Naagu's Engineering Tutorial
Sunday, 5 January 2014
Tuesday, 17 December 2013
NETWORK SCHEDULING BY PERT/CPM
PERT- Program Evaluation Review Technique
CPM-Critical Path Method
NETWORK SCHEDULING
Its a technique used for planning and scheduling large project.Its a method for minimizing troubleshoots such as delays, interruption.
There are two basic planning & Control techniques.:
PERT
CPM
NETWORK:
Its the graphical representation of logically and sequentially connected arrows and nodes representing activities and events in the project.Its an arrow diagram.
ACTIVITY:
An activity represents some action and its a time concuming effort necessary to complete a parrticular part of the overall project.
EVENT:
The beginning and the end points of an activity are called events/nodes.
MERGE EVENT:
Its not necessary for an event to be ending event for only one activity as it can be the ending event of two or more activity such a event is defined as merge event.
BURST EVENT:
If the event happens to be the beginning event of two or more activities then they are called as Burst event.
We have the following types of activities
1. Preceding
2. Succeeding
3.Concurrent
4.Dummy Activity
DUMMY ACTIVITY:
Certain activities which neither consume type or resources but are used simply to represent a connection/link between events are known as Dummy Activity.
CONSTRUCTION OF NETWORK:
Construct the network for the project whose activities and preceding relationships are shown below
ACTIVITIES A B C D E F
IMMEDIATE
PREDECESSOR - A A - D B,C,E
CPM-Critical Path Method
NETWORK SCHEDULING
Its a technique used for planning and scheduling large project.Its a method for minimizing troubleshoots such as delays, interruption.
There are two basic planning & Control techniques.:
PERT
CPM
NETWORK:
Its the graphical representation of logically and sequentially connected arrows and nodes representing activities and events in the project.Its an arrow diagram.
ACTIVITY:
An activity represents some action and its a time concuming effort necessary to complete a parrticular part of the overall project.
EVENT:
The beginning and the end points of an activity are called events/nodes.
MERGE EVENT:
Its not necessary for an event to be ending event for only one activity as it can be the ending event of two or more activity such a event is defined as merge event.
BURST EVENT:
If the event happens to be the beginning event of two or more activities then they are called as Burst event.
We have the following types of activities
1. Preceding
2. Succeeding
3.Concurrent
4.Dummy Activity
DUMMY ACTIVITY:
Certain activities which neither consume type or resources but are used simply to represent a connection/link between events are known as Dummy Activity.
CONSTRUCTION OF NETWORK:
Construct the network for the project whose activities and preceding relationships are shown below
ACTIVITIES A B C D E F
IMMEDIATE
PREDECESSOR - A A - D B,C,E
DOMINANCE PROPERTY
Q1. Solve
Player B
B1 B2 B3
Player A A1 [ 1 7 2
A2 6 2 7
A3 5 1 6]
Soln:
STEP 1:
Find out the maximin and minimax values
[ 1 7 2 1
6 2 7 2 maximin=2
5 1 6] 1
6 7 7
minimax=6
maximin != minimax
Therefore no saddle point.
STEP II:
Now compare each row and check whether they are minimum values. If so then delete that row.
Compare A1 and A2 , here few values are minimum and few are maximum.
Now compare A2 and A3, A3 has minimum values therefore delete A3.
By dominance Property, B1 B2 B3
A1 1 7 2
A2 6 2 7
STEP III:
Now compare each column and check whether they are maximum values. If so then delete that column.
Compare B1 and B2 , here few values are minimum and few are maximum.
Now compare B1 and B3, B3 has maximum values therefore delete B3.
B1 B2
A1 1 7 1
A2 6 2 2 maximin=2
6 7
minimax=6
maximin!=minimax
Therefore no saddle point
STEP IV:
a22-a12
p1=--------------------------
(a22+a11)-(a12+a21)
p1=1/2 p2=1/2
a22-a21
q1=--------------------------
(a22+a11)-(a12+a21)
q1=2/5 q2=3/5
a11a22-a21a12
value of game=-------------------------- =4
(a22+a11)-(a12+a21)
SA= [ A1 A2 A3
1/2 1/2 0]
SB=[B1 B2 B3
2/5 3/5 0]
Player B
B1 B2 B3
Player A A1 [ 1 7 2
A2 6 2 7
A3 5 1 6]
Soln:
STEP 1:
Find out the maximin and minimax values
[ 1 7 2 1
6 2 7 2 maximin=2
5 1 6] 1
6 7 7
minimax=6
maximin != minimax
Therefore no saddle point.
STEP II:
Now compare each row and check whether they are minimum values. If so then delete that row.
Compare A1 and A2 , here few values are minimum and few are maximum.
Now compare A2 and A3, A3 has minimum values therefore delete A3.
By dominance Property, B1 B2 B3
A1 1 7 2
A2 6 2 7
STEP III:
Now compare each column and check whether they are maximum values. If so then delete that column.
Compare B1 and B2 , here few values are minimum and few are maximum.
Now compare B1 and B3, B3 has maximum values therefore delete B3.
B1 B2
A1 1 7 1
A2 6 2 2 maximin=2
6 7
minimax=6
maximin!=minimax
Therefore no saddle point
STEP IV:
a22-a12
p1=--------------------------
(a22+a11)-(a12+a21)
p1=1/2 p2=1/2
a22-a21
q1=--------------------------
(a22+a11)-(a12+a21)
q1=2/5 q2=3/5
a11a22-a21a12
value of game=-------------------------- =4
(a22+a11)-(a12+a21)
SA= [ A1 A2 A3
1/2 1/2 0]
SB=[B1 B2 B3
2/5 3/5 0]
Sunday, 15 December 2013
GRAPHICAL METHOD FOR GAME THEORY
Q1.Solve the following 2 x 3 game graphically
Player B
Player A [ 1 3 11
8 5 2]
STEP I :
A's expected pay off against B's pure move is given by:
B's PURE MOVE A's EXPECTED PAY OFF E(P)
B1 E1(P)=8-7P1
B2 E2(P)=5-2P1
B3 E3(P)=2P1+2
STEP II:
Now for the graph consider Axis 1 as A2 and Axis 2 as A1
STEP III:
The reduced Pay off matrix is given by
B2 B3 --------> the intersection point
A1 [ 3 11 3
A2 5 2] 2 maxmin=3
5 11
minimax=5
maxmin != minmax
Therefore there exists no saddle point
P1=9/11 p2=2/11
q1=3/11 q2=8/11
Value of game = 49/11
The optimal Strategy is given by
[ A1 A2
SA = 9/11 2/11]
[ B1 B2 B3
SB = 0 3/11 8/11]
NOTE:
The left out matrix value is given as '0'
For 2 x n matrix find lower envelope, find maximin (maximum values)
For m x 2 matrix find upper envelope, find minimax (minimum values)
Player B
Player A [ 1 3 11
8 5 2]
STEP I :
A's expected pay off against B's pure move is given by:
B's PURE MOVE A's EXPECTED PAY OFF E(P)
B1 E1(P)=8-7P1
B2 E2(P)=5-2P1
B3 E3(P)=2P1+2
STEP II:
Now for the graph consider Axis 1 as A2 and Axis 2 as A1
STEP III:
The reduced Pay off matrix is given by
B2 B3 --------> the intersection point
A1 [ 3 11 3
A2 5 2] 2 maxmin=3
5 11
minimax=5
maxmin != minmax
Therefore there exists no saddle point
P1=9/11 p2=2/11
q1=3/11 q2=8/11
Value of game = 49/11
The optimal Strategy is given by
[ A1 A2
SA = 9/11 2/11]
[ B1 B2 B3
SB = 0 3/11 8/11]
NOTE:
The left out matrix value is given as '0'
For 2 x n matrix find lower envelope, find maximin (maximum values)
For m x 2 matrix find upper envelope, find minimax (minimum values)
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