Precedence Graphs

March 15, 2016

Example

You want to remodel a chemistry lab...

Example

I Dependencies become more complex as project progresses

I If we did every task sequentially (in series), the project wouldtake 93 days to complete

I What can we do in parallel?

I How fast could project be finished?

Precedence Graph

I One node for each activity

I A directed edge (u, v) means activity v depends on u

Example

Example

I If a task depends on 2 and 4, dependence on 2 is redundantsince task 4 depends on 2

I No clear beginning or end

Example

After removing redundant edges, and adding a start and end node:

Example

Sweeping from start-to-finish, recording earliest start and earliest finish

times in form (ES,EF ):

Example

Sweeping from finish-to-start, recording latest start and latest finish times

in form (LS,LF ):

Free-float vs. Total-float

I The free float of a activity i is the number of days thatactivity i can be delayed without affecting the start of anyactivity that depends directly on activity i.

I The total float of activity i is the number of days thatactivity i can be delayed without affecting the completion ofthe entire project.

I Every free float is a total float, but not all total floats are freefloats. Day k is a total float for activity i and a free float foractivity i provided performing activity i on day k does notdelay any activities that depend directly on i.

Example - Gantt Chart

Total floats are marked with a T; Free floats are marked with an F

Example - Gantt Chart

Notice some activities have zero float. These activities are known as

“critical activities” and their collection forms the “critical path” in the

graph.

The critical path is activity 15, 16, and 19.