This is an example of an Activity On Arrow $($AOA$)$ Network Chart for a set of activities. Each activity is represented by an Arrow, and identified by a Letter, and its duration $[$ex$.$ F $(3)].$ "Start" and "Finish" represent the beginning and end of the total set of activities. The Circles between arrows represent the start and completion milestones for each activity. Note that an activity to the right of any milestone cannot start until the activities to the left of the milestone are done. This represents the precedence of activities. The "Critical Path" $($CP$)$ through the network is the longest path through the network, and establishes the duration for the whole network. The CP is determined on an AOA chart by tracing individual paths through the network, adding up the individual durations for each path, and then finding the path whose duration is the longest. This becomes the CP$.$ In this example, the CP is FKLPQRO,

and its duration is $19$ days. This means that the duration of the entire project is $19$ days, and no shorter.

CP analysis is very useful because of the concept of "slack" time $($also known as "float"$).$ Slack is the amount of time an activity or set of activities can be extended, or delayed, before it affects downstream activities. For example, note that ABIJ has duration of $10$ days. Compared to the CP$,$ ABIJ has a total of $9$ days of slack time. However, what if task I needed more time? How could a Project Manager determine how long task I could be extended without increasing the duration of the current CP$?$ Assuming task J will stay at $5$ days, task I could be extended up to $9$ additional days, for a total of up to $11$ days. If this happened, there would be two Critical Paths through the network now $($both would be the longest path$),$ but the overall duration $(19$ days$)$ would stay the same. What if Task I could not start on schedule $($right after task B$)?$ How many days could task I$\text{'}$s start be delayed without changing the CP duration $($assuming task A & B run as scheduled and J still required $5$ days$)?$ Task I require $2$ days, and the overall slack

available to ABIJ is $9$ days. If task I was delayed to the $13$th and $14$th day of the project, task J would then run from the day $15$ to day $19$ and the projects CP would not change. Therefore, I$\text{'}$s start could be delayed $9$ days $($the slack time$),$ and completed on time $(2$ days later$),$ for an overall duration $($delay time

plus task time$)$ of $11$ days. This would not increase the current duration of the CP$.$ Another way of phrasing this is that task I$$s start date could "float" $9$ days to the right along path ABIJ.

Critical Path analysis on small AOA networks is easier to understand and work with than CP analysis on Activity On Node $($AON$)$ networks $($as described in detail by Lewis$).$ You can "see" the paths through the network, and in many cases see the slack periods and how they effect other activities. Unfortunately, as the network gets more complicated, AOA and CP analysis becomes complex to work with, and is unpopular.

Questions:

$1.$How much of an increase in duration could task M have without affecting the current critical path

duration?

$2.$ How much longer could task M$\text{'}$s start be delayed without affecting the current duration of the

CP$?$

$3.$ The manager of the group responsible for task B believes he underestimated the resource

requirement $(1$ day$)$ to complete the task in the allocated time. This manager feels strongly that he

needs more project time, and is asking for a total $3$ days to complete. This manager may be "fishing"

for extra budget to cover some other activity at your expense, but this does not matter to you, the PM$.$

You are focused on the total project duration. You do not need to go investigate the manager's

accuracy in his revised estimate, and you tell the manager to use the existing

resources and take the $2$ extra days. What is the basis for your decision?