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- 1. The PERT approach
- 2. CPM vs PERT <ul><li>Difference how “task duration” is treated </li></ul><ul><li>CPM assumes time estimates are deterministic </li></ul><ul><li> - Obtain task duration from previous projects </li></ul><ul><li>- Suitable for “construction”-type projects </li></ul><ul><li>PERT treats durations as probabilistic </li></ul><ul><li>- PERT = CPM + probabilistic task times </li></ul><ul><li>- Better for R&D type projects </li></ul><ul><li>- Limited previous data to estimate time durations </li></ul><ul><li>- Captures schedule (and implicitly some cost) risk </li></ul>
- 3. PERT <ul><li>Project Evaluation and Review Technique </li></ul><ul><li>Task time durations are treated as uncertain </li></ul><ul><li>A - optimistic time estimate </li></ul><ul><li>- minimum time in which the task could be completed </li></ul><ul><li>- everything has to go right </li></ul><ul><li>M - most likely task duration </li></ul><ul><li> - task duration under “normal” working conditions </li></ul><ul><li> - most frequent task duration based on past experience </li></ul><ul><li>B - pessimistic time estimate </li></ul><ul><li>- time required under particularly “bad” circumstances </li></ul><ul><li>- most difficult to estimate, includes unexpected delays </li></ul><ul><li>-should be exceeded no more than 1% of the time </li></ul>
- 4. A-M-B time estimate
- 5. Beta distribution
- 6. Expected time and variance
- 7. Characteristics of a Normal Probability Distribution <ul><li>The normal curve is bell-shaped and has a single peak at the exact center of the distribution. </li></ul><ul><li>The arithmetic mean, median, and mode of the distribution are equal and located at the peak. Thus half the area under the curve is above the mean and half is below it. </li></ul>
- 8. Characteristics of a Normal Probability Distribution <ul><li>The normal probability distribution is symmetrical about its mean. </li></ul><ul><li>The normal probability distribution is asymptotic . That is the curve gets closer and closer to the X -axis but never actually touches it. </li></ul>
- 9. Characteristics of a Normal Distribution Mean, median, and mode are equal Normal curve is symmetrical Theoretically, curve extends to infinity a - 5 0 . 4 0 . 3 0 . 2 0 . 1 . 0 x f ( x r a l i t r b u i o n : = 0 , = 1
- 10. The Standard Normal Probability Distribution <ul><li>The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. </li></ul><ul><li>It is also called the z distribution. </li></ul><ul><li>A z- value is the distance between a selected value, designated X , and the population mean μ , divided by the population standard deviation, σ . The formula is: </li></ul>
- 11. EXAMPLE 1 <ul><li>The bi-monthly starting salaries of recent MBA graduates follows the normal distribution with a mean of Rs.20,000 and a standard deviation of Rs.2,000. What is the z- value for a salary of Rs.2,2000? </li></ul>
- 12. EXAMPLE 1 continued <ul><li>A z-v alue of 2 indicates that the value of Rs22,000 is one standard deviation above the mean of Rs.20,000. A z-v alue of –2.50 indicates that Rs.17,000 is 2.5 standard deviation below the mean of RS.20,000 . </li></ul><ul><li>What is the z-value of Rs.17,000. </li></ul>
- 13. Areas Under the Normal Curve <ul><li>About 68 percent of the area under the normal curve is within one standard deviation of the mean. </li></ul><ul><li>About 95 percent is within two standard deviations of the mean. </li></ul><ul><li>Practically all is within three standard deviations of the mean. </li></ul>
- 14. Areas Under the Normal Curve - 5 0 . 4 0 . 3 0 . 2 0 . 1 . 0 x f ( x r a l i t r b u i o n : = 0 , = 1
- 15. EXAMPLE 2 <ul><li>The daily water usage per person in a city is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons. About 68 percent of those living in the city will use how many gallons of water? </li></ul><ul><li>About 68% of the daily water usage will lie between 15 and 25 gallons. </li></ul>
- 16. EXAMPLE 3 <ul><li>What is the probability that a person selected at random will use between 20 and 24 gallons per day? </li></ul>
- 17. Example 3 continued <ul><li>The area under a normal curve between a z -value of 0 and a z -value of 0.80 is 0.2881. </li></ul><ul><li>We conclude that 28.81 percent of the residents use between 20 and 24 gallons of water per day. </li></ul><ul><li>See the following diagram. </li></ul>
- 18. -4 -3 -2 -1 0 1 2 3 4 P(0 <z<. 8) =.2881 EXAMPLE 3 0 <x<. 8 - 5 0 . 4 0 . 3 0 . 2 0 . 1 . 0 x f ( x r a l i t r b u i o n : = 0 ,
- 19. EXAMPLE 3 continued <ul><li>What percent of the population use between 18 and 26 gallons per day? </li></ul>
- 20. Example 3 continued <ul><li>The area associated with a z- value of –0.40 is .1554 </li></ul><ul><li>The area associated with a z -value of 1.20 is .3849 </li></ul><ul><li>Adding these areas, the result is .5403 </li></ul><ul><li>We conclude that 54.03 percent of the residents use between 18 and 26 gallons of water per day. </li></ul>
- 21. Calculating Probabilistic Activity Times <ul><li>Three Time Estimates </li></ul><ul><ul><li>pessimistic ( a ) </li></ul></ul><ul><ul><li>most likely ( m ) </li></ul></ul><ul><ul><li>optimistic ( b ) </li></ul></ul>
- 22. The Statistical Distribution of all Possible Times for an Activity
- 23. Activity Expected Time and Variance
- 24. 95 Percent Level <ul><li>Task will be a or lower 5 percent of the time </li></ul><ul><li>Task will be b or greater 5 percent of the time </li></ul>
- 25. 90 Percent Level <ul><li>Task will be a or lower 10 percent of the time </li></ul><ul><li>Task will be b or greater 10 percent of the time </li></ul>
- 26. 95 Percent Level (Alternative Interpretation) <ul><li>Task will be between a and b 95 percent of the time </li></ul>
- 27. 90 Percent Level (Alternative Interpretation) <ul><li>Task will be between a and b 90 percent of the time </li></ul>
- 28. An AON Network
- 29. An MSP Version of a Sample Problem Network
- 30. A Pert/CPM Network for the Day Care Project
- 31. An MSP Calendar for the Day Care Project, 4/16/00 to 5/27/00
- 32. The Probability of Completing the Project on Time =NORMDIST( D , , ,TRUE)
- 33. The Statistical Distribution of Completion Times of the Path a-b-d-g-h
- 34. Selecting Risk and Finding D NORMINV(probability, , ,TRUE)
- 35. SIMULATION
- 36. Traditional Statistics Versus Simulation <ul><li>Similarities </li></ul><ul><ul><li>must enumerate alternate paths </li></ul></ul><ul><li>Differences </li></ul><ul><ul><li>simulation does not require assumption of path independence </li></ul></ul>
- 37. EXTENSIONS TO PERT/CPM
- 38. Precedence Diagramming <ul><li>Finish-to-start linkage </li></ul><ul><li>Start-to-start linkage </li></ul><ul><li>Finish-to-finish linkage </li></ul><ul><li>Start-to-finish linkage </li></ul>
- 39. Figure 5-27 Precedence Diagramming Conventions
- 40. Other Methods <ul><li>Graphical Evaluation and Review Technique (GERT) </li></ul><ul><ul><li>combines flowgraphs, probabilistic networks, and decision trees </li></ul></ul><ul><ul><li>allows loops back to earlier events and probabilistic branching </li></ul></ul>
- 41. 3 2 1 6, 7 Inspection and testing 8 11 4 3 4, 5 Install air pollution device 7 9 2 1 3 Install control system 6 7 4 1 3 Build high-temperature burner 5 6 4 2 2 Pour concrete and install frame 4 3 2 1 1 Construct collection stack 3 4 3 2 none Modify roof and floor 2 3 2 1 none Build internal components 1 Pessimistic Duration Most Likely Duration Optimistic Duration Prerequisites Task Description Task ID

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