ANRS LEADERSHIP ACADEMY

SCHOOL OF LEADERSHIP AND POLITICAL ECONOMY

DEPARTMENT OF PROJECT MANAGEMENT

Group assignment two

Course title; Quantitative Methods for Project Management (PM6312)

Group member’s Name ID. No.

1. Endale Alebachew ----------AMI140178ps

2. Honelign Kasie-------------- AMI140185ps

3. Aderajew Fentabil -------- AMI140167ps

4. Chanyalew Aychew ------ AMI140170ps

5. Hailu Tamir -------------- AMI140184ps

6. Yimen Aytenew ---------- AM 140188ps

Submmited To:- Amare Mabrie.(Assistant Professor)

August/ 2023

Bahir Dar, Ethiopia

Q1. Assume ABC Company wants to assign four machines for the appropriate five persons to
maximize his profits and he requested you to do this assignment. Based on the given determine
the optimal allocation of the machines and the maximum possible profit for the company.

A B C D E
Machines 1 65 102 105 98 121
2 87 57 78 103 85
3 95 63 85 106 74
4 58 76 131 82 67

Answer
Step 1: Row maximums have been identified. All values are to be subtracted from These
values

A B C D E Row maxim
Machines 1 65 102 105 98 121 121
2 87 57 78 103 85 103
3 95 63 85 106 74 106
4 58 76 131 82 67 131

Reduced time matrix is obtained thus ( revised time matrix I)

A B C D E
Machines 1 56 19 16 23 0
2 16 46 25 0 18
3 11 43 21 0 32
4 73 55 0 49 64
Column minima 11 19 0 0 0
Step 2: Now we obtain minimum values of each column from the above reduced time
matrix I and subtract these from each respective column elements to achieve revised
matrix II

A B C D E
Machines 1 45 0 16 23 0
2 5 27 25 0 18
3 0 24 21 0 32
4 62 36 0 49 64

Step 3: Now drawing minimum number of horizontal/ vertical lines to cover all zero
elements we get matrix III below
A B C D E
Machines 1 45 0 16 23 0
2 5 27 25 0 18
3 0 24 21 0 32
4 62 36 0 49 64
If the number of lines drawn is 4=n, but the number of rows and columns are not equal
1
the solution is not optimal. Thus we add an artificial rows to balance rows and columns
we obtain a new table as follows;

A B C D E
Machines 1 45 0 16 23 0
2 5 27 25 0 18
3 0 24 21 0 32
4 62 36 0 49 64
5 0 0 0 0 0

Since, the number of lines drawn is 5=n, the solution is optimal, so we proceed step 6

A B C D E
1 45 0 16 23 0
2 5 27 25 0 18
Machines 3 0 24 21 0 32
4 62 36 0 49 64
5 0 0 0 0 0

Step 6: making assignments on zero elements , we obtain

A B C D E
Machines 1 45 0 16 23 0
2 5 27 25 0 18
3 0 24 21 0 32
4 62 36 0 49 64
5 0 0 0 0 0

Hence, person have been assigned on zero elements 3A,1B,4C,2D,5E

Total machine =95+102+131+103+0 = 431 maximum possible profit (optimal).

2
 Q2. Consider that we have two players with four alternative stages and payoff matrix of
the game is given as follows. . Based on the given determine the equilibrium of the game
for the case of
A. dominate strategy equilibrium

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

In question A, we would expect an equilibrium outcome in which player1 plays A, receiving an

equilibrium pay off of 7, and Player 2 plays F, receiving an equilibrium pay off of 4. Therefore,
(A, F) is the DSE (7, 4) is DSE outcome or DSE payoff.

B. Nash equilibrium

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

Steps 1 and 2 if Player 2 chooses E, what is Player 1’s best choice?

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

Step 3: Underline best payout, given the choice of the player 2 Choose A, since 6> 5 and 2
underline 6

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

3
Step 4; if Player2 chooses F, what is Player 1’s best choice? Underline best payout, given the
choice of the player 2 Choose A, since 7> 6, 3 and 2 underline 7.

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

If Player 2 chooses G, what is Player 1’s best choice? Underline best payout, given the choice of
the player 2 Choose A, since 4> 3 and 0 underline 4

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

If Player 2 chooses H, what is Player 1’s best choice? Underline best payout, given the choice of
the player 2 Choose A, since 6> 5 and 4 underline 6

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

Step 5; if Player 1 chooses A, what is Player 2’s best choice? Underline best payout, given the
choice of the player2 Choose F, since 4 > 3, 1 and 0 underline 4

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0

B 5,4 6,3 0,2 5,1

C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

4
If Player 1 chooses B, what is Player 2’s best choice? Underline best payout, given the choice of
the player2 Choose E, since 4 > 3, 2 and 1 underline 4

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0

B 5,4 6,3 0,2 5,1

C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

If Player 1 chooses C, what is Player 2’s best choice? Underline best payout, given the choice of
the player2 Choose F, since 2> 0 and 1 underline 2

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C
5,0 3,2 3,1 4,0

D 2,0 2,3 3,3 6,1

If Player 1 chooses D, what is Player 2’s best choice? Underline best payout, given the choice of
the player2 Choose F and G, since 3> 0 and 1 underline 3

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D
2,0 2,3 3,3 6,1

Step 6; which box has underlines under both numbers? Player 1 chooses A, and Player 2 chooses
F this is the only Nash Equilibrium.

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1
5
 C. Eliminating the dominant strategy
Four by four games

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

For player 1, A and D dominates B and C we can eliminate the B and C players1 as a
dominated strategy.

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1

After B and C are eliminated, E and H will be dominated by F and G for the player2

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1
At this stage, the player1 dominant strategy is A, so we can eliminate the D strategy.

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1
Finally, player2 chooses F dominate G, yielding a unique outcome after the iterated elimination
of dominated strategies “G”, which is (A, F).

Player 2
Player 1 E F G H
A 6,3 7,4 4,1 5,0
B 5,4 6,3 0,2 5,1
C 5,0 3,2 3,1 4,0
D 2,0 2,3 3,3 6,1
The equilibrium of the game is (A, F) with a payoff (7, 4) for both players.