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Approximating the Criticality Indices of the Activities in PERT Networks

Author

Listed:
  • Bajis M. Dodin

    (Graduate School of Management, University of California, Riverside, California 92521)

  • Salah E. Elmaghraby

    (School of Engineering, North Carolina State University, Raleigh, North Carolina 27650)

Abstract

A stochastic PERT network is a directed acyclic network in which the arc lengths are independent random variables with known distributions. A fundamental problem in PERT networks is to identify the activities which are critical to the achievement of the project objectives. In an activity network if the duration of each activity is not a random variable, then it is easy to identify the criticality of each activity represented by its float time. However, when the duration of any activity is a random variable, it is not easy to identify the criticality of each activity. In this case the criticality of an activity is known as the "criticality index," which is defined as the sum of the criticality indices of the paths containing it. The criticality index of a path is the probability that the duration of the path is greater than or equal to the duration of every other path in the network. Clearly, the criticality index of an activity can be obtained by determining the criticality indices of the paths, which requires identifying all the paths, determining their criticality indices, then identifying the paths containing the activity. In this paper we develop a theory which leads to a procedure to approximate the criticality indices of all the activities without going through the above three steps. The procedure has been applied to large size PERT networks generated at random, and the results are found to be very close to those obtained by extensive Monte Carlo sampling.

Suggested Citation

  • Bajis M. Dodin & Salah E. Elmaghraby, 1985. "Approximating the Criticality Indices of the Activities in PERT Networks," Management Science, INFORMS, vol. 31(2), pages 207-223, February.
    • Handle: RePEc:inm:ormnsc:v:31:y:1985:i:2:p:207-223
    DOI: 10.1287/mnsc.31.2.207
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    Citations

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    Cited by:

    1. Williams, Terry, 1995. "A classified bibliography of recent research relating to project risk management," European Journal of Operational Research, Elsevier, vol. 85(1), pages 18-38, August.
    2. Zhichao Zheng & Karthik Natarajan & Chung-Piaw Teo, 2016. "Least Squares Approximation to the Distribution of Project Completion Times with Gaussian Uncertainty," Operations Research, INFORMS, vol. 64(6), pages 1406-1421, December.
    3. Fernando Acebes & Javier Pajares & José M. González-Varona & Adolfo López-Paredes, 2021. "Project risk management from the bottom-up: Activity Risk Index," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(4), pages 1375-1396, December.
    4. Fernando Acebes & Javier Pajares & Jose M Gonzalez-Varona & Adolfo Lopez-Paredes, 2024. "Project Risk Management from the bottom-up: Activity Risk Index," Papers 2406.00078, arXiv.org.
    5. Li, Xiaobo & Natarajan, Karthik & Teo, Chung-Piaw & Zheng, Zhichao, 2014. "Distributionally robust mixed integer linear programs: Persistency models with applications," European Journal of Operational Research, Elsevier, vol. 233(3), pages 459-473.
    6. Tao, Liangyan & Wu, Desheng & Liu, Sifeng & Lambert, James H., 2017. "Schedule risk analysis for new-product development: The GERT method extended by a characteristic function," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 464-473.
    7. Elmaghraby, Salah E., 2000. "On criticality and sensitivity in activity networks," European Journal of Operational Research, Elsevier, vol. 127(2), pages 220-238, December.
    8. Elmaghraby, S. E. & Fathi, Y. & Taner, M. R., 1999. "On the sensitivity of project variability to activity mean duration," International Journal of Production Economics, Elsevier, vol. 62(3), pages 219-232, September.
    9. Bregman, Robert L., 2009. "A heuristic procedure for solving the dynamic probabilistic project expediting problem," European Journal of Operational Research, Elsevier, vol. 192(1), pages 125-137, January.
    10. Tereso, Anabela P. & Araujo, M. Madalena T. & Elmaghraby, Salah E., 2004. "Adaptive resource allocation in multimodal activity networks," International Journal of Production Economics, Elsevier, vol. 92(1), pages 1-10, November.
    11. Williams, Terry, 1999. "Towards realism in network simulation," Omega, Elsevier, vol. 27(3), pages 305-314, June.
    12. Cho, Sungbin, 2009. "A linear Bayesian stochastic approximation to update project duration estimates," European Journal of Operational Research, Elsevier, vol. 196(2), pages 585-593, July.
    13. Fatemi Ghomi, S. M. T. & Teimouri, E., 2002. "Path critical index and activity critical index in PERT networks," European Journal of Operational Research, Elsevier, vol. 141(1), pages 147-152, August.
    14. Fatemi Ghomi, S. M. T. & Rabbani, M., 2003. "A new structural mechanism for reducibility of stochastic PERT networks," European Journal of Operational Research, Elsevier, vol. 145(2), pages 394-402, March.
    15. Vanhoucke, Mario, 2010. "Using activity sensitivity and network topology information to monitor project time performance," Omega, Elsevier, vol. 38(5), pages 359-370, October.
    16. Daniel Reich & Leo Lopes, 2011. "Preprocessing Stochastic Shortest-Path Problems with Application to PERT Activity Networks," INFORMS Journal on Computing, INFORMS, vol. 23(3), pages 460-469, August.
    17. Rabbani, M. & Fatemi Ghomi, S.M.T. & Jolai, F. & Lahiji, N.S., 2007. "A new heuristic for resource-constrained project scheduling in stochastic networks using critical chain concept," European Journal of Operational Research, Elsevier, vol. 176(2), pages 794-808, January.
    18. Bowers, J., 1996. "Identifying critical activities in stochastic resource constrained networks," Omega, Elsevier, vol. 24(1), pages 37-46, February.
    19. Badinelli, Ralph D., 1996. "Approximating probability density functions and their convolutions using orthogonal polynomials," European Journal of Operational Research, Elsevier, vol. 95(1), pages 211-230, November.
    20. Zarghami, Seyed Ashkan & Dumrak, Jantanee, 2021. "Aleatory uncertainty quantification of project resources and its application to project scheduling," Reliability Engineering and System Safety, Elsevier, vol. 211(C).
    21. R. Alan Bowman, 2003. "Sensitivity curves for effective project management," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(5), pages 481-497, August.
    22. Madadi, M. & Iranmanesh, H., 2012. "A management oriented approach to reduce a project duration and its risk (variability)," European Journal of Operational Research, Elsevier, vol. 219(3), pages 751-761.

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    Keywords

    project management: PERT;

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