## PERT method

The task information is summarized in a schedule like the following example:

task | precedence | duration |

TO | – | 6 |

B | – | 5 |

VS | TO | 4 |

D | B | 6 |

E | VS | 5 |

F | A, D | 6 |

G | E, F | 4 |

## Step 1: construction of the graph from the timeline

*PERT method - Determination of task levels:*

We will assign the level **0** to tasks that have no previous task.

We will assign the level **1** to tasks whose previous tasks are level **0**.

**k + 1**will be the tasks whose previous tasks are lower level with at least one level k task among them.

We will build the graph by plotting the tasks in order of increasing level.

*PERT method - Beginning, ending, convergent tasks:*

Before embarking on the construction of the graph, it will often be useful to detect the so-called starting, ending or converging tasks.

**1**of the graph.

terminal vertex of the graph.

**2**and

**5**is called

*dummy task*and is always of zero degree. Its purpose is to model the fact that the task

**TO**must be completed to start the task

**F**. The ridge between

**1**and

**2**means that the task

**TO**starts as is

**1**and ends as is

**2**.

It is important to place the tasks in order of execution. Task **F** can only be placed after the task **TO** and **D** placed, and the task **D** can only be placed after the task **B**. This explains the fictitious edge between **2** and **5** (**TO** and at a distance **1** while **F** is in the distance **3** from the start).

## Step 2: determine the dates and margins

Once the graph has been constructed, we will determine the dates at the earliest and at the latest for the

different vertices and free and total margins for tasks.

**PERT method - Earliest dates:**For a summit, the earliest date (noted:**t**) represents the minimum time required to reach this peak. It will be determined step by step, in ascending order of vertex, from the entry of the graph, thanks to Ford's algorithm for finding the longest path.

Thereby :

t_{1} = 0 and t_{j} = Max (t_{i} + d_{ij} ) on all **i** preceding **j** with_{ij} = time between peak **i** and **j**.

In the example, t_{1} = 0, t_{2} = 0 + 6 = 6, t_{3} = 0 + 5 = 5, t_{4} = 6 + 4 = 10, t_{5} = max (6 + 0, 5 + 6) = 11, t_{6} = max (11 + 6, 10 + 5) = 17, t_{7} = 17+4 = 21.

The earliest date of the graph output represents the minimum duration achievable for

the whole project (in the example, t_{7}= 21, so the project will last 21 days at best).

*PERT method - Dates at the latest:*For a summit, the latest date (noted:**T**) concretely represents the date on which this state must be reached if we do not want to increase the total duration of the project. It will be determined in a manner analogous to**t**, but in descending order of vertex, from the output of the graph to the input.

Thereby :

T_{not} = t_{not} = Duration of the project and T_{i} = Min (T_{j} - d_{ij} ) on all j preceding i.

In the example, T_{7} = 21, T_{6} = 21 - 4 = 17, T_{5} = 17 - 6 = 11, T_{4} = 17 - 5 = 12, T_{3} = 11 - 6 = 5, T_{2} = min (11-0, 12-4) = 8, T_{1} = min (8-6, 5-5) = 0.

We will always have t_{1} = T_{1} = 0 and **t** less than or equal to** T** for any summit. We call Tt la

top float margin.

**PERT method - Task margins:**The free margin of a task represents the maximum possible delay of a task without delaying the start of subsequent tasks, note**ML**. The total margin of a task represents the maximum possible delay for the completion of a task without delaying the entire project, it will be noted**MT**: ML_{ij}= t_{j}- t_{i}- d_{ij}and MT_{ij}= T_{j}- t_{i}- d_{ij}.

Taking into account the calculation mode, the margins will always be positive or zero and the free margin of a task will always be less than or equal to its total margin.

We will qualify as critical, a task whose total margin is zero. A critical task should not be delayed if you do not want to increase the total duration of the project.

If the duration of a non-critical task increases, part of this increase will be absorbed by the task margin, only the surplus will affect the duration of the project.

## Aside

Vertices can contain several pieces of information at the same time: