1. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 1 Chapter 3 part 2 Project Management

2. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 2 CPM With 3 Time Estimates (PERT)  In past, time estimate is firm  Now, uncertain as to task duration –Natural variance  Use 3 time estimates to deal with uncertainty –a = optimistic estimate –m = most likely –b = pessimistic

3. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 3 Critical Path  Path length is sum of expected times of activities on path (not sum of most likely times)  Longest expected path length is critical –ET = expected time of project  Path length is uncertain, and so is project duration  Path has variance equal to sum of variances of individual activities on path

4. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 4 Formulas  ET (expected time) = (a + 4m + b)/6  σ 2 (variance) = [(b - a)/6] 2  σ p (st dev of project) = sqrt(  (variances on CP))  Z = (D - ET)/σ

5. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 5 PERT Probability Example You’re a project planner for Apple. A new Ipod project has an expected completion time of 40 weeks, with a standard deviation of 5 weeks. What is the probability of finishing the sub in 50 weeks or less?

6. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 6 T = 40 s = 5 50X Normal Distribution Z XET = - = - = s 5040 5 20. m z = 0 s Z = 1 Z 2.0 Standardized Normal Distribution Converting to Standardized Variable

7. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 7 m z = 0 s Z = 1 Z 2.0 Z.00.01 0.0.50000.50399 : ::: 2.0.97725.97784.97831 2.1.98214.98257.98300 Standardized Normal Probability Table (Portion) Probabilities in body Obtaining the Probability.02.50798.97725

8. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 8 Example 1. CPM with Three Activity Time Estimates

9. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 9 Example 1. Expected Time Calculations ET(A)= 3+4(6)+15 6 ET(A)= 3+4(6)+15 6 ET(A)=42/6=7

10. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 10 Ex. 1. Expected Time Calculations ET(B)=32/6=5.333 ET(B)= 2+4(4)+14 6 ET(B)= 2+4(4)+14 6

11. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 11 Ex 1. Expected Time Calculations ET(C)= 6+4(12)+30 6 ET(C)= 6+4(12)+30 6 ET(C)=84/6=14

12. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 12 Example 1. Network A(7) B (5.333) C(14) D(5) E(11) F(7) H(4) G(11) I(18) Duration = 54 Days

13. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 13 Example 1. Probability Exercise What is the probability of finishing this project in less than 53 days? What is the probability of finishing this project in less than 53 days? p(t < D) T E = 54 t D=53

14. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 14 (Sum the variance along the critical path.)

15. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 15 There is a 43.8% probability that this project will be completed in less than 53 weeks. (from the z table) p(Z < -.156) =.438, or 43.8 % T E = 54 p(t < D) t D=53

16. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 16 Ex 1. Additional Probability Exercise  What is the probability that the project duration will exceed 56 weeks?

17. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 17 Example 1. Additional Exercise Solution t T E = 54 p(t < D) D=56 p(Z >.312) =.378, or 37.8 %

18. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 18 Activity List for Example Problem Activity Description Immediate Predecessors Required Activity Time (weeks) A B C D E F G H I J Select office site Create organization and financial plan Determine personnel requirements Design facility Construct the interior Select personnel to move Hire new employees Move records, key personnel, etc. Make financial arrangements Train new personnel - - B A,C D C F F B H,E,G 3 5 3 4 8 2 4 2 5 3

19. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 19 Example 2 ET = (a + 4m + b)/6 σ 2 = (b - a) 2 /36 = [(b - a)/6] 2

20. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 20 Example 2 T = (a + 4m + b)/6 σ 2 = (b - a) 2 /36 = [(b - a)/6] 2

21. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 21 Example 2 T = (a + 4m + b)/6 σ 2 = (b - a) 2 /36 = [(b - a)/6] 2

22. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 22 Calculations ET = 23 weeks Σσ 2 CP = σ 2 B + σ 2 C + σ 2 D + σ 2 E + σ 2 J = 5.8056 Z = (D - ET)/ σ CP = (22-23)/2.4095 = -0.42 Take.42 to the Z table => 1 – Z[.42] =.337 When negative – subtract from 1 Beware - never add standard deviations!

23. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 23 0 0.1 0.2 0.3 0.4 ET = 23 22 34% chance < 22 50% chance > 23 16% chance between 22 and 23 (from table) 22 is 0.42 SD below Mean

24. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 24 Variability of Completion Time for Noncritical Paths  Variability of times for activities on noncritical paths must be considered when finding the probability of finishing in a specified time.  Variation in noncritical activity may cause change in critical path.