Module 5 – Networks Module 5 – Networks and Decision

and Decision MathematicsMathematics

Chapter 24 – Directed Graphs

24.3 Critical Path Problems

0Developing and manufacturing products frequently involves many interconnected activities.

0Some activities cannot be started until other activities have been completed.

0Weighted digraphs can be used and are often created using a table.

Important facts about critical paths

0The weight of the critical path is the minimal length of time required to complete the project.

0Increasing the time required for any critical activity will also increase the time necessary to complete the project.

0A critical activity is any task, that if delayed, will also delay the entire project.

Drawing Weighted DiagramsWhen drawing weighted diagrams for critical path problems, the following conventions apply:0The edges (arcs) represent the activities

0The vertices (nodes) represent events

0The ‘start vertices’ have no immediate predecessors (activities that must take place before them)

0A vertex (finishing node) representing the completion of the project, must be included in the network

0No multiple edges (an activity can only be represented by one edge)

02 vertices can be connected by, at most, one edge.

In order to satisfy the last two conventions, it is sometimes necessary to include a dummy activity that takes zero time.

Insert diagram

Earliest Starting Time (EST)

0The earliest time an activity can be commenced.

0The EST for activities without predecessors is zero.

Insert diagram (page 653)

Latest Starting Time (LST)

0The time an activity can be left if the whole project is to be completed on time.

0LSTs are found by working backwards through the network.

Insert diagram (page 653)

Float

0The float or slack of a non-critical activity is the amount by which the latest starting time is greater than its earliest starting time.

Float time = latest starting time – earliest starting time

0The existence of a float means that an activity can start later than its earliest start time, or the duration of the activity can be increased.

0For critical activities, the float time is zero.

0For non-critical activities this is the difference between the LST & EST.

Insert diagram

Critical Path = B – D – E – F

Float times:

Activity A = 3-0 = 3 Activity B = 9-9 = 0

Activity C = 18-9 = 9 Activity D = 9-9 = 0 etc…

Project Crashing

0Project Crashing is the process of shortening the length of time in which a project can be completed by completing some activities more quickly.

0This can usually only be done by increasing the cost of the project.

Insert diagram