Project Management - PERT/CPM

1
Project Management - PERT/CPM
What is project management? Consider building a
house Step A Prepare site. (5 days) Step B
Build foundation. (8 days) Step C Frame walls
and roof. (15 days) Step D Rough in Plumbing
(12 days) Step E Rough in Electrical (10
days) Step F HVAC Venting (8 days) Step G
Drywall (11 days) Step H Finish Electrical (5
days) Step I Finish Plumbing (4 days) Step M
Paint (5 days) Step J Finish HVAC (2 days) Step
N Landscape (5 days) Step K Install Kitchen (8
days) Step L Install Baths (14 days)
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Project Management - PERT/CPM
Let each node represent a project event/milestone
(node 1 is start of project, node 11 is end of
project). Let each arc represent a project
task/job. Each arc is identified by a job
letter and duration. Note the dummy jobs
indicating precedence that jobs H and I must
complete before K or L begins.
J,2
7
H,5
M,5
D,12
K,8
0
A,5
B,8
C,15
G,11
E,10
11
2
1
3
4
5
6
9
10
0
L,14
N,5
F,8
I,4
8
 
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Project Management - PERT/CPM

• What questions might project managers be
  interested in?
• How long will the project take?
• Can I add manpower or tools to reduce the
  overall project length?
• To which tasks should I add manpower?
• What tasks are on the critical path?
• Is the project on schedule?
• When should materials and personnel be in place
  to begin a task?
• Other?

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Project Management - Examples

• University Convocation Center
• Windsor Engine Plant
• Other major construction projects
• Large defense contracts
• NASA projects (space shuttle)
• Maintenance planning of oil refineries, power
  plants, etc
• other

5
Project Management Minimum Completion Time
A,3
C,4
E,5
2
1
4
5
0
D,2
B,1
3
LP Solution Let ti be the time of event
i. Min Z t5 t1 s.t. t2 t1 gt 3
t3 t2 gt 0 t3 t1 gt 1 t4 t2 gt
4 t4 t3 gt 2 t5 t4 gt 5 ti gt
0 for all i
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Project Management Critical Path
A,3
C,4
E,5
2
1
4
5
0
D,2
B,1
3
LP Solution insert Lindo Solution here How
do you find the critical path from the Lindo
solution?
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Project Management Minimum Completion Time and
Critical Path
A,3
C,4
E,5
2
1
4
5
0
D,2
B,1
3
Solution by Network Analysis Let earliest time
of node j, Uj, be the earliest time at which
event j can occur. Set U1 0 then U2
U1 t12 0 3 3 U3 MaxU1 t13 , U2
t23 Max1,3 3 U4 MaxU3 t34 , U2
t24 Max5,7 7 U5 U4 t45 12
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Project Management Minimum Completion Time and
Critical Path
A,3
C,4
E,5
2
1
4
5
0
D,2
B,1
3
Solution by Network Analysis Let latest time of
node j, Vj, be the latest time at which event j
can occur while still completing project by
minimum the minimum completion time, Um . Set
V5 U5 12 then V4 V5 - t45 12 - 5
7 V3 V4 - t34 7 2 5 V2 MinV4
- t24 ,V3 - 0 3 V1 MinV2 - t12 ,V3
t13 0
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Project Management Minimum Completion Time and
Critical Path
A,3
C,4
E,5
2
1
4
5
0
D,2
B,1
3
Solution by Network Analysis To find the
critical path, solve for slack time Vj - Uj.
All events with slack time equal to 0, and tasks
connecting these events are on the critical
path. V5 - U5 12 12 0 V4 - U4 7
7 0 V3 - U3 5 3 2 V2 - U2 3 3
0 V1 - U1 0 0 0 Critical Path
1-gt2-gt4-gt5
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CPM Critical Path Method

• Can normal task times be reduced?
• Is there an increase in direct costs?
• Additional manpower
• Additional machines
• Overtime, etc
• Can there be a reduction in indirect costs?
• Less overhead costs
• Less daily rental charges
• Bonus for early completion
• Avoid penalties for running late
• Avoid cost of late startup
• CPM addresses these cost trade-offs.

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CPM Critical Path Method
Example
Overhead cost 5/day
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CPM Critical Path Method
Enumerative Approach Reduce job H by 1 day
Total Cost improves by 5 - 4 1. Reduce job
A by 2 days Total cost improves by 10 - 8
2. Reduce job A by an additional day, and job B
by a day? Total cost improves by 5 - 4 - 2
-1. Therefore do not take this action. Reduce
job A by an additional day, and job C by a day?
Total cost improves by 5 - 4 - 2 -1.
Therefore do not take this action. Evaluate
combinations of reducing path 3-4-6 and 3-5-6 by
one day. D E 5 - 3 - 3 -1 F E 5 -
5 - 3 -3 D G 5 - 3 - 1 1 F G
5 - 5 - 1 -1 Therefore, reduce job D G by
1 day TC improves by 5 - 3 -1 1. Overall
improvement 1 2 1 4.
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CPM Critical Path Method
LP Approach Let tij decision variable for
time to complete task connecting
events i and j. kij normal completion
time of task connecting events i and j. lij
minimum completion time of task connecting
events i and j. Cij incremental cost of
reducing task connecting events i and j. Model
I Given project must be complete by some time T,
which tasks should be reduced to minimize the
total cost? Min s.t.
for all jobs (i,j) for all jobs (i,j) for all i
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CPM Critical Path Method
LP Approach Model II Given an additional
budget of B for crashing tasks, what minimum
project completion time can be obtained while
staying within your budget? Min s.t.
for all jobs (i,j) for all jobs (i,j) for all i