Skip to main content
Log in

How adding new information modifies the estimation of the mean and the variance in PERT: a maximum entropy distribution approach

  • Original Research
  • Published:
Annals of Operations Research Aims and scope Submit manuscript


This paper presents an alternative method to estimate the mean and the variance when using the program evaluation and review technique (PERT). Different levels of information, provided by an expert, are considered in the PERT scenario to obtain the values of the mean and the variance by means of a maximum entropy distribution approach. In our opinion, the information to be taken into account should be only that supplied by the expert. We also perform a numerical analysis to examine how the estimates vary if new information is added. From this, we conclude that the inclusion of new information (even for analytic purposes) produces significant changes in the estimates proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others


  • Abdelkader, Y. H. (2004). Evaluating project completion times when activity times are weibull distributed. European Journal of Operational Research, 157(3), 704–715.

    Article  Google Scholar 

  • Azaron, A., Katagiri, H., & Sakawa, M. (2007). Time-cost trade-off via optimal control theory in Markov PERT network. Annals of Operational Research, 150, 47–64.

    Article  Google Scholar 

  • Chae, K. C. (1990). A geometric interpretation of the PERT assumptions on the activity time. International Journal of Mathematical Education in Science and Technology, 21(2), 283–288.

    Article  Google Scholar 

  • Chae, K. C., & Kim, S. (1990). Estimating the mean and variance of PERT activity time using likelihood-ratio of the mode and the midpoint. IEEE Transactions on Engineering Management, 22(3), 198–203.

    Article  Google Scholar 

  • Farnum, N. R., & Stanton, L. W. (1987). Some results concerning the estimation of beta distribution parameters in PERT. Journal of Operational Research Society, 38(3), 287–290.

    Article  Google Scholar 

  • Feller, W. (1968). Introduction to probability theory and its applications (Vol. 1). New York: Wiley.

    Google Scholar 

  • Gallagher, C. (1987). A note on PERT assumptions. Management Science, 33(10), 1360.

    Article  Google Scholar 

  • García, C. B., García, J., & Cruz, S. (2010). Proposal of a new distribution in PERT methodology. Annals of Operational Research, 181, 515–538.

    Article  Google Scholar 

  • Golenko-Ginzburg, D. (1988). On the distribution of activity time in PERT. Journal of the Operational Research Society, 39(8), 767–771.

    Article  Google Scholar 

  • Golenko-Ginzburg, D. (1989). PERT assumptions revisited. OMEGA International Journal of Management Science, 17(4), 393–396.

    Article  Google Scholar 

  • Herrerías, R., García, J., & Cruz, S. (2003). A note on the reasonableness of PERT hypothesis. Operations Research Letters, 31, 60–62.

    Article  Google Scholar 

  • Kamburowski, J. (1997). New validations of PERT times. OMEGA International Journal of Management Science, 25(3), 323–328.

    Article  Google Scholar 

  • Keefer, D. L., & Bodily, S. E. (1983). Three-point approximations for continuous random variables. Management Science, 39(9), 1086–1091.

    Article  Google Scholar 

  • Kim, S. D., Hammond, R. K., & Bickel, J. E. (2014). Improved mean and variance estimating formulas for PERT analyses. IEEE Transactions on Engineering Management, 61(2), 362–369.

    Article  Google Scholar 

  • Kotiah, T. C. T., & Wallace, N. D. (1973). Another look at the Pert assumptions. Management Science, 20(1), 44–49.

    Article  Google Scholar 

  • Lau, A., Lau, H., & Zhang, Y. (1996). A simple and logical alternative for making PERT time estimates. IEEE Transactions on Engineering Management, 28(3), 183–192.

    Article  Google Scholar 

  • Littlefield, J. R., & Randolph, P. H. (1987). An answer to Sasieni’s question on PERT times. Management Science, 33(10), 1357–1359.

    Article  Google Scholar 

  • MacCrimmon, K. R., & Ryavec, C. A. (1964). An analytical study of the PERT Assumptions. Operations Research, 12, 16–37.

    Article  Google Scholar 

  • Malcolm, D. G., Roseboom, J. H., Clark, C. E., & Fazar, W. (1959). Application of a technique for research and development program evaluation. Operations Research, 7, 646–669.

    Article  Google Scholar 

  • Moders, J. J., & Rodgers, E. G. (1968). Judgement estimates of the moments of PERT type distributions. Management Science, 15(2), B76–B83.

    Article  Google Scholar 

  • Premachandra, I. M. (2001). An approximation of the activity duration distribution in PERT. Computers and Operations Research, 28, 443–452.

    Article  Google Scholar 

  • Sasieni, M. W. (1986). A note on PERT times. Management Science, 32(12), 1652–1653.

    Article  Google Scholar 

  • Shankar, N. R., Rao, S. N., & Sireesha, V. (2010). Estimating the mean and variance of activity duration in PERT. International Mathematical Forum, 5(18), 861–868.

    Google Scholar 

  • Shankar, N. R., & Sireesha, V. (2009). An approximation for the activity duration distribution, supporting original PERT. Applied Mathematical Sciences, 3(57), 2823–2834.

    Google Scholar 

Download references


AHB thanks to the Project ECO2017-85577-P. The authors gratefully acknowledge the helpful comments of the anonymous reviewers and the editor-in-chief.

Author information

Authors and Affiliations


Corresponding author

Correspondence to M. P. Fernández-Sánchez.

Ethics declarations

Conflict of interest

The authors report no conflicts of interest. The authors are solely responsible for the content and writing of this paper.



The results shown in this Appendix are from “Maximum entropy distributions with position and unimodality restrictions”, an unpublished working paper, University of Las Palmas de Gran Canaria (2009) available upon request to authors.

Given two distributions with probability density function (pdf) \( f(x) \) and \( g(x) \), the Kullback–Leibler divergence or relative entropy of \( f \) with respect to \( g \) is defined by \( D_{KL} \left[ {f:g} \right] = \smallint f(x) \cdot log\frac{f(x)}{g(x)} dx \), where \( g(x) \) is the reference distribution. In our case, in a bounded domain the reference distribution is \( Uniform_{{\left[ {a,b} \right]}} \) distribution. If we consider restrictions written as \( E_{f} \left[ {g_{k} (t)} \right] = \mu_{k} \), for \( k = 1, \ldots ,m \), where \( g_{k} (t) \) are known functions and \( \mu_{k} \) are fixed and known values. Then, \( \mathop {\hbox{max} }\limits_{f} \left\{ { - D_{KL} [f:g]} \right\} \), subject to \( E_{f} \left[ {g_{k} (t)} \right] = \mu_{k} \), for \( k = 1, \ldots ,m \); has the solution \( f\left( t \right) \propto exp\left\{ {\mathop \sum \limits_{1}^{m} \lambda_{k} g_{k} (t)} \right\} \), where the constant of proportionality \( f(t) \) is a density function and \( \lambda_{k} \) are given by the restrictions.

The lowest divergence from the \( Uniform_{[a,b]} \) distribution is the least possible information subject to the restriction of belonging to class \( Q_{**} \).

  • A.1. Obtaining\( f^{(3)} (t) \) In this case the assumptions are \( g_{0} (t) = 1_{[a,b]} (t) \) with \( \mu_{0} \) = 1, and, \( g_{1} (t) = 1_{{[a_{1} ,b_{1} ]}} (t) \) with \( \mu_{1} = \varepsilon . \)

  • A.2. Obtaining\( f^{(2)} (t) \).To obtain \( f^{(2)} (t) \) the following affirmations made (without proof) are necessary.

The next Proposition 1 is known as the Khintchine characterisation.

Proposition 1

(see Feller 1968). A random variable\( T \)is unimodal with mode at\( M \)if and only if T − M = U.Z where\( U \)is a random variable that follows a\( Uniform \)distribution at\( \left[ {0,1} \right] \)and Z is a random variable independent of\( U \).

Henceforth, \( g(z) \) denotes the unknown density function of the random variable \( Z \). The following proposition connects the pdf of \( T \) with that of \( Z \).

Proposition 2

Let\( T \)be a random variable, unimodal with mode at\( M \), then\( f(t) \)can be written as:

$$ f(t) = \left\{ {\begin{array}{*{20}l} {\mathop \int \limits_{t - M}^{\infty } g(z) \cdot \frac{1}{z} dz;} \hfill & {if\; t \ge M} \hfill \\ {\mathop \int \limits_{ - \infty }^{t - M} g(z) \cdot \frac{1}{ - z} dz;} \hfill & {if\; t < M} \hfill \\ \end{array} } \right.. $$

Proposition 3, below, then allows us to specify restrictions on \( T \) as restrictions on \( Z \).

Proposition 3

Let\( T \)be a random variable, unimodal with mode at\( M \), i.e., T − M = U.Z as in Proposition 1. Hence, \( E[h(t)] = \mu \), where\( h( \cdot ) \)is a known function and\( \mu \)is a known constant, if and only if\( E[h^{*} (z)] = \mu \), where

$$ h^{*} (z) = \left\{ {\begin{array}{*{20}l} {\frac{1}{z}\mathop \int \limits_{0}^{z} h(M + u)du;} \hfill & {z > 0} \hfill \\ {\frac{1}{ - z}\mathop \int \limits_{z}^{0} h(M + u)du;} \hfill & {z < 0} \hfill \\ \end{array} } \right. $$

From these propositions we can obtain maximum entropy distributions with restrictions of unimodality with known mode. The method for doing so consists of using Proposition 3 and then specifying the maximum entropy distribution for \( Z \) under the restrictions. From this distribution and using Proposition 2 we obtain the distribution of \( T \). Specifically, to obtain \( f^{(2)} (t) \) the restriction on \( {\text{T}} \) is given by \( h(t) = 1_{[a,b]} (t) \) and \( \mu \) = 1 which becomes a restriction on Z given by \( h^{*} (z) = 1_{[a - M, b - M]} (z) \).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernández-Bastida, A., Fernández-Sánchez, M.P. How adding new information modifies the estimation of the mean and the variance in PERT: a maximum entropy distribution approach. Ann Oper Res 274, 291–308 (2019).

Download citation

  • Published:

  • Issue Date:

  • DOI: