08/11/2010
A.AMRANI IMS L b University IMS-Lab, U i it Bordeaux1 B d 1
1
PERT method 1. How to draw PERT network? Kind
of constraints Concepts Steps to follow margins calculation
Total margin Free margin
2. Advanced PERT Probabilistic PERT PERT costs
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How to draw PERT networks? Kind of constraints Execution of a p project, j , often,, requires q succession of tasks linking g by some constraints. - Constraints of time: Leadtime to respect in performing the task (taking account the time of using resources) - Constraints of anteriority: Some tasks must be executed befor others - Constraints of simultaneity: Some tasks are performed in the same time Î PERT (Program Evaluation and Review Technique) charts represents the project schedule as an activity network.
How to draw PERT networks? Concepts - Task of project is represented by node - Duration of task is represented by an arc (branch) - Start node and Finish node are also represented For each activity, these values are estimated ES – Earliest start time EF – Earliest finish time LS – Latest start time LF – Latest finish time The ‘length’ of each path has to be calculated The LONGEST path in the project is the CRITICAL PATH
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How to draw PERT networks? Steps to follow To draw PERT, 6 Big steps are required 1. Establish a list of tasks
2. Determine anteriority conditions 3. Draw PERT network 4 Calculate the earliest and latest dates 4. 5. Calculation of margins 6.Found the critical path of the project
How to draw PERT networks? Steps to follow 1. Establish a list of tasks
Enounce a list of tasks to perform Assess the durations of tasks (processing times) to determine the required resources Assign a codification to tasks to make easier the construction of the network
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How to draw PERT networks? 1. Establish a list of tasks Exemple : The building of storage warehouse Tasks
Duration (t.u)
A
Study, realization and acceptance of plans
4
B
Preparation of the ground
2
C
Order materials (wood, bricks, cement, sheet for the roof)
1
D
Digging of the foundations
1
E
D Doors, Windows Wi d orders d
2
F
Delivery of materials
2
G
Casting of the foundations
2
H
Delivery of doors, windows
10
I
Construction of the walls, the roof
4
J
Installation doors and windows
1
How to draw PERT networks? Steps to follow To draw PERT, 5 big steps are required 2. Determine anteriority conditions By answering these questions :
Which task must be ended before another could start ? Which task have to follow some tasks ?
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How to draw PERT networks? 2. Determine anteriority conditions Previous tasks
Task
Following tasks
-
A
C,D,E D
-
B
A
C
F
A,B
D
G
A
E
H
C
F
G
D,F
G
I
E
H
J
G
I
J
H,I
J
-
How to draw PERT networks? Steps to follow 3. Draw PERT network Network is made of enounced tasks. Tasks are the node and arcs are constraints of precedence
E
4
Start
0
A
4
C
2
1
H F
10
2
G
10
2
I
4
J
1
End
4 0
B
2
D
1
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How to draw PERT networks? 4. Calculate the earliest and latest dates
Task i
Earliest start date of task i:
ESi
The earliest date on which task i could start taking account the required time to process previous tasks
ESi = max( ES j + p ji ) j∈ P(i)
Latest start date of task j:
LSi
The latest date on which task i must absolutely start in order to not disturb and delay the overall project.
Task i
LS i = min ( LS j − pij ) j∈ F(i)
How to draw PERT networks? 4. Calculation of ES and LS
4/4 E
4
0/0 Start
0
0/0 A
4/7 C
4
2
1
6/6 H 5/8 F
4 0
B 0/7
2
D 4/9
ESi: Forward computation
2
10
7/10 2 G
I 9/12
4
1
J 16/16
17/17 End
1
LSi: Backward computation
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How to draw PERT networks? 5. Calculation of margins {
EEi
EEi
g Total margin
Allowed flexibility for task without changing project duration
Task i ESi
M i = LS i − ESi ≥ 0 T
MT
LSi MT
Free margin Allowed flexibility of task i without delaying following tasks {
Task i
ESj
EEi
M i = min( ES j − pij − ESi ) ≥ 0 F
Task j MF
How to draw PERT networks? 6. Find a critical path 4/4 2 4 E 0/0 Start
0
0/0 A
4/7 C
4
1
6/6 H 5/8 F
2
10
7/10 2 G
4 0
B 0/7
2
D 4/9
1
Find the critical pathÎ each task whose MT=O Project duration (critical path) = 17 t.u Critical tasks are: A, E, H, J
I 9/12
1
4
J 16/16
17/17 End
TASK
MT
ML
A
0
0
B
7
2
C
3
0
D
5
2
E
0
0 0
F
3
G
3
0
H
0
0
I
3
3
J
0
0
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P.E.R.T method 1. How to draw PERT network? Kind
of constraints
Concepts Steps
to follow calculation
margins
Total margin Free margin
2. Advanced P.E.R.T Probabilistic PERT PERT costs
Probabilistic PERT
Inside project appears some difficulties to get the exact durations of tasks
Probabilistic PERT considers the uncertainity about the dates and durations of tasks Uncertainity of durations ?
Necessity to taking account delay variation in margins computations
For each task, it is important to define to: Optimistic time, tr: realistic time (the most probabilistic) tp: pessemistic time
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Probabilistic PERT An assessment of random duration of tasks often follows probability distribution of type β Frequency of time existence
β distribution is characterized by these parameters:
The mean m Variance V Standard deviation σ
to + 4tr + tp 6 2 ⎡ ( tp − to ) ⎤ V=
moyenne Mean = tm = ⎢ ⎣
6
to tr
⎥ ⎦
tm
tp Time
Probabilistic PERT Objectives
Probabilistic PERT allows to determine the probability of fulfilling project in certain duration with variable task’s durations What
is the probability that project would be performed in x units of time?
Total duration of project is distributed according «normale distribution » gaussian curve, with a mean m equal to the sum of average durations of critical tasks
The variance of sum of random variables is equal to the sum of variables. It becomes possible to determine standard deviation of critical path.
σ=
n
∑σ i =1
2 i
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Probabilistic PERT Once the mean and standard deviation calculated, the Î probability of realization is deduced The probability of realizing project of duration T in x unit of times P(T≤X)
u
⎛ ⎛ X − tm ⎞ ⎞ p (T ≤ X ) = p ⎜ T ≤ ⎜ F(u) ((α)) ⎟⎟ = ∏ ⎝ σ ⎠⎠ ⎝ F(u) π(α) is a value to find in the table of normal distribution (it is a probability)
Probabilistic PERT Normale distribution u
Distribution function
F(u)
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Probabilistic PERT: exercise The project manager of a Warehouse building has just made the assessment of durations. Task
Durations (u.t) Optim
Real
Pess
Previous tasks C, G, L
A
3
4
5
B
2
3,5
8
-
C
3
4
11
B, H
D
2,5
3,5
7,5
B
E
13
15
23
B, G, L
F
5
6
13
A D A,
G
2
2
2
D
H
2
2,5
6
B
I
1,5
2,5
6,5
J
J
2
3
4
G, L
K
1
2
9
A, F, I
L
1
1,5
5
B, C, H
Probabilistic PERT: exercise Build
the network of the project Calculate the average durations for each task Determine margins (free and total) for each task Deduce the critical path of this project What is the probability to realize the construction in 35 days ? Probability to realize it in 29 days? What would be the limit path duration that ensures probability of 95% to realize the building
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Probabilistic PERT : exercise
The
mean of the values (tm) follows β distribution
moyenne = tm =
to + 4tr + tp 6
Tâche
to
tr
tp
tm
A
3
4
5
4
B
2
3,5
8
4
C
3
4
11
5
D
2,5
3,5
7,5
4
E
13
15
23
16
F
5
6
13
7
G
2
2
2
2
H
2
2,5
6
3
I
1,5
2,5
6,5
3
J
2
3
4
3
K
1
2
9
3
L
1
1,5
5
2
Continue to solve the problem…
P.E.R.T method 1. How to draw PERT network? Kind
of constraints Concepts Steps to follow margins calculation
Total margin Free margin
2. Advanced P.E.R.T Probabilistic PERT PERT cost
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In the classic PERT newtork, durations are defined by the enterprise by assigning to each task constant resources disregarding the costs of their realization By assigning additional resources, these durations may vary… How the decider may now whether the assignment of resources to perform a task is competitive?
Two possibilities: A. Minimizing the cost with delay constraint B. Minimizing the delay with cost constraint
It is possible to determine for each operation the following parameters: Normal cost(Cn) : it is the lowest cost for the enterprise to lead the task with minimum of resources. Normal time (Dn) : It is the time corresponding to the normal cost (realized with the minimum resources) Accelerated time (ta) : It is the minimum time potentialy granted to perform the task with sufficient resources. Accelerated cost (Ca) : it corresponds to the minimum time of realization.
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The evolution of cost with duration may pursue different schemes To simplify, we admit that the cost is proportional to the time Cost
Accelerated cost
Normal cost
time
Normal time
Accelerated time
For decreasing the overall delay: Reduce durations of critical tasks It may be interesting to diminish the duration of a critical task, task but it is unuseful to spend additional budget for another task
Critical task
B How i reduce time?
F can become critical?
When reducing the duration of a task another can become a critical task !
F B duration
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Normal program
Accelerated program Cot ($)
6000
Duration (day) 1
12600
3
14000
4
12000
2
15000
L
7
22000
5
25000
E
L
7
22000
6
26000
F
L
14
44000
11
47000
G
A
22
70000
18
80000
H
D
7
22000
6
24000
I
G
7
22000
6
24000
J
I
7
22000
6
25000
K
J
7
22000
6
24000
L
I
21
66000
15
72000
Operations
Previous
Duration
A
-
2
B
-
4
C
-
D
Cost
342 600
8000
384 000
A. Minimizing costs with duration Improvement constraint focus Total cost of the project Total duration of the project
Duration: constraint to respect absolutely!
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A. Minimizing costs with delay constraint This is the procedure of reducing costs with delay constraint
1. Establish PERT network of accelerated project 2. Identify the critical path 3. Extend the duration of non-critical tasks 3 (without creating a new critical path) in way to reduce the cost of project maintaining its minimal duration
A. Minimizing costs with delay constraint Accelerated project level1
Level 2
level 3
level 4
level 5
level 6
A; B; C
G
I
J; L
D; E; F; K
H
PERT of accelerated project ES EE
0 A 0 1
0
start
0 0
1 G 1 18
duration 0 B
19 I 19 6
25 J 39 6
31 K 45 6
25 L
40 D 40 5
25 15
48 3
45 H
51
45 6
51 0
end
40 E 45 6 40 F 40 11 0 C 49 2
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A. Minimizing costs with delay constraint
Task
Prev
Durati
ES
EE
A
-
1
0
1
B
-
3
0
3
C
-
2
0
2
D
L
5
40
E
L
6
F
L
G
LS
LE
MT
MF
0
1
0
0
48
51
48
48
49
51
49
49
45
40
45
0
0
40
46
45
51
5
5
11
40
51
40
51
0
0
A
18
1
19
1
19
0
0
H
D
6
45
51
45
51
0
0
I
G
6
19
25
19
25
0
0
J
I
6
25
31
39
45
14
0
K
J
6
31
37
45
51
14
14
L
I
15
25
40
25
40
0
0
7 critical tasks: A, G, I, L, D, F et H 2 critical paths: A-G-I-L-D-H et A-G-I-L-F
A. Minimizing costs with delay constraint Costs of the project : 384000 $ Duration of the project: 51 days.
Is it possible de reduce the costs without extending the duration of the project To reduce the cost Î Extend certain activities
But conserve the delay Î Not change the critical tasks
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A. Minimizing costs with delay constraint Tasks to extend: B,C,E,J,K. To calculate UGE: Unitay gain of extension
UGE =
Ca − Cn tn − ta
This is the economy realized when we extend with 1 day the duration of task (1 unit of time).
A. Minimizing costs with delay constraint 1st step: Extend the duration of tasks whose MF>MaxExtension MaxExtension=Dn-Da Tasks
Max Extension
UGE
MF
A
1
2000
0
B
1
1400
48
C
2
1500
49
D
2
1500
0
E
1
4000
5
F
3
1000
0
G
4
2500
0
H
1
2000
0
I
1
2000
0
J
1
3000
0
K
1
2000
14
L
6
1000
0
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A. Minimizing costs with delay constraint 1st step: Extend the duration of tasks whose MF>Maximum extension MaxExtension=Dn-Da
0 A
1 G 1 18
0 1 0
deb
19 I 19 6
0 B
0 0
25 J 39 6
31 K 45 7
25 L
40 D 40 5
25 15
48 4
45 H
51
45 6
51 0
fin
40 E 45 7 4 F 40 40 11 0 C 49 4
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A. Minimizing costs with delay constraint Gain calculation
MaxExtension
Tasks
UGE
A
2000
B
1400
1
C
1500
2
D
1500
E
4000
F
1000
G
2500
H
2000
I
2000
J
3000
= 1400 + 2*1500 + 4000 + 2000
K
2000
L
1000
= 10400 euros of savings
1
1
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A. Minimizing costs with delay constraint 2st step: Extend the duration of other non critical tasks
Can we extend other tasks (non critical ones) without modifiying project duration?
It is possible to extend J with 1 day without consequence on the overall duration. TOTAL GAIN : 10400 + 3000 = 13400 euros Projet cost = 384000 – 13400 = 370600 euros
A. Minimizing costs with delay constraint Final result 0 A 0 1 0
deb
0 0
1 G 1 18
0 B
19 I 19 6
25 J 39 7
32 K 45 7
25 L
40 D 40 5
25 15
48 4
45 H
51
45 6
51 0
fin
40 E 45 7 40 F 40 11 0 C 49 4
Total cost of the project = 370600 Total duration of the project = 51 days 40
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A. A Minimizing duration with cost constraint Total cost of the project Total duration of the project
Cost: constraint to respect absolutely!
A. Minimizing duration with cost constraint 1.
Establish normal PERT network
2 2.
Identify critical path
3.
Select the tasks whose UGE is the lowest
4.
Reduce the duration of tasks to reduce overall duration of the project
If several critical path exists, you must consider each task of critical path simultaneously
5.
Go on 3
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A. Minimizing duration with cost constraint 1. PERT network of normal project 0 A 0 2 0
deb
2 G 2 22
0 B
0 0
24 I 24 7
31 J 52 7
38 K 59 7
31 L
52 D 52 7
31 21
62 4
59 H
66
59 7
66 0
fin
52 E 59 7 52 F 52 14 0 C 62 4
Project cost= 342600 euros Duration of project= 66 days.
How to reduce the duration without exceeding the cost ?
A. Minimizing duration with cost constraint 2. Select the lowest UGE = UCA (Unitary cost of acceleration) Tasks
Max Diminution Dn- Da
Unitary cost of acceleration
A
1
2000
B
1
1400
C
2
1500
D
2
1500
E
1
4000
F
3
1000
G
4
2500
H
1
2000
I
1
2000
J
1
3000
K
1
2000
L
6
1000
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A. Minimizing duration with cost constraint
The project as it was described costs 342600 euros for 66 days duration. duration How to reduce the duration in such way to not exceed 360000 euros ?
Decreasing the durationxIncreasing of costs!!
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Tasks F et L are accelerated at first. 2 critical pathsÎ A-G-I-L-D-H or A-G-I-L-F. Accelerate F does not reduce the project duration if D and H are not accelerated simultaneously We accelerate L with 6 days (max diminution). ÆEnhancement of 6*1000 = 6000 euros A and I are accelerated with 1 day ÆEnhancement of 2000 +2000 =4000 euros G is accelerated To respect cost constraint, G is accelerated with 2 days ÆEnhancement of 2*2500 = 5000 euros Cost= 342600 + 6000 + 2000 + 5000 = 357600 euros Duration of the projet = 66 – (6 +1 +1 + 2) = 56 jours.
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Sources (according to lecture of Y.Ducq, S.Sperandio)
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