Project Planning and Scheduling Examples

Problem Statement: An activity has the following time estimates: Optimistic ($t_o$) = 4 days, Most Likely ($t_m$) = 7 days, and Pessimistic ($t_p$) = 16 days. Calculate the expected duration ($t_e$) and the variance ($v$) of the activity.

Problem Statement: A project has an expected critical path length ($T_e$) of 40 days and a project standard deviation ($\sigma_p$) of 2 days. What is the probability that the project will be completed in 43 days or less?

Problem Statement: Activity D has an Early Start ($\text{ES}$) of day 10, a duration ($t$) of 5 days, a Late Finish ($\text{LF}$) of day 20. Calculate the Early Finish ($\text{EF}$), Late Start ($\text{LS}$), Total Float ($\text{TF}$), and Free Float ($\text{FF}$) if its only successor, Activity E, has an $\text{ES}$ of day 17.

Problem Statement: An activity normally takes 10 days and costs $5,000. It can be "crashed" (expedited) to 7 days at a total cost of $6,500. Calculate the cost slope. If the project manager needs to save 2 days on the critical path, how much extra will it cost?

Problem Statement: A project begins with Activity A (duration = 4 days) and Activity B (duration = 6 days). Both start on day 0. Activity C (duration = 5 days) depends on Activity A. Activity D (duration = 3 days) depends on both A and B. Calculate the Early Start ($\text{ES}$) and Early Finish ($\text{EF}$) for all four activities.

Problem Statement: Continuing from the previous example, the project ends after Activities C and D are completed. The required project duration is 9 days. Calculate the Late Finish ($\text{LF}$) and Late Start ($\text{LS}$) for all four activities.

Problem Statement: A project consists of three activities on its critical path: Activity X, Activity Y, and Activity Z. Their expected durations ($t_e$) are 5, 8, and 12 days, respectively. Their variances ($v$) are 1.5, 2.0, and 3.5. Calculate the expected project duration ($T_e$) and project standard deviation ($\sigma_p$).

Problem Statement: A project has two parallel critical paths, Path 1 and Path 2, both with a current duration of 20 days. To reduce the project duration to 19 days, you must crash activities on both paths simultaneously. On Path 1, Activity M can be crashed for $400/day. On Path 2, Activity N can be crashed for $300/day, and Activity P can be crashed for $500/day. Additionally, there is a common Activity Q that lies on both paths and can be crashed for $800/day. What is the most cost-effective way to save 1 day on the project schedule?