arrow computation 130107203943 phpapp01 (1)
TRANSCRIPT
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Spring 2008,King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
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CPM Network Computation
Computation Nomenclature
The following definitions and
subsequent formulas will be given in terms
of an arbitrary activity designed as (i-j) as
shown below:
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Arrow DiagrammingDr. Khalid Al-Gahtani
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Computation Nomenclature
kl
Li
Ei Ej
Lj
l
k
i
ACT (ESij, EFij)
Dij(LSij, LFij)
Predecessors
Activities
Successors
Activities
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Forward Pass Computations
STEP 1: E1= 0
STEP 2: Ei= Max all l(El+ Dli) 2 i n.
STEP 3: ESij= Ei all ijEFij= Ei+ Dij all ij
STEP 4: The (Expected) project duration can be
computed as the last activity (En) event time.
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Backward Pass Computations
STEP 1: Ln= Tsor En
STEP 2: Lj= Minall k(LkDjk) 1 j n-1
STEP 3: LFij= Lj all ij
LSij= LjDij all ij
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Example 1:
Activity ID Depends on Time ES EF LS LFA (1-2) 5
B (2-3) A 15
C (2-4) A 10
Dummy (3-4)
D (3-5) B 15E (4-5) B, C 10
F (5-6) D, E 5
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Example 1:
1 A
5
B
15
3
4
2 5
C
10
D
15
E
10
F
5
6
50
20
20
35 40
4035
20
25
50
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Arrow DiagrammingDr. Khalid Al-Gahtani
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Example 2:
Activity Description Predecessors Duration
AB
CD
EF
GH
I
Site clearingRemoval of trees
General excavationGrading general area
Excavation for trenchesPlacing formwork and reinforcement for concrete
Installing sewer linesInstalling other utilities
Pouring concrete
------
AA
B, CB, C
D, ED, E
F, G
43
87
912
25
6
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Example 2:
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Forward pass calculations
Step 1 E0= 0Step 2
j = 1 E1= Max{E0+ D01} = Max{ 0 + 4 } = 4j = 2 E2= Max{E0+ D02; E(1) + D12} = Max{0 + 3; 4 + 8} = 12
j = 3 E3= Max{E1+ D13; E(2) + D23} = Max{4 + 7; 12 + 9} = 21j = 4 E4= Max{E2+ D24; E(3) + D34} = Max{12 + 12; 21 + 2} = 24
j = 5 E5= Max{E3+ D35; E(4) + D45} = Max{21 + 5; 24 + 6} = 30
the minimum time required to complete the project is 30 since E5 = 30
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Spring 2008,King Saud University
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Backward pass calculations
Step 1 L5= E5= 30Step 2
j = 4 L4= Min {L5- D45} = Min {30 - 6} = 24j = 3 L3= Min {L5- D35; L4- D34} = Min {30 -5; 24 - 2} = 22
j = 2 L2= Min {L4- D24; L3- D23} = Min {24 - 12; 22 - 9} = 12j = 1 L1= Min {L3- D13; L2- D12} = Min {22 - 7; 12 - 8} = 4
j = 0 L0= Min {L2- D02; L1- D01} = Min {12 - 3; 4 - 4} = 0
E0 = L0, E1 = L1, E2 = L2, E4 = L4,and E5 = L5.
As a result, all nodes but node 3 are in the critical path.
Activities on the critical path include:
A (0,1), C (1,2), F (2,4) and I (4,5)
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Final Results of Example 1
ActivityDuration
Dij
Earlieststart time
ESij=Ei
Earliestfinish time
EFij=ESij+Dij
Lateststart time
LSij= LFijDij
Latestfinish time
Li=LFij
A (0,1)B (0,2)
C (1,2)D (1,3)
E (2,3)F (2,4)
G (3,4)H (3,5)
I (4,5)
43
87
912
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0*0
4*4
1212*
2121
24*
4*3
12*11
2124*
2326
30*
09
415
1312
2225
24*
4*12
12*22
2224*
2430
30*
*Activity on a critical path since Ei+ Dij= Lj.
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Float and their Management
Float Definitions:
Floator Slackis the spare time available or
not critical activities.
Indicates an amount of flexibility associated
with an activity.
There are four various categories of activity
float:
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1. Total Float:
Total Floator Path Floatis the maximumamount of time that the activity can be delayed
without extending the completion time of the
project.
It is the total float associated with a path.
For arbitrary activity (ij), the Total Float can
be written as:
Path FloatTotal Float (Fij) = LSijESij
= LFijEFij
= LjEFij
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2. Free Float
Free FloatorActivity Floatis equal to the amountof time that the activity completion time can bedelayed without affecting the earliest start oroccurrence time of any other activity or event in the
network. It is owned by an individual activity, whereas path
or total float is shared by all activities a long slackpath.
can be written as:Activity FloatFree Float (AFij) = Min (ESjk) EFij
= EjEFij
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3. Interfering Float:
That if used will effect the float of other
activities along its path (shared float).
For arbitrary activity (ij), the Interfering
Float can be written as:
Interfering Float (ITFij) = FijAFij
= Lj
Ej
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4. Independent Float
It is the amount of float which an activity willalways possess no matter how early or late itor its predecessors and successors are.
Float that is owned by one activity. In all cases, independent float is always lessthan or equal to free float.
can be written as:
Independent Float (IDFij) = Max (0, EjLiDij)
= Max (0, Min (ESjk)-Max (LFli) Dij)
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ESij EFij ESjk LFij
AF ITF
F
IDF
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Float Computations
Path FloatTotal Float (Fij) = LSijESij
= LFijEFij
= LjEFij
Activity FloatFree Float (AFij) = Min (ESjk) EFij
= EjEFij
Interfering Float (ITFij) = FijAFij
= LjEj
Independent Float (IDFij) = Max (0, EjLiDij)
= Max (0, Min (ESjk)Max (LFli) Dij)
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Example 3:
Activity Description Predecessors DurationAB
CD
EF
G
Preliminary designEvaluation of design
Contract negotiationPreparation of fabrication plant
Final designFabrication of Product
Shipment of Product to owner
---A
---C
B, CD, E
F
61
85
912
3
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Example 3:
A
C
B
X
0
1
2
3
4 5 6D
E
F G
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Example 3:
NodeEarliest Time
Ei
Latest Time
Li
0
12
34
5
6
0
68
817
29
32
0
78
817
29
32
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Example 3:
Activity
Earlieststart time
ESij
Lateststart time
LSij
TotalFloat
Fij
FreeFloat
AFij
InterferingFloat
ITFij
IndependentFloat
IDFij
A (0,1)
B (1,3)
C (0,2)D (2,4)
E (3,4)F (4,5)
G (5,6)X (2,3)
0
6
08
817
298
1
7
012
817
298
1
1
04
00
00
0
1
04
00
00
1
0
00
00
00
0
0
04
00
00
The minimum completion time for the project is 32 days
Activities C,E,F,G and the dummy activity X are seen to lie on the critical path.
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Critical Path Identifications
The critical path is continues chain of activities from thebeginning to the end, with zero float (if the zero-floatconvention of letting Lt = Et for terminal network event isfollowed).
The critical path is the one with least path float (if thezero-float convention of letting Lt = Et for terminalnetwork event is NOT followed).
The longest path through the network.
T = ti*, where T = project Completion Time
ti* = Duration of Critical Activity
There may be more than one critical paths in a network
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Identify CP activities & path(s)
1. Critical Activity:
An activity for which no extra time is available
(no float, F = 0). Any delay in the completion of
a critical activity will delay the project duration.
2. Critical Path:
Joins all the critical activities. Is the longest time path in the network?
CPs could be multiple in a project network.
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Ownership of floatFloat
Float OwnershipOwnership issues
concepts
Allow
Flexibilityfor Resource
leveling
AllowFlexibilityto include
changeorder
Prevent
disentitledfloat
consumption
PreventScheduleGames
Ability to
Distribute TFamong project
parties
Solve TFchanging
issues
Contractor
Owner
Project # # * *
Bar1
50/502 # # * *
Contract Risk3
Path Distribution4
Commodity5 *
Day-by-day
Contract Risk +Path Distribution +
Commodity +Day-by-day