# Understanding the basics of CPM calculations

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## James L. Jenkins, Purdue University

### Abstract

You've used a computer to generate a schedule, but have you ever wondered what the terms Early Start, Late Finish, Total Float, or Critical Path really mean? This paper offers a look at basic scheduling terminology by guiding the reader through basic schedule calculations. Example schedules are presented to show how durations and activity relationships are used to calculate start/finish dates and float values for activities. AON (Activity-on-Node) logic networks will be discussed.

### Introduction

Schedules are used as management tools to communicate work sequence, dictate start/finish dates for activities, and to track progress on just about every type of project. In today's business world of complex projects, managers can get lost in the plethora of information provided by the project schedule. While important terms such as the Early Start Dates, Late Finish Dates, Total Float, or Critical Path commonly appear on computer generated schedules, project managers might not fully understand what they really mean! As a result, decisions made by management based on this lack of knowledge may not be as effective as they could have been. It is the belief of the authors, that in order for construction management personnel to understand schedules and become successful in their careers, they need a thorough understanding of basic scheduling terms. And to fully understand these basic terms, one must learn how their values are calculated. This paper will discuss the basic terminology of scheduling and illustrate how values are calculated using the Activity-on-Node (AON) calculation methods.

### Terms and Abbreviations

Since the advent of computers, managers have generated schedules for their projects through computer scheduling software packages. Critical Path Method (CPM) schedules have evolved into valuable management and communication tools for today's complex projects. Activity-on-Node (AON) schedules show the Critical Path of the schedule, and thus are considered to be CPM Schedules. It is through these schedules that the logical flow of the work sequence is graphically illustrated. In addition, schedules also show the planned start/finish dates for activities, schedule duration, and float values for the individual activities on the project. One should note that schedules could also be structured to show other types of information such as budgeted costs or labor quantities. However, these options are not addressed in the basic scheduling discussion of this paper.

It is the belief of the authors that personnel in their respective construction companies must become knowledgeable in the schedule calculation process in order to fully understand the importance of schedules. By doing so, project managers will recognize how activity-sequencing affects planned start/finish dates for activities and the duration of the schedule. Also, through comprehension of float values and their definitions, managers will be able to apply this knowledge in their schedule analysis should a delay occur on the project.

A schedule graphically illustrates the intended flow of the work sequence and will designate the planned start/finish dates for activities once they are calculated. However before the discussion on the calculation process used to determine planned start/finish dates for activities, schedule duration, and float values for each activity is initiated, an explanation of basic scheduling terminology is in order. Terms and abbreviations, which are shown in parenthesis, are listed below.

• Activity-on-Node (AON) diagram: A basic type of a logic diagram used in scheduling
• Critical Activity: Any activity in the schedule that does not possess any float; Total Float=0
• Critical Path: The continuous string(s) of critical activities in the schedule between the Start and Finish of the project. The sum of the activity durations in the Critical Path is equal to the Project's Duration; therefore, a delay to any Critical Activity will result in a delay to the Project Completion Date.
• Critical Path Method (CPM): Any calculation method that shows the Critical Path in the schedule
• Duration: The amount of time required to complete a schedule activity
• Early Start (ES): Earliest date the activity can start
• Early Finish (EF): Earliest date that the activity can finish
• Free Float (FF): The maximum number of days the activity can be delayed without delaying any succeeding activity
• Lag: Planned wait time between activities
• Late Finish (LF): Latest date that the activity can finish without causing a delay to the project completion date.
• Late Start (LS): Latest date that the activity can start without causing a delay to the project completion date.
• Predecessor: The “before” activity; immediately precedes
• Successor: The “after” activity; immediately follows
• Total Float (TF): The maximum number of days the activity can be delayed without delaying the project completion date.

### Activity-on-Node Diagrams

Each project, whether it is a simple assignment or a large complicated job, can be broken down into a list of individual work activities that need to be completed in a specific sequence. This sequence of activities, which is usually pre-planned by management prior to project commencement, will dictate the logical order in which activities start or finish. A common way to express the sequence, or logical order, of the activities is through the means of a logic diagram. In a logic diagram, “nodes” (squares or rectangles) represent each activity with arrows representing the relationships between these activities. Hence, the names of logic diagrams are commonly referred as “Activity-on-Node” (AON) diagrams.

A common relationship in schedules is known as the “Finish-to-Start” relationship and is shown in Exhibit 1. With this type of relationship, the preceding activity (Activity A) must be 100% complete (or finished) before the succeeding activity (Activity B) can start. As shown in Exhibit 1, the relationship is shown by an arrow “exiting” the finish side of Activity A and “entering” the start side of Activity B (see Exhibit 2 for start and finish sides). Therefore, it is called a Finish-to-Start relationship.

Exhibit 1. Finish-to-Start Relationship

### AON Calculation Method

Once all activities are arranged in logical order and relationships are set, the logic diagram is ready for the calculation process. The format used to indicate values for each activity is shown in Exhibit 2. TT and FF indicate where to place float values. ES, EF, LS, LF and the Duration indicate where to place their respective values.

Exhibit 2. Activity Node Format

#### Forward Pass

The first step in the calculation process is known as the Forward Pass. In the forward pass, the Early Start and Early Finish values for each activity, along with the overall Project Duration, are calculated. To facilitate schedule calculations, an “end of day” notation is used for both the Early Start and the Late Start values. By doing this, the start of the network diagram is the “end of Day Zero”. In other words, the calculation process begins with placing a zero in the Early Start (ES) position of the first activity. The rest of the calculation continues with the use of the following formulas:

• Early Start = Maximum (or Highest) EF value from immediate Predecessor(s)
• Early Finish = ES + Duration

An example of a Forward Pass calculation is shown in Exhibit 3. Generic activities are used in the examples in this paper to help illustrate the different calculation processes.

Exhibit 3. AON Forward Pass

In the diagram shown in Exhibit 3, note that the early start value for Activity E is based on the maximum early finish from its two predecessors (Activity B and Activity C). Based on the Finish-to-Start relationships shown, both Activity B and Activity C must both be finished before Activity E can start. In other words, Activity E is waiting on whichever predecessor finishes the latest (in this case, Activity B). The same situation exists for Activity F. The duration of the schedule is shown in the EF of the latest activity in the network diagram. Exhibit 3 shows an EF value of “16” for Activity F. Therefore, assuming that the time-unit of the example schedule is based on days, the schedule duration is sixteen days.

#### Backward Pass

The second step in the calculation is comprised of the Backward Pass. Through this pass, the Late Start and Late Finish values are calculated. The formulas for the backward pass are shown below:

• Late Start = LF – Duration
• Late Finish = Minimum (or Lowest) LS value from immediate Successor(s)

As the name implies, this calculation step starts at the last activity in the schedule and proceeds backward through the schedule until the Late Start value is computed for the schedule's beginning activity. An example of the backward pass is shown in Exhibit 4.

Exhibit 4. AON Backward Pass

To start the backward pass calculation, the EF value in the last activity is “dropped” down to the LF value. Now the backward pass formulas for late start and late finish can be applied. Note that in diagram above, the LF value for Activity B is based on the lowest LS shown by its two successors (Activity D and Activity E).

#### Float Calculations

The importance of calculating Total Float and Free Float are found it the definitions of these terms. As stated earlier, float values indicate how much each individual activity can be delayed before affecting successor activities or the planned project completion date. The float calculations for the sample schedule are shown in Exhibit 5. Formulas for calculating Total Float and Free Float are as follows:

• Total Float = LS – ES (it is also calculated by LF – EF)Free Float = Lowest ES of successors – EF
• Free Float = Lowest ES of successors – EF

Exhibit 5. Total Float and Free Float Calculations for AON Calculations

Note that Total Float shows the difference between the earliest date that the activity can start and latest date the activity can start before the completion date is delayed. Total float can also be calculated as the difference between Late Finish and Early Finish, as LS minus ES and LF minus EF calculate the exact same number.

Free Float, per definition, is the amount of time that the activity can be delayed before any successors will be delayed. Early Start/Finish times are used to calculate the Free Float values. So in basic terms, the finish date (Early Finish) of the activity is compared with the planned start (Early Start) of succeeding activities. As shown in Exhibit 5, Activity C has three days of float before the start of Activity E will be delayed.

#### Critical Path

Once the float values are calculated, the string of critical activities will be identified. This continuous string of critical activities is called the Critical Path. Critical activities are those that do not posses any float. Note that the summation of the critical activity durations (Activities A, B, D, and F in Exhibit 5) is equal to the overall project duration (which was calculated in the forward pass). This reinforces the fact that a delay (or additional time added to the activity duration) in any critical activity will cause a subsequent delay in the completion date. While it doesn't appear in the example problem, a schedule can have more than one branch or string of activities that make up the critical path. See Exhibit 6 for an example of a computer-generated CPM schedule.

Exhibit 6. Example of a computer-generated CPM schedule

#### Calculation Rules

There are a few common checks to conduct to help gauge whether or not the manual calculations have been performed error-free. First, an activity's value for Total Float can either be greater or equal to Free Float. Or in other words, Free Float can never be larger than its value for Total Float. Second, the Critical Path must be continuous from the first activity in the schedule all the way through to the last activity. Third, negative numbers should not appear for any calculated value.

### Conclusion

This paper presented the basic terms of scheduling and illustrated how their values are calculated using the Activity-on-Node (AON) method of CPM scheduling. Schedules are used as management tools to communicate work sequence, determine start/finish dates for activities, and to track progress on just about every type of project. With the increasing demand for construction project managers, many companies are hiring individuals who have not had any formal scheduling classroom instruction. In addition, computers and scheduling software have become extremely inexpensive and CPM software has incorporated many “bells & whistles” that may get young assistant project managers into trouble if they do not understand the underlying concepts of scheduling calculations. Explanations presented in this paper will be of benefit to ALL project managers, young and old alike, associated with the design and construction of the built environment, and make them informed users of scheduling software.

Originally published as a part of 2006 PMI Global Congress Proceedings – Madrid, Spain

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