PERT is a project management technique used to plan, schedule, and control complex projects. This Workspirited article discusses what the PERT formula is, and how it is used to estimate a project duration, with examples.

PERT was developed by the US Navy in 1958, to plan and control the Polaris missile program.

A project is a set of interrelated activities that must be executed in a certain logical sequence to complete the entire task. Every activity in a project requires time and resources for its completion. It cannot be initiated unless the other preceding activities are completed.

Managing an entire project is a challenging job performed by a project manager; however, before the project goes on the floor, he has to perform a crucial task of providing an estimation of the project cost and duration to the client. For doing this, the PERT (Program Evaluation and Review Technique) comes to the rescue. We shall discuss what is PERT, and how it is used to calculate the expected project completion time.

PERT is a project scheduling technique which uses three estimates per activity to get the final estimate for the project duration.

- Pessimistic (T
_{p}): the longest time that an activity might require to complete - Optimistic (T
_{o}): the shortest time in which an activity can be completed - Most likely (T
_{m}): the completion time of an activity having the highest probability

The PERT technique is used in conjunction with the Critical Path Method (CPM), therefore, it is also referred to as the PERT-CPM method. Before we begin to use PERT, we need to be familiar with the following keywords:

Also known as the PERT diagram, it is a pictorial representation of all the activities involved in the project. The diagram is easy to understand, since it describes the logical relationships among the activities and the milestones they reach upon completion. The activities are denoted by arrows, while the milestones are denoted by nodes on the network diagram. Generally, the activities are named alphabetically, while the milestones are named numerically.

It is the longest time path on the network, which indicates that it is the longest time it takes to complete a project. Delay in any activity along the critical path will delay the project completion time. All activities on this path have a slack time of **zero**.

It is the amount of time an activity can be delayed before the project finish date is delayed. Formula to calculate the slack time is:

- S < 0: indicates the amount of time that must be saved so that the project finish date is not delayed.
- S > 0: indicates the acceptable delay so that the project finish date is not delayed.
- S = 0: indicates that the activity is a critical task.

The estimated time to complete a project can be computed if we have the pessimistic (T_{p}), optimistic (T_{o}), and most likely (T_{m}) estimates of all the activities involved in the project. The estimated time for an activity to be completed can be calculated using the following PERT equation:

_{e}= (T

_{o}+ 4 T

_{m}+ T

_{p}) ÷ 6

Where:

T

_{o}= Optimistic Estimate

T

_{m}= Most Likely Estimate

T

_{p}= Pessimistic Estimate

The PERT equation is a weighted average, where the most likely estimate is weighted 4 times more heavily than the optimistic and pessimistic estimates. This prevents the PERT output from being too heavily inclined in one direction.

Consider a scenario where you estimate that a task will get completed in 6 hours (T_{o}) if everything goes right. The same task is estimated to be completed in 24 hours (T_{p}) if everything goes wrong, while under normal circumstances, it is mostly likely to get completed in 12 hours (T_{m}). Using the PERT formula, we get:

_{e}= (6 + (4 × 12) + 24) ÷ 6 = 13

We observe that the estimated duration of an activity (13) is just slightly inclined towards the pessimistic value (24), since the result is still weighted heavily towards the most likely value (12).

You can also compute the standard deviation (σ_{e}) and variance (V_{e}) for an activity. The formulae are given below:

_{e}= (T

_{p}– T

_{o}) ÷ 6

V

_{e}= σ

_{e}

^{2}

According to statistics and probability theory, the standard deviation (σ) is the variation from the mean, which can either be a simple average or a weighted average. A low σ indicates that data points are close to the mean, whereas a high σ indicates that data points are spread out over a large range. The value of standard deviation can be subsequently used to calculate the probability of completing an activity within a specified range. The higher the probability, the higher is the confidence level of completing the activity, and vice versa.

So, the standard deviation of the activity will be,

σ_{e} = (24-6) ÷ 6 =2

It means that there is a 2 hours of variation from the mean. Using the standard deviation, we can conclude that the duration estimate is ±2 hours. Or, we can say that it is likely to take between 11 hours to 15 hours to complete the task.

Similarly, you can calculate the standard deviation, variance, and estimated time to complete an entire project. The formulae are given below.

_{E}= Sum of Estimated Time of Individual Activities

Project V

_{E}= Sum of Variances of Individual Activities

Project σ

_{E}= Square Root of Project Variance

Once you have the aforementioned values related to a project, i.e., T_{E} and σ_{E}, you can calculate the chances of meeting a specific project scheduled time (T_{S}) using the probability theory formula:

_{S}– T

_{E}) ÷ σ

_{E}

Where:

Z = Number of Standard Deviations from the Mean

T

_{S}= Project Scheduled Time

T

_{E}= Expected Earliest Project Time

Using the computed Z, refer to the standard probability table for normal distribution functions to determine the probability of meeting the scheduled time T_{S}.

Z | Probability |

+ 2.0 | 0.98 |

+ 1.5 | .93 |

+ 1.3 | .90 |

+ 1.0 | .84 |

+ 0.9 | .82 |

+ 0.8 | .79 |

+ 0.7 | .76 |

+ 0.6 | .73 |

+ 0.5 | .69 |

Z | Probability |

– 2.0 | 0.02 |

– 1.5 | .07 |

– 1.3 | .10 |

– 1.0 | .16 |

– 0.9 | .18 |

– 0.8 | .21 |

– 0.7 | .24 |

– 0.6 | .27 |

– 0.5 | .31 |

- Identify the activities in the project.
- Determine the logical sequence of the activities.
- Construct a network diagram.
- Calculate T
_{e}and V_{e}for every activity. - Determine the critical path.
- Calculate T
_{E}by adding up the values of T_{e}for every activity on the critical path. This value is the overall expected completion time for the project. - Similarly, calculate the V
_{E}by adding up the values of V_{e}for every activity on the critical path. This value is the variance for the entire project. - Calculate the standard deviation of the project (σ
_{E}), which is equal to the square root of the variance (V_{E}). - Calculate the project completion time with the desired probability by using the normal probability distribution.

The following calculator will enable you to compute the PERT estimate value.

Activity | Immediate Predecessor | Optimistic Time (T_{o}) |
Most Likely Time (T_{m}) |
Pessimistic Time (T_{p}) |

a | — | 3 | 6 | 15 |

b | — | 2 | 5 | 14 |

c | — | 6 | 12 | 30 |

d | a | 2 | 5 | 8 |

e | a | 5 | 11 | 17 |

f | b | 3 | 6 | 15 |

g | c | 3 | 9 | 27 |

h | d | 1 | 4 | 7 |

i | e, f | 4 | 19 | 28 |

Based on the data given above,

- Calculate the expected project duration (unit of time is weeks).
- What is the probability that the project will be completed in 38 weeks?
- What project duration will have 95% chance of completion?

**Answer:**

Step 1:

Construct the network diagram.

Step 2:

Calculate the expected time, standard deviation, and variance for every activity using the PERT formulae. Create a table depicting the calculated data.

Activity | Expected Time T_{e} = (T_{o}+4×T_{m}+T_{p})÷6 |
Standard Deviation σ_{e} = (T_{o}-T_{p})÷6 |
Variance V_{e} = σ_{e}^{2} |

a | 7 | 2 | 4 |

b | 6 | 2.4 | 5.76 |

c | 14 | 4 | 16 |

d | 5 | 1 | 1 |

e | 11 | 2 | 4 |

f | 7 | 2 | 4 |

g | 11 | 4 | 16 |

h | 4 | 1 | 1 |

i | 18 | 4 | 16 |

Step 3:

Calculate the values for Earliest Start time (ES), Earliest Finish time (EF), Latest Start time (LS), and Latest Finish time (LF) for all the activities by traversing through the network diagram and using the following formulae.

EF = ES + T_{e}

LS = LF – T_{e}

Where:

T_{e} is the expected time for that activity.

Rules for calculating the aforementioned values:

- The ES is calculated forward, starting from the first node till the last node.
- The ES for an activity leaving a particular node is equal to the largest of the EF for all activities entering the node.
- The LF is calculated backward, starting from the last node till the first node.
- The LF for an activity entering a particular node is equal to the smallest of the LS for all activities leaving the node.

Step 4:

Calculate the slack time of every activity, and make a table that depicts all these values.

Slack Time = LS – ES

Mark all those activities that have a slack time of ‘0’, and determine their path. This is the critical path.

Activity | ES | EF (ES+T _{e}) |
LS (LF-T _{e}) |
LF | Slack (LS-ES) |
Critical |

a | 0 | 7 | 0 | 7 | 0 | Yes |

b | 0 | 6 | 5 | 11 | 5 | No |

c | 0 | 14 | 11 | 25 | 11 | No |

d | 7 | 12 | 29 | 34 | 21 | No |

e | 7 | 18 | 7 | 18 | 0 | Yes |

f | 6 | 11 | 11 | 18 | 5 | No |

g | 14 | 25 | 25 | 36 | 11 | No |

h | 12 | 16 | 34 | 36 | 22 | No |

i | 18 | 36 | 18 | 36 | 0 | Yes |

**Answer 1:**

Here, the critical path is a-e-i, and the mean critical path duration (sum of all T_{e} on the critical path) is T_{E} = 7 + 11 + 18 = 36 weeks. This is the estimated project completion time.

**Answer 2:**

V_{E} = 4 + 4 + 16 = 24 (sum of all V_{e} on the critical path)

σ_{E} = √V_{E} = √24 = 4.89

Z = (38 – 36) ÷ 4.89 = 0.40

According to the Probability Table,

Z_{0.40} = 66%

Therefore, the probability that the project will be completed in 38 weeks is 66%.

**Answer 3:**

According to the Probability Table,

Z_{0.95} = 1.65

According to Probability Theory Formula,

T_{S} = (Z × σ_{E}) + T_{E} = (1.65 × 4.89) + 36 = 44.78

Therefore, the project duration that will have 95% chance of completion is 45 days.