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Network CalculationsPart 4

Construction Project

Management

(CE 110401346)

Hashemite University

Department of Civil Engineering

➢Early Activity Start (ES): Earliest time an activity can start—as

determined by the latest of the early finish times of all immediately

preceding activities (IPAs)

➢Early Activity Finish (EF): Earliest time an activity can finish—

determined by adding the duration of the activity to the early start

time

➢Late Activity Start (LS): Latest time an activity can start without

delaying the project completion

➢Late Activity Finish (LF): Latest time an activity can be finished

without delaying project completion

Calculations On a Precedence Network

Early Event Occurrence Time: Earliest an event can occur—determined by the latest early finish

Late Event Occurrence Time: Latest an event can occur

Calculations On a Precedence Network

Step 1: Perform Forward Pass Calculations to determine:➢Early Start (ES) and Early Finish (EF) of each activity.

ES (initial activities) = S

ES (x) = Latest (EF (all predecessors of x))

EF (x) = ES(x) + D(x)

Where,

S= Project start time.

D(x) = Duration of activity x.

ES(x) = Earliest start time of activity x.

EF(x) = Earliest finish time of activity x.

Calculations On a Precedence Network

Early Times (Early Start [ES] and Early Finish [EF])

Calculations On a Precedence Network

1. Assign 1 as the early start date of the first activity

2. Calculate the early finish time for the activity

3. The early start of activities will be determined by the early

finish times of preceding activities

➢Other than the first activity or activities

4. Repeat steps 2 and 3 for each network activity until ES &

EF are determined for the last activity

Calculations On a Precedence Network

Step 5: Perform Backward Pass Calculations to determine:

➢Late Start (LS) and Late Finish (LF) of each activity.

LF (end activities) = T

LF (x) = Earliest (LS (all successors of x))

LS (x) = LF(x) - D(x)

Where,

T = Project completion time.

D(x) = Duration of activity x.

LS(x) = Latest start time of activity x.

LF(x) = Latest finish time of activity x.

Calculations On a Precedence Network

Calculations On a Precedence Network

Calculations On a Arrow Network

▪When an activity start date is fixed in this way, the activity is

said to have no float , Such activities are said to be “critical”

▪If the activity starts later than the assigned date, or takes

longer to complete than the assigned duration, the project

completion date will be extended by the same amount of time

Identify the Critical Path

▪Total Float (TF): maximum amt of time that the activity can be delayed without delaying the completion time of the project.

▪Float (FF): maximum amount of time that the activity can be delayed without delaying the early start of any of its successors, assuming its predecessors were completed early.

▪Free Float: Amount of time an activity can be delayed before it impacts the start of any succeeding activity

Activity Floats

▪Independent Float (IF): maximum amount of time that the activity can be delayed without delaying the early start of any of its successors, assuming its predecessors were completed late.

Activity Floats

Successors Started

Early Late

Predecessors

Completed

Early Free Float Total

Float

Late Independent

Float

Activity Floats

Float Type Calculation

Total FloatTF = LS – ESTF = LF – EF

Free Float FF = Min (ES of all successors – EF)

Independent

Float

IF =Min (ES of all successors

-LF of the same activity)

Activity Floats

▪Once the early and late start times, early and late finish times, free

float, and total float of all activities are determined, the

calculations are completed

Calculations On a Precedence Network

Activity TF FF IF

Mobilize 0 0 0

Remove old Cabinets 0 0 0

Buy materials 3 0 0

Install new flooring 0 0 0

Hang wallpaper 3 0 -3

Install new cabinets 0 0 0

Touch-up paint 3 3 0

Demobilize 0 0 0

Calculations On a Precedence Network

Critical Path

Critical Path – series of interconnected critical activities through the network diagram. The delay of any of the critical activities will delay the project completion date (A critical activity is an activity with a total float value equals to zero).

10 - 17

10 - 18

A

EF

LS

ES

LF

Example 2: Bridge Construction

Activity Duration

(O, M, P)

Mean

Duration

Predecessors

Pile and foundation W

Pile and foundation C

Pile and foundation E

Substructure W

Substructure C

Substructure E

Cast-in-place span

Precast span

Bridge deck

Finishes

(9,10,11)

(4,5,6)

(6,8,10)

(20,23,26)

(16,17,18)

(19,20,21)

(28,30,32)

(4,5,6)

(4,5,6)

(12,14,16)

10

5

8

23

17

20

30

5

5

14

--

--

--

Pile and foundation W

Pile and foundation C

Pile and foundation E

Substructures W and C

Substructures C and E

Cast-in-place and precast

spans

Bridge deck

Start

Pile foundation W

Pile foundation C

Pile foundation E

Substructure W

Substructure C

Substructure E

Cast-in-place span

Precast span

Bridge deck Finishes

0Start

10Pile foundation W

5Pile foundation C

8Pile foundation E

23Substructure W

17Substructure C

20Substructure E

30Cast-in-place span

5Precast span

5Bridge deck

14Finishes

ES D EFDescription

LS TF LF

Forward Pass

33 or 22?

63 or 33?

22 or 28?

0 0 0Start

0 10 10Pile foundation W

0 5 5Pile foundation C

0 8 8Pile foundation E

10 23 33Substructure W

5 17 22Substructure C

8 20 28Substructure E

33 30 63Cast-in-place span

28 5 33Precast span

63 5 68Bridge deck

68 14 82Finishes

ES D EFDescription

LS TF LF

58 or 33?

Backward Pass

ES Dur EFDescription

LS TF LF

0 0 0Start

0 0

0 10 10Pile foundation W

0 10

0 5 5Pile foundation C

11 16

0 8 8Pile foundation E

30 38

10 23 33Substructure W

10 33

5 17 22Substructure C

16 33

8 20 28Substructure E

38 58

33 30 63Cast-in-place span

33 63

28 5 33Precast span

58 63

63 5 68Bridge deck

63 68

68 14 82Finishes

68 82

Total Float

TF=LST-EST=LFT-EFT

ES D EFDescription

LS TF LF

0 0 0Start

0 0 0

0 10 10Pile foundation W

0 0 10

0 5 5Pile foundation C

11 11 16

0 8 8Pile foundation E

30 30 38

10 23 33Substructure W

10 0 33

5 17 22Substructure C

16 11 33

8 20 28Substructure E

38 30 58

33 30 63Cast-in-place span

33 0 63

28 5 33Precast span

58 30 63

63 5 68Bridge deck

63 0 68

68 14 82Finishes

68 0 82

Activity Floats

• In addition to total float, the following floats

can be estimated for each activity:

– Free Float (FF)

– Interfering Float (IF)

– Independent Float (Ind. F)

Free Float (FF)

• FF is the total time that an activity can be

delayed without causing any delay to the

early start of the following activities.

• FFA = min. (ES activities following A - EFA)ESB=13

A

C

B

EFA=10

ESC=16FFA = min (3,6) = 3

Interfering Float (IF)

• IF is the total time utilized in the current

activity that interferes with its following

activities:

– IFA = TFA - FFA

• Example: Assume TFA is 3 days and FFA is 0

days. IFA = 3 – 0 = 3. This implies that

although 3 days of delay can occur on activity

A without impacting the project duration, each

day of IF for A will interfere with the float

available for following activities.

Independent Float (Ind. F)

• Ind. F is the total time between an activity

late finish and the early start of the

following activities.

• Ind. FA = min. (ES activities following A - LFA)ESB=13

A

C

B

ESC=16Ind. FA = min. (1,4) = 1

LFA=12

THE CPM EXPLAINED THROUGH EXAMPLES

•Example 3: Draw the logic network and perform the CPM calculations for the schedule shown next.

Solution: The Forward Pass

• The project starts with activity A, which starts at the beginning of day 1 (end of day 0).

• It takes 5 days to finish activity A; it finishes on day 5 (end of the day).

• At this point, activities B and C can start. Activity B takes 8 days; it can start on day 5 (directly after activity A finishes), so it can finish as early as day 13.

• Similarly, activity C can finish on day 11 (5 + 6).

• Activity D follows activity B. It can start on day 13 (end of B) and end on day 22.

• Activity E must wait until both activities B and C are finished.

Solution: The Forward Pass

• Activity C finishes on day 11, but activity B does not finish until day 13. Thus, activity E cannot start until day 13. With 6 days’ duration, activity E can then finish on day 19.

• Activity F depends on activity C only. Thus, it can start on day 11 and finish on day 14.

• The last activity, G, cannot start until activities D, E, and F are finished.

• Through simple observation, we can see that activity G cannot start until day 22 (when the last activity of D, E, and F finishes).

• Activity G takes 1 day, so it can finish on day 23.

Solution: The Forward Pass

Solution: The Forward Pass

• For this example, we have calculated two types of dates:

1. The expected completion date of the project: day 23

2. The earliest date when each activity can start and finish

• These dates are called the early start (ES) and the early finish (EF) dates for each activity. As you will soon learn, an activity cannot start earlier than its ES date and cannot finish earlier than its EF date, but it may start or finish later than these dates.

Solution: The Forward Pass

• In mathematical terms, the ES time for activity j (ESj) is as follows:

ESj = max(EFi) (4.1)

• where (EFi) represents the EF times for all immediately preceding activities.

• Likewise, the EF time for activity j (EFj) is as follows:

EFj = ESj + Durj (4.2)

• where Durj is the duration of activity j.

• The forward pass is defined as the process of navigating through a network from start to finish and calculating the early dates for each activity and the completion date of the project.

Solution: The Backward Pass

• Now let us start from the end of the project and work our way back to the start.

• We already know the end-of-project date: day 23.

• Activity G must finish by day 23.

• Its duration is only 1 day, so it must start no later than day 22 (23 − 1) so that it does not delay the project.

• Similarly, activities D, E, and F must finish no later than day 22 so that they will not delay activity G. Through simple computations, we can find their late start dates: activity F: 22 − 3 = 19; activity E: 22 − 6 = 16; and activity D: 22 − 9 = 13.

• Activity C must finish before activities E and F can start. Their late start dates are 16 and 19, respectively.

Solution: The Backward Pass

• Clearly, activity C must finish by the earlier of the two dates, day 16, so that it will not delay the start of activity E.

• Thus, its late start date is day 10 (16 − 6). Similarly, activity B must finish by the earlier of its successors’ late start dates: day 13 for D and day 16 for E. Therefore, the late finish date for activity B is day 13, and its late start date is day 5 (13 − 8).

• The last activity (from the start) is A: It must finish by the earlier of the late start dates for activities B and C, which are day 5 for B and day 10 for C. Consequently, the late finish date for activity A is day 5, and its late start date is day 0 (5 − 5).

Solution: The Backward Pass

• In mathematical terms, the late finish (LF) time for activity j (LFj) is as follows:

LFj = min(LSk) (4.3)

• where (LSk) represents the late start times for all succeeding activities.

• Likewise, the late start (LS) time for activity j (LSj) is as follows:

LSj = LFj − Durj (4.4)

• The backward pass is defined as the process of navigating through a network from finish to start and calculating the late dates for all activities.

• This pass, along with the forward-pass calculations, helps identify the critical path and the float for all activities.

Solution: The Backward Pass

Solution: The Backward Pass

• If you refer to the Figure on the previous slide, you can see that for some activities (light lines), the late dates (shown under the boxes) are later than their early dates (shown above the boxes). For other activities (thick lines), late and early dates are the same.

• For the second group, we can tell that these activities have strict start and finish dates. Any delay in them will result in a delay in the entire project. We call these activities critical activities.

• We call the continuous chain of critical activities from the start to the end of the project the critical path.

• Other activities have some leeway. For example, activity C can start on day 5, 6, 7, 8, 9, or 10 without delaying the entire project. As mentioned previously, we call this leeway float.

Solution: The Backward Pass

• There are several types of float. The simplest and most important type of float is total float (TF):

TF = LS − ES or TF = LF − EF or TF = LF − Dur − ES

Solution: The Backward Pass

• With the completion of the backward pass, we have calculated the late dates for all activities. With both passes completed, the critical path is now defined and the amount of float for each activity is calculated.

A

Example 4: Draw the Activity-on-Arrow network for the following

activities. Remove any redundant dependencies and label dummy activities as dummy1, dummy2, etc.

B

C

D F H

I

G

J K L

E

Dummy1 Dummy2

Dummy3

Activity A B C D E F G H I J K L

Predecessor - - A, B B C C, D F F F G H, I, J E, K

▪LAG: The amount of time that exists between the early finish of an activity and the early start of a specified succeeding activity

Calculations On a Precedence Network

Example 5: Network Calculations

Activity Description Predecessors Duration

A Preliminary design - 6

B Evaluation of design A 1

C Contract negotiation - 8

D Preparation of fabrication plant C 5

E Final design B, C 9

F Fabrication of Product D, E 12

G Shipment of Product to owner F 3

Activity Duration ES EF LS LF

A 6

B 1

C 8

D 5

E 9

F 12

G 3

Activity Duration ES EF LS LF

A 6 1 7 2 8

B 1 7 8 8 9

C 8 1 9 1 9

D 5 9 14 13 18

E 9 9 18 9 18

F 12 18 30 18 30

G 3 30 33 30 33

Activity Duration ES EF LS LF TF FF IF

A 6 1 7 2 8

B 1 7 8 8 9

C 8 1 9 1 9

D 5 9 14 13 18

E 9 9 18 9 18

F 12 18 30 18 30

G 3 30 33 30 33

Activity Duration ES EF LS LF TF FF IF

A 6 1 7 2 8 1 0 0

B 1 7 8 8 9 1 1 0

C 8 1 9 1 9 0 0 0

D 5 9 14 13 18 4 4 4

E 9 9 18 9 18 0 0 0

F 12 18 30 18 30 0 0 0

G 3 30 33 30 33 0 0 0

Activity Description Predecessors Duration

A Site clearing - 4

B Removal of trees - 3

C General excavation A 8

D Grading general area A 7

E Excavation for utility trenches B, C 9

F Placing formwork and

reinforcement for concrete

B, C 12

G Installing sewer lines D,E 2

H Installing other utilities D,E 5

I Pouring concrete F,G 6

Example 6: Network Calculations

Example 6: Network Calculations

Activity Duration

Early

Start

Early

Finish

Late

Start

Late

Finish

A 4

B 3

C 8

D 7

E 9

F 12

G 2

H 5

I 6

Example 6: Network Calculations

Activity Duration

Early

Start

Early

Finish

Late

Start

Late

Finish

A 4 1 5 1 5

B 3 1 4 10 13

C 8 5 13 5 13

D 7 5 12 16 23

E 9 13 22 14 23

F 12 13 25 13 25

G 2 22 24 23 25

H 5 22 27 26 31

I 6 25 31 25 31

Example 6: Network Calculations

Activit

y

Duration ES EF LS LF TF FF IF

A 4 1 5 1 5

B 3 1 4 10 13

C 8 5 13 5 13

D 7 5 12 16 23

E 9 13 22 14 23

F 12 13 25 13 25

G 2 22 24 23 25

H 5 22 27 26 31

I 6 25 31 25 31

Example 6: Network Calculations

Activit

y

Duration ES EF LS LF TF FF IF

A 4 1 5 1 5 0 0 0

B 3 1 4 10 13 9 9 9

C 8 5 13 5 13 0 0 0

D 7 5 12 16 23 11 10 10

E 9 13 22 14 23 1 0 0

F 12 13 25 13 25 0 0 0

G 2 22 24 23 25 1 1 0

H 5 22 27 26 31 4 4 3

I 6 25 31 25 31 0 0 0

Example 6: Network Calculations

▪Types of Logical Precedence Relationships in PDM

➢ Finish to Start (with or without lag):

Each activity depends on the completion of its preceding activity

Precedence Diagram Relationships

Typical Sequence of Finish-to-Start Relationships

Finish-to-Start Relationship with a 28-Day Delay (lag)

Precedence Diagram Relationships

Types of Logical Precedence Relationships in PDM

➢ Start to Start (with or without lag).

Precedence Diagram Relationships

Activities with Start-to-Start RelationshipsPrecedence Diagram Relationships

Activities with Start-to-Startwith a Delay Relationships

Precedence Diagram Relationships

Activities with Start-to-Startwith a Delay Relationships

Precedence Diagram Relationships

Types of Logical Precedence Relationships in PDM

➢ Finish to Finish (with or without lag).

Precedence Diagram Relationships

Finish-to-Finish (FF)

Precedence Diagram Relationships

Finish-to-Start Relationshipfor Window Installation

Finish-to-Finish (FF) — with Delay

Precedence Diagram Relationships

Activities with Finish-to-Finishwith a Delay (lag) Relationships

Precedence Diagram Relationships

Types of Logical Precedence Relationships in PDM

➢ Start to Finish (with or without lag).

Precedence Diagram Relationships

Start-to-Finish with a Delay — RelationshipsPrecedence Diagram Relationships

Start-to-Finish with a Delay — RelationshipsPrecedence Diagram Relationships

Start-to-Finish with a Delay — RelationshipsPrecedence Diagram Relationships

▪Advantages of PDM Over CPM:

➢ Easier to construct & modify network.

➢ No need for dummies.

➢ Less activities in presentation.

➢ Precedence relationships with lag times are more effective in modeling project activities.

Precedence Diagram Method