Planning of Barus

1
Planning of Barus Holley Addition
Activity Duration, Days
Predecessors
215
Procurement
60
Site Preparation
SP
31
Pour Foundation
P, PF
15
Erect Steel
ES
40
Roof, Ext. Wall
ES
50
Fabricate Glass
REW
85
Int Walls Gl.
REW
55
Landscaping
280
Acquire Furn.
IWG, AF, FG
15
Install Furn.
2
Forward Pass Find ES(I)
ES(I) Earliest Start of Activities emanating
from node I
60
270
2
5
PF
L
REW
SP
IWG
P
ES
FG
9
3
4
1
6
7
370
215
230
280
355
0
AF
IF
8
355
3
Backward Pass Find LC(I)
LC(I) Latest Completion of Activities
terminating at Node I.
184
270
60
270
2
5
PF
L
REW
SP
IWG
P
ES
FG
9
3
4
1
6
7
370
215
230
280
355
0
215
370
230
355
0
355
AF
IF
8
355
355
4
Identify Critical Path
184
270
60
270
2
5
PF
L
REW
SP
IWG
P
ES
FG
9
3
4
1
6
7
370
215
230
355
280
0
370
215
230
355
355
0
AF
IF
355
355
5
Determine Slack Times
184
270
60
270
2
5
SP(124)
L (45)
PF(124)
REW
IWG
ES
P
P
FG(75)
9
3
4
1
6
7
370
215
230
355
280
0
370
215
230
355
355
0
AF(75)
IF
355
355
6
Two-Person, Zero-Sum Game The Campers
Column Player Carol (j)
Matrix of Payoffs to Row Player
(1) (2) (3) (4)
7 2 5 1 2 2 3 4 5 3 4 4 3 2 1 6
(1) (2) (3) (4)
Row Player Ray (i)
7
Two-Person, Zero-Sum Game The Campers
Column Player Carol (j)
Matrix of Payoffs to Row Player
Row Minima
(1) (2) (3) (4)
7 2 5 1 2 2 3 4 5 3 4 4 3 2 1 6
(1) (2) (3) (4)
1 2 3 1
Row Player Ray (i)
Column Maxima
7 3 5 6
8
Two-Person, Zero-Sum Game The Campers
Column Player Carol (j)
Matrix of Payoffs to Row Player
Row Minima
(1) (2) (3) (4)
7 2 5 1 2 2 3 4 5 3 4 4 3 2 1 6
(1) (2) (3) (4)
1 2 3 1
Row Player Ray (i)
MaxiMin
Column Maxima
7 3 5 6
Game has a saddle point!
MiniMax
9
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
0 TV N TVN
0 -.6 -.4 -1 .6 0 .2 -.4 .4
-.2 0 -.6 1 .4 .6 0
0 TV N TVN
Row Player
10
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
Row Minima
0 TV N TVN
0 -.6 -.4 -1 .6 0 .2 -.4 .4
-.2 0 -.6 1 .4 .6 0
0 TV N TVN
-.6 -.4 -.6 0
Row Player
Column Maxima
1 .4 .6 0
11
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
Row Minima
0 TV N TVN
0 -.6 -.4 -1 .6 0 .2 -.4 .4
-.2 0 -.6 1 .4 .6 0
0 TV N TVN
-.6 -.4 -.6 0
Row Player
MaxiMin
Column Maxima
1 .4 .6 0
MiniMax
Game has a saddle point!
12
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
Row Minima
0 TV N TVN
0 .2 -.4 -.2 0 -.6
.4 .6 0
0 TV N TVN
-.4 -.6 0
Row Player
MaxiMin
Column Maxima
.4 .6 0
Game has a saddle point!
MiniMax
13
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
Row Minima
0 TV N TVN
0 -.6 .6 0
0 TV N TVN
-.6 0
Row Player
MaxiMin
Column Maxima
.6 0
Game has a saddle point!
MiniMax
14
Two-Person, Zero-Sum Game Advertising
Matrix of Payoffs to Row Player
Column Player
Row Minima
0 TV N TVN
0
0 TV N TVN
0
Row Player
MaxiMin
Column Maxima
0
Game has a saddle point!
MiniMax
15
Two-Person, Zero-Sum Game Mixed Strategies
Column Player
Matrix of Payoffs to Row Player
Row Minima
C1 C2
0 5 10 -2
R1 R2
0 -2
Row Player
10 5
Column Maxima
16
Two-Person, Zero-Sum Game Mixed Strategies
Column Player
Matrix of Payoffs to Row Player
Row Minima
C1 C2
MaxiMin
0 5 10 -2
R1 R2
0 -2
Row Player
10 5
Column Maxima
MiniMax
VC
VR
No Saddle Point!
17
Two-Person, Zero-Sum Game Mixed Strategies
Column Player
Matrix of Payoffs to Row Player
Row Minima
Y1 Y2
C1 C2
MaxiMin
0 5 10 -2
X1 R1 X2 R2
0 -2
Row Player
10 5
Column Maxima
MiniMax
MiniMax
MaxiMin
No Saddle Point!
18
Graphical Solution
VR
10
VR lt 10(1-X1)
VR lt -2 7X1
50/17
Optimal Solution X112/17, X25/17 VRMAX50/17
0
1
X1
12/17
19
Graphical Solution
VR
10
VR lt 10(1-X1)
Y11
Y1.75
Y10
VR lt -2 7X1
Y1.5
50/17
Optimal Solution X112/17, X25/17 VRMAX50/17
Y1.25
0
1
X1
12/17
20
Two-Person, Zero-Sum Games Summary

• Represent outcomes as payoffs to row player
• Evaluate row minima and column maxima
• If maximinminimax, players adopt pure strategy
  corresponding to saddle point choices are in
  stable equilibrium -- secrecy not required
• If maximin minimax, use linear programming to
  find optimal mixed strategy secrecy essential
• Number of options to consider can be reduced by
  using iterative dominance procedure

21
The Minimax Theorem
Every finite, two-person, zero-sum game has a
rational solution in the form of a pure or mixed
strategy.
John Von Neumann, 1926