International Mathematical Forum, 5, 2010, no. 18, 861 - 868
Estimating the Mean and Variance of
Activity Duration in PERT
N. Ravi Shankar1, K. Surya Narayana Rao2, V. Sireesha1
1
Department of Applied Mathematics ,GIS
GITAM University, Visakhapatnam, India
drravi68@gmail.com
2
Department of Mathematics, Al-Ameer College of Engineering & IT
Visakhapatnam,India
Abstract
The traditional PERT (Program Evaluation and Review Technique) model uses
beta distribution as the distribution of activity duration and estimates the mean
and the variance of activity duration using “pessimistic”, “most likely” and
“optimistic” time estimates proposed by an expert. In the past several authors
have modified the original PERT estimators to improve the accuracy. In this
paper, on the basis of the study of the PERT assumptions, we present an
improvement of these estimates. At the end of the paper, an example is presented
to compare with those obtained using the proposed method as well as other
method. The comparisons reveal that the method proposed in this paper is more
effective in determining the activity criticalities and finding the critical path.
Keywords: Estimating the mean and the variance of activity times; Beta
distribution; PERT; Critical Path
1. Introduction
In recent years, the range of project management applications has greatly
expanded. Project management concerns the scheduling and control of activities
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N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha
in such a way that the project can be completed in as little time as possible [1,2].
PERT [12] is a well known technique with proven value in managing large-scale
projects. In 1959, the creators of PERT [12] considered beta distribution
Γ(α + β ) ( y − a) α −1 (b − y ) β −1
f y ( y) =
, a < y < b, α , β > 0.
(1)
Γ(α )Γ( β )
(b − a ) α + β −1
as an adequate distribution of the activity duration y where α and β are
parameters of the beta distribution. They suggested the estimates of the mean and
variance values
1
μ = (a + 4m + b),
(2)
6
1
σ 2 = (b − a) 2 ,
(3)
36
where a ,m and b are the “optimistic”, “most likely” and “pessimistic” activity
time estimates respectively determined by a specialist. In the last five decades,
numerous attempts have been made to improve the PERT analysis based on the
subjective determination of a , m and b. By using PERT, managers are able to
obtain[9,10,11] :
(i)
A graphical display of project activities.
(ii)
An estimate of how long the project will take.
(iii) An indication of which activities are the most critical for timely project
completion.
(iv)
An indication of how long any task can be delayed without delaying
the project.
There are, indeed, few areas as open until now to such a sharp criticism as in
PERT applications.One of the criticism about PERT estimates as pointed by
Clark[10] , Grubbs[11] and Sasieni[8] is that the estimates eq.(2) and eq (3),
which are based on the three activity times a , m and b cannot be obtained directly
from (1), implying a lack of a sound theoretical basis. Ginzburg [6] assumed p+q
= z (constant) as an extension of earlier assumptions, saying, on the basis of
statistical analysis and some other intuitive arguments, the creators of PERT [12]
assumed that p + q ≅ 4. In [13], Ravi Shankar and Sireesha generalize the
assumption on parameters in original PERT and obtained new approximation for
the mean and variance of a PERT activity duration distribution. Based on beta
activity time distribution, we assumed p = qz and obtained new approximations
for the mean and the variance of activity time in PERT. By comparison with
actual values, it was shown that the proposed approximations have the lowest
average absolute error compared to the existing ones.
The rest of this paper is organized as follows. In Section 2, we briefly
review Original and Ginzburg’s PERT approximations. In Section 3, we proposed
PERT approximation. In Section 4, we given a numerical example for PERT
approximations. In section 5 we summarizes contributions of this paper.
Estimating the mean and variance
863
2 Traditional and Ginzburg’s PERT approximations
2.1 Traditional PERT approximation
Since in PERT applications a and b of the density function (1) are either known
or subjectively determined, we can always transform the density function to a
standard form,
Γ(α + β ) α −1
f ( x) =
x (1 − x) β −1 ,0 < x < 1, α , β > 0,
(4)
Γ(α )Γ( β )
y−a
where x =
.
b−a
Note that simple relations
μx =
μy − a
b−a
, σx =
σy
b−a
, mx =
my − a
b−a
(5)
hold.
Let α-1 = p , β-1 = q. The density function (4) becomes
Γ( p + q + 2)
x p (1 − x) q ,0 < x < 1, p, q > −1,
Γ( p + 1)Γ(q + 1)
with the mean, variance and mode as follows :
f ( x) =
(6)
p +1
,
(7)
p+q+2
( p + 1)(q + 1)
σ x2 =
,
(8)
( p + q + 2) 2 ( p + q + 3)
p
mx =
.
(9)
p+q
From (6) and (9) we obtain
Γ( p + q + 2)
f ( x) =
(10)
x p (1 − x) p (1 / mx −1) .
Γ( p + 1)Γ(q + 1)
Thus value mx, being obtained from the analyst’s subjective knowledge, indicates
the density function. On the basis of statistical analysis and some other intuitive
arguments, the creators of PERT assumed [2] that
μx =
p+q ≅ 4.
(11)
It is from that assertion that estimates (2) and (3) were finally obtained, according
to (6) – (9).
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N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha
2.2 Ginzburg’s [6] PERT approximation
Ginzburg[6] showed that the PERT assumptions (11) is poor because the
actual standard deviation may be considerably smaller than 1/6 , especially in the
tails of the distribution[7,8]. In order to make the assumption more flexible, he
assumed that the sum p+q in (6) is approximately constant but not predetermined;
i.e., relation
p + q ≅ Z = constant
(12)
From (9) we obtain
p= Zmx ,
(13)
and values μx and σx2 are
Zm x + 1
μ x (m x ) =
,
(14)
Z +2
1 + Z + Z 2 m x − Z 2 m x2
σ x2 (m x ) =
(15)
( Z + 2) 2 ( Z + 3)
To satisfy the main PERT assumption the average value σ x2 (mx ) for 0 < mx <1
has to be equal to 1/36; i.e.,
1
1
2
(16)
∫0 σ x (m x ) dm x = 36
Substituting (15) in (16), integrating and solving (16) for Z, obtained Z = 4.55.
Approximating Z to 4.5 and getting
(17)
p = 4.5 mx, q = 4.5 (1-mx)
from (12) and (13) , finally obtained
Γ(6.5)
f ( x) =
x 4.5 mx (1 − x) 4.5(1− mx ) . ,
(18)
Γ(4.5m x + 1)Γ(5.5 − 4.5m x )
with the mean and variance
9m + 2
μx = x
,
(19)
13
1
σ x2 =
(22 + 81m x − 81m x2 ),
(20)
1268
For the general beta distribution of the activity time, estimates (19) and (20) are
transformed to
2a + 9m + 2b
,
13
2
(b − a) 2 ⎡
m−a
⎛m−a⎞ ⎤
2
σy =
− 81⎜
⎟ ⎥.
⎢22 + 81
1268 ⎣⎢
b−a
⎝ b − a ⎠ ⎦⎥
μy =
(21)
(22)
Estimating the mean and variance
865
2a + b
, he further improved these estimates
3
when the estimated mode of the activity time is located in the tail of the
distribution as follows.
By assuming that p =1, q = 2 and m=
μ y = 0.2(3a + 2b)
(23)
40
(b − a) 2 ≅ 0.04(b − a) 2
1268
Thus estimates (2) and (3) are replaced by estimates (21) and (24).
σ y2 =
(24)
3. Proposed PERT approximation
p
= z (constant)where p and q are equal to α-1 and β-1 respectively. By
q
substituting p = qz in (7) - (9) we obtain
qz + 1
μx =
(25)
qz + q + 2
z
(26)
mx =
z +1
(qz + 1)(q + 1)
σ x2 =
(27)
(qz + q + 2) 2 (qz + q + 3)
From (27) ,
z
σ x2 ≅
(28)
q(1 + z ) 3
1
We assume that original PERT assumption, σ x = , to solve (28) using (26) to
6
obtain the following values for p and q.
p = 36mx2 (1 − mx ),
(29)
Let
q = 36m x (1 − m x ) 2
Substituting p and q values in (7) and (8) we obtain
36m x2 (1 − m x ) + 1
μx =
36m x (1 − m x ) + 2
σ x2 =
(36m
2
x
)
(1 − m x ) + 1 (36m x (1 − m x ) + 1)
(36m x (1 − m x ) + 2)2 (36m x (1 − m x ) + 3)
Using relations (5) in (31) and (32) and also substituting m y =
(30)
(31)
(32)
2a + b
which is
3
used by Ginzberg[7] in variance σ y2 , we obtain
μy =
36(m − a )(b − m)m + (b + a)(b − a) 2
36(m − a )(b − m) + 2(b − a) 2
(33)
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N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha
σ y2 = 0.03 (b − a) 2
(34)
Thus estimates (2) and (3) are replaced by estimates (33) and (34).
4. Numerical Example
The data for activities is represented in table I including mean and variance
estimates for original , Ginzburg and proposed approximations. The estimated
project duration has approximately same value by using the original, Ginzburg
and proposed methods.
Table I.
Mean and variance estimates
Original
approximation
Activity
A
B
C
D
E
F
G
H
I
J
a
5
8
9
5
9
14
21
8
14
6
m
12
10
11
8
11
18
25
13
17
9
Ginzburg
approximation
Proposed
approximation
b
17
13
12
9
13
22
30
17
21
12
μ
11.66
10.16
10.83
7.66
11.00
18.00
25.16
12.83
17.00
9.00
σ2
4.00
0.69
0.25
0.44
0.44
1.78
2.25
2.25
1.30
1.00
μ
11.69
10.15
10.84
7.69
11.00
18.00
25.15
12.84
17.15
9.00
σ2
4.49
0.78
0.28
0.50
0.50
2.01
2.55
2.55
1.54
1.13
μ
11.81
10.57
10.65
7.88
11.00
18.00
25.09
12.90
17.09
9.00
σ2
4.32
0.75
0.27
0.48
0.48
1.92
2.43
2.43
1.47
1.08
5. Conclusion
Based on beta activity time distribution, we obtained new approximations for the
mean and the variance of activity time in PERT. By comparison with actual
values, it was shown that the proposed approximations are identical with the
existing ones. The estimated project duration has approximately same value by
using the original, Ginzburg and proposed methods.
Estimating the mean and variance
867
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N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha
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Received: October, 2009