Project Risk Analysis Using an Integrated Probabilistic Beta-S Model and Multi-Parameter Copula Function

Document Type : Research Article


Construction Engineering and Management, Science and Research Branch, Islamic Azad University, Tehran, Iran


One of the key attributes of any project is its time and cost constraints. Nowadays project-oriented organizations are looking for more advanced, accurate and efficient methods rather than traditional project management practices. Earned Value Management (EVM) is a well-accepted methodology to combine time, cost and scope of the projects reported in terms of Cost Performance Index (CPI) and Schedule Performance Index (SPI). However, a deterministic EVM cannot consider uncertainties of time and cost of activities of project and the correlation structure amongst them, which are inevitable and prevalent in any project. Therefore, an advanced probabilistic EVM is needed. A literature review reveals that there are only very limited studies in this area with different levels of complexity, maturity and limitations. In this study after defining the probability Number density functions, i.e., pdfs, of time and cost of every activity of project’s scheduled plan and their correlation structure, using Primavera Risk Analysis®, as a commercially available project risk analysis software, and Monte Carlo Simulation (MCS), in every iteration a time based cumulative cost of project, the socalled S-curve is created and normalized to be the inputs for curve fitting into a four parameter Beta-S function. Hence, for every iteration the corresponding values of these parameters can be calculated and the best performing marginal pdfs, be derived. In this paper in a novel approach, copula functions are employed to bind together these pdfs with a high level of efficiency and accuracy in terms of reliving the limitation of their belongingness to the same parametric group of marginal pdfs, e.g., multivariate Gaussian joint probability distribution. The proposed model collates all propagated uncertainties of the project activities in a single probabilistic closed form function. This copula function can be used in estimation of the performance indices of a probabilistic EVM and more importantly fed into a Bayesian updating scheme to estimate the project future performance more accurately.


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