Download - Project Scheduling -- Probabilistic PERT
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Project Scheduling
Probabilistic PERT
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PERT Probability Approach to
Project Scheduling• Activity completion times are seldom known with
cetainty.
• PERT is a technique that treats activity completion times as random variables.
• Completion time estimates can be estimated using the Three Time Estimate approach. In this approach, three time estimates are required for each activity:
– Results from statistical studies
– Subjective best estimates
a = an optimistic time to perform the activity P(Finish < a) < .01
m = the most likely time to perform the activity (mode)
b = a pessimistic time to perform the activity P(Finish > b) < .01
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3-Time Estimate Approach
Probability Distribution• With three time estimates, the activity completion time
can be approximated by a Beta distribution.
• Beta distributions can come in a variety of shapes:
a m b ba mm a b
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Mean and Standard Deviation for
Activity Completion Times
• The best estimate for the mean is a weighted
average of the three time estimates with weights
1/6, 4/6, and 1/6 respectively on a, m, and b.
• Since most of the area is with the range from a to
b (b-a), and since most of the area lies 3 standard
deviations on either side of the mean (6 standard
deviations total), then the standard deviation is
approximated by Range/6.
6
a-b=deviation standard the=
6
b+4m+a= timecompletionmean the=
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• Assumption 2– There are enough activities on the critical path so that
the distribution of the overall project completion time can be approximated by the normal distribution.
PERT Assumptions
• Assumption 1– A critical path can be determined by using the mean
completion times for the activities.
– The project mean completion time is determined solely by the completion time of the activities on the critical path.
• Assumption 3– The time to complete one activity is independent of the
completion time of any other activity.
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The three assumptions imply that the
overall project completion time is normally
distributed, with:
The Project Completion Time
Distribution
= Sum of the ’s on the critical path
2 = Sum of the 2 ’s on the critical path
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Activity Optimistic Most Likely Pessimistic
A 76 86 120
B 12 15 18
C 4 5 6
D 15 18 33
E 18 21 24
F 16 26 30
G 10 13 22
H 24 28 32
I 22 27 50
J 38 43 60
The Probability Approach(76 + 4(86) +120)/6 (120-76)/6
2
90 7.33 53.73
15 1.00 1.00
5 0.33 0.11
20 3.00 9.00
21 1.00 1.00
25 2.33 5.43
14 2.00 4.00
28 1.33 1.77
30 4.67 21.81
45 3.67 13.47
2
90 7.33 53.73
15 1.00 1.00
5 0.33 0.11
20 3.00 9.00
21 1.00 1.00
25 2.33 5.43
14 2.00 4.00
28 1.33 1.77
30 4.67 21.81
45 3.67 13.47
2
90 7.33 53.73
15 1.00 1.00
5 0.33 0.11
20 3.00 9.00
21 1.00 1.00
25 2.33 5.43
14 2.00 4.00
28 1.33 1.77
30 4.67 21.81
45 3.67 13.47
(7.33)2
2
90 7.33 53.73
15 1.00 1.00
5 0.33 0.11
20 3.00 9.00
21 1.00 1.00
25 2.33 5.43
14 2.00 4.00
28 1.33 1.77
30 4.67 21.81
45 3.67 13.47
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Distribution For Klone Computers
• The project has a normal distribution.
• The critical path is A-F-G-D-J.
45 20 14 25 90
μμμμμμ JDGFA
194
13.44 9 4 5.44 53.78
σσσσσσ 2
J
2
D
2
G
2
F
2
A
2
85.66
85.66σσ 2
9.255
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Standard Probability Questions1. What is the probability the project will be finished
within 194 days? • P(X < 194)
2. Give an interval within which we are 95% sure of completing the project.
• X values, xL, the lower confidnce limit, and xU, the upper confidnce limit, such that P(X<xL) = .025 and P(X>xU) = .025
3. What is the probability the project will be completed within 180 days?
• P(X < 180)
4. What is the probability the project will take longer than 210 days.
• P(X > 210)
5. By what time are we 99% sure of completing the project?
• X value such that P(X < x) = .99
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Excel Solutions
NORMDIST(194, 194, 9.255, TRUE)
NORMINV(.025, 194, 9.255)
NORMINV(.975, 194, 9.255)
NORMDIST(180, 194, 9.255, TRUE)
1 - NORMDIST(210, 194, 9.255, TRUE)
NORMINV(.99, 194, 9.255)
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Using the PERT-CPM Template for
Probabilistic Models
• Instead of calculating µ and by hand,
the Excel template may be used.
• Instead of entering data in the µ and
columns, input the estimates for a, m ,
and b into columns C, D, and E.
– The template does all the required
calculations
– After the problem has been solved,
probability analyses may be performed.
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Enter a, m, b instead of
Call Solver
Click Solve
Go to PERT OUTPUT worksheet
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Call Solver
Click Solve
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To get a cumulative
probability, enter
a number here
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P(Project is completed in less than 180 days)
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Cost Analysis Using the
Expected Value Approach
• Spending extra money, in general
should decrease project duration.
• But is this operation cost effective?
• The expected value criterion can be
used as a guide for answering this
question.
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Suppose an analysis of the competition
indicated:
– If the project is completed within 180 days,
this would yields an additional profit of $1
million.
– If the project is completed in 180 days to
200 days, this would yield an additional
profit of $400,000.
Cost Analyses Using Probabilities
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• Completion time reduction can be achieved by
additional training.
• Two possible activities are being considered.
– Sales personnel training: (Activity H)
• Cost $200,000;
• New time estimates are a = 19, m= 21, and b = 23 days.
– Technical staff training: (Activity F)
• Cost $250,000;
• New time estimates are a = 12, m = 14, and b = 16.
• Which, if either option, should be pursued?
KLONE COMPUTERS -Cost analysis using probabilities
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Sales personnel training (Activity H) is
not a critical activity.• Thus any reduction in Activity H will not affect
the critical path and hence the distribution of
the project completion time.
Analysis of Additional
Sales Personnel Training
This option should not be
pursued at any cost.
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Analysis of Additional
Technical Staff Training
• Technical Staff Training (Activity F) is on
the critical path so this option should be
analyzed.
• One of three things will happen:
– The project will finish within 180 days:
• Klonepalm will net an additional $1 million
– The project will finish in the period from 180
to 200 days
• Klonepalm will net an additional $400,000
– The project will take longer than 200 days
• Klonepalm will not make any additional profit.
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The Expected Value Approach
• Find the P(X < 180), P(180 < X < 200), and P(X > 200) under the scenarios that– No additional staff training is done
– Additional staff is done
• For each scenario find the expected profit:
• Subtract the two expected values. If the difference is less than the cost of the
training, do not perform the additional training.– Caution: These are expected values (long run average
values). But this approach serves as a good indicator for the decision maker to consider.
Expected Additional Profit
1000000(P(X<180)) + 400000(P(180<X<200)) + 0(P(X>200))
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The Calculations• The PERT-CPM template can be used to
calculate the probabilities.No Additional
TrainingAdditional
Training
µ = 194
= 9.255
µ = 189
= 9.0185
.065192P(X < 180)
P(180 <X < 200) .676398
P(X > 200) .258410
X $1000000
$ 0
$ 65,192
$270,559X $400000
X $0
Total = $335,751
$159,152
$ 0
$291,824
.159152
.729561
.111287
Total = $450,976
Net increase = $450,976-$335,751 = $115,225
This is less than the $250,000 required for training.
Do not perform the
additional training!
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Review• 3-Time Estimate Approach for PERT
– Each activity has a Beta distribution
– Calculation of Mean of each activity
– Calculation Variance and Standard Deviation for each activity
• Assumptions for using PERT approach
• Distribution of Project CompletionTime
– Normal
– Mean = Sum of means on critical path
– Variance = Sum of variances on critical path
• Using the PERT-CPM template
• Using PERT in cost analyses