No o 37 PROGRAMME EVALUATION AND REVIEW TECHNIQUE

(An initiation to PERT-TIME)

by Jacques Hallak

IIEP/TM/57/69 Paris, April I97O

INTERNATIONAL INSTITUTE FOR EDUCATIONAL PLANNING Э» rue Eugène-Delacroix

Paris l6e, France

PROGRAMME EVALUATION AND REVIEW TECHNIQUE

(An initiation to PERT-TIME)

by

Jacques Hallak

This lecture is part of 'Fundamentals of Educational Planning! Lecture^ Discussion Series', a controlled experiment undertaken by the International Institute for Educational Planning in collaboration with a limited number of organizations and individuals aiming at the development of efficient teaching materials in the field of educational planning« By their very nature these materials, which draw upon tape recordings, transcriptions and summary notes of seminars, lectures and discussions conducted by HEP as part of its training and research programme <, are informal and not subject to the type of editing customary for published documents« They are therefore not to be considered as 'official publications'»

The opinions expressed in this lecture are those of the author and do not necessarily represent the views of the Institute.

The use2 adaptation or reproduction^ in whole or in part, of these materials is limited to institutions and persons specifically authorized by IIEP.

i

IIEP/TM/37/69

CONTENTS Pases

I. GENERAL PRINCIPLES 1 le Definition 1 2«, Programme analysis 2 3« The construction of the network 6 Conclusion 9

II. PERT-TIME 10 1. Estimating the duration of the activities

of a project 11 2e Time calculation 13 3«. The concept of slack or float 16 Ц-* The critical path 18 5 » The probability of respecting a timetable 20 6e An example of application: research project

on education administration 23 (a) Preliminary over-all analysis 23 (b) Detailed analysis 25

Selected Bibliography 33

ii

XIEP/TM/37/69 ~ PaSe 1

I. GENERAL PRINCIPLES

Programme Evaluation and Review Technique,, commonly called PERT, used on a large scale by the United States Navy Department in 195'У* has since been extended into other fields and in other countries. Thus, in France, for example, the PERT Method, and its variants (Critical Path Method, CPM) are beginning to be widely used»

The need to introduce these methods was originally felt owing to the increasing complexity of the projects which had to be handled in certain spheres* Substantial savings of time and money have thus been made possible and the field of application of methods of time and cost control is progressively widening and extending into the most diverse sectors of activity.

This is borne out, for example, by the recent adoption of PERT for major educational research and development programmes in the United States.

While recognizing that, at first sight, Programme Evaluation and -Review Technique might seem too complicated for general application to all education programmes in the developing countries, the fact nevertheless remains that it can be profitably applied in certain circumstances. For example, the programming of school building projects can usefully be based on the PERT method! similarly, the organization of examinations on a national scale, owing to its cost and complexity, is worth basing on a PERT network^ furthermore, the introduction of a reform or innovation into a school system should be carefully planned, and simplified versions of PERT methods will probably prove very valuable in planning the timetable of operations. One last example of the potential use of PERT methods is often suggested, namely for the preparation of education plans.

The object of this series of two talks is to introduce the PERT method in such a way as to show its possible applications and its limits. This first part is thus devoted to presenting the general principles of the PERT method, and the second will contain an initiation to PERT-TIME5 which is one of the variations of Programme Evaluation and Review Technique.

L Definition

Like all new techniques, PERT is the subject of animated discussion in business management circles, the public service and wherever its appli­cation is a topical issue. Some people regard it as an ideal solution to all planning problems, and the remedy against inopportune decisions, while others are less categorical. These conflicting attitudes often arise out of a faulty definition of PERT.

IIEP./PM/37/69 ~ Page 2

PERT is an instrument of management designed to define and co­ordinate what has to be done to achieve the objective set within the time fixed. It is therefore a decision-making tool.

The method integrates a considerable number of statistical datai it brings out the uncertainties attached to the accomplishment of any specific task and shows, thanks to the technique of simulation, the possibilities of adjustments to meet the time requirements or to economize resources; it is based on strict mathematical principles, while not de­manding from its users any particularly advanced scientific training»

PERT is based on the theory of graphsj it is expressed in graphic diagrams in the form of a network of arrows resulting from the analysis of a programme. These diagrams support the elements used to establish the calculations.

It follows from this definition that, in order to introduce the general principles on which the PERT method is based, it is necessary to consider (i) the programme analysis phase and (ii) the phase of construct­ing the network and the significance of these different elements.

2. Programme analysis

Analysis consists essentially in specifying the precise objectives of the programme and breaking them down into successive stages, increasingly detailed, until the breakdown shows all the phases of advancement. The task therefore is to define a general objective of the programme and sub­divide it into partial objectives or successive phases of advancement, as shown in the following diagram.

DIAGRAM No. 1

Objective of the programme

Partial Objective 1

Partial Objective 2

Etc

Partial Objective J>

Etc

Stage 1 Stage 2 Stage 3

Etc

ИЕР/ГМ/37/69 - Page 3

Naturally, the breakdown can be carried as far as desired, but what is essential at the level of the Programme Director is to be able clearly to specify at the start what is the objective and what major elements he desires to analyse, while leaving the analysis of each element to the person responsible. The levels and content of this objective., the struc­ture of the company or public authority, the key elements which the person responsible has picked out, existing commitments, etc, are among the fac­tors which determine the choice of the breakdown sections and the assign­ment of responsibilities.

The scheme of analysis is therefore not limited to defining objec­tives and bringing out their relations, it also shows various degrees of complexity for each degree of responsibility. At the bottom of the scale, at the level of the execution of elementary activities, is found what, in PERT jargon, is called a ?pre-stage?„ The pre-stage is the stage which precedes the phase of advancement of an elementary event in the programme.

We would note here and now, and we can confirm it later, that once the activities reach a certain number, a programme cannot be handled with­out the help of electronic machines. It is precisely owing to computers that the field of application of PERT has developed in all spheres.

In brief, programme analysis means answering the following questions?

(i) What is the objective and what is the starting point towards achieving it?

(ii) What are the major intermediate events to be achieved in order to carry out the programme?

(iii) What activities are necessary to achieve these events?

Analysis therefore amounts to defining the key points of the pro­gramme, that is to say, its origin, its objective, its events and the activities to be carried out.

It is useful, at this stage3 to define very precisely what is meant by events and activities in PERT terminology.

An event is the beginning or end of a task, and an activity is the execution proper of the task. Thus 'drafting a report' is an activity and not an event, while, on the other hand 'report completed' is an event and not an activity.

IIEP/PM/57/69 - page 4

PERT events, therefore, unlike activities, consume neither time nor resources. Furthermore, each event except the first event (or events) and the last event (or events) are bound to be the beginning and the end of one or more activities,, Similarly, every activity starts with an event and ends with an event.

In a sense, therefore, activities constitute a constraint in the achievement of events, since an event cannot be deemed to be achieved (or attained) unless the activities which lead to it (or predecessor activities) are accomplished. The result is constraints between activities starting from (or successor to) an event and the predecessor activities,(l)

The analysis phase is precisely designed to arrange events and activities and to bring out the constraints between the different elements in the analysis scheme. The best way of checking the arrangement is to make sure that the order of succession adopted is logical. In this connexion, there are two types of difficulty to overcome. In the first place, certain constraints may correspond neither to activities nor to events. They must nevertheless be taken into consideration in the analysis, in so far as they govern the achievement of subsequent events. These are what are called 'dummy* activities, which consume neither time nor resources; they represent constraints of liaison.

Secondly, it must be made certain that the constraints defined do not make it impossible to achieve the objective of the programme. This happens, for example, when the successor activity to event В is at the same time predecessor to event A, which precedes B.

Generally, the critical analysis of the lists of events and activi­ties makes it easy to find the logical order of succession. But as soon as there are a great many events or activities, there is a danger of errors or omissions and a certain discipline then becomes essential. Several procedures have been suggested; the simplest is to prepare a double entry table, or matrix, interlinking events and activities.

More precisely, the first thing is to find the initial event (or events). It is easy to identify. It is the one which has no predecessor events. The next thing is to look for all the activities subsequent to the initial event. Each of these activities necessarily leads to an event. The question then is, what is the starting event from which that activity can be undertaken? We thus get a table of events completed by immediately preceding or following activities. The table is checked (and, if necessary, completed) by examining the activities preceding each event.

(l) The event or events immediately following another event with no inter­mediate event or events, are called Successor events?. The event or events immediately preceding another event with no intermediate event or events are called fpredecessorч events.

ИЕР/ГМ/37/69 - Pag© 5

This very rapid process allows the construction of a matrix showing the immediate predecessor activities and the immediate successor activities of each event.

An example will help to illustrate the method. The following table lists the events and activities of a sub-project without necessarily arranging them in logical order.

Events

A. Final English examination passed.

B. Final history examination passed.

С Enrolments in English and history classes completed.

D. Begin enrolments in English and history classes.

E. Examiners' awards completed,

F. Certificates issued.

Activities

a. Attend history courses.

b. Enrol students.

c. Correct English papers.

d. Attend English classes.

e. Correct history papers,

f. Prepare certificates.

The following matrix shox s the final result of the analysis of the sub-project. It indicates the logical sequence of events and activities.

D

С

A

В

E

F

D

Objective

С

b

A

d

В

a

E

с

e

F

f

We have described the analysis of the project. The next phase in the method is to prepare the graph or construct the network.

ИЕР/ГМ/37/69 - page б

3. The construction of the network

As we have seen, РЕЕТ is based on a network of arrows prepared by using the results of the analysis.

The principle of constructing a network is very simple, It uses the properties of vectors. It may be recalled that a vector is a segment of a continuous line* having a beginning,, an end, a direction and a value (often measured by the length of the vector). In the PERT method, the vector represents an activity, its beginning the predecessor event, its end the successor event and its value both time and cost«

The graph of a project is a network made up of a succession of vectors with common points of origin and termination«, In this network the events which are the static elements of the project are represented by squares or circles. The order in which the activities must take place determines the arrangement of the vectors on the graph.

It is quite evident that the transcription of activities and events on the graph would be very clumsy without the aid of a coding system.

The habit has been adopted, for ease of presentation, of using letter or figure codes. In simple cases, moreover, one single code, for events only, can be usedj under the conventions adopted under the PERT method, all activities can be identified by the code from the initial event to the terminal event. For example, the activity which starts from event X and finishes at event Y is coded X-Y. Generally, however, a very elaborate coding system must be worked out, and it is usual to prepare very carefully a reference document explicitly clarifying the code and to make it available to all participants in the programmed project.

To construct the graph of a programme, it is very convenient to use the double entry table prepared in the programme analysis phase. It is enough to indicate the sequence of events and activities in the order established in the analysis table or matrix. It is also possible to pro­ceed by way of arranging the events in generations.

First generation events are achieved at the end of a single series of activities starting from the origin of the programme. Second genera­tion events are achieved at the end of two series of activities starting from the origin of the programme. The events of the x th generation are those which are reached by a path consisting of x series of activities.

ИЕР/ГМ/37/69 - page 7

By way of illustration,, the example of a sub-project given under (2) is represented by Diagram No. 2.3 under the method of arranging events in generations.

Origin 1st generation 2nd generation Jvd. generation 4th generation D C A E F

DIAGRAM No, 2

D b

Ш J^.

A more simplified graph is generally accepted and we shall use it in the following pages,

DIAGRAM No. 3

(with or without express mention of activities)

The following may be noted on the graph thus constituted?

two sequences of activities leading from the initial event to the objective event. These sequences constitute paths (DCAEF and DCBEF);

event E is not achieved (or passed, since it is of nil duration) unless activities с and e are accomplished!

- that all events and all activities are clearly shown on the graph.

To sum up,, once the construction of the graph is completed, it is possible to proceed to check the project analysis«

IXEP/PM/37/69 - page 8

In the first place, the paths plotted on the graph must be followed from the origin to the objective to check the logic of the sequence adopted.

Secondly, in checking the paths it will be easy to check that no impossibility is being introduced. This impossibility might be reflected on the graph by a loop. Diagram No, 4 illustrates a case of impossibility (or loop); this case is obviously very easily brought out on the graph.

DIAGRAM No. 4

Furthermore, the checking of the graph sometimes suggests an adjust' ment of the scheme of analysis, in so far as it leads to replacing the presentation of a sub-set of events 'in seriesf by its presentation ?in parallel s. This substitution generates savings and amounts to a realloca­tion of the resources of the project. Diagram No, 5 shows the time saved by such a substitution,

DIAGRAM No, 5

,0 ,© ,0 0) *0- - -

.0^ ^0 *© O I 2 3 4 5 Months

The check of the paths may also show that some of them do not lead to the objective event. This means;

either that the objective achieved is a partial objective of the programme and that there is at least one dummy activity which must be associated with it, either towards another event or towards the objective?

or that the objective attained bears no relation to the final objective, in which case certain activities must be eliminated.

IIEP/tDM/37/69 - page 9

DIAGRAM No, 6

0 - 0 •0-0-0-0-0

.0-0. ->0—0—0—0—0

Finally5 a 'reverse check% of the graph must be made, going back­

wards from the objective event to the initial event«, centred mainly on the necessary conditions for the start of each activity.

Conclusion

Those are the broad outlines of the general principles of the PERT method., From these principles., it clearly follows that PERT consists essentially of analysing logically and in detail all the events and activi­ties which go towards the execution of a project in order to programme them with optimum effect«, This is enough to introduce the potential uses of PERT in educational planning. It is quite certain., for example that school building projects may constitute a preferential field of application for PERT«, Another field is the execution of applied research or experimen­tation projects. Similarly., the preparation of education plans., owing to its increasing complexity and the fact that it calls upon a considerable number of participating agencies^ may be built around planning based on PERT, Finally} the application of techniques allied to PERT to the adminis­tration and management of higher education establishments seems also destined to extend and to become progressively more firmly established«,

For all these reasons., we have thought it valuable to introduce this method by indicating its general principles and concepts«, In order to carry this initiation a little further., we think it worth going rather more deeply into one of the variants of the PERT method, the application of which raises relatively few methodological difficulties^ namely PERT-TIME5 which is dis­cussed in Part II,

-> Objective

Dummy activity

-> Objective

IIEP/TM/37/69 - page 10

II. PERD-TIME

A building promoter has lost control of his site when he can no longer respect the time and cost of its execution» A project is properly controlled when its director can answer with precision the two questions? When will it be finished? and What will it cost?

The two questions are obviously interdependent, since direct costs are linked to the time taken to execute the project, and the most natural idea is to use PERT as a method of controlling time and cost, known as PERT-COST,

Time data can, however, be dissociated from cost data, since in practice, it is often more important, or at least more significant, to check execution time, once costs have been precisely established,, Techni­cally, in particular, the method of controlling time is less complicated to apply than that relating to costs» In the following pages, therefore, we shall confine ourselves to PERT-TIME, explaining the method and illustrating its use by a concrete example.

It is obviously no part of our intention to train PERT-TIME specialists, but merely to give a few indications about the method so that the educational planner can assess its possibilities and, where appropriate, call in its help in executing the projects for which he is responsible» For this reason, this part will not be an exhaustive account of PERT-TIME, but will be centred on certain specially important aspects of its applica­tion, namelyj

(i) estimating the duration of the activities of a projects

(ii) calculating timej

(iii) the concept of ?slack* or ffloat?j

(iv) the 'critical path?i

(v) the probability of keeping to a timetable»

Lastly, and to conclude this study of the PERT method, we shall consider a concrete case by way of illustration»

IIEP/PM/37/69 - page 11

1„ Estimating the duration of the activities of a project

After the project has been analysed and the network or graph has been constructed (see Part l) it is valuable to complete this network by indicating the duration of the various activities identified in the project. In practice, as has been shown, while events consume neither time nor money, activities, on the other hand (other than dummy activities) consume time«

It is obvious that a difficult and fundamental task in the applica­tion of PERT is to obtain accurate information on the duration of activities.

It is difficult., because the technicians in charge of elementary activities may in fact not know the true duration of the activities for which they are responsible) they may also be reluctant to give this kind of information,, either because they are afraid of committing themselves to time limits without being able to observe them, or because, rightly or wrongly, they see signs of a desire for closer control on the part of the central agencies and the Direction.

It is fundamental because, in the last analysis, if the information about the duration of activities is wrong, the whole PEHP-TIME method be­comes ineffective» What, indeed, is the use of drawing up timetables and programming activities, if these timetables and programmes are more academic than practical? If duration is systematically over-estimated, to ensure that the timetable is respected, whatever happens, does this not amount to making the application of PEPO? meaningless?

PERT-TIME specialists therefore attach great importance to the phase of estimating the duration of activities and suggest that this information should be collected 'at source1, that is to say, by direct inquiry from the executants of the project, on the principle that the best estimate of the duration of an activity comes from the person responsible for carrying out the activity.

In order to overcome the uncertainties inherent in any estimate, it is recommended that three different values should be calculated for the duration of each activity«,

Optimistic estimate; this is the shortest time needed to carry out the activity on the assumption that everything goes better than planned,

Pessimistic estimate; this is the maximum time in which the activity can be completed assuming чeverything goes badly1.(l)

(l) Not allowing for major eatastrophies.

IIEP/tPM/37/69 - Pa§e 12

Most likely estimate; this is the most likely time for the com­pletion of the activity. It is the time which would have been given in any event if the executants had been asked to give a single estimate instead of three.

Each of these times is expressed in days., weeks or months. The estimates represent calendar days and not working days. Once established,, these estimates are firm and cannot, in principle be varied, unless the nature of the activity changes or the time for the supply of resources is varied.

In spite of their somewhat subjective character, the three time estimates can be used as a basis for calculating a magnitude - the standard time - which obeys a law of probability! it is therefore possible to analyse the chances of completing a project within the available time; we shall revert to this beloi*.

If fa? is the optimistic estimate, %\>4 the pessimistic estimate and 'mf the most likely estimate, the standard time, fte' is given by the formulai

te = a + 4m 4- b (l)

In other words, the standard time represents a statistical mean between the three estimates faf, ?mf and 'b!. For the statisticians this means that ?te* is the standard time which the activity would take if it were repeated a great many times. It should be noted that if the values given to 'a' and ?b? are such that; m - a = b - m, the standard time will be equal to the most likely time. The standard time will lie between 'a* and *т? or between ?m' and *Ъ* according as m - a is greater or less than b - m. Lastly, it is quite clear that the closer the values of !a' and *b( to 'mf the more precise will be the time estimates. The following graphs illustrate the remarks made on ?a?, ! m !b* and 4e

DIAGRAM No. 7

(l) If a = 9, b = 19 and m =

IÏEP/DM/37/69 - page 13

Uncertainty about time is greater in case SC' than in case ?A'. Furthermore* to measure dispersion., or uncertainty, in statistics, it is usual to calculate what is called the variance^ which is given by the equation tf 2 _ / i2

The higher the variance, the greater the uncertainty,

2. Time calculation

Once the standard times have been estimated for all the activities of the project, they are plotted on the network above the activity arrows, as shown in the following diagrams

(a) Early start date (or fearliest expected time')? once the times are shown for each operation on the graph, it is possible, by following the paths which make it up, to calculate the dates at which the various events of the project can be achieved. If there is more than one path leading to an event, the event is obviously not achieved until all the activities pre­ceding it have been completed. The date which corresponds to the time of the activities on all paths is the !earliest expected time* of the event. For example, in the above diagram, the earliest expected date of event k is 9 (i.e. 2 + 7 ) and not 7 (i.e. k + 3).

Thus, the ?early start date' or 'earliest expected time1 for reaching an event is obtained by taking the maximum time for the activities on all the paths leading to the event.

If ?ТД' is the earliest expected time of ?A? and ?d? is the duration of activity fABf, the earliest date of В is obviously T. + d. But if the two activities fAB' and ?CB? lead to ?B*, the activity of fABf will be com­pleted at the date T. + d , and the activity !СВ? at the date T + d j the earliest date for event В will be the later of these two dates.

IIEP/TM/37/69 - page 14

It is thus possible, step by step* to calculate the earliest dates for all the events in the project, and thereby, for the objective. These times, calculated in days, weeks or months, can equally be expressed in calendar dates, subject to the proviso that the starting date of the pro­ject is fixed»

Exercise

What is the earliest expected time of event 251 in the following network?

Te=0

There are three paths from event 2 to event 251,

(i) (T) (ZL) (2b) (251) (ii) (2^) (22 J [2bj (251.

(iii) ( 2 ) [2-yj (25J (251)

T°l = T2 + 1 = 1 T22 = T2 + б = б T23 = T2 + 12 = 12 T24 r= Maximum of (T21 + 13 or T22 + 11) = 17 T25 = T23 + 10 = 22 T25I = Maximum of (T24 + 9 or T25 + 7) = 29„

11ЕРД,М/37/б9 - page 15

(b) Late start date (or 9latest allowable time*)J in general, the question of time and completion dates arises differently. A time limit for completing the project is decided upon, and it then remains to fix the flatest allowable times* for the successive events of the project«, In other words, the completion date for a project being specified by contract, what is the latest date for reaching each event? Those responsible for the dif­ferent activities are all concerned by this question since they want to know the time limit within which they must complete the activities for which they are responsible«, To arrive at the latest allowable time., or late start dates for each event, it is necessary to start from the objective date and work back to the originj by subtracting from the duration of each preceding activity its latest allowable date, we arrive at the latest allowable oíate of the preceding event. If, however, more than one value is obtained for the latest allowable time of an event, the lowest is obviously chosen* For example, if two activities start from event SA? the latest date for the start of fACs will be T?c - dc and the latest date for the start of fAB?

will be T'B - dB, The latest date at which event 9A* must be completed is therefore the earlier of these two dates.(l)

It is thus possible, step by step, to arrive at the latest dates for all the events of the project.

Exercise

Taking the network given above, assume that the directed date for 5251* is 50. In order words, Tf 251 = 30, Calculate the latest dates for event 2„

0

(l) The symbol T is generally used for the earliest dates and the symbol T for the latest dates.

TIEP/ÏM/37/69 - Page 1 б

The following table shows the procedures

Event No. Subtract te T?

251

25 ?A 21 22

23 2

30-

30-7 30-9 21-13 21-11

23-10 8-1

10-6

13-12

30

23 21

8

10

13 1

3. The concept of slack or float

Once the latest date and the earliest date for each event have been assessed., sufficient information is available to calculate the slack or ?floatf available for achieving each event. If the earliest date and the latest date coincide, there is no margins the slack is nil, and any delay in reaching the event will have a repercussion on following activities.

If, on the other hand, the latest date is subsequent to the earliest date for an event, the difference between these two dates constitutes a margin of manoeuvre for the project manager, since he has a certain latitude in carrying out the activity in question or supervising the event. The value of the slack T* - T is then positive, and measures the magnitude of the available time margin.

This concept of slack applied to events, may also apply to the activi­ties of a programme. There may thus be a slack between the earliest starting date and the latest starting date of an activity. Similarly, there may be slack betitfeen the earliest finishing date and the latest finishing date. The simultaneous intervention of different types of slack very soon makes the network of a PERT project very complicated* Without wishing to go too deeply into this idea of slack, I nevertheless think it useful to show where its application may lead in two important special cases.

IIEP/ÏM/37/69 - page 17

Î-2Î (a) Two activities 1-3 and 2-3 lead to event f3

T1 = 2 T = 7

© — ! ^ T = 1 2

Since the earliest date of event *2S is 1, and of event *1* is 2, and the time of the two activities 1-3 and 2-3 is successively 4 and 6S the earliest date for event 93f is 7» But it is clear that if activity 1-3 starts at the earliest date of event sl', which is also the earliest date for starting this activity, then the earliest finishing date for activity 1-3 does not coincide with the earliest date of event *3?e There is a float of 7-6 « lj this is called the ffree float* of activity 1-3.

It will be noted at once that if the duration of activity 1-3 is modi­fied so as to absorb the free float, the rest of the programme is not modi­fied in any wayi this means that resources can be economized on activity 1-3, up to the limit of the free float, as we shall see below,

(b) Two activities 1-3 and 1-4 follow event 'l1.

XIEP/TM/37/69 - page 18

Since the latest dates of events 'З1 and f4 f and successively 8 and 9* and the duration of activities 1-3 and 1-4 4 and 7* the latest date of event fl* is obviously 2. But it is clear that, even if activity 1-3 does not start at the latest date of event ?1* but two time units later, there will be no change in the rest of the programme. There is therefore a slack between the latest finishing date for event flf and the latest starting date of activity 1-3« This kind of slack is called findependent'. Here again,, resources can be economized by eliminating the independent slack, thus making the achievement of the project more efficient»

4„ The critical path

As defined, the slack of an event is the difference between its latest date and its earliest date« The value of the slack may therefore be positive, negative or nil« A positive slack obviously indicates an ad­vance on the programme (excess resources) a negative slack indicates a delay in the programme (lack of resources).,

On the other hand, a nil slack means that the times are strictly observed (adequate resources) and that the earliest and latest dates of the event coincide. The event is then said to be critical. Any delay in reaching a critical event has the effect of delaying all subsequent activi­ties, and may therefore imperil the completion of the project within the time fixed. It is quite clear that if the value of the slack of an event is negative, the event is also critical.

In the execution of a programme there are a number of different paths leading from the initial event to the objective event. They are not all critical to the same extent, but one of them is generally more critical than the others! this is called the critical path. More precisely, the critical path is the one which, in the aggregate, shows the least slack. There may therefore be more than one critical path on a network.

Experience shows that these paths, which brook no delay in execution without imperilling the timetable for the project, rarely affect more than 10 per cent of all activities. But it is on them that attention and effort should be concentrated. They give a realistic idea of the limits within which certain states of the system can be achieved. They thus make it possible to enter into commitments and, if necessary, to refuse certain clauses of the contract which are manifestly impossible to satisfy. Lastly, the supervision of the critical path (or paths) makes it possible automati­cally to assess the consequences of a mistake which may have been made, or if delay in an activity becomes inevitable, the consequences on all the

XIEP/TM/37/69 - page 19

events of the programme, thus allowing action to be taken in sufficient time to be effective« In practice, it is possible to act well in advance perhaps even before the project is launched - by proceeding by simulation« Computers afford very vast possibilities of calculating variants and thus bringing to light the weak points of the project,,

Exercise

Calculate the latest times and the slack of the events on the following network. Indicate the critical path of the programme»

T„=8

Тг=31

Events

6 5 4 7 3 2 1

T»

31 27 15 20 16 7 0

T

31 27 8 8 16 7 0

Slack

0 0 7 12 0 0 0

The critical path is therefore; l-2-3~5~6.

IIEP/ÏM/37/69 - page 20

5. The probability of respecting a timetable

As we have seen, the identification of the critical path is a very important phase in the application of PERT-TIME, since it allows, among other things, the definition of the limits or the possibilities of executing the project«,

Thus, in negotiations on the timing of objectives, special attention is paid to events on the critical path. The question which naturally has to be answered iss What is the possibility of reaching (or passing) a given event in the project by a given date in the timetable?

The use of statistical mathematics is necessary to answer this question»

(i) Let the variable Z = s - T

У 2 T, "E where;

t is the timetable date in question s - T is the earliest date of the objective event e

the denominator of the fraction measures the standard varia­tion, i.e. the dispersion of the variable which equals the square root of the sum of the variances of all the activities preceding the objective event. It will be remembered that the variance of an activity is given by the formulas

(ii) The variable Z follows a law of probability known as the Laplace-Gauss Law (Normal Law)» This law is tabulated! it is therefore possible for each value of Z to determine the probability by simple reference to the table on page 22«,

(iii) To calculate the numerator of Z it is enough to calculate the earliest date of the objective event. To obtain the denominator of Z it is necessary to list all the activities on the critical path leading to the objective and to calculate the variances for the duration of each activity,,

Let us take an example ?

IIEP/PM/37/69 - page 21

The elements of a project are tabulated as follows;

Activities

1 - 2 1 - 3 1 - 4 4 - 6

a

7 16 7 14

П1

10 18 8 18

b

13 20 9 28

Activities

2 » 5 3 - 5 5 - 6

a

10 20 2

m

12 24 3

b

17 34 7

Assume that the data are in dayse The agreed time for reaching event 5 is 45 days from the start of the work. Calculate the probability of respecting this date. The average duration of the 7 activities obtained by the equation te = (a + 4m + b)/6 are as follows: 1-2 ; 10; 1-3 % 18; 1-4 : 8; 2-5 : 12.5l 3-5 % 25; 4-6 % 19; 5-6 : 3.5 The earliest dates for the different events are therefore: T_ = 0 Т л = 10 T, = 18 T,. = 8 1 d 3 4

T = Max /"10 + 12,5 or 18 + 25J7 = 43 T 6 = Max /~43 + 3.5 or 8 + 19_7 = 46 „5 The numerator of Z is therefore 45-43 = 2 The critical path leading to event 5 is:

1 - 3 and 3 - 5 The variances of these activities are given by the equation:

*' -fcy - С -(*T -¿a » -(«У 116

The sum of the variances is therefore —=7* and the standard variation of the variable is: V •==•?» orXZTl»8

ИЕР/ГМ/37/69 - page 22

The table shows that the probability of observing the time limit of 45 days is 0.8643 or more than 86 per cent.

This exercise completes the particulars lie wish to give on PERT-TIME. It only remains,, by way of conclusion,, to study a concrete example,, applying the PERT technique.

NORMAL DISTRIBUTION (Laplace-Gauss Law)

z 0

0.1 0,2 0,3 0.4 0.5 0,6 0,7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2,3 2.4 2.5 2.6 2.7 2.8 2.9

PR

0.5000 5З98 5793 6179 6554 6915 7257 7580 7881 8159 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 9821 9861 9893 9918 9938 9953 9965 9974 9981

z -3.0 -2.9 -2,8 -2.7 -2.6 -2.5 -2.4 = 2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

PR

0.0013 0019 ОО26 ОО35 oo47 0062 0082 0107 0139 0179 0228 0287 0359 0446 0548 0668 0808 0968 1151 1357 1587 1841 2119 2420 2743 3085 3446 3821 4207 4602

3.0 0,9987

IIEP/FM/37/69 - Pa§e 23

6, An example of application; research project on education administration

The research project is to study the administrative structures of four different education systems and to make a general review in a con­solidated report. The duration of the project is 18 months.

(a) Preliminary over-all analysis

The objective OB is to publish and circulate a sufficient number of copies of a consolidated report of a study of the administrative structures of four different education systems.

Four sub-objectives (h, B, C, D) can be distinguished straight away, namely 'reports prepared for countries А, В* С and D*.

Four important predecessor events (M., M_5 M_ and 1VL) govern the achievement of these sub-objectives: 'Missions effected in countries A, В, С and D*.

A key predecessor event is obviously 'meeting of expert consultants held* (R).

An event which governs the projects 'credits and grants obtained' (S).

Finally5 the initial event OR: 'field and limits of the project established'„

The first outline analysis of the project therefore takes the following forms

IIEP/TM/57/69 - Page 24

DIAGRAM No, 8

The Main Events

N/

R

ч/-

M/-

MC

v/ D

v

IIEP/FM/J7/69 - page 25

(b) Detailed analysis

In the next phase the intermediate events between the main events must be analysed in greater detail;

Between OR and S

Events

Activities

Between S and R

Events

fNote of proposals to aid agencies prepared* 1.

fNote of proposals despatched' 2.

'Contracts negotiated8 5»

'Contracts signed' 4„

fPrepare a note of proposals' OR-1

'Submit note to various aid agencies1 1-2

'Handle correspondence with aid agencies' 2-3

'Discuss clauses of aid contract' J5-4

'Sign contract' 4-S

'Preliminary bibliographical study made* 5.

'Bibliographic note* 6»

'Methodological study made' 7.

'Methodological study reproduced' 8„

'Correspondence carried out with group of experts' 9°

'Date of meeting fixed and practical arrangements made' 10.

'Notes for submission to participants prepared9 11,

'Notes reproduced' 12.

IIEP/TM/37/69 - Page 2б

'Meeting started1 r о

'Bibliography discussed1 r

*Studies discussed* r

'Notes discussed* r_

Activities ; 'Make preliminary study* OR-5

'Draft bibliographical note* 5-S

'Reproduce bibliographical note* S-6

*Prepare methodological study* S~7

'Reproduce methodological study* 7-8

'Prepare correspondence with expert groups' 5-9

*Make practical arrangements for meeting' S-10

'Dummy activity' 9-Ю

'Prepare notes for submission to participants' 7

'Reproduce notes' 11-12

'Discuss studies' r-]~To

*Discuss bibliography' vrTT-\

'Confirm experts' l°"ro

'Discuss notes' r -R

Between R and м«, L* M , ¥L

Events t 'Minutes of meeting prepared' YJ>*

*Methodological study revised' l4.

'Brief for participants in project drafted' 15,

ИЕР/ГМ/37/69 - Page 27

'Brief reproduced8 16. 8Circular letter sent to a sample of countries* 17.

'Answers processed8 l8.

"Criteria for the selection of education systems finalized8 19.

'First selection of countries completed' 20„

'Correspondence with selected countries completed9 21,

'Second selection of countries., A, B5 Cs Bs completed'

22,

'Agreement of countries obtained' 23*

'Mission teams assembled' 24.

'Mission timetables established5 25. (A, B9 C, D)

'Correspondents informed' 26. (A, B5 Cs D)

'Missions sent out' 27. (A, B, C, D)

'Information compiled' 28, (A, B, C, D)

'Missions return' (M. ... WL)

Activities % 'Prepare minutes of meetings' R-13

'Revise methodological study' R-l4

'Draft brief 14-15

'Reproduce brief' 15-16

'Prepare circular letter' R-17

'Process answers' 17-l8 'Establish criteria for the selection of education systems' 13-19

ÏIEP/TM/37/69 - page 28

'Make first selection of countries' 9-20

'Dummy activity1 18-20

'Send complementary information to selected countries8

20-21

'Select A, Bs C, D* 21-22

8Ask for agreement' 22-23 (A, B, C, D)

•Constitute mission teams' 23-24 (A, Bs C, D)

'Distribute briefs' 16-24

'Set timetables and organize missions' 24-25 (A-D)

'Send timetables to correspondents' 25-26 (A-D)

'Confirm by correspondents* 26-27 (A-D)

'Collect information' 27-28 (A-D)

'Despatch documentation' 28 (Мл-Мп) A U

Between M.-1VL and A-D

Events s 'First versions of reports drafted' 29 (A, B, C, D)

'Comments by project chief supplied' 30 (A-D)

'Second version of reports drafted' 31 (A-D)

'Second version reproduced' 32 (A-D)

'Second version commented on by national authorities' 33 (A-D)

'Third version drafted' As Bs C, D,

Activities ; 'Analyse the documentation submitted' IYL-IVL~29 (A--D)

'Submit reports to project chief' 29 (A-D)-30 (A-D)

'Prepare second versions' 30-31 (A-D)

ИЕР/ГМ/37/69 - page 29

Between A-D and OB

Events

Activities

Reproduce second versions1 ^1^-^2,-^)2.-^2-„ 32_~32„ and 32C~32D 'Submit reports to national authorities' 32-33 (A-D)

'Process comments received' 33 (A-D)

'Plan of consolidated report established1 3 » 1Complementary enquiries made' 35»

'First version of report drafted' 36«

'Comments and suggestions received from group of experts' 37.

'Second version of report drafted' 38.

OB,

'Establish plan of report' 30 (A-D)~3^

'Make complementary enquiries8 3^-35

'Prepare first version of report? 35-36

'Submit consolidated report to a group of experts' 36-37

'Dummy activity' A Bs Cs D~38

'Prepare final version of consolidated report' 37-38

'Arrange for distribution and publication' 38--OB

IIEP/TM/37/69 - page 30

ИЕР/ГМ/37/69 - Ра§е 51

The PERT network of the project can be constructed from the elements of the detailed analysis. In the following table the standard times of the activities of the project have been expressed in weeks (one year = 45 weeks).

Standard times

Activity te Activity te OR-1 1-2 2-3 3-4 4-S OR-5 5-S S-6 S-7 7-8 S-9 S-10 10-r о 7-11 11-12 Г +Гт 0 1 г -г 1 2

Iu-Tp-R 15-16 R-13 R~l4 14-15 R~17 17-18

4

4

1 1

0.5 2

2

1 4

1

0.5 1

1

1.5 1 0,2

0.3

0o5 1

0,5 1 p

0e2

4

13-19 19-20 20-21

21-22

22-23 23-24

16-24

24-25 25-26

26-27 27-28

28-М. 1 M-29

29-30

ЗО-З1 З1-З2

32-33 33-AsB5C5D

ЗО-34

34-35 35-36

36-37 37-З8

38-OB

0.5 1 1 4 0,2 1 0 1 0,2 2 2 0,2 4 2 2

1 x 4 0,2 4 1 3 6 2 3 12

И Е Р / Г М / 3 7 / 6 9 - page 32

(1) Show that the critical path is the following?

0 R - l - 2 - 5 - ^ - S - 7 - l l - 1 2 - r - r 1 - r 2 - R - 1 7 -18 - 20 - 21 - 22 - 23 - 24 - 25 - 26 - 27 - 28 - M - 29 - 30 -34 , 35 . 36 - 37 - 38 - oB

(2) Given the duration of predecessor activities as shown in the following table, calculate the probability that the missions will have returned by the 34th week of the project.

Activity

0R-1 1-2 2-3 3-4 4-s S-7 7-11 11-12 r -r. 0 1 r —r 1 2 r2-R R-17 17=18 20-21

a

1 2 0.5 0,5 0„2 3*5 0,5 0.5 0,2 0*3 0,5 0*2 3 0*8

b

7 6 1-5 1.5 oe8 4.5 4.5 1.5 0,2 0,3 0,5 0,2 3 1.2

Activity

21-22 22-23 24-25 25-26 26-27 27-28 28-М 23-24

a

3 0.2 1 0.1 1.5 1.8 0.2 0.8

b

5 0.2 1 0.3 2.5 2.2 0,2 1.2

IIEP/TM/37/69 - page 33

Cook, Desmond L., Program Evaluation and Review Technique ; Applications Education, Washington, U.S. Government Printing Office, 1966

Federal Electric Corporation, A Programmed Introduction to PERT; Program, Evaluation and Review Technique, New York, John Wiley & Sons, I963

PERT orientation and Training Center, PERT Fundamentals, (Vols. I, II and III) Washington,U.So Government Printing Office-, 1963»

These documents are available in the HEP Library.