a$.$ Draw PERT Chart with earliest and latest start and finish times$/$activity$.$ Fill in the slack time in the table for each activity. Show the slack time for each path for all paths and show Critical Path $($CPM$)$ and CPM time.

b$.$ What is the expected time for $68,95$ and $99$ percent Confidence Interval $($CI$)$ for the project?

c$.$ Show the $($ID $5=50\%),$ Confidence Interval for the project $($use section $7\mathrm{.}4\mathrm{.}3.$ page $225$ for parts $2/3)$ for only one percentage corresponding to your ID $($id$=5),$ not all percentages

$7\mathrm{.}4\mathrm{.}3$

The total project duration time can be estimated from the critical path by the addition of all critical activities times. In order to account for the uncertainties in estimating these times, two probability based methods can be used:

$1.$Determine three different estimates for each activity time: realistic $($most likely$),$ optimistic, and pessimistic. These would represent three probability levels based on linear approximations of the normal curve. The realistic estimate is $4/6$ or $66\mathrm{.}7$ percent of the total area under the curve, corresponding to $\backslash$pm $($one$)\backslash$sigma activity. Optimistic and pessimistic linear approximations are $1/6$ or $16\mathrm{.}7$ percent of the area under the normal curve, corresponding to outside of the $\backslash$pm $($one$)\backslash$sigma activity. All three estimates will account for the total area under the linear approximation of the normal curve as follows:

$$The expected activity time $($EAT$)$ formula calculates an activity time based on the PM$$s estimate of the extremes of activity completion variance around the original time estimate. The total project duration time can be calculated from the sum of all expected critical activity times.

$2.$Alternatively$,$ the PM could estimate the standard deviation $(\backslash$sigma $)$ of critical activity time distribution, based on their own experience. The PM can then determine the confidence interval $($CI$)$ of project duration by calculating the project variability $(\backslash$sigma project$)$ from all critical activities$\backslash$sigma using root mean square $($RMS$)$ methodology. The CI produces a minimum and a maximum project duration estimate based on the desired confidence. The $\backslash$sigma project can be calculated as follows:

$$Once the $\backslash$sigma project is calculated, then CI may be estimated for a selected confidence percentage. Examples are $68$ percent confidence $(\backslash$pm $\backslash$sigma project$)$ and $95$ percent confidence $(\backslash$pm $2*\backslash$sigma project$).$ The $95$ percent CI for the project estimated duration is equivalent to declaring that a project completion will fall within $\backslash$pm $2*\backslash$sigma project CI in $95$ percent of the time. The end points represent the most optimistic and pessimistic times consequently.

$$For example, let$$s assume a project with a critical path of $10$ weeks with four critical activities that the PM has estimated to have standard deviations $(\backslash$sigma $)$ of $1,0\mathrm{.}5,1\mathrm{.}25,$ and $0\mathrm{.}5.$ At $95$ percent confidence, the minimum project duration $($optimistic time$)$ will be $6\mathrm{.}5$ weeks and the maximum project duration $($pessimistic time$)$ will be $13\mathrm{.}5$ weeks, as follows:

$\backslash$sigma project $=(12+0\mathrm{.}52+1\mathrm{.}252+0\mathrm{.}52)=(1+0\mathrm{.}25+1\mathrm{.}56+0\mathrm{.}25)=3\mathrm{.}06=1\mathrm{.}7595$ percent CI $=10\backslash$pm $2*\backslash$sigma project $=10\backslash$pm $3\mathrm{.}5->6\mathrm{.}5($minimum$/$optimistic$)$ and $13\mathrm{.}5($maximum$/$pessimistic project duration$).$ To double check, EAT $=(6\mathrm{.}5+4*10+13\mathrm{.}5)/6=10$ weeks.

$$The most common confidence intervals used for project duration estimates are $50$ and $90$ percent, corresponding to $\backslash$pm $0\mathrm{.}685*\backslash$sigma project $=50$ percent CI and $\backslash$pm $1\mathrm{.}645*\backslash$sigma project $=90$ percent CI of the project duration estimate.