)

5. Develop a case in which you apply DEA to your performance evaluation related to your job.

(a) Specify your case study

(b) Formulate a DEA model

(c) Solve the DEA formulation by LP

(d) Discuss its managerial implications

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.

Player 2 Strategy

1 2 3 4 1 3 0 2 1 2 2 -1 3 2 Player 1 3 3 5 0 3

7. (a)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.

EMGT 501

MID-TERM EXAM

ANSWER

Start

Finish

A EC G K

B FD H

I

L

S=(0, 12)F=(25, 37)

S=(0, 0)F=(0, 0)

S=(0, 17)F=(10, 27)

S=(0, 5)F=(10, 15)

S=(0, 0)F=(15, 15)

S=(0, 15)F=(15, 30)

S=(15, 30)F=(25, 40)

S=(15, 15)F=(40, 40)

S=(10, 27)F=(23, 40)

S=(25, 37)F=(40, 52)

S=(40, 40)F=(52, 52)

S=(52, 52)F=(62, 62)

S=(62, 62)F=(82, 82)

S=(82, 82)F=(82, 82)

2510 10

1515

10251315

10

12

20

1. (a)(b)(c)(d)

The critical pathS=(40, 40)F=(52, 52)

12I

Earlieststart time

Lateststart time

Earliestfinish time

Latestfinish time

Duration

J

G - F - I - J - L

TaskSlack

( LF – EF )On Critical

Path?A 12 NoB 12 NoC 17 NoD 17 NoE 5 NoF 0 YesG 0 YesH 15 NoI 0 YesJ 0 YesK 15 NoL 0 Yes

(e)

10 delay – 5 slack = 5 min delay.

The dinner will still be delayed for 2 minutes. (2 = 7 - 5)

(f)

.0,0,0,0,004.06.0025.025.0

000,50000,50000,100

s.t.045.0075.0064.0103.0073.0

Max

54321

21

543

43

21

54321

54321

xxxxxxx

xxxxx

xxxxxxx

xxxxx

bond Government into investment The :SteelHuber into investment The :

SteelMidwest into investment The :Oil Pacific into investment The :Oil Atlantic into investment The :

5

4

3

2

1

xxxxx

3.

000,504321 xxxx

Option: this is also acceptable.

Investment Amount ReturnAtlantic Oil $20000 $1,460Pacific Oil $30000 $3,090Midwest Steel 0 0Huber Steel $40000 $3,000Government bond $10000 $450

$100000 $8000

The expected return is $8000.

The portfolio recommendations-investments

4.

The efficiency scores

A 0.5838 F 0.2239

B 0.8233 G 0.4398

C 0.2073 H 0.1407

D 0.5733 I 0.3477

E 1

• Minimax Criterion:

Minimize y5

Subject to 3y1 + 0y2 + 2y3 + y4 y5

2y1 - y2 + 3y3 + 2y4 y5

3y1 + 5y2 + 0y3 + 3y4 y5

y1 + y2 + y3 + y4 = 1

y1 , y2 , y3 , y4 0

6. (a)

• Solving the problem gives:

y1 = 0 y4 = 0

y2 = 0.3333 y5 = 1.6667

y3 = 0.6667

Objective Function Z = 1.6667

6. (b)

• Maximin Criterion for Player 1:

Maximize x4

Subject to 3x1 + 2x2 + 3x3 x4

0x1 - x2 + 5x3 x4

2x1 + 3x2 + 0x3 x4

x1 + 2x2 + 3x3 x4

x1 + x2 + x3 = 1

x1 , x2 , x3 , x4 0

7. (a)

• We can rewrite the equations as:

Minimize x4

Subject to 3x1 + 2x2 + 3x3 + x4 0

0x1 - x2 + 5x3 + x4 0

2x1 + 3x2 + 0x3 + x4 0

x1 + 2x2 + 3x3 + x4 0

x1 + x2 + x3 = 1

x1 , x2 , x3 , x4 0

• If we take the dual of this problem:

Maximize y5

Subject to 3y1 + 0y2 + 2y3 + y4 + y5 0

2y1 - y2 + 3y3 + 2y4 + y5 0

3y1 + 5y2 + 0y3 + 3y4 + y5 0

y1 + y2 + y3 + y4 = 1

y1 , y2 , y3 , y4 , y5 0

• We can rewrite the dual problem as:

Minimize y5

Subject to 3y1 + 0y2 + 2y3 + y4 - y5 0

2y1 - y2 + 3y3 + 2y4 - y5 0

3y1 + 5y2 + 0y3 + 3y4 - y5 0

y1 + y2 + y3 + y4 = 1

y1 , y2 , y3 , y4 , y5 0

This is equal to minimax strategy of Player II. If we solve it, we get the same objective function (Z = 1.6667). This proves that maximin strategy of Player I is equal to minimax strategy of Player II.

• 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 x4 Redundant Constraint 0x1 - x2 + 5x3 = x4 Binding Constraint 2x1 + 3x2 + 0x3 = x4 Binding Constraint x1 + 2x2 + 3x3 x4 Redundant Constraint

x1 + x2 + x3 = 1 Binding Constraint

x1 , x2 , x3 , x4 0

Since solution of minimax strategy of Player II gives:

y1 = 0 y4 = 0 y2 = 0.3333 y5 = 1.6667 y3 = 0.6667

This allows us to ignore the redundant strategies and only concentrate on the binding ones (no slack).

7. (b)

• 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 y5 Redundant Constraint

2y1 - y2 + 3y3 + 2y4 = y5 Binding Constraint

3y1 + 5y2 + 0y3 + 3y4 = y5 Binding Constraint

y1 + y2 + y3 + y4 = 1 Binding Constraint

y1 , y2 , y3 , y4 0

Since solution of maximin strategy of Player I gives:

x1 = 0 x3 = 0.4444

x2 = 0.5556 x4 = 1.6667 This allows us to ignore the redundant strategies and only concentrate on

the binding ones (no slack).