Download - Arrow computation
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
1
CPM Network ComputationComputation Nomenclature• The following definitions and
subsequent formulas will be given in terms of an arbitrary activity designed as (i-j) as shown below:
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
2
Computation Nomenclature
k l
Li
Ei Ej
Lj
l k
j i ACT (ESij, EFij)
Dij (LSij, LFij)
PredecessorsActivities
SuccessorsActivities
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
3
Forward Pass Computations
STEP 1: E1 = 0
STEP 2: Ei = Max all l (El + Dli) 2 ≤ i ≤ n.
STEP 3: ESij = Ei all ij
EFij = Ei + Dij all ij
STEP 4: The (Expected) project duration can be computed as the last activity (En) event time.
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
4
Backward Pass Computations
STEP 1: Ln = Ts or En
STEP 2: Lj = Minall k (Lk Djk) 1 ≤ j ≤ n-1
STEP 3: LFij = Lj all ijLSij = Lj Dij all ij
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
5
Example 1: Activity ID Depends on Time ES EF LS LF
A (1-2) 5 B (2-3) A 15 C (2-4) A 10
Dummy (3-4) D (3-5) B 15 E (4-5) B, C 10 F (5-6) D, E 5
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
6
Example 1:
1 A
5
B
15
3
4
2 5
C
10
D
15
E
10
F
5 6
5 0
20
20
35 40
40 35
20
25
5 0
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
7
Example 2: Activity Description Predecessors Duration
A B C D E F G H I
Site clearing Removal of trees
General excavation Grading general area
Excavation for trenches Placing formwork and reinforcement for concrete
Installing sewer lines Installing other utilities
Pouring concrete
--- --- A A
B, C B, C D, E D, E F, G
4 3 8 7 9 12 2 5 6
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
8
Example 2:
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
9
Forward pass calculations
Step 1 E0 = 0 Step 2
j = 1 E1 = Max{E0 + D01} = Max{ 0 + 4 } = 4 j = 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} = 21 j = 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
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
10
Backward pass calculations Step 1 L5 = E5 = 30
Step 2 j = 4 L4 = Min {L5 - D45} = Min {30 - 6} = 24 j = 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} = 12 j = 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)
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
11
Final Results of Example 1
Activity Duration Dij
Earliest start time ESij =Ei
Earliest finish time
EFij=ESij +Dij
Latest start time
LSij= LFij Dij
Latest finish 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)
4 3 8 7 9
12 2 5 6
0* 0 4* 4 12
12* 21 21
24*
4* 3
12* 11 21
24* 23 26
30*
0 9 4
15 13 12 22 25 24*
4* 12
12* 22 22
24* 24 30
30*
*Activity on a critical path since Ei + Dij = Lj.
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
12
Float and their Management• Float Definitions:
– Float or Slack is 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:
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
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1. Total Float:• Total Float or Path Float is the maximum
amount 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) = LSij ESij
= LFijEFij
= Lj – EFij
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
14
2. Free Float• Free Float or Activity Float is equal to the amount
of time that the activity completion time can be delayed without affecting the earliest start or occurrence 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 slack path.
• can be written as:Activity FloatFree Float (AFij) = Min (ESjk) EFij
= Ej EFij
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
15
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) = Fij AFij = Lj Ej
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
16
4. Independent Float• It is the amount of float which an activity will
always possess no matter how early or late it or its predecessors and successors are.
• Float that is “owned” by one activity.• In all cases, independent float is always less
than or equal to free float.• can be written as:
Independent Float (IDFij) = Max (0, Ej Li –Dij)
= Max (0, Min (ESjk) - Max (LFli) Dij)
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
17
ESij EFij ESjk LFij
AF ITF
F IDF
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
18
Float ComputationsPath FloatTotal Float (Fij) = LSij ESij
= LFijEFij = Lj – EFij
Activity FloatFree Float (AFij) = Min (ESjk) EFij = Ej EFij
Interfering Float (ITFij) = Fij AFij = Lj Ej
Independent Float (IDFij) = Max (0, Ej Li –Dij)= Max (0, Min
(ESjk) Max (LFli) Dij)
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
19
Example 3:Activity Description Predecessors Duration
A B C D E F G
Preliminary design Evaluation of design Contract negotiation
Preparation of fabrication plant Final design
Fabrication of Product Shipment of Product to owner
--- A --- C
B, C D, E
F
6 1 8 5 9 12 3
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
20
Example 3:
A
C
B
X 0
1
2
3
4 5 6 D
E
F G
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
21
Example 3:
Node Earliest Time Ei
Latest Time Li
0 1 2 3 4 5 6
0 6 8 8
17 29 32
0 7 8 8 17 29 32
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
22
Example 3:
Activity Earliest
start time ESij
Latest start time
LSij
Total Float
Fij
Free Float AFij
Interfering Float ITFij
Independent Float 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 0 8 8
17 29 8
1 7 0
12 8
17 29 8
1 1 0 4 0 0 0 0
0 1 0 4 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 4 0 0 0 0
• 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.
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
23
Critical Path Identifications • The critical path is continues chain of activities from the
beginning to the end, with zero float (if the zero-float convention of letting Lt = Et for terminal network event is followed).
• The critical path is the one with least path float (if the zero-float convention of letting Lt = Et for terminal network 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
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
24
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?• CP’s could be multiple in a project network.
Spring 2008, King Saud University
Arrow DiagrammingDr. Khalid Al-Gahtani
25
Ownership of float Float Float Ownership Ownership issues concepts
Allow Flexibility
for Resource leveling
Allow Flexibility to include
change order
Prevent disentitled
float consumption
Prevent Schedule Games
Ability to Distribute TF among project
parties
Solve TF changing
issues
Contractor ✓ ✕ ✕ ✓ ✕ ✕
Owner ✕ ✓ ✕ ✕ ✕ ✕
Project # # * * ✕ ✕
Bar1 ✕ ✕ ✓ ✕ ✕ ✕
50/502 # # * * ✕ ✕
Contract Risk3 ✓ ✕ ✓ ✓ ✕ ✕
Path Distribution4 ✓ ✕ ✓ ✓ ✓ ✕
Commodity5 ✓ ✓ * ✓ ✕ ✕
Day-by-day ✕ ✕ ✕ ✕ ✕ ✓ Contract Risk +
Path Distribution + Commodity + Day-by-day
✓ ✓ ✓ ✓ ✓ ✓