Project Crashing Optimization Strategy with Risk Consideration

Read this article. The study develops a comprehensive evaluation strategy for project management. Section 2.1, Schedule Method-CPM/PERT, suggests that CPM does not consider risk or uncertainty. What would you add to a sensitivity analysis such that it could address risks or uncertainties?

Background

Monte Carlo Simulation

Hulett stated that CPM scheduling tools, which include manual and software-based systems, cannot handle the uncertainty that exists in the real world regarding project activity durations because these tools assume that activity durations are with certainty as single-point numbers. A stochastic risk analysis technique called Monte Carlo simulation can be applied to evaluate project uncertainties. Monte Carlo simulation is suitable for determining the project completion date because the date is determined by the uncertainty in the duration of many activities that have already been linked logically in the CPM schedule. Monte Carlo simulation is a statistical sampling technique that operates with random components as input variables subject to uncertainties and presents a set of results in terms of probabilities after several iterations. As Covert stated, a probability density function (PDF) is used to define the probability distributions for continuous distributions, which can be expressed in terms of the mathematical formula of f_{y}(x), where f_{y}(x) is the PDF defined over the range, x. Any point estimate (c) has some probability to be sufficient or to be exceeded. The probability that an estimate will be exceeded (i.e., overrun) is the risk, and the probability that the estimate will be sufficient is the opportunity. Therefore, the risk is the integral of the PDF from the point estimate, c, to infinity (\infty), which can be expressed as formula (3). Opportunity represents the area under the curve from -\infty to c, which is expressed as formula (4).

\text { Risk }=\int_{c}^{\infty} f_{y}(x) d x=1-\int_{-\infty}^{c} f_{y}(x) d x=1-F_{y}(c)       (3)

\text { Opportunity }=\int_{-\infty}^{c} \mathrm{f}_{\mathrm{y}}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{F}_{\mathrm{y}}(\mathrm{c})      (4)

A triangular distribution model is frequently used in project risk analyses, and three-point scenarios are applied in the analyses. The parameters of a triangular distribution are estimated using the lowest possible value (L), the highest possible value (H), and the most likely value (M). In Monte Carlo simulation, each project activity has a respective range and a pattern of duration possibilities. Figure 1 is the typical triangular distribution.

Triangular distribution: f_{y}(x ; L, M, H)=\mathrm{T}(\mathrm{L}, \mathrm{M}, \mathrm{H})

where L is the lowest possible value (optimistic value)

           M is the most likely value

           H is the highest possible value (pessimistic value)


Figure 1

Triangular distribution.


The PDF of the triangular distribution T(L, M, H) is

\mathrm{f}_{\mathrm{y}}(\mathrm{x})= \begin{cases}\frac{2(x-L)}{(H-L)(M-L)} & \text { if } L \leq x < M \\ \frac{2(H-x)}{(H-L)(H-M)} & \text { if } M \leq x \leq H\end{cases}

Formulas (7), (8), and (9) combine the optimistic, pessimistic, and most likely time to estimate the average time, variance, and standard deviation of project activity i. respectively.

Triangular average time \left(t_{i}\right)=\frac{\left(t_{o}+t_{m}+t_{p}\right)}{3}                            (7)

Triangular variance \left(\mu_{i}\right)                                                                                           (8)

=\frac{\left(t_{p}-t_{m}\right)^{2}+\left(t_{p}-t_{m}\right) \times\left(t_{m}-t_{o}\right)+\left(t_{m}-t_{o}\right)^{2}}{18}

Triangular standard deviation \left(\sigma_{\mathrm{i}}\right)=\sqrt{\sum \mu_{\mathrm{i}}}, where \mathrm{i}=1,2,3, \ldots, \mathrm{n}                             (9)

Experts familiar with the project tasks should accomplish them to provide estimates in the workshop. If these uncertainties are identified early in the project, plans that minimize or prevent risks can be formulated. Project managers can accurately and confidently estimate the overall completion time for the project under consideration by dealing with a range of probable durations. The results of the Monte Carlo simulation show the logical consequences of a particular set of risk assumptions, which can include the range estimates of durations, resource variations, and correlations among project categories. Monte Carlo simulation provides quantitative results for decision-making and determines the key risk factors that can make planned activities meet the scheduled milestones.