Project Scheduling

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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 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
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3-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

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Mean 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.

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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 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.

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The 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
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The Probability Approach
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Distribution For Klone Computers

• The project has a normal distribution.
• The critical path is A-F-G-D-J.

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Standard 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

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Excel Solutions
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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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Call Solver Click Solve Go to PERT OUTPUT
worksheet
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Call Solver Click Solve
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Cost 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.

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Cost 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.

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KLONE 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?

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Analysis 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.
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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 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))
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The 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!
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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