### Project Scheduling and Critical Path Analysis

• PERT: Project Evaluation and Review Technique

• PERT:

 The Project Evaluation and Review Technique (PERT) is a model for project management designed to analyze and represent the tasks involved in completing a given project.

• PERT is used to:

 find the minimum time needed to complete the total project. identify projects on a critical path Projects on the critical path must be completed in time or else, the duration of the project will increase

• PERT: Example Project Scheduling

• Example: Custom automotive assembly business

• Activities to accomplish the ordering of a custom automtive frame and the time needed to finish the activity:

Activities   Completion Time
• (A): initial paper work
• (B): build body of car
• (C): build frame of car
• (D): finish body
• (E): finish frame
• (F): final paperwork
• (G): mount body to frame
• (H): room washdown
• 3
• 3
• 2
• 3
• 7
• 3
• 6
• 2

• Dependency of the various activities:

Note:

• The meaning of A → B is:

 Activity B cannot begin until activity A has been completed

• Definitions

• Terminolgy:

• Earliest Start Time (ES) = the earliest time that an activity can begin

• Earliest Finish Time (EF) = the earliest time that an activity can be completed

 EF = ET + duration of activity

• Lastest Start Time (LS) = the latest time that an activity can begin without lengthening the minimum project duration

• Latest Finish Time (LF) = the earliest time than an activity can be completed without lengthening the minimum project duration

 LF = LS + duration of activity

• Duration of the project = the difference between the maximum value of the "lastest finish time" of projects and the minimum value of the "earliest start time" of projects

 ``` Project duration = max(LF) - min(ES) ```

• Slack = the amount of time that a project can be delayed without increasing the duration of the project

• Critical activity = an activity in the project with slack = 0

I.e.: critical activities have no slack, when such project is delayed, the duration of the project will increase

• Critical Path Method (CPM)

• CPM:

• The Critical Path Method is a (simple) analysis technique used to find:

 Compute ES, EF, LS and LF in a PERT network Identify the critical activities

• Computing Earliest Start and Earliest Finish times (ES and EF)

• ES and EF are computed by a forward pass through the PERT network

• Example:

• Initial PERT network:

• ES(A) = 0 and EF(A) = 0 + 3 = 3:

• ES(B) = EF(A) = 3 and EF(B) = EF(B) + 3 = 6
ES(C) = EF(A) = 3 and EF(C) = EF(C) + 2 = 5

• Note: ES(F) = max(EF(B), EF(C)) = 6 and EF(F) = EF(F) + 3 = 9

• Note: ES(G) = max(EF(D), EF(E)) = 12 and EF(F) = EF(F) + 6 = 18

• Duration of the project:

 ``` Project duration = max(LF) - min(ES) = 18 - 0 = 18 ```

• Computing Latest Start and Latest Finish times (LS and LF)

• LS and LF are computed by a backward pass through the PERT network

• Example:

• Initial PERT network:

• The latest time that F, G and H can finish without lengthening the duration of the project = 18

Therefore:

 LF(F) = 18 LF(G) = 18      LF(H) = 18

That means that the latest time that G and H can start without lengthening the duration of the project are:

 EF(F) = 18 − 3 = 15 EF(G) = 18 − 6 = 12      EF(H) = 18 − 2 = 16

Therefore:

• The latest time that G must start = 12

Therefore: the latest time that D must finish without pushing the start of G further = 12

Therefore:

 LF(D) = 12

That means that the latest time that D must start without lengthening the duration of the project is:

 EF(D) = 12 − 3 = 9

(Similar reasoning hold for activity E)

Therefore:

• The latest time that:

 D must start = 9 F must start = 15

Therefore: the latest time that B must finish without pushing the start of D or F further = 9 !!!

Therefore:

 LF(B) = 9

That means that the latest time that D must start without lengthening the duration of the project is:

 EF(D) = 9 − 3 = 6

(Similar reasoning hold for activity C)

Therefore:

• The latest time that:

 B must start = 6 C must start = 3

Therefore: the latest time that A must finish without pushing the start of B or C further = 3 !!!

Therefore:

 LF(A) = 3

That means that the latest time that D must start without lengthening the duration of the project is:

 EF(A) = 3 − 3 = 0

Result:

• Interpreting the PERT network

• Example:

Notes:

• Activity B:

 Earliest Start time = 3 Lastest Start time = 6 (without causing project delay)

Activity B can start any time between [3..6] without causing project delay

We say that:

 Activity B has slack = 3

• Activity E:

 Earliest Start time = 5 Lastest Start time = 5 (without causing project delay)

Activity E can start any time between [5..5] without causing project delay

We say that:

 Activity B has slack = 0

Such an activity is called citical

• Critical Activities and Critical Path

• The critical activities in the PERT example are:

• The critical activities form a (critical) path through the PERT network