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

1Department of Applied Mathematics ,GIS GITAM University, Visakhapatnam, India

drravi68@gmail.com

2Department 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

862 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

.0,,,)(

)()()()()()( 1

11

>

Estimating the mean and variance 863 2 Traditional and Ginzburgs 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,

,0,,10,)1()()()()( 11 >

864 N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha 2.2 Ginzburgs [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 x and x2 are

,2

1)(

++

=Z

Zmm xxx (14)

)3()2(

1)( 2

2222

++++

=ZZ

mZmZZm xxxx (15)

To satisfy the main PERT assumption the average value )(2 xx m for 0 < mx

Estimating the mean and variance 865

By assuming that p =1, q = 2 and m= 3

2 ba + , he further improved these estimates

when the estimated mode of the activity time is located in the tail of the distribution as follows. )23(2.0 bay += (23)

222 )(04.0)(1268

40 ababy = (24)

Thus estimates (2) and (3) are replaced by estimates (21) and (24).

3. Proposed PERT approximation Let z

qp= (constant)where p and q are equal to -1 and -1 respectively. By

substituting p = qz in (7) - (9) we obtain

2

1++

+=

qqzqz

x (25)

1+

=z

zmx (26)

)3()2(

)1)(1(2

2

++++++

=qqzqqz

qqzx (27)

From (27) ,

32

)1( zqz

x + (28)

We assume that original PERT assumption, 61

=x , to solve (28) using (26) to

obtain the following values for p and q. ),1(36 2 xx mmp = (29) 2)1(36 xx mmq = (30) Substituting p and q values in (7) and (8) we obtain

2)1(361)1(36 2

++

=xx

xxx mm

mm (31)

( )( )( ) ( )3)1(362)1(36

1)1(361)1(362

22

++

++=

xxxx

xxxxx mmmm

mmmm (32)

Using relations (5) in (31) and (32) and also substituting 3

2 bamy+

= which is

used by Ginzberg[7] in variance 2y , we obtain

22

)(2))((36))(())((36

abmbamababmmbam

y +++

= (33)

866 N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha 22 )(03.0 aby = (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

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.

Activity

a

m

b

Original approximation

Ginzburg approximation

Proposed approximation

2 2 2 A 5 12 17 11.66 4.00 11.69 4.49 11.81 4.32 B 8 10 13 10.16 0.69 10.15 0.78 10.57 0.75 C 9 11 12 10.83 0.25 10.84 0.28 10.65 0.27 D 5 8 9 7.66 0.44 7.69 0.50 7.88 0.48 E 9 11 13 11.00 0.44 11.00 0.50 11.00 0.48 F 14 18 22 18.00 1.78 18.00 2.01 18.00 1.92 G 21 25 30 25.16 2.25 25.15 2.55 25.09 2.43 H 8 13 17 12.83 2.25 12.84 2.55 12.90 2.43 I 14 17 21 17.00 1.30 17.15 1.54 17.09 1.47 J 6 9 12 9.00 1.00 9.00 1.13 9.00 1.08

Estimating the mean and variance 867

References

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868 N. Ravi Shankar, K. Surya Narayana Rao, V. Sireesha [13] Ravi Shankar N.,and Sireesha V.,(2009), An Approximation for the Activity Duration Distribution, Supporting Original PERT ,Applied Mathematical Sciences, Vol. 3, no. 57, 2823 2834. Received: October, 2009