Title: Project Scheduling
1Project Scheduling
2PERT 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 lt a) lt ?.01 m the most likely time to
perform the activity (mode) b a pessimistic
time to perform the activity P(Finish gt b) lt ?.01
33-Time Estimate ApproachProbability 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
4Mean and Standard Deviation forActivity
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.
5PERT 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 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.
- Assumption 3
- The time to complete one activity is independent
of the completion time of any other activity.
6The Project Completion Time Distribution
- The three assumptions imply that the overall
project completion time is normally distributed,
with
? Sum of the ?s on the critical path ??2
Sum of the ??2 s on the critical path
7The Probability Approach
8Distribution For Klone Computers
- The project has a normal distribution.
- The critical path is A-F-G-D-J.
9Standard Probability Questions
- What is the probability the project will be
finished within 194 days? - P(X lt 194)
- 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(XltxL)
.025 and P(XgtxU) .025 - What is the probability the project will be
completed within 180 days? - P(X lt 180)
- What is the probability the project will take
longer than 210 days. - P(X gt 210)
- By what time are we 99 sure of completing the
project? - X value such that P(X lt x) .99
10Excel Solutions
11Using 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.
12Call Solver Click Solve Go to PERT OUTPUT
worksheet
13Call Solver Click Solve
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16Cost Analysis Using theExpected 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.
17Cost Analyses Using Probabilities
- 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.
18KLONE COMPUTERS - Cost analysis 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?
19Analysis of Additional Sales Personnel Training
- 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.
This option should not be pursued at any cost.
20Analysis 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.
21The Expected Value Approach
- Find the P(X lt 180), P(180 lt X lt 200), and
P(X gt 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(Xlt180))
400000(P(180ltXlt200)) 0(P(Xgt200))
22The Calculations
- The PERT-CPM template can be used to calculate
the probabilities.
µ 189 ? 9.0185
µ 194 ? 9.255
Total 450,976
Total 335,751
Net increase 450,976-335,751 115,225 This
is less than the 250,000 required for training.
Do not perform the additional training!
23Review
- 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