Critical Path Analysis – Introduction

Critical path analysis allows you to determine the best way of arranging activities. The typical question to answer is “what is the minimum time required for a process?”

Performing activities one after the other as you come to them might not be the best way of organising a project. If more than one activity can be worked on at the same time you will need to decide when to start each one. Below, in table 1, are the activities needed to build a house with durations given in days. We are going to perform a critical path analysis on the process of house construction.

AExcavate foundations2
BPour concrete1
CBuild walls3
DConstruct roof sections7
EPut up roof sections1
FTile roof3
GInstall plumbing4
HInstall electrics3
IInstall windows and doors2
JPaint ceilings2
KPaint walls3
LLay carpets2
Table 1: Activities involved in building a house

The first thing that we need to do is to decide which of the activities depend on other activities:

  • B requires that A be complete.
  • C requires that B be complete.
  • E requires that C and D be complete.
  • F requires that E be complete.
  • G, H, and I all require that F be complete.
  • J and K require that G, H, and I be complete.
  • L requires that J and K be complete.

With this information we can draw an activity network. In an activity network the edges represent activities, such as those listed above. The nodes represent events. An event is the start and/or finish of one or more activities. The activity network for the house building example is shown in figure 1.

Figure 1: The activity network for house building.
Figure 1: The activity network for house building.

The activity network shows that there are 12 events involved in the building of this house, and that 12 activities (plus 3 dummy activities) are required before the house is complete.

As you can see, there are a few activities that can occur concurrently. Later we will use this example to find the critical path through the operation (finding a critical path). It is the longest path and determines the minimum length of time that is necessary to complete the house.

Activity Networks

As we saw in the introduction to critical path analysis, an activity network is a graphical representation of activities and events that are needed for a process. There are several rules that have to be obeyed when creating accurate networks.

These rules are:

  1. An edge is used to represent an activity.
  2. A vertex represents an event, which is the start or end of one or more activities.
  3. Events are numbered so that the activity ends at a higher numbered event than it started.
  4. Events are numbered so that no activity starts and ends at the same event. An activity can therefore be uniquely identified by its start and end events.
  5. In order to avoid ambiguity between activities you must use ‘dummy’ activities. These are activities that have zero length and are inserted to ensure that precedence requirements are met. They allow two activities that ought to have the same start and end events to be distinguished.
  6. There should only be one start event and one finish event.
  7. The activity detail (or code) and duration are written along the edges.
  8. The edge length has no meaning.

Using these rules you should be able to create activity networks for any process that involves more than one activity.

Earliest and Latest Times

Once you have drawn out an activity network the next step is to work out the earliest and latest possible times for each event. This will let you find the minimum time necessary for the tasks to be completed taking activity precedence into account.

In order to make the identification of earliest and latest times easier, they are often put into a pair of boxes. The left box will be the earliest time and the right box will be the latest time. When working out the earliest times for each event you must start at the first event and assign an earliest time of 0. Work through each event in turn. Add the duration of the activity to the earliest time of the start event to get the earliest time of the end event. If there is more than one way of arriving at the end event then take the largest value for the earliest time.

Figure 1: Earliest times for the house building activity network.
Figure 1: Earliest times for the house building activity network.

Once you have gone through the entire network assigning earliest times you can begin to work out the latest times. This is done by starting at the last event and working backwards though the network. To start the process let the latest time of the last event be equal to the earliest time of that event. By subtracting the duration of an activity from the event that it ends at you will be able to find the latest time for the start event for that activity. If there is more than one way to arrive at a start event then take the smallest value for the latest time.

Figure 2: Earliest and latest times for the house building activity network.
Figure 2: Earliest and latest times for the house building activity network.

By looking at the earliest / latest times for the last event you will know the minimum duration for the overall process.

Allocating Resources

Once you have decided on the critical path for an activity network you will need to allocate resources to each task.

Resources could take the form of equipment, people, money, or any number of other things. Most often we will use people. The question you need to ask yourself is “how many people do I need so that the project can be finished without any delay?” In answering this question you are halfway to allocating resources.

In the example we have been looking at, there are at most three things that can be happening at any one time. Do we need to use three people, or will two people be enough (have we got enough time to complete H and I before G is completed)? Looking at the activity network we can see that there will not be enough time to do this.

Figure 1: Earliest and latest times for the house building activity network.

Another way of showing the resources needed is to draw a cascade chart and then a resource histogram.

The cascade chart in figure 2 shows the critical path on the lower row and has a row for each of the other activities, starting at their earliest time and showing float at the end. This allows the viewer to see where activities can be collapsed down to be allocated to a single worker. The collapsed form, as a resource histogram, is shown in figure 3.

Figure 2: Cascade chart showing the activities in the house building example. Critical path is the lower row.
Figure 3: Resource histogram for the house building example. A total of 3 workers are needed.