1. SE503 Advanced Project Management Dr. Ahmed Sameh, Ph.D. Professor, CS & IS Project Uncertainty Management

2. Background There are no facts about the future Project Uncertainty – Task durations – Costs and funding sources – Resources – Environmental conditions “Statistical variations and dependent events”

3. Project Uncertainty Management Identify individual task and resource uncertainties Determine their effect on project outcomes

4. Probability Relative frequency of random outcomes of an event – Toss coin – Roll dice – Deal cards – Purchase a lottery ticket Decimal (0.35) or percent (35%)

5. Definitions Joint Probability Conditional Probability Expected value Discrete distribution Continuous distribution

6. Uniform Distribution All outcomes equally likely Computers generate uniform distributions Most useful for discrete outcomes

7. Normal Distribution Gaussian “bell curve” Mean and Standard Deviation Infinite in extent Symmetrical Useful to model combinations of many variables

8. Beta Distribution Can be skewed Finite extent Four parameters (α, β, a, and b) (α, β) determine shape (a, b) determine extent Assumed for PERT analysis

9. Triangle Distribution Can be skewed Finite extent Approximates normal or beta Three parameters: min, max, most likely Useful to represent three-point estimates

10. Probability Density Function (pdf) Displays the characteristic shape of the distribution (bell, triangle, etc.) Area beneath the curve is 1.0 For task duration, this is the probability that the task will finish at this time.

11. Triangle pdf

12. Cumulative Density Function (cdf) Integral of the pdf Starts at 0.0, must reach 1.0 at maximum S-curve shape For task duration, this represents the probability of the task being complete by now.

13. Triangle cdf

14. PERT Model of Uncertainty Individual task durations are assumed to be Beta distributed Three-point estimates are used Outcomes are assumed to be normally distributed (central limit theorem) Expected value of duration is:

15. PERT Characteristics Incorporated into software packages Good “first approximation” of uncertainty Assumptions have not been shown to be valid Ignores dependencies in the network

16. Monte-Carlo Simulation A distribution is selected for each task Task durations are selected randomly from distributions Durations are combined as defined by the network to determine outcomes Process is repeated many times to build a statistical record of the outcomes

17. Monte-Carlo Characteristics Time consuming – must be done thousands of times to get a complete statistical picture Results accurately reflect all dependencies Standard software is available Usually used as a planning aid

18. Project Duration Triangle (4,5,7) Triangle (5,6,8)

19. Duration Calculation Traditional: 5+5+max(5,6)=16 Expected Value Using (a+4m+b)/6: 5.17+5.17+max(5.17,6.17)=16.5

20. Monte-Carlo Results

21. Why Projects Finish Late Traditional 16 days is 18% probable EV 16.5 days is 33% probable 90% confidence is at 18.5 days

22. Computational Techniques Possible alternative to Monte-Carlo simulation Higher accuracy Less time Wider application (budget, resources, Critical Chain) Useful throughout the project

23. Fuzzy Critical Path Method Fuzzy task durations Fuzzy arithmetic Fuzzy slack times Degree of criticality for each task, project

24. Fuzzy Arithmetic Issues Linguistic variables apply well to criticality, not duration and slack Fuzzy arithmetic does not imitate (or even estimate well) probabilistic combination Fuzzy “possibility theory” is related to, but not the same as, probability theory

25. Rosandich: Research Goals Develop a computational alternative to Monte Carlo simulation Fully integrate uncertainty management with project management Test the method against real project data

26. Approach Develop computational operators to combine: – Uncertain start time and uncertain duration to determine uncertain end time (series operator) – Multiple uncertain end times of predecessors into a single uncertain start time (parallel operator)

27. Combining Start Time and Duration (series)

28. Verification by Simulation

29. Start Time Results (parallel)

30. Verification by Simulation

31. How do we solve a network? At each step we did the calculation with independent variables!

32. Duration Results

33. Results vs. Simulation IterationsMean Absolute Difference 1,0000.001722 10,0000.000515 100,0000.000194

34. Example Project Network

35. Calculating Slack Time and Criticality Need to compare the uncertain duration of one path with all other paths in the network (backward pass) Needed to develop a negate operator – Reverses time direction of duration – Reverses time direction of uncertainty – Allows uncertain durations to be subtracted in series rather than added

36. Task B Slack Computation

37. Task B Slack Time Results

38. Criticality Results

39. Limitation: Path Dependencies

40. Solution: Replicate Tasks

41. Computation with replicated tasks Replicate tasks that are in two or more paths Assign a fixed duration to replicated tasks Use series/parallel operators to solve the network and determine outcome Repeat this for all possible durations of the replicated task to determine the probability of each outcome This is computationally intensive

42. Computational results with replicated tasks

43. Example Gantt Chart

44. Network diagram for example

45. Replicating multiple tasks

46. Computational results with multiple replicated tasks

47. Computing time with multiple replicated tasks Replicated TasksComputing time 13 sec. 21.5 min. 345 min. 41 day 51 month Good Monte-Carlo Simulation: 200,000 iterations takes a few minutes

48. Conclusions Accurate computation of: – Duration uncertainty – Slack time uncertainty – Criticality Series slack time uncertainties are identical Criticality divides between parallel tasks Total criticality for a project is 100% Computational time is exponential in number of replicated tasks