Estimating the Mean and Variance of. Activity Duration in PERT

Transcription

1 International Mathematical Forum, 5, 010, no. 18, Estimating the Mean and Variance of Activit Duration in PERT N. Ravi Shankar 1, K. Sura Naraana Rao, V. Sireesha 1 1 Department of Applied Mathematics,GIS GITAM Universit, Visakhapatnam, India [email protected] 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 activit duration and estimates the mean and the variance of activit duration using pessimistic, most likel and optimistic time estimates proposed b an epert. In the past several authors have modified the original PERT estimators to improve the accurac. In this paper, on the basis of the stud of the PERT assumptions, we present an improvement of these estimates. At the end of the paper, an eample 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 activit criticalities and finding the critical path. Kewords: Estimating the mean and the variance of activit times; Beta distribution; PERT; Critical Path 1. Introduction In recent ears, the range of project management applications has greatl epanded. Project management concerns the scheduling and control of activities

2 86 N. Ravi Shankar, K. Sura Naraana Rao, V. Sireesha in such a wa that the project can be completed in as little time as possible [1,]. PERT [1] is a well known technique with proven value in managing large-scale projects. In 1959, the creators of PERT [1] considered beta distribution α 1 β 1 α + β ) ( a) ( b ) f ( ), a < < b, α, β > 0. (1) α + β 1 α) β ) ( b a) as an adequate distribution of the activit duration where α and β are parameters of the beta distribution. The suggested the estimates of the mean and variance values 1 ( a + 4m + b), () 6 1 ( b a), (3) 36 where a,m and b are the optimistic, most likel and pessimistic activit time estimates respectivel determined b a specialist. In the last five decades, numerous attempts have been made to improve the PERT analsis based on the subjective determination of a, m and b. B using PERT, managers are able to obtain[9,10,11] : (i) A graphical displa of project activities. (ii) An estimate of how long the project will take. (iii) An indication of which activities are the most critical for timel project completion. (iv) An indication of how long an task can be delaed without delaing 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 b Clark[10], Grubbs[11] and Sasieni[8] is that the estimates eq.() and eq (3), which are based on the three activit times a, m and b cannot be obtained directl from (1), impling a lack of a sound theoretical basis. Ginzburg [6] assumed p+q z (constant) as an etension of earlier assumptions, saing, on the basis of statistical analsis and some other intuitive arguments, the creators of PERT [1] assumed that p + q 4. In [13], Ravi Shankar and Sireesha generalize the assumption on parameters in original PERT and obtained new approimation for the mean and variance of a PERT activit duration distribution. Based on beta activit time distribution, we assumed p qz and obtained new approimations for the mean and the variance of activit time in PERT. B comparison with actual values, it was shown that the proposed approimations have the lowest average absolute error compared to the eisting ones. The rest of this paper is organized as follows. In Section, we briefl review Original and Ginzburg s PERT approimations. In Section 3, we proposed PERT approimation. In Section 4, we given a numerical eample for PERT approimations. In section 5 we summarizes contributions of this paper.

3 Estimating the mean and variance 863 Traditional and Ginzburg s PERT approimations.1 Traditional PERT approimation Since in PERT applications a and b of the densit function (1) are either known or subjectivel determined, we can alwas transform the densit function to a standard form, α + β ) α 1 β 1 f ( ) (1 ),0 < < 1, α, β > 0, (4) α) β ) a where. b a Note that simple relations a m a,, m (5) b a b a b a hold. Let α-1 p, β-1 q. The densit function (4) becomes p + q + ) p q f ( ) (1 ),0 < < 1, p, q > 1, p + 1) q + 1) with the mean, variance and mode as follows : (6) p + 1, (7) p + q + ( p + 1)( q + 1), (8) ( p + q + ) ( p + q + 3) m p. p + (9) q From (6) and (9) we obtain p + q + ) p p( 1/ m 1) f ( ) (1 ). (10) p + 1) q + 1) Thus value m, being obtained from the analst s subjective knowledge, indicates the densit function. On the basis of statistical analsis and some other intuitive arguments, the creators of PERT assumed [] that p+q 4. (11) It is from that assertion that estimates () and (3) were finall obtained, according to (6) (9).

4 864 N. Ravi Shankar, K. Sura Naraana Rao, V. Sireesha. Ginzburg s [6] PERT approimation Ginzburg[6] showed that the PERT assumptions (11) is poor because the actual standard deviation ma be considerabl smaller than 1/6, especiall in the tails of the distribution[7,8]. In order to make the assumption more fleible, he assumed that the sum p+q in (6) is approimatel constant but not predetermined; i.e., relation p + q Z constant (1) From (9) we obtain p Zm, (13) and values and are Zm + 1 ( m ), (14) Z + 1+ Z + Z m Z m ( m ) (15) ( Z + ) ( Z + 3) To satisf the main PERT assumption the average value ( m ) for 0 < m <1 has to be equal to 1/36; i.e., 1 1 ( m ) dm (16) 36 0 Substituting (15) in (16), integrating and solving (16) for Z, obtained Z Approimating Z to 4.5 and getting p 4.5 m, q 4.5 (1-m ) (17) from (1) and (13), finall obtained 6.5) 4.5m ( ) 4.5(1 m ) f (1 )., (18) 4.5m + 1) m ) with the mean and variance 9 m +, (19) 13 1 ( m m ), (0) 168 For the general beta distribution of the activit time, estimates (19) and (0) are transformed to a + 9m + b, 13 (1) ( b a) m a m a b a b a ()

5 Estimating the mean and variance 865 a + b B assuming that p 1, q and m, he further improved these estimates 3 when the estimated mode of the activit time is located in the tail of the distribution as follows. 0.(3a b) (3) Thus estimates () and (3) are replaced b estimates (1) and (4). ( b a) 0.04( b a) (4) 3. Proposed PERT approimation Let p z (constant)where p and q are equal to α-1 and β-1 respectivel. B q substituting p qz in (7) - (9) we obtain qz + 1 (5) qz + q + z m (6) z +1 ( qz + 1)( q + 1) (7) ( qz + q + ) ( qz + q + 3) From (7), z (8) 3 q( 1+ z) 1 We assume that original PERT assumption,, to solve (8) using (6) to 6 obtain the following values for p and q. p 36m (1 m), (9) q 36m (1 m ) (30) Substituting p and q values in (7) and (8) we obtain 36m (1 m ) + 1 (31) 36m (1 m ) + ( 36m (1 m ) + 1)( 36m (1 m ) + 1) (3) ( 36m (1 m ) + ) ( 36m (1 m ) + 3) a + b Using relations (5) in (31) and (3) and also substituting m which is 3 used b Ginzberg[7] in variance, we obtain 36( m a)( b m) m + ( b + a)( b a) (33) 36( m a)( b m) + ( b a)

6 866 N. Ravi Shankar, K. Sura Naraana Rao, V. Sireesha 0.03 ( b a) (34) Thus estimates () and (3) are replaced b estimates (33) and (34). 4. Numerical Eample The data for activities is represented in table I including mean and variance estimates for original, Ginzburg and proposed approimations. The estimated project duration has approimatel same value b using the original, Ginzburg and proposed methods. Table I. Mean and variance estimates Activit a m b Original approimation Ginzburg approimation Proposed approimation A B C D E F G H I J Conclusion Based on beta activit time distribution, we obtained new approimations for the mean and the variance of activit time in PERT. B comparison with actual values, it was shown that the proposed approimations are identical with the eisting ones. The estimated project duration has approimatel same value b using the original, Ginzburg and proposed methods.

7 Estimating the mean and variance 867 References [1] Chen,C.T. and Huang,S.F.,(007), Appling fuzz method for measuring criticalit in project network, Information Sciences,177, [] Azaron, A., Katagiri, H., Sakawa,M., Kato,K. and Memariani,A.,(006), A multiobjective resource allocation problem in PERT networks, European Journal of Operational Research, 17, [3] Stevenson,W.J.(00),Operation Management, seventh edition, McGraw-Hill. [4] Ro, G. Nava, P. and Israel, S. (00), Integrating sstem analsis and project management tools, International Journal of Project management,0, [5] Avraham, S., (1997), Project segmentation- a tool for project management, International Journal of Project Management, 15, [6] Ginzberg, DG.,(1988), On the distribution of activit time in PERT, Journal of the Operations Research Societ,39, [7] Farnum, N.R. and Stanton, L.W.,(1987), Some results concerning the estimation of beta distribution parameters in PERT, Journal of Operations Research Societ, 38, [8] Sasieni, M.W.,(1986), A note on PERT times, Management Science, 3, [9] Moderi, J.J. and Rodgers, E.G., (1968), Judgment estimates of the moments of PERT tpe distributions, Management Science, 15, B76-B83. [10] Clark, CE.,(196), The PERT model for the distribution of an activit time, Operations Research, 10, [11] Grubbs, FE.,(196), Attempts to validate certain PERT statistics or picking on PERT, Operations Research, 10, [1] Malcolm D.G., Roseboom J.H., and Clark C.E., (1959), Application of a technique of research and development program evaluation, operations research, 7,

8 868 N. Ravi Shankar, K. Sura Naraana Rao, V. Sireesha [13] Ravi Shankar N.,and Sireesha V.,(009), An Approimation for the Activit Duration Distribution, Supporting Original PERT,Applied Mathematical Sciences, Vol. 3, no. 57, Received: October, 009