UNIT 12 PERTICPM -

Structure 12.1 Introduction

Objectives

12.2 Network Analysis

12.3 Guidelines , , ,. for Constructing Network Diagrams

12.4 ~ e t e r r n i n k ~ c .. . =me Estimates

12.5 Developing a Project Network

12.6 Project Duration and Critical Path

12.7 Forward Pass (Earliest Expected Time)

12.8 Backward Pass (Latest Allowable Time)

12.9 Float

12.10 Probabilistic rime Estimates

12.11 Probability of Project Completion by a Target Date

12.12 Summary

12.13 Key Words

12.14 Answers to SAQs

12.1 INTRODUCTION -- - - - - - - -

Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM) are two of the most widely used techniques in project management. The objectives of project management can be described in terms of a successful completion of project on time within the budgeted cost and adhering to technical specifications which satisfy the end users. A project is any human undertaking with a clear beginning and a clear ending. Planning, scheduling and controlling the work during any worth-while project is the main task for a project manager. Project planning calls for detailing the project into activities, estimating resource requirements and time for each activity and describing activity inter-relationships. Scheduling requires the details of starting and completion dates for each activity. Control requires not only current status information but also insight into possible trade-offs when difficulties arise. Normally, for any project, we may be interested in answering questions, such as (i) when do we expect the project to be completed; (ii) if any activity is delayed, what effect will this have on the overall completion time of the project; (iii) if there are additional funds available to reduce the time to perform certain activities, how should they be spent, and (iv) what is the probability of completing the project by the scheduled date.

Prior to the development of PERT and CPM, the most popular technique for project scheduling was the bar or Gantt Chart developed by Henry L. Gantt in around 1900. These Charts show a graphic representation of work on a time scale. A typical BarIGantt Chart has been shown in Figure 12.1. The primary limitation of this technique is its inability to show

, the inter-relationships and inter-dependencies among the many activities which control the progress of the project. Although it is possible to redraw the chart to show the inter-relationships, the confusion arises as the size of the project increases. To overcome such limitations, PERT and CPM were proposed in the late 1950s.

Project A I Project B --

Project c 1 project D ~

Figure 121 : BarIGantt Chart

Historically speaking, PERT and CPM developed independently out of research studies conducted by the U. S. Navy and DuPont company. While PERTwas an outgrowth of the U. S. Navy's Polaris submarine-missile programme, CPM was developed for planning the construction of chemical plants.

THe difference between PERT and CPM arose primarily because of the original job for which each technique was developed. Initially the PERT technique was applied to research and development projects while the CPM was used in construction projects. Both of them share the notion of a critical path and are based on the network analysis that determines the most critical activities to be controlled so as to meet completion dates. However, since the use of either technique is based on individual characteristics, the main difference is that PERT is particularly useful for non-repetitive and complex projects in which time estimates are uncertain. CPM is best utilised for repetitive and non-complex projects when time estimates can be made with some measure of certainty. The selection of the technique depends on the degree of uncertainty associated with time estimates and the cost of non-accomplishing time estimates. Actually, PERT restricts its attention to the time variable whereas CPM includes time cost trade-offs. For our purpose, we will not differentiate between the two techniques but we can further define PERT and CPM as the process of employing network techniques to optirnise the use of scarce project resources.

Objectives After studying this unit, you should be able to

appreciate problems involved in planning, scheduling and controlling of projects,

list and discuss the special terms developed for this unit, namely activity, event, dummy activity, critical activity, slack, critical path and float,

develop simple network diagrams with activities and events,

identify critical path through the calculation of the earliest expected time and the latest allowable time,

compute slack and float, and

estimate the probability of completing a project by a certain target date.

12.2 NETWORK ANALYSIS

A fundamental ingredient in both PERT and CPM is the use of network systems as a means of graphically depicting the current problems or proposed project. Because of its importance to a basic understanding of both PERT and CPM, the network concept will be examined. When a network is being constructed, certain conventions are followed to represent a project graphically. In a network diagram, it is essential that the relationship between activities a@ events are correctly depicted. Before illustrating the network representation, it is necessary to define some of the key concepts,

Activity

All projects may be viewed as being composed of operations or tasks called activities which require the expenditure of time and resources for their accomplishment. An •

activity is depicted by a single arrow ( -, ) on the project network. The activity arrows are called arcs. The activity arrow is not scaled; the length of the activity time is only a matter of convenience and clarity, and does not represent the importance of time. The head of the arrow shows the sequence or flow of activities. An activity c ot begin until the completion of its preceding activities. It is "(" important tha activities should be well-defined so that beginning and end of each activity can be identified clearly.

Event

An event represents a specific accomplishment in the project and takes place at a particular instant of time, and does not, therefore, consume time or resources. An event in a network is a time-oriented reference point that signifies the end of one activity and the beginning of another. Events are usually represented in the project network by circles (0). The event circles are called nodes. Therefore, the major difference between activities and events is that activities represent the passage of

time whereas events are points in time. All activity arrows must begin and end with event nodes as shown in Figure 12.2.

First Activity

Event

I figure 12.2 : Event-Activity Representation

1 Predecessor Activity

Activities that must be completed immediately prior to the start of anotheractivity are called predecessor activities.

Successor Activity

Activities that cannot be started until one or more of the other activities are completed, but immediately succeed them are called successor activities.

Cor~current Activity

Activities which can be accomplished concurrently are known as concurrent activities. It may be pointed out that an activity can be a predecessor or a successor to an event or it may be concurrent with one or more of the other activities.

Dummy Activity

In most projects many activities can be performed concurrently or simultaneousiy. It is possible that two activities could be drawn by the same beginning and end events. In situations where two or more activities can be performed concurrently, Ule concept of dummy activity is introduced to resolve this problem. Therefore there will be only one activity between two events. As a result of using the dummy activity, other activities can be identified by unique end events. Dummy activities consume no time or resources. By convention, dummy activities are represented by a dashed arrow on the project network. In Figure 12.3, both activities A and B have the same beginning and end events.

The network diagram given in Figure 12.3 is incorrect because it breaks the rule of assigning unique numbers to each activity for the purpose of identification. Vie network diagram shown in Figure 12.3 demonstrates the principle of using a dummy activity for overcoming the problem of parallel activities with identical start and finish events.

figure 124 : Use &Dummy Activities

Therefore, a dummy activity is created to make activities with common starting and finishing events distinguishable and also to identify and maintain the proper precedence relationship between activities.

OptiaizPtioa Techoiqmea-LI 12.3 GUIDELINES FOR CONSTRUCTING

NETWORK DIAGRAMS

(1) Each activity is represented by one and only one arrow in the network. Therefore, no single activity can be represented twice in the network.

(2) No two activities can be identified by the same beginning and cnd events. In such cases, a dummy activity is introduced to resolve the problem.

(3) Two events are numbered in such a way that the event of higher number can happen only after the event of a lower number is completed.

(4) Dangling must be avoided in a network diagram. This happens when precedence and inter-relationships of the activities are not properly identified.

(5) To ensure the correct logical sequence and inter-relationships, one has to answer the following questions satisfactorily.

(i) Which activities precede this activity? That is, what other activities must be completed before this activity can be started ?

(ii) Which activities follow this activity ? Or, what activities cannot be started until this activity is completed ?

(iii) Which activities can take place concurrently with this activity ? Or, what activities can be worked on at the same time when this activity is being performed?

12.4 DETERMINISTIC TIME ESTIMATES

The main determinant of the way PERT and CPM networks are analysed and interpreted is whether activity time estimates are deterministic or probabilistic. If time estimates can be made with a high degree of confidence so that actual time will not differ significantly from estimated time, we say, the time estimates are deterministic. On the other hand, if estimated times are subject to variation, we say the time estimates are probabilistic. First, we shall describe the analysis of network with deterministic time estimates and at a later stage with probabilistic time estimates.

12.5 DEVELOPING A PROJECT NETWORK

Networks of PERTICPM consist of two basic elements : activities and events. The network clearly shows the sequence and inter-relationships of all activities in the project. To illustrate how a project network can be developed, let us consider an example where a project involves several activities which are listed in Table 12.1 alongwith their predecessor activities.

Table 12.1

All the activities associated with the project can be combined into an integrated network of events and activities as shown in the network diagram in Figure 12.5.

Activity

A

B

C

D

E

F

G

The network diagram in Figure 12.5 gives the complete description of the project. For example, activities A and B have no predecessor activities, they can begin immediately and are shown coming out at the start event 1. You may 0 b s e ~ e that activity G has two predecessor activities D and E. Similarly other relationships are also satisfied.

Predecessor Activity

- - A

A

B

C

D, E

Figure 125 t Network Diagram t

SAQ 1 A car manufacturing company has decided to redesign its fuel pump for their new car model. This project involves several activities which are listed in Table 12.2. First activity is that the engineering department must evolve the design of fuel pump. Second, the marketing department must develop the marketing strategy for its promotion. Third, a new manufacturing process must be designed. Fourth, advertising media must be selected. Fifth, an initial production run must be completed. Finally, the pump must be released to the market.

Table 123 - - - -

Activity Description of Activity Predecessor Time Estimate Activity (Weeks)

A Evolve the design of fuel pump - 5

B Develop marketing strategy A 4

C Design manufacturing process A 7

D Select advertising media B 8

E Initial production run . C 9

I F Release fuel pump to market D, E 4 I Draw a network diagram for the given project.

12.6 PROJECT DURATION AND CRITICAL PATH

The longest path in the network is called the critical path. ldentifying the critical path is of great importance as it determines the duration of the entire project. If any activity on the critical path is delayed, then the entire project will be delayed. Every network has a critical path. It is possible to have multiple critical paths if there are ties among the longest paths. For finding the project duration and critical path, let us consider the example discussed earlier (Table 12.1). For this example, the time estimates (in weeks) for each activity are as shown in Table 12.3.

Table 123

Activity

G

Predecessor Activity . 1 Time Esthates (Weeks)

In the network diagram given in Figure 12.6, the time estimates (weeks) are specified inside the bracket alongwith the activity.

There are three possible paths for this network. For this simple network, the critical path is found by enumerating all of the possible paths to the completion time. These paths are listed in Table 1 2.4.

Table 12.4

(a) A-+C+F 4 + 2 + 4 = 10 weeks

(b) A + D + G 4 + 7 + 2 = 13 weeks

(c) B + E + G 3 + 6 + 2 = 11 weeks

The second path ( A 3 D -+ G ) is the critical path because it requires the longest period of time, i.e. 13 weeks for completion of the project. For this network, the project duration time to complete the project is 13 weeks. The activities on the critical path are called critical activities because a delay in any of these activities results in a delay of the entire project. In other words, there is no slack time in the activities on the critical path. Slack time is defined as the latest time an activity can be completed without delaying the project minus the earliest time ~e activity can be completed. Therefore, slack time is the amount of time an activity can be delayed without delaying the entire project.

For this small network, it is a simple process to identify the critical path by comparing all possible paths. As the number of activities increases, it may become very difficult and time consuming to find the critical path by complete enumeration or inspection. Therefore, we need to develop an algorithm (a systematic approach) to determine the critical path. The critical path calculations proceed in two phases. The first phase or forward pass begins from left to right through the network. The calculation begins at the start event and moves towards the end event of the project network. The second phase or backward pass begins from right to left through the network. In this phase, the calculation begins from the end event and moves backward to the start event.

- - - - - - - -

12.7 FORWARD PASS (EARLIEST EXPECTED TIME) . In forward pass, we compute the earliest time an event can be expected to occur which in turn depends upon the latest completion time of an activity terminating at that point. Thus, the longest path, in terms of duration times, is the earliest expected time for that event to occur. During the forward calculations, we need to compute the earliest expected time (ET) for each of the events which can be calculated as follows :

where,

ETj = the earliest expected time of event j

ETi = the earliest expected time that an activity can be started leading to event j

dij = duration time of an activity from event i to event j

Max = Maximum (of)

Let us use this computational procedure to determine the earliest expected time for eqch event for the network diagram shown in Figure 12.6. The use of this procedure suggests that the earliest expected time for a given event is primarily a function of the previous events plus the activity times of all prior activities. Applying this procedure, we get,

ETl = 0 ( starting event set at zero )

ET2 = E T i + d i z = 0 + 4 = 4

ET3 = ETi+dls = 0 + 3 = 3

= Max[11,9] = 11

ET6 = Max[ET4+&6,ETs+ds6] = M a x [ 6 + 4 , 1 1 + 2 ]

= Max [ 10,13 1 = 13

Note that the end event 6 occurs at the end of 13 weeks.

12.8 BACKWARD PASS (LATEST ALLOWABLE TIME)

In backward pass, we compute the latest allowable time (LT). The LT for an event is the latest time that the event can be delayed without delaying the completion of the entire project. The procedure we use in computing LT is to start from the end event of the network and proceeding backward to the starting event. The latest allowable completion time (LT) for a given event is calculated by subtracting the duration times of all activities coming into the event. In cases, where two or more activities start from an event, we must select the smaller of LT values. The LT value for an event in a network can be calculated as follows :

LTi = Min ( LTj - dij )

where,

LTi = the latest allowable time of event i

LTj = the latest allowable time of event j towards which activity ( i, j ) is headed

dij = duration time of an activity from event i to event j.

Min = Minimum (of)

Let us use this procedure to determine the LT value for each event in the network shown in Figure 12.6. To find LT values we begin at the end event of the network. The expected completion time for the project is 13 weeks.

In our example, since event 6 is project completion, it must occur no later than 13 weeks or else the project will be completed later than expected. Therefore, for the end event, ET = LT = 13 weeks. The computation of other LT values are shown below :

LT6 = 13

LT5 = LT6-d56 = 1 3 - 2 = 11

LT2 = Min [ LT4 - d u , LT5 - d25 ]

= M i n [ 9 - 2 , l l - 7 1 = Min[7 ,4 ] = 4

LTi = Min [ LT2 -,d12, LT3 - dl3 ]

= Min[4-4,5-31 = Min[0,2] = 0

It should be noted here that, by definition, at the beginning point of the network, i.e. event 1, we must have ET1 = LT 1 = 0.

Once the values of ET and LT for all the events are determined, we can easily identify the critical path. These values of ET and LT are listed around each event in Figure 12.7. If the values of ET and LT of an event are equal, then such an event is referred to as the critical event. If the values of ET and LT of an event are not equal, then such an event is referred to as noncritical events.

Figure 12.7

Critical activities can also be identified from the project network diagram. A critical activity is an activity which joins two critical events and has a duration which equals the difference between the times of these critical events. A critical path consists only of such critical activities. It may be pointed out again that critical activities are important because if they exceed their estimated durations, the whole project will be delayed to that extent.

An event that is not critical is said to have slack. Slack is the calculated time span within which the event must occur. The term slack is used only for referring to events.

As you would expect, every event on the critical path has no slack time. The critical path is shown by thj& lines in Figure 12.7. The importance of identifying the critical path is that it points out those activities and events which are critical and as such, must be carefully monitored and controlled.

SAQ 2 Consider the data of SAQ 1. Compute the earliest expected time and latest allowable time for the events in the given project, Also determine the critical path and slack time. Interpret your slack time values.

12.9 FLOAT

The concept of float is of great importance for a project manager. It is the time available for an activity in addition to its duration time. Since both start and end events of an activity have earliest and latest times, an activity has four associated times. Thus, there are four possible types of float but in practice only three of these are used.

Total Float

This is the time by which an activity may be delayed or extended without affecting the total project duration. This is computed as follows :

where,

TFij = total float for actidy ( i, j ),

LTj = latest allowable time for event j ,

ETi = earliest expected time for eyent i, and

dij = the time duration for activity ( i, j ).

Free Float

This is the time by which an'activity may be delayed or extended without delaying the start of any succeeding activity. This is calculated as follows:

FFij = ETj - ETi - dij

where, 1

FFl, = free float for activity ( i, j ),

I ETj = the earliest expected time for event j ,

ETl = the earliest expected time for event i , and

diJ = the time duration for activity ( i, j ) . Independent Float

This is the time by which an activity may be delayed or extended without affecting the preceding or succeeding activities in any way. This is obtained as follows:

I F I j = ETj - LTI - dq

where,

i FFiJ = Independent float for activity ( i, j ),

ET, = the earliest expected time for event j,

LTl = the latest allowable time for event i , and

dij = the time duration for activity ( i, j ).

SAQ 3 Compute total float, free float and independent float from the results you obtained SAQ 2.

12.10 PROBABILISTIC TIME ESTIMATES

Uptil now, we have discussed cases where the activity duration times were known with certainty. It is obvious that for most projects these activity times are random variables. PERT is more effective in handling cases in which activity duration times are uncertain. The PERT technique makes the following basic assumptions:

(1) Activity times are statistically independent and usually associated with a 'beta' distribution.

(2) There are enough activities involved in the network that the totals of activity times based on their means and variances will be 'normally' distributed.

(3) The three estimates of the activity-duration can be obtained for each activity.

The three time estimates are referred to as

(i) Optimistic time estimate denoted by a . (ii) Most likely time estimate denoted by m

(iii) Pessimistic time estimate denoted by b

The useful property of the beta distribution is that if we know the three time estimates (a, m and b) for an activity, we can compute mean or expected duration time ( t, ) and the

variance of duration ( o:, ) as follows :

a + 4 m + b te = 6 and ot = [y)

The shape of the beta distribution is skewed. It can either be skewed left or skewed right. The Figure 12.8 depicts a beta distribution which is skewed to the right.

To demonstrate the use of PERT, let us continue with the same example. Instead of activity times to be known with certainty, let the three time estimates beas shown in Table 12.5.

Probability

t

Optimistic Most likely tc -

Activity time

Figure 12.8 : Beta Distribution Curve

Table 12.5

1 I I Time Estimates (weeks) I

In order to find the critical path, we need to determine the mean or expected duration for each activity. Once this is done, the procedures you have already learnt can be applied to find critical path. The expected time (b), is shown in Table 12.6.

Activity

Table 12.6

Predecessor Activity

Nste that the expected time ( t, ) for each activity turned out to be the same as the single time estimate used earlier in this example. Obviously, no coincidence, this was intentiona done for the sake of simplicity. The calculations for standard deviation and variance are shown in Table 12.7.

Table 12.7

Optimistic

a + 4 m + b t, =

6

4

3

2

7

6

4

2

Activity

A

B

C

D

E

F

G

Activity Expected (mean) Standard Deviation Variance Time ( t, )

Most likely

Predecessor Activity

- -

A

A

B

C

D, E

Pessimistic a I m

E

F

(i *

b

Time (weeks)

a

2

2

1

4

4

3

1

* Critical activity

6

4

2

m

3

3

2

6

5

4

1

8/6 = 4/3

2/6 = 1 4

6/6 = 1

b

10

4

3

14

12

5

7 '

16/9

1 /9

1

The distribution of each activity completion time is 'normally' distributed. Thus, the expected activity times for critical activities are also normally distributed. The following

! table shows the critical activities along with expected time (k) , standard deviation (otJ and 2 variance (ate).

I

Table 12.8

12.11 PROBABILITY OF PROJECT COMPLETION BY A TARGET DATE

Critical Activity

A

D

G

Sometimes, the management would also like to know the probability of completing the project by a particular date. Let us assume that in our example, we are required to complete the project within 11 weeks.

We know that the expected activity times for critical activities are also normally distributed (central limit theorem).

Expected Time (04)

4

7

2

ot, = 13

Therefore, in order to find the probability of project completion by a target date, we can use the following formula :

where

Standard Deviation (otJ

4/3

5/3

1

x = target project completion time,

Variance

( ol)

16/9

25/9

1

E 02 =' 50/9

te = expected project comgletion time, and

ot, = standard deviation of activities on the critical path

In our example expected activity times follow a normal distribution with mean time ate = 13 weeks and standard deviation t, = 2.357 weeks. The target due date is 11 weeks. Using the formula, we get

Now, we can find the probability to any value of Z from the standard normal distribution table. The probability for the value of Z = - 0.85 is 0.8023. Since Z = - 0.85, we must subtract - 0.8023 from 1.0. Thus, we obtain 1 - 0.8023 = 0.1977. Therefore the required probability of completing the project within 11 weeks is 0.1977 or 19.77 percent.

Suppose now we are interested in finding the probability of completing the project in 16 weeks. Again using the formula, we get

The probability for the value of Z = 1.27 is 0.8980 from the standard normal distribution table. Thus, the probability of completing the project in 16 weeks is 0.8980 or 89.8 percent.

SAQ 4 Consider a project having activities and their associated time estimates as given in Table 12.9.

(a) Draw the project network diagram.

(b) Identify the critical path md compute the expected project completion time.

0pthnbrti00 T ~ ~ M ~ I I W - I I (c) What is the probability that the project will be completed on or before 55 days?

(d) What is the probability that the project will be completed after 70 days ?

Table 12.9

12.12 SUMMARY

Activity

A

B

C

D

E

F

G

H

PERT/CPM is a network technique that is very useful to a project manager throughout all phases of a project. An understanding of events and activities and an appreciation of the inter-relationships between them are necessary before a network for the project can be constructed. A network can provide information such as earliest expected time, latest allowable time, slack and critical path. Activity times may be deterministic or probabilistic in nature. PERT introduces probabilistic aspects to the project network. It uses three time estimates : Optimistic, most likely, and pessimistic. llle random characteristics of activity times are considered to follow beta distribution. The use of normal distribution assists the manager in determining the probability of project completion within a certain specified time period.

12.13 KEY WORDS

Predecessor Activity

-

A

A

B

B, C

E

D

F, G

Activity : A clearly definable portion of a project that requires for its completion, the consumption of resources and time in particular.

Time (days)

Critical Activity : An activity becomes critical, if delay in its estimated time duration delays the whole project to that extent.

Optimistic

2

8

14

4

6

6

18

8

Critical Path : The longest path through the network, consisting of critical activities. The length of the critical path is the shortest time allowable for project completion.

Dummy Activity : Dummy activity is an activity which does not consume any resource or time. It is used in network to show logical links between other real activities.

Most likely

4

12

16

10

12

8

18

14

Event

Pessimistic

6

16

30

16

18

22

30

32

: An event represents a specific accomplishment in the project and takes place at a particular instant of time and therefore does not consume resources or time.

Earliest Expected : The earliest time that an event can occur is on the latest Time completion of an activity.

Float : It is the amount of time available for an activity in addition to its duration time. Float is computed in relation to activity.

Latest Allowable Time : The latest time that the event can be delayed without delaying the completion of the entire project.

Slack : The amount of time by which the start of an activity may be delayed without affecting the overall duration of the project. Slack is computed in relation to events.

12.14 ANSWERS TO SAQs

SAQ 1

Please refer Section 12.5.

SAQ 2

Please refer Sections 12.7 and 12.8.

48 SAQ 3

Please refer Section 12.9.

I ~ T c Q d q U a - I I

FURTHER READING

(1) Buffa,E. S., 1990, Modern Production/Operation Management, Wiley Eastern Limited.

(2) Everen E. Adam, Jr and Ronald J. Ebert, 1986, Productions and Operations Management :.Concepts, Models and Behaviour, Prentice Hal1 International.

(3) Hadley, G. and Whitin, T. M., 1963, Analysis of Inventory System, Prentice Hall, New Jersy, USA.

(4) Levin, R. and Kirkpatrik, C. A., 1978, Quantitative Approaches to Management, McGraw Hill, Kogakusha Ltd., International Students Edition.

( 5 ) Mustafi, C. K, 1988, Operations Research, Methods and Practice, Wiley Eastern Limited.

(6) Peterson R. and Silver, E. A., 1979, Decision System for Inventory Munugement and Production Planning, Wiley, New York, USA.

(7) Taha, H. A., 1976, Operations Research :An Introduction, MaCMillan Publishing Co. inc., New York.

(8) Rao, S. S., 1984, Optimizarion 'Ilreory and Applications, Wiley Eastern Ltd., New Delhi.

(9) Saaty, T. L., 1961, Elements of Queueing Theory with Applications, McGraw Hill, New Yo*.

(10) Gupta, M. P. and J. K. Sharma, 1987, Operations Research for Management, National Publishing House, New Delhi.