Arrow Computation 130107203943 Phpapp01 (1)

8/12/2019 Arrow Computation 130107203943 Phpapp01 (1)

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Spring 2008,King Saud University

Arrow DiagrammingDr. Khalid Al-Gahtani

1

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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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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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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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

25

6

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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Spring 2008,King Saud University Arrow DiagrammingDr. Khalid Al-Gahtani 23

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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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?

CPs could be multiple in a project network.


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Spring 2008,King Saud University Arrow DiagrammingDr Khalid Al-Gahtani 25

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

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Arrow Computation 130107203943 Phpapp01 (1)