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Spring 2008, King Saud University

Arrow DiagrammingDr. Khalid Al-Gahtani

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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:



2

Computation Nomenclature

k l

Li

Ei Ej

Lj

l k

j i ACT (ESij, EFij)

Dij (LSij, LFij)

PredecessorsActivities

SuccessorsActivities



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.



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



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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



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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



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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



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Example 2:



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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



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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)



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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.



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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:



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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



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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



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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) = Fij AFij = Lj Ej



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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)



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ESij EFij ESjk LFij

AF ITF

F IDF



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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)



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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



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Example 3:

A

C

B

X 0

1

2

3

4 5 6 D

E

F G



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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



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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.



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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



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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?• CP’s could be multiple in a project network.



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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

✓ ✓ ✓ ✓ ✓ ✓

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