Modern Construction Project Management, Second Edition

S.L. Tang Francis K.W. Wong S.W. Poon Syed M. Ahmed

Published by Hong Kong University Press, HKU

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11 CRITICAL

1. Introductio n

The representation o f activities in a civil engineering projec t b y a network diagram was discussed in the previous chapter. In this chapter, we are going to see tha t ever y project wit h activitie s tha t ar e to be programmed ha s a t least on e sequence o f activities which i s critical t o the completion o f tha t project. Any of the activities on this critical path which are not completed in the estimated period o f duration wil l cause the overal l project lengt h t o be extended. The expected project completion time can be computed from thi s path.

2.1

Six Steps in the Critica l Path Metho d

There ar e six steps in the analysis of a network diagra m using th e Critica l Path Method. They will be thoroughly discussed in this section.

Identification o f Projec t Activities If we are given a project an d are asked t o plan a programme o f work fo r it , the first step is to analyse the objective of the project. What are the activities that should be done in order to accomplish the project? The n we should list

1 58 MODER N CONSTRUCTION PROJEC T MANAGEMENT

the activities on a piece of paper; the activities listed may not necessarily be in a proper order . This is just a brainstorming exercise . (Se e Table 10. 1 of Chapter 10. )

2.2 Networ k Formulatio n Next, label the activities listed in Step 1 and then determine a logical sequence for th e activitie s an d for m a network diagra m fo r th e list . The technique s involved were discussed in Chapter 10 .

2.3 Duratio n Estimation Then, based on individual experience, estimate the time duration (t e) of each activity in the network. Let us use Example 1 in Chapter 1 0 as an illustration. Suppose the estimated tim e durations, t , fo r th e activities are as shown i n Table 11.1.

Activity

1-2

1-3

1-4

2-3

4-5

3-6

4-7

5-8

6-7

6-8

7-9

8-9

Description

Obtain materia l

Obtain mixe r

Dig hol e 1

(Dummy)

Dig hole 2

Mix concret e

Set up pylon 1

Set up pylon 2

(Dummy)

(Dummy)

Concreting of pylon 1 by

pouring concrete int o hol e 1

Concreting of pylon 2 by

pouring concrete int o hol e 2

t e

(in days) 1/2

1

1

0

1

1/4

3/4

3/4

0

0

1/4

1/4

Table 11.1 Estimate d time durations o f the activities .

CRITICAL PATH METHOD 1 5 9

So now the estimated tim e durations can be put onto th e network diagra m (Fig. 11.1) , which is similar to Fig. 10.11 of Chapter 10 .

Fig. 11.1 Networ k wit h estimate d time durations .

Forward Pass Computation This step involves the computation o f the earliest start time of each activity. The computation process is called the forward pass .

The earliest start time is the earliest possible time by which the activity under consideration can be completed. That is to say, it is the time by which all the preceding activitie s mergin g int o th e star t nod e o f the activit y have bee n completed.

To assist computation, th e earliest start time (the result of the forward pas s computation) i s entered in a small square box beside each node, as shown in Fig. 11.2.

To calculate the earliest start time for an activity in a network, start with the start node with th e smallest node number (th e initia l activity) . Set this t o zero (se e Fig. 11.3) . This means tha t activit y 1- 2 start s a t zero time . This activity can be completed in half a day, so 1/2 is entered into the square next to node 2 (i.e. end node of activity 1-2 or start node of dummy activity 2-3).

1 60 MODER N CONSTRUCTION PROJECT MANAGEMENT

Fig. 11.2 Squar e boxes are put beside the node s for entr y o f

earliest star t times.

So, in general, add the t o f an activity to the figure in the square at the tail of the arrow (i.e . start node) an d put the sum in the square at the head of the arrow (i.e . end node).

Let us consider some more examples. The te of activity 1-4 is 1; hence we put 1 (0 + 1 = 1) in the square box next to node 4. The te of activity 4-5 is also 1, so we put 2 (1 + 1 = 2) in the box next to node 5.

Now let us consider activities which merge into a single node. Both activities 1-3 and 2-3 merge to node 3. Taking the path 1-2-3 , node 3 can be reached in 1/ 2 day because activity 2-3 has zero time duration, but when path 1- 3 is taken, node 3 can onl y be reached i n 1 day. In thi s case , the larger figure prevails. That is, 1 is put in the box beside node 3 instead of 1/2. This means that activity 3-6 can only commence one day the earliest after th e start of the project.

Using this method, the earliest start time for al l activities can be computed . The earliest start time of an activity is the figure in the square box next to the start node o f that activity .

CRITICAL PATH METHOD 1 61 > •HUH

Fig. 11.3 Networ k with earliest start times.

Note that the earliest start time represented in the square box next to the last node (node 9 in this example) represents the duration of the project. I n this example, the project duratio n i s three days.

Before reading further, chec k that the computations in Fig. 11.3 are correct .

2.5 Backwar d Pass Computation While the forward-pass computation s provide us with the earliest start time of each activity the backward pas s computations provide us with the latest finish tim e of the activity. The latest finish tim e is the latest possible time by which a n activit y mus t b e complete d i f ther e i s t o be n o dela y i n th e completion o f the project .

We usually draw a small circle beside each node fo r entr y o f the results of the backward pass computation (Fig . 11.4) .

1 62 MODER N CONSTRUCTIO N PROJEC T MANAGEMEN T

Fig. 11.4 Networ k wit h circle s fo r the entry o f lates t finish times .

In the backward-pass computation , we start with the last node (i.e. , node 9 in this case).

The lates t finis h tim e o f th e las t activit y i s equa l t o th e projec t duratio n found fro m th e forward-pass computation .

So 3 is put inside the circle next to node 9 (Fig. 11.5) . Then the duration of the preceding activity is subtracted from thi s and the difference i s put in the circle next to the preceding node. For example, the te of activity 8-9 is V . We put 23/4 (i.e., 3 - V4 = 23/4) in the circle next to node 8. The te of activity 5-8 is 3/4; so we put 2 (i.e. 23/4 - 3/4 = 2) in the circle next to node 5 . This means that if the project i s to be completed without delay , the latest finish tim e of activity 5-8 is 23/4 days after the project starts; the latest finish time of activity 4-5 is two days from th e commencement dat e of the project an d so on.

Now consider th e situation where more than on e activity proceeds from a node. Let us take the example of node 4. Activities 4-5 and 4-7 proceed fro m node 4. Taking path 4-5, the latest finish tim e for node 4 is 1 , but when th e path 4- 7 i s taken, th e lates t finis h tim e a t node 4 is 2 . In such a case, th e smaller figur e i s chosen. Tha t is , 1 is put insid e th e circl e beside nod e 4 instead of 2.

CRITICAL PATH METHOD 1 63

Fig. 11.5 Networ k wit h lates t finish times .

Using this method, the latest finish time s for each activity (i.e. , represented by the circle next to the end node of the activity) ca n be computed .

Note that the figure in the circle next to the initial node is always zero.

Before reading further, chec k that the computations in Fig. 11.5 are correct .

Tracing the Critical Path In Step 4, the project duratio n tim e is found b y forward pas s computation . There is a path which determines the shortest project duration. This longes t path in the network, which also represents the shortest projec t duration , is called the critical path . Any delay in an activity which lies on the critical path will result in a delay of the project .

Comparing the networks in Figs. 11.3 and 11.5 shows that some nodes have earliest star t time s equa l t o th e lates t finis h times . (Se e Fig . 11. 6 whic h combines Figs . 11. 3 and 11.5. ) Others , however, have different figures . I n the former case , the earliest start time of an activity represented by the star t node which can be possibly achieved is the same as the latest finish tim e for the precedin g activit y (th e sam e node representin g th e en d nod e o f tha t preceding activity) to be completed in order not to delay the overall schedule.

1 64 MODER N CONSTRUCTION PROJECT MANAGEMENT

The cas e i s therefor e critica l sinc e ther e mus t b e n o dela y betwee n it s achievement and the start of the next activity. By a similar argument, therefore , the critical path must pass through all nodes which have the same figures in their respective square and circle boxes.

The critical path is traced from th e starting node through a series of nodes each of which has the earlies t start time equal to the latest finish time . The critical path contains the series of activities which defines the project duration.

In our present example , the critical path i s therefore give n by 1-4-5-8-9 . It can be seen tha t thi s is the path with the longest duration . Activities alon g this path (i.e . 1-4, 4-5, 5-8, 8-9) ar e called critical activitie s while all other activities are called non-critica l activities .

Fig. 11.6 Tracin g the critical path.

3. Th e Float of an Activit y

In Fig . 11.5 , we have seen tha t th e activities given in Table 11. 2 form th e critical path.

CRITICAL PATH METHOD 1 65

Activity

1-4

4-5

5-8

8-9

Description

Dig hole 1

Dig hole 2

Set up pylon 2

Concreting of pylon 2

te (i n days)

1

1

3/4

1/4

Table 11.2 Critica l Activities o f the Example .

These activitie s ar e critica l activitie s whic h mak e u p th e overal l projec t duration. They must be very well controlled i n the process of constructio n because the completion o f the project wil l be delayed if any one of them is delayed.

Delays of non-critical activities , however, may not affect th e overall projec t completion. Let us now look into this point in more detail .

If we draw the critical activities along a time axis in days (Fig. 11.7) , we get what is known as the critical activity chain, the duration of which is exactly equal to the time required for project completio n (i n this case three days).

1 1 o 1-4 4- 5 5- 8 8- 9

I I I I I I 0 V 2 1 Vh 2 27 2 3

Time Duration (Days) Fig. 11.7 Critica l pat h chain : paths of critica l activitie s draw n

along a time axis .

There are other paths joining the starting node to the finishing nod e which are non-critical in the network (Fig . 11.6) . They are listed below:

Path Path Path Path Path

1: 2: 3: 4. 5:

1-3-6-7-9 1-3-6-8-9 1-2-3-6-7-9 1-2-3-6-8-9 1-4-7-9

1 66 MODER N CONSTRUCTION PROJECT MANAGEMENT

Let us no w examin e Pat h 2 (i.e . 1-3-6-8-9 ) an d se e how i t relate s t o th e critical path.

Earliest start of Activity 1-3

Path 2 (Non-critical)

Critical Path Chain

0

Earliest finish of Activity 1-3 an d also v earliest \ start of \ Activity ^ 3-6

Earliest finish of Activity 3-6 \

\ \ \

\ \ \ \ >» *

Latest start of Activity 1-3 /

/ /

/

/ /

f * _

Latest Latest finish of finis h of Activity Activit y 1-3 an d 3- 6 also latest start of Activity 3-6

/ /

/

• V

I I ! ! 1-3

1-4

__L 1/2

3-6

_L 1

-g Float

4-5

_1_ V/2

1-3

= r

I 2

3-6

ii days

8-9

5-8 8- 9

Vii 3 Time Duration (Days )

Fig. 11.8 Path s of al l critica l an d some non-critica l activitie s draw n

along a time axis .

Some activities in Path 2 (i.e., 1-3 and 3-6) d o not represent activities on the critical path but some (e.g . activity 8-9) for m par t of the critical path.

Let us consider activity 3-6, which lies entirely outside the critical path.

Associated with each activity is its latest finish tim e (given at the end of the activity) an d it s earliest star t tim e (give n a t the star t o f tha t activity) . Th e latest finis h tim e o f activity 3- 6 i s 23/4 days (indicate d a t node 6 ) an d it s earliest start time is 1 day (indicated at node 3). We do not know the earliest finish tim e of activity 3-6, but this is easy to find. I f the activity commence s at the earliest start time, then it will arrive at its earliest finish tim e after th e duration o f the activity. That is to say:

Earliest finis h time of an activity

Earliest star t time of an activity

Duration of the activity

The earliest finish tim e of activity 3-6 is therefore 1 + V4 = 1V4 days.

CRITICAL PATH METHOD 1 6 7

Now we define the difference between the latest finish time and earliest finish time of an activity to be the float of that activity. The float indicates the spare time or leeway for a n activity . Such a float tim e is not availabl e along th e critical path. I n the above example , activity 3-6 ha s a float 2 3/4 - 1V 4 = 1V2

days. We can see that delay in the completion o f the activities along a non-critical path within the float available will not affect th e overall completio n time of the project .

This statement may be written:

_, Lates t finish tim e Earlies t finish tim e Float = r . . - r . . ol an activity o i an activity

The lates t star t tim e i s again eas y t o find . I t i s th e lates t tim e t o star t a n activity so that the activity finishes a t its latest finish time . That is to say:

Latest start time Lates t finish tim e Duratio n of an activity o f an activity o f the activity

The latest start time of activity 3-6 is therefore 2 3/4 - V = 2V2 days.

In summary, the float o f an activity can be found by :

either (i ) Floa t = Lates t Start Time - Earlies t Start Time or (ii ) Floa t = Lates t Finish Time - Earlies t Finish Time

The readers can verify th e following statements by themselves:

(i) floa t o f a critical activity = 0; an d (ii) floa t o f a non-critical activity > 0

Let us consider activity 3-6 and see how we actually calculat e its float . As explained in the backward-pass computatio n i n Section 11.2 , the figure i n the circle at node 6 (i.e., 23/4) is the latest finish tim e of activity 3-6. Also, as explained in the forward pas s computation, th e figure in the square at node 3 (i.e. , 1) is the earliest start time of the activity.

However, the floa t o f an activity is the difference betwee n th e latest finis h time and the earliest finish time . And the earliest finish time can be found by just adding the duration o f the activity to the earliest start time.

1 68 MODER N CONSTRUCTION PROJECT MANAGEMENT

In general then , the float o f an activity i-j can be computed a s follows:

(Float). = (Lates t finish tim e of i-j) - (Earlies t finish tim e of i-j )

= (Lates t finish time of i-j) - (Earlies t start time of i-j + Duration ofi-j(te))

= ( ( J o f node j) - ( I I of node i + te of activity i-j )

The float o f activity 3-6 is therefore: ( 2 3/4) - ( 1 + V4) = 1 V2 days.

4. Presentin g Activities in a Bar Char t

After analysin g a network using the critical path method, the activities of a project can be presented in a bar chart. The example in Section 11. 2 will be used t o illustrate this . Fig. 11. 9 is a bar char t representing th e activities of the project under consideration .

Activity

1-2

1-3

1-4

4-5

3-6

4-7

5-8

7-9

8-9

Description

Obtain materia l

Obtain mixer

Dig hole 1

Dig hole 2

Mix concrete

Set up pylon 1

Set up pylon 2

Concreting of

pylon 1

Concreting of

pylon 2

Days 1

I

2

I

I

3

1 Z~3

1

i

1 1

Fig. 11.9 Ba r chart indicatin g the schedule of the activities an d their floats .

Note tha t al l non-critical activitie s have been assumed t o be carried ou t a t their respective earlies t star t times . Floats of the activities are indicated o n the chart. Note also that critica l activities have no floats .