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

- 60 Downloads

## Abstract

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.

## Keywords

PERT Unimodality Maximum entropy Beta distribution## Notes

### Acknowledgements

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

### Compliance with ethical standards

### Conflict of interest

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

## References

- Abdelkader, Y. H. (2004). Evaluating project completion times when activity times are weibull distributed.
*European Journal of Operational Research,**157*(3), 704–715.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - García, C. B., García, J., & Cruz, S. (2010). Proposal of a new distribution in PERT methodology.
*Annals of Operational Research,**181,*515–538.CrossRefGoogle Scholar - Golenko-Ginzburg, D. (1988). On the distribution of activity time in PERT.
*Journal of the Operational Research Society,**39*(8), 767–771.CrossRefGoogle Scholar - Golenko-Ginzburg, D. (1989). PERT assumptions revisited.
*OMEGA International Journal of Management Science,**17*(4), 393–396.CrossRefGoogle Scholar - Herrerías, R., García, J., & Cruz, S. (2003). A note on the reasonableness of PERT hypothesis.
*Operations Research Letters,**31,*60–62.CrossRefGoogle Scholar - Kamburowski, J. (1997). New validations of PERT times.
*OMEGA International Journal of Management Science,**25*(3), 323–328.CrossRefGoogle Scholar - Keefer, D. L., & Bodily, S. E. (1983). Three-point approximations for continuous random variables.
*Management Science,**39*(9), 1086–1091.CrossRefGoogle 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.CrossRefGoogle Scholar - Kotiah, T. C. T., & Wallace, N. D. (1973). Another look at the Pert assumptions.
*Management Science,**20*(1), 44–49.CrossRefGoogle 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.CrossRefGoogle Scholar - Littlefield, J. R., & Randolph, P. H. (1987). An answer to Sasieni’s question on PERT times.
*Management Science,**33*(10), 1357–1359.CrossRefGoogle Scholar - MacCrimmon, K. R., & Ryavec, C. A. (1964). An analytical study of the PERT Assumptions.
*Operations Research,**12,*16–37.CrossRefGoogle 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.CrossRefGoogle Scholar - Moders, J. J., & Rodgers, E. G. (1968). Judgement estimates of the moments of PERT type distributions.
*Management Science,**15*(2), B76–B83.CrossRefGoogle Scholar - Premachandra, I. M. (2001). An approximation of the activity duration distribution in PERT.
*Computers and Operations Research,**28,*443–452.CrossRefGoogle Scholar - Sasieni, M. W. (1986). A note on PERT times.
*Management Science,**32*(12), 1652–1653.CrossRefGoogle 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