# Random PERT: application to physical activity/sports programs

**ABSTRACT** This paper illustrates the variety of PERT technique known as random PERT. The aim of this technique is to help plan the duration

of activities, something which can be particularly difficult in psychosocial programs. Thus, this task is often carried out

by experts, who know that there are many events which may modify the proposed calendar. The paper includes an empirical illustration

of random PERT applied to a physical activity/sports program for elderly people.

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**ABSTRACT:**Planning, implementing and evaluating an intervention program all hinge around time. A program’s actions are planned according to a forecast of the time required to achieve certain objectives, and the program’s implementation among a group of users is conditioned by its real time application. Similarly, program evaluation needs to take into consideration the time resource when analysing objectively the extent to which a program’s targets have been reached, and when conducting a cost analysis of the program. In limited resource programs, any disparity between the scheduled time and the real time available can have serious consequences, and even undermine a program’s efficacy. Time management, above all where resources are limited, is therefore the linchpin in the planning, implementation and evaluation of an intervention program. In this study we analyse the utility of PERT and CPM as basic tools for the efficient time management of limited resource programs.Quality and Quantity 07/2005; 39(4):391-411. · 0.76 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**”This paper investigates PERT networks with independent and exponentially distributed activity durations. We model such networks as finite-state, absorbing, continuous-time Markov chains with upper triangular generator matrices. The state space is related to the network structure. We present simple and computationally stable algorithms to evaluate the usual performance criteria: the distribution and moments of project completion time, the probability that a given path is critical, and other related performance measures. In addition, we algorithmically analyze conditional conditional performance measures - for example, project completion time, given a critical path - and present computational results. We then study extensions both to resource- constrained PERT networks and to a special class of nonexponential PERT networks.”Operations Research 10/1986; 34(5):769-781. · 1.50 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper builds upon earlier work from the decision/risk analysis area in presenting simple, easy-to-use approximations for the mean and variance of PERT activity times. These approximations offer significant advantages over the PERT formulas currently being taught and used, as well as over recently proposed modifications. For instance, they are several orders of magnitude more accurate than their PERT counterparts in estimating means and variances of beta distributions if the data required for all methods are obtained accurately. Moreover, they utilize probability data that can be assessed more reliably than those required by the PERT formulas, while still requiring just three points from each activity time probability distribution. Using the proposed approximations can significantly improve the accuracy of probability statements about project completion time, and their use complements ongoing efforts to improve PERT analyses of networks involving multiple critical paths.Management Science 09/1993; 39(9):1086-1091. · 2.52 Impact Factor

Page 1

Qual Quant

DOI 10.1007/s11135-007-9124-0

ORIGINAL PAPER

Random PERT: application to physical activity/sports

programs

Verónica Morales-Sánchez · Antonio Hernández-Mendo ·

Pedro Sánchez-Algarra · Ángel Blanco-Villaseñor ·

María-Teresa Anguera-Argilaga

Received: 15 July 2006 / Accepted: 15 January 2007

© Springer Science + Business Media B.V. 2007

Abstract

The aim of this technique is to help plan the duration of activities, something which can

be particularly difficult in psychosocial programs. Thus, this task is often carried out by

experts, who know that there are many events which may modify the proposed calendar. The

paper includes an empirical illustration of random PERT applied to a physical activity/sports

program for elderly people.

This paper illustrates the variety of PERT technique known as random PERT.

Keywords

Random PERT · Physical Activity · Sport Programs

1 Introduction

Since the 1950s the strategy known as PERT has, along with other critical path techniques,

been regarded as useful for planning and scheduling various kinds of program. An essential

feature of the technique is uncertainty, and therefore it has adopted a probabilistic approach,

incontrasttothedeterministicapproachtakenbythecloselyrelatedtechniqueofCPM(Pillai

andTiwari1995).InatraditionalPERTanalysis,themainobjectivewastoscheduleaproject

assumingdeterministicdurations.However,eachoneofthevariousactivitiesrequirestheuse

of available resources (equipment, materials, etc.) and, as the resource combinations them-

selves also imply a cost, this may create logistical problems (Sánchez-Algarra and Anguera,

in press).

Indeed, one of the most controversial issues in the history of PERT has been the ap-

proachtoactivitytimes(Sasieni1986;LittlefieldandRandolph1987,1991;Gallagher1987;

Farnum and Stanton 1987; Golenko-Ginzburg 1988, 1989; Sculli 1989; Chae 1990; Chae

and Kim 1990; Rhiel 1990; Keefer and Verdini 1993; Somarajan et al. 1992; Williams 1995;

V. Morales-Sánchez · A. Hernández-Mendo (B )

University of Malaga, Malaga, Spain

e-mail: mendo@uma.es

P. Sánchez-Algarra · Á. Blanco-Villaseñor · M.-T. Anguera-Argilaga

University of Barcelona, Barcelona, Spain

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V. Morales-Sánchez et al.

Kamburowski 1997). In the specific case of social intervention programs, especially those

concerningphysicalactivity,numerousdifficultiesmustoftenbeovercomeinordertodecide

realistically on the times for different activities, whether these be concurrent or diachronic;

furthermore, a schedule must be explicitly set out in such a way that it enables the proposed

intervention plan to be organized on the basis of the interrelationship between points in

time and activities (Collantes 1982; Kerzner 1998; Prado 1988; Yu 1989; Alberich 1995;

Muscatello 1988; Sánchez-Algarra and Anguera 1993; Hernández Mendo and Anguera

2001).

The recent use of computational optimization techniques has led to genetic algorithms

being proposed (Feng et al. 1997; Chau et al. 1997), although during project implementation

certain unknown variables may affect the planned durations and, therefore, the costs. This

affects the temporal proposal of PERT, a problem which only recently has been resolved

by Azaron et al. (in press); these authors, whose work is based on the earlier proposal by

Kulkarni and Adlakha (1986), assume that the mean duration of each activity is a non-

increasing function and that the direct cost of each activity is a non-decreasing function of

the assigned resources.

2 Random PERT

When using the PERT technique it is necessary to calculate what are termed the earliest

times (optimistic times) and latest times (pessimistic times). The use of random PERT in

any social intervention program is especially recommended when the duration of activities is

not known very precisely. Therefore, it is assumed that the durations are random variables,

for which the probability distributions are known, and hence that the activity durations are

randomvariablesthatfollowbetaprobabilitylaws.Thedensityfunctionofarandomvariable

t with a beta probability distribution in an interval [a,b] is as follows:

f (t) = 0 for t < a

f (t) = 0 for t > a

f (t) = K(t − a)α(b − t)β

for a < t < b

where K is a constant which depends on the values of a and b, and on the parameters α

and β.

The above expression represents a family of beta density curves which will be asymmet-

rical toward the right if

(a + b)

2

> m

and toward the left if

(a + b)

2

< m

Moreover, the curves are not asymptotic with respect to the abscissa axis, but rather cross it

at the end points of the distribution,a and b. However, they resemble, to an extent, the curves

of a normal distribution.

The mean and variance of these distributions will be:

D =a + (α + ϕ)m + b

α + ϕ + 2

(1)

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Random PERT: application to physical activity/sports programs

v2=

(b − a)2(α + 1)(ϕ + 1)

(α + ϕ + 2)2(α + ϕ + 3)

(2)

In order to estimate which of the possible curves shows the best fit to the representation of

the activity durations, it is assumed, for the PERT technique, that the standard deviation of

the distribution is one sixth of the total, such that:

v =1

6(b − a)

(3)

which fits beta distributions adequately. If we introduce this condition we will establish a

single beta distribution, in other words, we will determine single values for α and φ from the

end values a and b and the mode value m of the distribution. The mode of the distribution

can be calculated by deriving the density function with respect to t and making the result

equal to zero:

α(b − t) − ϕ(t − a) = 0(4)

The mode is obtained by finding the value of t from the previous equation and substituting t

by m:

m =aϕ − bα

α + ϕ

(5)

If we consider the standard deviation Eq. 3, the expression of variance can be taken as:

(α + 1)(ϕ + 1)

(α + ϕ + 2)2(α + ϕ + 3)=

1

36

(6)

Solving the set of equations formed by (6) and (3) yields the values of α and ϕ which deter-

minethecurvethatmustbeusedinthePERTtechnique.Thevaluesofthemeanandvariance

of t will also be estimated as the activity duration. The values of α and ϕ that are normally

used are as follows:

√2,ϕ = 2 −

Or alternatively:

√2,ϕ = 2 +

Substituting these values in the expressions of the mean (1) and variance (6) of the beta

distribution yields:

α = 2 +

√2,

if m ≥a + b

2

(7a)

α = 2 −

√2,

if m ≤a + b

2

(7b)

D =a + 4m + b

6

(8)

and also,

v2=(b − a)2

36

(9)

which leads us back to the previous formula of activity time and enables its justification.

However, there are actually no strong grounds for assuming that the distribution of activity

durations takes a beta form. The statistical assumptions used in the PERT technique may

generate absolute errors of up to 33% of the total for the mean of the duration and 17% for

its standard deviation.

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V. Morales-Sánchez et al.

The above equations can provide complementary information which is useful in schedul-

ing and controlling programs and, in particular, it is possible to determine the ‘probability’

of adhering to the estimates proposed by PERT on the basis of the earliest times.

If we call the activities i, j,..., where ai,mi, and bi, are the three estimates of activity

durationi, and we take a randomvariable that measures the duration of an activity andwhich

belongs to the project’s critical path, the mean and variance of this variable ξiwill be given

by:

Di=ai+ 4mi+ bi

6

(10)

v2

i=

?bi− ai

6

?2

(11)

If we define η as a new random variable, such that:

η = ξ1+ ξ2+ ··· + ξi+ ··· + ξn=

?

ξi

(12)

then, following the central limit theorem, the distribution of the sum of n random variables,

which are distributed in the same way and independently, converges into a normal distribu-

tion whose mean and variance is the sum of the means and variances of the n variables, when

n tends towards infinity.

?

Thus, the duration of a program will be a normal variable whose parameters will be the sum

of the different critical path activities. In order to determine the probability of completing a

project in time T, it is necessary to calculate:

η α NM =

?

Di;V2=

?

V2

i

?

(13)

P (η ≤ T) = F (T)

(14)

which corresponds to the function:

f

?

t

−∞≤t≤∞

?

=

1

?√2π.

?

V

e(−1/2)(t−M/V)2

(15)

Therefore, the probability will be obtained by solving the integral between −∞ and t;

P (η ≤ T) = F (T) =

1

?√2π

?

V

t ?

−∞

e(−1/2)(t−M/v)2dt

(16)

Eq. 16 is difficult to integrate, and thus the probability is estimated from an equivalent

expression:

?η − M

The above expression thus becomes:

?

123

P

V

≤T − M

V

?

= F

?T − M

V

?

(17)

P

η?≤T − M

V

?

= F

?T − M

V

?

(18)

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Random PERT: application to physical activity/sports programs

In Eq. 18, η?is a normal variable with a mean of 0 and a variance of 1. This operation is

the typification of a normal variable (or estimate of a score z). This strategy makes it easy to

calculate the probabilities.

z =T − M

V

=xi− X

SX

(19)

The earliest time of the project’s final event is the sum of the mean values of the activity

durations which make up the critical path. In other words, the earliest time of the final event

will be the mean value of the random variable η which measures the program duration. Fur-

thermore, given that the normal distribution is symmetrical with respect to its mean value M,

in a random context the probability of completing a project by the earliest time will be 50%.

The basic application of this mathematical reasoning enables us to determine the number

of time units X that are necessary for there to be a given probability β of completing the

program. This is valuable information for a project manager to have, and the value of X can

be derived from the equation:

P (η ≤ X) = β

(20)

Typifying the variable gives

P

?

η?≤X − M

V

?

= β

(21)

Having found X, normal distribution tables can be used to obtain the abscissa which leaves

area β to its left. If this abscissa is ?, the duration will be given by:

X − M

V

= φ

(22)

or the equivalent expression,

X = φV + M

(23)

The variance of an activity duration is an index which estimates the likelihood of being able

to carry out the activity in the predicted time. Greater variance means a greater dispersion of

times, in other words, less likelihood. Therefore, when there is more than one critical path

in a PERT calculation, they will be organized into a hierarchy of ‘criticality’ on the basis of

the corresponding variances.

However, in reality it should not be assumed that the expected duration is the sum of the

mean durations of the different activities. In fact, this assumption results in estimates that are

lower than the real times achieved.

3 Application of random PERT to intervention programs: an example

Let us consider the case of a physical activity/sports program for elderly people. Table 1

shows the Priorities Chart, which comprises four columns. The first includes all the activi-

tiesofwhichtheprojectiscomprised.Thesecondcolumnshowstheactivitieswhichprecede

their counterpart in the first column. The initial activities are identifiable by the fact that they

have no preceding activity; the final activities are identifiable as they do not appear in the

second column. The third column shows the duration, and the fourth describes the activity

(Fig. 1).

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V. Morales-Sánchez et al.

Table 1 Priorities chart

ActivityPredecessorDuration (days)Description

A

B

C

D

E

F

G

H

I

J

K

L

–

–

A–B

A

A

D

D

G

F

E

C

H–I–J

14

14

Program registration

Handing out material

Introduction to the program

General mobility activities

Joint mobility activities

Endurance activities Basic aerobics

Basic anaerobic endurance activities

Basic strength activities

Advanced aerobic endurance activities

Activities combining basic aspects

Closing activities

End of program party

2

24

16

24

6

6

12

24

1

1

fí=0

L=1

I=12

9 10

12

1164

5

873

21

D=24

C=2

A=14

B=14

E=16

F=24

G=6

H=6

J=24

K=1

f’=0

Fig. 1 Graph of the program with numbered nodes and activity durations

Using the term t(i, j) to refer to the time of the activity that links event i with event j,

the earliest time will be given by the following expression:

t(j) = max[t(i) + t(i, j)]

The minimum time of the program, which indicates its total duration, is given by the value

of the earliest time (or latest time) of the project’s final event (Fig. 2).

Thus, the earliest times of the PERT in Fig. 1 will be as follows:

1. The earliest time of the program’s initial event is zero:

Node 1: t(1) = 0

2. The earliest times of the nodes reached by a single arrow are as follows:

Node 2: t(2) = t(1) + t(1,2) = 0 + 14 = 14

Node 3: t(3) = t(1) + t(1,3) = 0 + 14 = 14

Node 4: t(4) = t(2) + t(2,4) = 14 + 24 = 38

Node 5: t(5) = t(2) + t(2,5) = 14 + 16 = 30

Node 6: t(6) = t(4) + t(4,6) = 38 + 24 = 62

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Random PERT: application to physical activity/sports programs

Node 7: t(7) = t(3) + t(3,7) = 14 + 2 = 16

Node 8: t(8) = t(7) + t(7,8) = 16 + 1 = 17

Node 9: t(9) = t(4) + t(4,9) = 38 + 6 = 44

Node 10: t(10) = t(9) + t(9,10) = 44 + 6 = 50

3. The earliest time of the node reached by two arrows is as follows (only node 11 fulfils

this condition):

Node 11: t(11) = max[t(6) + t(6,11),t(5) + t(5,11)] = max[62 + 12,30 + 24] =

max[74,54] = 74

4. The earliest time of the node reached by three arrows is as follows (only node 12 fulfils

this condition):

Node12:t(12) = max[t(10)+t(10,12),t(11)+t(11,12),t(8)+t(8,12)] = max[50+

0,74 + 1,17 + 0] = max[50,75,17] = 75

5. Therefore, the total duration of the program is t(12) = 75

Once the earliest times have been calculated we can move on to calculate the latest times.

The latest or slow time is the longest time which may be taken in order to reach a given event

without delaying the overall program.

The latest time of the program’s final event will be equal to the value of its earliest time.

For the remaining nodes, and in decreasing order of their assigned number, the latest time

will be calculated by taking the lowest value among all the activities related to each node,

given by subtracting the time of each activity from its final event time. Therefore, the latest

time will be given by the following expression:

t(i) = min[t?(j) + t(i, j)]

The latest times of the program will thus be as follows:

1. The latest time of the program’s final event is equal to its earliest time:

Node 12: t?(12) = 75

2. The latest times of the nodes reached by a single arrow are as follows:

Node 11: t?(11) = t?(12) − t(11,12) = 75 − 1 = 74

Node 10: t?(10) = t?(12) − t(10,12) = 75 − 0 = 75

Node 9: t?(9) = t?(10) − t(9,5) = 75 − 6 = 69

Node 8: t?(8) = t?(12) − t(8,12) = 75 − 0 = 75

Node 7: t?(7) = t?(8) − t(7,8) = 75 − 1 = 74

Node 6: t?(6) = t?(11) − t(6,11) = 74 − 12 = 62

Node 5: t?(5) = t?(11) − t(5,11) = 74 − 24 = 50

Node 3: t?(3) = t?(7) − t(3,7) = 74 − 2 = 72

3. The latest times of nodes 4 and 2 are:

Node 4:t?(4)=min[t?(9)−t(4,9),t?(6)−t(4,6)]=min[69−6,62−24]=min[63,38]=38

Node2:t?(2)= min[t?(4)−t(2,4),t?(5)−t(2,5)]= min[38−24,50−16]= min[14,24]

=14

4. The latest time of node 1 or the initial event will be:

Node1:t?(1)= min[t?(2)−t(1,2),t?(3)−t(1,3)]= min[14−14,72−14]= min[0,58]

=0

Once the various earliest and latest times have been estimated, a final temporal consideration

can be made regarding the differences between them; this is known as the slack. The slack

refers to the degree of flexibility available according to the early/delayed implementation of

certain activities, or the early/delayed completion of others, or the interplay between these

(Fig. 3).

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V. Morales-Sánchez et al.

Fig. 2 Representation of the

node showing the order number

and the earliest and latest times

Earliest

time

Latest

time

Node

number

74 74

11

16 74

7

1775

8

7575

12

44 69

9

50 75

10

62 62

6

3050

5

33

4

17

3

11

2

00

1

F=24

f’=0

L=1

I=12

D=24

C=2

A=14

B=14

E=16

G=6

H=6

J=24

K=1

f’=0

Fig. 3 Graph of the program showing numbered nodes, activity durations and values of earliest and latest

times

If we consider that an activity has an initial event and a final event, then the activity will

be represented by:

Earliest time of the initial event: t(i)

Latest time of the initial event: t?(i)

Earliest time of the final event: t(j)

Latest time of the final event: t?(j)

Two basic kinds of slack can be considered: event slack and activity slack.

1. Event slack: Event slack is the difference between its earliest and latest times. This slack

indicates how much delay is permitted in carrying out a given event without delaying the

whole program. Within this category two sub-types of slack can be considered:

1.1 Start slack: H(i) is equal to the latest time of the initial event minus the earliest time

of the initial event [H(i) = t?(i) − t(i)]. This indicates how long the start of an

activity may be delayed. If H(i) = 0 the start of the activity cannot be delayed.

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Random PERT: application to physical activity/sports programs

1.2 End slack: H(j) is equal to the latest time of the final event minus the earliest time of

the final event [H(j) = t?(j) − t(j)]. This indicates how long the end of an activity

may be delayed. If H(j) = 0 the end of an activity cannot be delayed.

2. Activity slack: This has three types: total activity slack [HT(i, j)], free slack [HL(i, j)]

and independent slack [HI(i, j)].

2.1 Total activity slack: [HT(i, j)] is equal to the latest time of the final event minus the

initial earliest time minus the activity time.

HT(i, j) = t?(j) − t(i) − t(i, j)

This slack indicates how long a given activity can be delayed without delaying the

whole program. When the total slack of an activity is equal to zero, it is termed a

critical activity. The set of critical activities from the initial program event to the final

event is known as the critical path, of which there may be more than one. Critical

activities are the key to avoiding delay in the overall program.

2.2 Free slack: [HL(i, j)] is equal to the earliest time of the final event minus the initial

earliest time minus the activity time.

HL(i, j) = t(j) − t(i) − t(i, j)

This slack indicates that part of the total slack can be used up without affecting

subsequent activities.

2.3 Independent slack: [HI(i, j)] is equal to the earliest time of the final event minus the

initial latest time minus the activity time.

HI(i, j) = t(j) − t?(i) − t(i, j)

This slack indicates the extent to which the total activity slack has been used up.

Critical path: Activities whose total slack is equal to zero are termed critical activities. By

linking the critical activities a path is formed from the initial event to the final event, and

this is known as the critical path. This path is essential for control of the project. The graph

shows this path represented by a double line for the activities of which it is comprised. A

necessary (although not sufficient) condition for an activity to be critical is that the slack of

its initial and final events is equal to zero (Table 2).

On the basis of the Priorities Chart, the sub-graph corresponding to the project’s critical

path will be as follows (Fig. 4):

ThePERTtimesandvariancesareasshown,thevarianceshavingbeencalculatedaccord-

ing to the formula:

?bi − a

Applying the above we can establish that the duration of the project is a normally distributed

random variable, with a mean and variance equal to:

Vi2=

6

?2

M = 14 + 24 + 24 + 12 + 1 = 75

V2= 2.32 + 4 + 4 + 2 + 0.16 = 12.48

It should be borne in mind that this is stretching the theory somewhat, as we are postulating

the convergence to a normal distribution with only five variables.

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V. Morales-Sánchez et al.

Table 2 Summary table with all the data of the PERT and critical paths

Activity

i − j

Identifica-

tion

Duration

t(i,j)

Earliest

start t(i)

Ear-

liest

finish

t(j)

Latest

start t?(i)

Latest

finish

t?(j)

Event

slack start

H(i)

Event

slack

finish

H(j)

Total ac-

tiv. Slack

HT

Free

slack

HL

Indep.

Slack

HI

Situation

1 – 2

A

14

0

14

0

14

0

0

0

0

0

Critical

1 – 3

B

14

0

14

0

72

0

58

58

0

0

3 – 7

C

2

14

16

72

74

58

58

58

0

2 – 4

D

24

14

38

14

38

0

0

0

0

0

Critical

2 – 5

E

16

14

30

14

50

0

20

20

0

0

4 – 6

F

24

38

62

38

62

0

0

0

0

0

Critical

4 – 9

G

6

38

44

38

69

0

25

25

0

0

9 – 10

H

6

44

50

69

75

25

25

25

0

6 – 11

I

12

62

74

62

74

0

0

0

0

0

Critical

5 –11

J

24

30

74

50

74

20

0

20

0

0

7 – 8

K

1

16

17

74

75

58

58

58

0

11–12

L

1

74

75

74

75

0

0

0

0

0

Critical

i − j

t(i,j)

t(i)

t(j)

t?(i)

t?(j)

H(i) = t?(i) − t(i)

H(j) = t?(j) − t(j)

HT(i, j) = t?(j) − t(i) − t(i, j)

HL(i, j) = t(j) − t(i) − t(i, j)

HI(i, j) = t(j) − t?(i) − t(i, j)

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Random PERT: application to physical activity/sports programs

A=14D=24F=24I=12 L=1

2.324420.16

2

4

6

11

12

1

Fig. 4 Critical path of the project

These estimates can be used to consider the likelihood of the program being completed

in 100 sessions. This likelihood can be calculated using the following formula:

?

Thus, the likelihood is that situated to the left of the abscissa 2.01 in a normal distribution of

mean 0 and variance 1, in this case 0.9555.

P

η ≤100 − 75

V

?

= P(≤ 2.01) = F(2.01)

4 Discussion

Sport and physical activity are increasingly regarded as characteristic of our society’s devel-

opment and numerous social intervention programs have been planned on this basis, partic-

ularly for elderly people. Naturally, these programs must be evaluated. The notion of sport

as a genuine mass phenomenon linked to the transmission of an identifiable set of values can

be witnessed every day in our streets, parks, gardens, sports centers, gyms and swimming

pools—places in which people exercise, run, cycle, swim and walk. Indeed, the leisure soci-

ety is characterized by the large-scale practice of sport, whose main objective may be health,

improved quality of life or the cult of the body.

One of the aims of this study was to illustrate the importance of random PERT and its

application to the field of program evaluation, in general, and physical activity/sports pro-

grams, in particular. As pointed out above, these programs are linked to the philosophy of

social intervention programs.

ThestudyhasappliedrandomPERTtoaphysicalactivity/sportsinterventionprogramfor

elderly people using simulated data. Starting from the proposed priorities and the predicted

durations for each one of the program’s activities, the calculation of the earliest and latest

times provided information about the different degrees of flexibility or slack, and enabled

the critical path to be obtained.

However, the specificity of the random PERT procedure in the proposed application has

an additional benefit, namely the possibility of knowing how likely it is that the program will

be completed in a fixed time period, that is, within a given number of sessions. Access to

this information clearly has enormous repercussions in terms of the financial evaluation of

intervention programs and, consequently, in determining their efficiency.

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