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Project Management with Certain
             Time Estimates
• Summary of steps:
  – Determine activities that need to be accomplished
  – Determine precedence relationships and completion
    times
  – Construct network diagram
  – Determine the critical path
  – Determine early start and late start schedules
Project Management: Scheduling
   Projects with Certain Time Estimates
• Create Network Diagram
  – Based on order of precedence
    among activities
Calculation of the Critical Path
• Network approach helps calculate project duration
    – A “path” is a sequence of activities that begins at the start of
      the project and goes to the end of the project
      • 1,2,3,5,6,7,8
      • 1,2,4,6,7,8
    – The “critical path” is the path that takes the longest to
      complete
      • and thus determines the minimum duration of the project

                        3             5
1         2                                        6              7      8

                              4
Calculation of the Critical Path
Calculation of the Critical Path
   • Critical Path
          – The path that
            takes the longest
            to complete




                                2 weeks         3 weeks

4 weeks       10 weeks            3               5       16 weeks   4 weeks   1 week

  1              2                    4 weeks
                                                             6         7         8

                                          4
Calculation of the Critical Path
   • Critical Path
          – The path that
            takes the longest
            to complete




                                2 weeks         3 weeks

4 weeks       10 weeks            3               5       16 weeks   4 weeks   1 week

  1              2                    4 weeks
                                                             6         7           8

                                          4                      C.P. = 40 weeks
Calculation of the Critical Path
 • It is possible for multiple
   Critical Paths to exist
      – New information suggests that
        Activity 4 will take 5 weeks
        instead of 4
                                                                            5
                                                                                 X




                             2 weeks         3 weeks

4 weeks     10 weeks           3               5       16 weeks   4 weeks       1 week

  1             2                  4 weeks
                                                          6         7             8

                                        4
Calculation of the Critical Path
 • It is possible for
   multiple Critical
   Paths to exist
                                                                             5
      – New information                                                           X
        suggests that Activity 4
        will take 5 weeks instead
        of 4


                           2 weeks            3 weeks

4 weeks     10 weeks          3                 5       16 weeks   4 weeks       1 week

  1            2                    5 weeks
                                                           6         7             8

                                      4                        C.P. = 40 weeks
Calculation of the Critical Path
 • Critical Path may also
   shift if non-critical
   activity is lengthened or
   Critical Path activity is                                                  6
   shortened                                                                        X
      – Another update indicates it
        will actually take 6 weeks for
        Activity 4


                             2 weeks           3 weeks

4 weeks     10 weeks            3                5       16 weeks   4 weeks       1 week

  1             2                    4 weeks
                                                            6         7             8

                                         4
Calculation of the Critical Path
• Critical Path may also
  shift if non-critical
  activity is lengthened or
  Critical Path activity is                                                    6
  shortened                                                                         X
   – Another update indicates it
     will actually take 6 weeks for
     Activity 4


                            2 weeks             3 weeks

4 weeks    10 weeks            3                  5       16 weeks   4 weeks       1 week

  1           2                       6 weeks
                                                             6         7             8

                                        4                        C.P. = 41 weeks
Determining Slack
• Slack - The amount of time an activity on a non-critical path
  can be delayed without affecting the duration of the project
  (i.e., without putting it on the critical path)
  – Uses four calculated values
     •   Early start - Earliest an activity can start (based on prerequisites)
     •   Early finish - Earliest it can finish (based on prerequisites & duration)
     •   Late start - Latest an activity can start and not delay the project
     •   Late finish - Latest an activity can finish and not delay the project
Calculating Early Start (ES) and
                   Early Finish (EF)
• Move from left to right in
  network
   – ES for 1st activity usually zero
   – EF equals ES plus activity
     duration
   – ES is latest of the EF times of
     an activity’s predecessors
THIS IS CALLED
THE FORWARD
PASS
Calculating Late Start (LS) and
•   Move from right to left in
                               Late Finish (LF)
    network
    – LF for last activity equals EF
      for last activity
       • Or target date if different
    – LS equals LF minus activity
      duration
    – LF is earliest of the LS times of
      an activity’s followers
    THIS IS CALLED THE
    BACKWARD PASS
Calculating Slack
     • Slack - The amount of time an        • Computed by either:
       activity on a non-critical path        Late Start - Early Start
       can be delayed without affecting            or
       the duration of the project (i.e.,
       putting it on the critical path)       Late Finish - Early Finish




• Activities that have zero slack are, by definition, on the critical path
Project Scheduling with Uncertain
          Time Estimates
• Summary of steps:
  – Determine activities that need to be accomplished
  – Determine precedence relationships and completion times
  – Construct network diagram
  – Determine the critical path
  – Determine early start and late start schedules
  – Calculate the variances for the activity times
  – Calculate the probability of completing by the desired due
    date
Project Scheduling with Time
             Uncertainty
• Take three time estimates
  – Optimistic - What is the (realistic) fastest we can
   get an activity done?
  – Pessimistic - What is the (realistic) worst case
   scenario for delay?
  – Most likely - What is our “most likely” estimate?
Project Scheduling with Time
               Uncertainty
• Calculate the “expected time” for the activity
                       o   4m     p
                 Te
                            6
Where:
 Te   =   Expected Time
 o    =   Optimistic estimate
 m    =   Most likely estimate
 p    =   Worst-case (pessimistic) estimate
Project Scheduling with Time
          Uncertainty
                                       E stim ates
   A ctivity #        O ptim istic     M o st L ikely   P essim istic
        1                  3                  4               5
        2                  8                 10              12
        3                  1                  2               4

                        3    4(4)      5
Activity #1      Te                          4 . 00 weeks
                               6

                        8    4 (10 )    12
Activity #2      Te                            10 . 00 weeks
                                   6

                        1    4(2)      4
Activity #3      Te                          2 . 17 weeks
                               6
Using Variation in Time Estimates to
         Assess Duration Probabilities
How do we interpret this estimate???
   Probability theory (“central limit theorem”) dictates that we assume that
         finish times across entire project are distributed “normally”

                 Probabilistically: 50% chance we will
                      finish faster, 50% chance
                         we will finish slower
                       2.17 weeks       3 weeks

4 weeks   10 weeks         3                5     16.17 weeks   4.17 weeks   1.17 weeks

  1          2                 4.17 weeks
                                                     6            7             8
                                    4
                                                  C.P. = 40.68 weeks
Using Variation in Time Estimates to
     Access Duration Probabilities
           D   Te                  Note that this is , not 2 ,
      Z                       We have to take the square root of the
                             “variance” to get the standard deviation



Where:
Z = Number of standard deviations D is from Te
Te = Expected Time
D = Project duration I am thinking about
Using Variation in Time Estimates to
      Access Duration Probabilities
• We recognize that there is variation around our estimates
  – We are estimating “most likely” or “expected” not “exact”
  – Commonly assume a “Beta” distribution
     • Another asymmetric distribution (because duration is never less than zero)


• Calculating variance of the activity’s duration estimate:

                                                2
                           2        p       o
                                        6
Using Variation in Time Estimates to
   Access Duration Probabilities
                                    E stim ates
  A ctivity #    O ptim istic       M o st L ikely      P essim istic
       1              3                    4                  5
       2              8                   10                 12
       3              1                    2                  4
                                        2
                       2   5        3
   Activity #1                                  0 . 11 weeks
                                6
                                            2
                   2       12       8
   Activity #2                                  0 . 44 weeks
                                6
                                        2
                       2   4        1
   Activity #3                                  0 . 25 weeks
                                6
Using Variation in Time Estimates to
      Assess Duration Probabilities
• Calculate Critical Path using Te
• Accumulate the variances of the individual activities
• Apply formula to estimate project duration probabilities
                            D      Te
                    Z

             Where:
             Z = Number of standard deviations D is from
                Te
             Te = Expected Time
             D = Activity duration I am thinking about
Using Variation in Time Estimates to
   Assess Duration Probabilities
Using Variation in Time Estimates to
         Assess Duration Probabilities
 What is the probability
 of finishing in 39 weeks
 or less?




                     2.17 weeks       3 weeks

4 weeks   10 weeks       3                5     16.17 weeks   4.17 weeks   1.17 weeks

  1          2               4.17 weeks
                                                   6            7             8

                                  4
Using Variation in Time Estimates to
         Assess Duration Probabilities
                                                Exhibit 8.22: Expected Values and Variances of Time Estimates

What is the probability of
finishing in 39 weeks or less?


 Critical Path = 40.68 weeks
 Sum of Variances = 2.56 weeks


                     2.17 weeks           3 weeks

4 weeks   10 weeks       3                  5               16.17 weeks           4.17 weeks          1.17 weeks

  1          2               4.17 weeks
                                                                  6                    7                    8

                                  4
Using Variation in Time Estimates to
       Assess Duration Probabilities
           D      Te     Where:
     Z                   Z = Number of standard deviations D is from Te
                         Te = Expected Time
                         D = Activity duration I am thinking about

      39       40 . 68
Z                          1 . 05 Standard deviation difference between
           1 .6
                                    “most likely” and desired duration
                                      1.6 = the square root of 2.56!!!



• So, I look up the cumulative probability [G(z)] of the activity
    being completed before -1.05 standard deviations from the
    “most likely” estimate…
     – … and find out I have a 85.314% chance of finishing within 39
       weeks!! FROM APPENDIX A PAGE A3
Using Variation in Time Estimates to
      Access Duration Probabilities
• So, what if I want to be 90 percent sure that I hit the
  project duration that I tell my boss I will (or she will fire
  me)?
  – Reconfigure “Z” equation to solve for D
  – Pick the value of “Z” that corresponds to the level of certainty I
                                         Find probability under G(z) in Appendix
    want (90% = +1.30)                    B and translate to number of standard

  – Solve
                                                        deviations
                       D Te         (Z )

                 D     40 . 68    1 .6 * 1 .3   42 . 76 weeks!!!

             What if I can stand to be only 80% sure??
Crashing Projects
• A methodical approach to reducing project duration
  – Focus on the time of activities on the critical path
  – Looking for greatest improvement with least cost
       • Additional labor, machinery
       • Overtime and temporary employees
       • Premiums paid to outside contractors for early delivery
• Steps
  –   Create network
  –   Identify critical path
  –   Identify costs of reducing each activity on path
  –   Reduce most cost effective activity
  –   Look for critical path changes
       • Beware of multiple critical paths
  – Crash next activity
Crashing Projects: Create the
         Network
Crashing Projects: Identify the Critical
                Path
          B                   C
          7                   7
      A             D                   F       H
      8             10                  8       5
                         E                  G
                         12                 9



                    A-B-C-F-H
              Critical Path = 35 days
Crashing Projects: Identify Costs of
      Crashing Each Activity
Crashing Projects: Reduce Most Cost
         Effective Activity
                           6
         B             C
         7             7
                       X
    A        D                 F       H
    8        10                8       5
                  E                G
                  12               9
Crashing Projects: Look for Critical
          Path Changes
                                        6
                B                 C
                7                 7
                                  X
       A                D                   F              H
       8                10                  8              5
                             E                  G
                             12                 9


Old Critical Path Completion = 35 days
Activity C Crashed by 1 day project completion = 34 days
Did that effect the critical path?
Crashing Projects: Look for Critical
          Path Changes
                                          6
                B                     C
                7                     7
                                      X
      A                  D                    F       H
      8                  10                   8       5
                               E                  G
                               12                 9



  Multiple Critical Paths Appear!!!

                        Critical Path = 34 days
Crashing Projects: Crash Next Activity
Exhibit 8.25: Crash Time and Costs




      Both C.P.                      Path 1 Only   Path 2 Only
Crashing Projects: Summary
              Exhibit 8.26: Crashing Summary
Solution
Caveats
•   Time estimates are frequently wrong
•   Early finishes are absorbed
•   Cushions get wasted
•   Resources aren’t always available when
    needed

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Critical Path Ppt

  • 1. Project Management with Certain Time Estimates • Summary of steps: – Determine activities that need to be accomplished – Determine precedence relationships and completion times – Construct network diagram – Determine the critical path – Determine early start and late start schedules
  • 2. Project Management: Scheduling Projects with Certain Time Estimates • Create Network Diagram – Based on order of precedence among activities
  • 3. Calculation of the Critical Path • Network approach helps calculate project duration – A “path” is a sequence of activities that begins at the start of the project and goes to the end of the project • 1,2,3,5,6,7,8 • 1,2,4,6,7,8 – The “critical path” is the path that takes the longest to complete • and thus determines the minimum duration of the project 3 5 1 2 6 7 8 4
  • 4. Calculation of the Critical Path
  • 5. Calculation of the Critical Path • Critical Path – The path that takes the longest to complete 2 weeks 3 weeks 4 weeks 10 weeks 3 5 16 weeks 4 weeks 1 week 1 2 4 weeks 6 7 8 4
  • 6. Calculation of the Critical Path • Critical Path – The path that takes the longest to complete 2 weeks 3 weeks 4 weeks 10 weeks 3 5 16 weeks 4 weeks 1 week 1 2 4 weeks 6 7 8 4 C.P. = 40 weeks
  • 7. Calculation of the Critical Path • It is possible for multiple Critical Paths to exist – New information suggests that Activity 4 will take 5 weeks instead of 4 5 X 2 weeks 3 weeks 4 weeks 10 weeks 3 5 16 weeks 4 weeks 1 week 1 2 4 weeks 6 7 8 4
  • 8. Calculation of the Critical Path • It is possible for multiple Critical Paths to exist 5 – New information X suggests that Activity 4 will take 5 weeks instead of 4 2 weeks 3 weeks 4 weeks 10 weeks 3 5 16 weeks 4 weeks 1 week 1 2 5 weeks 6 7 8 4 C.P. = 40 weeks
  • 9. Calculation of the Critical Path • Critical Path may also shift if non-critical activity is lengthened or Critical Path activity is 6 shortened X – Another update indicates it will actually take 6 weeks for Activity 4 2 weeks 3 weeks 4 weeks 10 weeks 3 5 16 weeks 4 weeks 1 week 1 2 4 weeks 6 7 8 4
  • 10. Calculation of the Critical Path • Critical Path may also shift if non-critical activity is lengthened or Critical Path activity is 6 shortened X – Another update indicates it will actually take 6 weeks for Activity 4 2 weeks 3 weeks 4 weeks 10 weeks 3 5 16 weeks 4 weeks 1 week 1 2 6 weeks 6 7 8 4 C.P. = 41 weeks
  • 11. Determining Slack • Slack - The amount of time an activity on a non-critical path can be delayed without affecting the duration of the project (i.e., without putting it on the critical path) – Uses four calculated values • Early start - Earliest an activity can start (based on prerequisites) • Early finish - Earliest it can finish (based on prerequisites & duration) • Late start - Latest an activity can start and not delay the project • Late finish - Latest an activity can finish and not delay the project
  • 12. Calculating Early Start (ES) and Early Finish (EF) • Move from left to right in network – ES for 1st activity usually zero – EF equals ES plus activity duration – ES is latest of the EF times of an activity’s predecessors THIS IS CALLED THE FORWARD PASS
  • 13. Calculating Late Start (LS) and • Move from right to left in Late Finish (LF) network – LF for last activity equals EF for last activity • Or target date if different – LS equals LF minus activity duration – LF is earliest of the LS times of an activity’s followers THIS IS CALLED THE BACKWARD PASS
  • 14. Calculating Slack • Slack - The amount of time an • Computed by either: activity on a non-critical path Late Start - Early Start can be delayed without affecting or the duration of the project (i.e., putting it on the critical path) Late Finish - Early Finish • Activities that have zero slack are, by definition, on the critical path
  • 15. Project Scheduling with Uncertain Time Estimates • Summary of steps: – Determine activities that need to be accomplished – Determine precedence relationships and completion times – Construct network diagram – Determine the critical path – Determine early start and late start schedules – Calculate the variances for the activity times – Calculate the probability of completing by the desired due date
  • 16. Project Scheduling with Time Uncertainty • Take three time estimates – Optimistic - What is the (realistic) fastest we can get an activity done? – Pessimistic - What is the (realistic) worst case scenario for delay? – Most likely - What is our “most likely” estimate?
  • 17. Project Scheduling with Time Uncertainty • Calculate the “expected time” for the activity o 4m p Te 6 Where: Te = Expected Time o = Optimistic estimate m = Most likely estimate p = Worst-case (pessimistic) estimate
  • 18. Project Scheduling with Time Uncertainty E stim ates A ctivity # O ptim istic M o st L ikely P essim istic 1 3 4 5 2 8 10 12 3 1 2 4 3 4(4) 5 Activity #1 Te 4 . 00 weeks 6 8 4 (10 ) 12 Activity #2 Te 10 . 00 weeks 6 1 4(2) 4 Activity #3 Te 2 . 17 weeks 6
  • 19. Using Variation in Time Estimates to Assess Duration Probabilities How do we interpret this estimate??? Probability theory (“central limit theorem”) dictates that we assume that finish times across entire project are distributed “normally” Probabilistically: 50% chance we will finish faster, 50% chance we will finish slower 2.17 weeks 3 weeks 4 weeks 10 weeks 3 5 16.17 weeks 4.17 weeks 1.17 weeks 1 2 4.17 weeks 6 7 8 4 C.P. = 40.68 weeks
  • 20. Using Variation in Time Estimates to Access Duration Probabilities D Te Note that this is , not 2 , Z We have to take the square root of the “variance” to get the standard deviation Where: Z = Number of standard deviations D is from Te Te = Expected Time D = Project duration I am thinking about
  • 21. Using Variation in Time Estimates to Access Duration Probabilities • We recognize that there is variation around our estimates – We are estimating “most likely” or “expected” not “exact” – Commonly assume a “Beta” distribution • Another asymmetric distribution (because duration is never less than zero) • Calculating variance of the activity’s duration estimate: 2 2 p o 6
  • 22. Using Variation in Time Estimates to Access Duration Probabilities E stim ates A ctivity # O ptim istic M o st L ikely P essim istic 1 3 4 5 2 8 10 12 3 1 2 4 2 2 5 3 Activity #1 0 . 11 weeks 6 2 2 12 8 Activity #2 0 . 44 weeks 6 2 2 4 1 Activity #3 0 . 25 weeks 6
  • 23. Using Variation in Time Estimates to Assess Duration Probabilities • Calculate Critical Path using Te • Accumulate the variances of the individual activities • Apply formula to estimate project duration probabilities D Te Z Where: Z = Number of standard deviations D is from Te Te = Expected Time D = Activity duration I am thinking about
  • 24. Using Variation in Time Estimates to Assess Duration Probabilities
  • 25. Using Variation in Time Estimates to Assess Duration Probabilities What is the probability of finishing in 39 weeks or less? 2.17 weeks 3 weeks 4 weeks 10 weeks 3 5 16.17 weeks 4.17 weeks 1.17 weeks 1 2 4.17 weeks 6 7 8 4
  • 26. Using Variation in Time Estimates to Assess Duration Probabilities Exhibit 8.22: Expected Values and Variances of Time Estimates What is the probability of finishing in 39 weeks or less? Critical Path = 40.68 weeks Sum of Variances = 2.56 weeks 2.17 weeks 3 weeks 4 weeks 10 weeks 3 5 16.17 weeks 4.17 weeks 1.17 weeks 1 2 4.17 weeks 6 7 8 4
  • 27. Using Variation in Time Estimates to Assess Duration Probabilities D Te Where: Z Z = Number of standard deviations D is from Te Te = Expected Time D = Activity duration I am thinking about 39 40 . 68 Z 1 . 05 Standard deviation difference between 1 .6 “most likely” and desired duration 1.6 = the square root of 2.56!!! • So, I look up the cumulative probability [G(z)] of the activity being completed before -1.05 standard deviations from the “most likely” estimate… – … and find out I have a 85.314% chance of finishing within 39 weeks!! FROM APPENDIX A PAGE A3
  • 28. Using Variation in Time Estimates to Access Duration Probabilities • So, what if I want to be 90 percent sure that I hit the project duration that I tell my boss I will (or she will fire me)? – Reconfigure “Z” equation to solve for D – Pick the value of “Z” that corresponds to the level of certainty I Find probability under G(z) in Appendix want (90% = +1.30) B and translate to number of standard – Solve deviations D Te (Z ) D 40 . 68 1 .6 * 1 .3 42 . 76 weeks!!! What if I can stand to be only 80% sure??
  • 29. Crashing Projects • A methodical approach to reducing project duration – Focus on the time of activities on the critical path – Looking for greatest improvement with least cost • Additional labor, machinery • Overtime and temporary employees • Premiums paid to outside contractors for early delivery • Steps – Create network – Identify critical path – Identify costs of reducing each activity on path – Reduce most cost effective activity – Look for critical path changes • Beware of multiple critical paths – Crash next activity
  • 31. Crashing Projects: Identify the Critical Path B C 7 7 A D F H 8 10 8 5 E G 12 9 A-B-C-F-H Critical Path = 35 days
  • 32. Crashing Projects: Identify Costs of Crashing Each Activity
  • 33. Crashing Projects: Reduce Most Cost Effective Activity 6 B C 7 7 X A D F H 8 10 8 5 E G 12 9
  • 34. Crashing Projects: Look for Critical Path Changes 6 B C 7 7 X A D F H 8 10 8 5 E G 12 9 Old Critical Path Completion = 35 days Activity C Crashed by 1 day project completion = 34 days Did that effect the critical path?
  • 35. Crashing Projects: Look for Critical Path Changes 6 B C 7 7 X A D F H 8 10 8 5 E G 12 9 Multiple Critical Paths Appear!!! Critical Path = 34 days
  • 36. Crashing Projects: Crash Next Activity Exhibit 8.25: Crash Time and Costs Both C.P. Path 1 Only Path 2 Only
  • 37. Crashing Projects: Summary Exhibit 8.26: Crashing Summary Solution
  • 38. Caveats • Time estimates are frequently wrong • Early finishes are absorbed • Cushions get wasted • Resources aren’t always available when needed