Chapter 9 : Project management
9.1
The PERT method (Program, Evaluation and Review Technique)
This method can also be applied to carry out studies or projects.
It was perfected in 1958 for the setting-up of the POLARIS programme that was to
coordinate the activities of several thousands of organizations. It has since then been a great
success and is now used in a lot of firms, small or big, private or public.
At the same period, other methods appeared such as the Critical Path Method (CPM)
and the potentials method. All these methods have a lot in common. In our course, we will
only focus on the PERT method.
9.1.1 Definition of the method
The PERT method is a scheduling method or a programming and control technique ; it
enables to schedule and control the carrying out of a number of activities leading to the
realization of a determined project according to a fixed objective. The problem in project
scheduling is to determine when each activity must begin or end so that the entire project can
be completed in time.
Examples : building a new hospital, development of advertising campaigns, and so on.
In this context, to programme or to schedule means to draw up a detailed timetable
concerning the beginning and the end of the different activities and, of course, the beginning
and the end of the whole project.
To control means to monitor the carrying out of this programme, to modify it
according to the real evolution of the project and to draw interesting conclusions.
These programming and control activities are done according to a determined
objective or to a criterion we have to optimize.
This criterion may be
-
the time : we have to minimize the global lead time of a project.
the resources : we have to minimize the necessary resources by balancing the loads.
the costs : we have to minimize the global cost of the operation (PERT COST).
In a general way, we will first try to optimize the global lead time and from that
programme, we will examine the other criteria.
9.1.2
Network representation
9.1.2.1 Preliminary convention
The problem elements are of two types :
-
The tasks or activities are the elements that take some time, consume time.
-
The events are the elements that do not consume time. They indicate a stage in the project
realization.
As the model used in this method is graphic, it is important to agree upon the used graphic
symbols :
-
the events are represented by nodes, they are identified by a number or a letter.
-
the activities are represented by arrows, the length of which has no significance. They are
identified by a number or a letter.
9.1.2.2 First stage : problem analysis – identification of the factors
As we have to develop a programme of activities, the first stage consists in identifying
all the tasks we have to carry out in order to meet the objective. We thus draw a list of all the
necessary tasks.
Practically, this problem is not as simple as it seems to be. Any omission at that level
may completely modify the solution.
9.1.2.3
Second stage : data quantification and network development
(a) data quantification, that is to say research concerning the lengths of the activities.
We try to determine the length of each identified task in the first stage. The way we
can determine the length depends on the field of the problem :
-
determined field : if we deal with an activity that has a precise operating mode with a
standard time, we know the length : t.
-
predictable field : some activities of a repair shop have predictable lengths (ex : the
repairing of a crankshaft depends on the number of fissured bearings we find when we
dismantle it). We can give a statistical estimate of the length : L(t).
-
uncertain field : the length of some works (research works, works influenced by
meteorological conditions, and so on.) cannot be determined, even in a predictable way.
We can then proceed by giving 3 estimates as regards the length :
-
optimistic estimate (t0) : we assume that all circumstances are favourable ;
-
pessimistic estimate (t p ) : we assume that all circumstances are unfavourable ;
-
average estimate (t m) : we assume that everything happens as usual : some
circumstances being favourable, others not.
These estimates are at least partially subjective. We then combine these 3 estimates by the
following formula in order to find out the length of the task :
t 0 + 4t m + t p
t=
6
(b) network development
In this case, the network is a logical representation of the sequence of activities. This
representation is given the form of a chart. We talk about PERT diagram or network.
We can find out these relations by asking the 3 following questions for each activity :
What are the activities that must be carried out before the one I consider ?
What are the activities that immediately follow the one I consider ?
What are the activities that can be carried out at the same time as the one I
consider ?
Actually, these 3 questions are 3 different ways of formulating the same question.
(c) let us take a simple example
We have identified 7 tasks for which we have estimated the length ; we have asked the
above questions for each one of them. All answers and data are to be found in the following
table :
We notice that the task n°5, that lasts 5 days, must be preceded by the tasks n°1 and 4
and followed by the task n°6.
We also notice that the process starts with the carrying out of the tasks n°1 and 2, and
ends with the carrying out of the tasks n°6 and 7.
Task n°
Length (days)
1
2
3
4
5
6
7
5
4
3
2
5
8
5
Tasks that must Tasks that must
precede
follow
2
2
1,4
3,5
3
5
3,4
6,7
5
6
-
Let us represent all these elements as a diagram.
5
C
1
E
t=5
6
t=5
t=8
A
t=2
4
F
t=0
7
2
t=4
3
B
t=5
D
t=3
The event A, beginning of the operations, indicates the beginning of the carrying out
of the tasks 1 and 2 that are not preceded by any other one.
In the nodes that represent the events, we should leave spaces where we can write the
time data.
A pragmatic work method for the representation is the following :
Start with the start and end events with the arrows that respectively go from them and
come to them.
1
6
F
A
7
2
Then draw intermediate nodes with the arrows that come to them and those that go
from them.
4
2
B
3
Try to make the ends and arrows meet by eventually moving the nodes so that these
arrows do not cross each other.
If we follow the inferior arc of the scheme, we notice that the task 2 is followed by the
task 3, being itself followed by the end task 7.
Just as on the superior arc, the task 1 is followed by the task 5, followed itself by the
task 6. But the task 5 must also be preceded by the task 4 that follows the task 2.
So we imagine a fictitious task of no length between the events D and E in order to
express the relation between the tasks 6 and 3. In order to indicate that the task 6 can only
begin if the task 3 is completed : we divide the first node of the preceding drawing in two with
an arrow between the two (well-oriented !).
5
E
6
F
7
3
D
Finally, we write its length on each arc.
9.1.2.4
Third stage : network exploitation
We will describe the network exploitation using the example above. We must though
choose a criterion : let us assume that our problem consists in carrying out the 7 tasks in a
minimum time.
(a) For each event A, B, C, we determine the earliest start time (ST) of happening.
-
event A : it is the beginning of the operations, so it can happen at the time 0. We write 0 in
the left space of the node.
-
event B : it is the end of the task 2, the beginning time of the tasks 3 and 4, it can happen
at the time 0 + 4 D. The earliest finishing time is actually equal to the earliest beginning
time + the activity length.
-
event C : it is the end of the tasks 1 and 4, the beginning of the task 5, it can happen at the
earliest after 6 D because, although the task 1 only lasts 5 days, the task 4 that follows the
task 2 can only be completed after 4 + 2 = 6 D. Both activities must be completed.
And so on, we then know that the whole operation cannot be completed in less than 19
days.
It is a first result but maybe not the most interesting one.
(b) Let us once again consider the events one by one, starting with the last one, and
calculate the latest start time (LS) of happening or, in other words, the moment when it can
still happen so that the whole operation is all the same completed in 19 days.
The event F can and must be completed in 19 D, we write 19 in the middle space of
the node.
In order to have completed F in 19 D,
-
the event E must happen at the latest after 11 D, we write 11 in the middle space ;
the event D must happen at the latest after 14 D as the activity 7 lasts 5 days but we must
not forget that D must also precede E that can at the latest happen after 11 D. Thus, the
latest time for D is also 11 ;
the event B must happen at the latest after the shortest of the 2 times given by the arrows
that go from it, that is to say 11 – 3 = 8 or 6 – 2 = 4, so after 4 D ;
the event A must happen at the latest after the shortest of the 2 times given by 4 – 4 = 0
and 6 – 5 = 1, that is to say 0 D.
C
1
t=5
6 6
E
5
0
t=5
t=8
A
1
0 0
6
11 110
F
t=2
4
0
2
t=4
t=0
7
B
4 4
3
0
t=3
D
19 190
t=5
7 110
(c) We notice that for some events, the earliest and latest times are the same.
There is no slack for these events : they are critical. Any delay in a critical event
automatically leads to a delay in the carrying out of the whole operation.
A task may be considered critical if it meets these 3 conditions :
-
the start event is critical ;
the end event is critical ;
the difference of time between its end event and its start event must be equal to its
carrying out length.
The whole of the critical activities constitutes the critical path of the project.
As regards the carrying out of the non-critical activities, we have a slack. In the
example, the activity 3 lasts 3 days. It can begin at the earliest after 4 D and end at the latest
after 11 D, we thus have a slack of 4 D.
C
1
t=5
6 6
E
5
0
t=5
11 11 0
t=8
A
1
0 0
F
t=2
4
0
t=0
7
2
B
t=4
4 4
9.1.2.5
6
3
0
t=3
D
19 19 0
t=5
7 11 4
Fourth stage : solution analysis
(a) The developed network provides us with management information:
- the minimum necessary time to carry out the whole project ;
- the critical path.
The model is thus a control tool. It indicates the critical activities, that is to say those
that require a particular attention. It also indicates the moment when each activity of
the critical path is to begin or to end.
(b)
As regards the non-critical activities, it is interesting for the person who organizes the
production to know the slacks, as they indicate the limits of his freedom to act
regarding the carrying out of the project. We therefore have to distinguish 3 slacks. If
we consider an activity ij limited by the events i and j, we have :
i
j
1
ti Ti 0
0
The total slack = mt
mt
t = tij
1
tj Tj 0
= Tj - ti - tij
The free slack : is the maximum delay we can accept for an activity, assuming that it has
begun at the earliest, without preventing the beginning at the earliest of the following
activities : mf = tj - ti - tij
The independent slack : is the maximum delay we can accept for an activity, assuming that it
has begun at the latest, without preventing the beginning at the earliest of the following
activity : mind = tj - Ti - dij
If we consume the total slack, it influences the two events i and j and can lead to the
appearance of a new critical path or to the modification of the critical path. It is thus the
coordinator of the different activities who can handle the total slacks.
The free slack does not influence the following activities but it is only completely
available if the preceding activity is carried out in time in order to enable ij to begin at the
earliest.
The independent slack completely belongs to its activity and is not influenced by the
preceding or following activities. It thus constitutes a freedom of action that may entirely be
given to the person in charge of the carrying out of an activity.
Slack calculation is as important for the planning of a project as for identifying the critical
path. A judicious study of the slacks will enable an optimal use of the available means.
During the carrying out, the person in charge knows where the reserves that can be directed
towards the critical areas are.
The model also shows us how we can modify the problem data in order to improve the
global result. For instance, if we allocate more resources to a task of the critical path by
removing them from another non-critical task, we can maybe shorten the length of the whole
operation.
9.2 GANT's diagram
9.2.1 Description
When we have finished calculating the PERT network, it is interesting to show our
results in the form of a diagram representing the activities according to time.
The time is represented on the horizontal axis, the activities on the Y-axis. The
diagram indicates the beginning at the earliest, the length and the end at the latest of each
activity.
The following diagram represents the activities of our example.
Tasks\Time
1 2 3 4 5 6 7
1
2
3
4
9.2.2
Optimization of the use of resources
GANT's diagram is a real timetable of the activities and is thus a very practical means
to control the carrying out of a project.
But it is even more a tool enabling the optimization of the use of resources by
balancing the loads (workforce, machines, transport means, and so on.).
EXAMPLE
-
Summary table of the tasks
Task
1
2
3
4
5
6
7
8
9
Length
6
4
5
3
7
4
4
6
2
Preceded by
2
1, 5
2
1, 5
3,4
6, 7
3, 4
Followed by
4, 6
3, 5
7, 9
7, 9
4, 6
8
8
-
-
PERT diagram
C
1
E
6
11 11 0
t=4
18 18 0
t=6
t=6
A
1
0 0
t=7
5
0
8
F
4
t=3
t=4
7
24 24 0
9
2
3
B
t=4
4 4
t=5
0
D
t=2
14 14 0
Minimum length of the project : 24 days.
Critical path : task 2, 5, 4, 7, 8.
-
GANT's diagram
Let us assume that the staff is able to carry out any of these tasks, the problem is to
spread these tasks over the time in order to minimize the necessary number of persons,
enabling a continuous rotation and respecting the fixed global lead time. GANT's diagram
will help us to solve this problem.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1
2
3
4
5
6
7
8
9
Task: 2
5
4
7
8
The activities of the critical path can follow one another on one time axis (1).
Concerning the other activities, we have to make them slide between the earliest and latest
moments in order to minimize the maximum number of activities on a same vertical, that is to
say to minimize the number of activities that must take place at the same time or, in other
words, minimize the number of teams.
In this example, we have been able to limit the number of teams to 2.
9.3 Extension of the method : PERT-COST
In the examples we have up to now seen, we have agreed that time was the criterion.
We may though choose other criteria and thus extend the application field of the method.
To minimize the costs is often an essential objective and many variants of the PERT
method also aim at this objective.
We will quickly describe the principle of a PERT-COST :
(a)
As in the PERT method, we first assess all the necessary tasks.
(b)
We then quantify by estimating the length of each identified task. Moreover,
for each task, we try to determine the relation between the cost of the task and
its carrying out length. In most cases, this curve looks like the one below.
Cost
Length
A length that is too short implies the use of numerous resources and expensive means
(overtime hours or shifts). A length that is too long brings about a bad use of resources and
waste. There is usually an optimal length to which a minimal cost corresponds.
(c)
We then establish the model as in the PERT method by selecting the optimal
lengths (except if there is a constraint for one length). By doing that, we
already have the minimum cost for all the tasks that belong to the critical path,
as they are carried out one after the other in their optimal length.
We then analyse the other tasks step by step in order to reduce the total cost.
Conclusion
The PERT method is a planning tool (setting-up of a programme of activities and
determination of the lead times), an organization tool (allocation and minimization of the
resources), a coordination tool (underscoring of the relations between the activities), a control
tool (assessment of the project progress in practice) and a management tool.
Project management : exercises
1.
Give the main characteristics of the PERT method.
2.
What is a critical activity ? What are the conditions for a task to be critical ?
3.
With the graphic method of the CPM, find out the minimum length of the project
having the following tasks :
Task
1
2
3
4
5
6
7
8
9
10
Length
5
4
3
5
6
2
3
4
5
4
Precedence
1
2
2
3, 4
3, 4
5
6
7, 8
Draw GANT's diagram and the critical path.
4.
During the carrying out of a project, what does happen if a machine that is
necessary for a task of the critical path breaks down ? What are your
solutions ?