Download - Project Scheduling -- Probabilistic PERT
Project Scheduling
Probabilistic PERT
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
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
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=
• 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.
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
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
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
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
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)
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.
Enter a, m, b instead of
Call Solver
Click Solve
Go to PERT OUTPUT worksheet
Call Solver
Click Solve
To get a cumulative
probability, enter
a number here
P(Project is completed in less than 180 days)
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.
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
• 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
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.
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.
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))
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!
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