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1 Chapter 3 part 2 Project Management 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 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 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 σp (st dev of project) = sqrt(S (variances on CP)) Z = (D - ET)/σ McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. PERT Probability Example 5 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? McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 6 Converting to Standardized Variable X - ET 50 - 40 = = 2 .0 Z = s 5 Normal Distribution Standardized Normal Distribution sZ = 1 s =5 T = 40 50 McGraw-Hill/Irwin X mz = 0 2.0 Z © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Obtaining the Probability 7 Standardized Normal Probability Table (Portion) Z .00 .01 .02 sZ =1 0.0 .50000 .50399 .50798 : : : : 2.0 .97725 .97784 .97831 .97725 2.1 .98214 .98257 .98300 mz = 0 2.0 Z Probabilities in body McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 8 Example 1. CPM with Three Activity Time Estimates Immediate Task Predecesors Optimistic Most Likely Pessimistic A None 3 6 15 B None 2 4 14 C A 6 12 30 D A 2 5 8 E C 5 11 17 F D 3 6 15 G B 3 9 27 H E,F 1 4 7 I G,H 4 19 28 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 9 Example 1. Expected Time Calculations ET(A)= 3+4(6)+15 Task A B C D E F G H I Immediate Expected Predecesors Time None 7 None 5.333 A 14 A 5 C 11 D 7 B 11 E,F 4 G,H 18 6 ET(A)=42/6=7 Immediate Task Predecesors Optimistic Most Likely Pessimistic A None 3 6 15 B None 2 4 14 C A 6 12 30 D A 2 5 8 E C 5 11 17 F D 3 6 15 G B 3 9 27 H E,F 1 4 7 I G,H 4 19 28 Opt. Time + 4(Most Likely Time) + Pess. Time Expected Time = 6 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 10 Ex. 1. Expected Time Calculations Task A B C D E F G H I Immediate Expected Predecesors Time None 7 None 5.333 A 14 A 5 C 11 D 7 B 11 E,F 4 G,H 18 ET(B)= 2+4(4)+14 6 ET(B)=32/6=5.333 Immediate Task Predecesors Optimistic Most Likely Pessimistic A None 3 6 15 B None 2 4 14 C A 6 12 30 D A 2 5 8 E C 5 11 17 F D 3 6 15 G B 3 9 27 H E,F 1 4 7 I G,H 4 19 28 Opt. Time + 4(Most Likely Time) + Pess. Time Expected Time = 6 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 11 Ex 1. Expected Time Calculations Task A B C D E F G H I Immediate Expected Predecesors Time None 7 None 5.333 A 14 A 5 C 11 D 7 B 11 E,F 4 G,H 18 ET(C)= 6+4(12)+30 6 ET(C)=84/6=14 Immediate Task Predecesors Optimistic Most Likely Pessimistic A None 3 6 15 B None 2 4 14 C A 6 12 30 D A 2 5 8 E C 5 11 17 F D 3 6 15 G B 3 9 27 H E,F 1 4 7 I G,H 4 19 28 Opt. Time + 4(Most Likely Time) + Pess. Time Expected Time = 6 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 12 Example 1. Network Duration = 54 Days C(14) E(11) H(4) A(7) D(5) F(7) I(18) B (5.333) McGraw-Hill/Irwin G(11) © 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? p(t < D) D=53 t TE = 54 Z = McGraw-Hill/Irwin D - TE 2 cp © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 14 A ctivity variance, Task A B C D E F G H I 2 Pessim . - O ptim . 2 = ( ) 6 Optimistic Most Likely Pessimistic Variance 3 6 15 4 2 4 14 6 12 30 16 2 5 8 5 11 17 4 3 6 15 3 9 27 1 4 7 1 4 19 28 16 (Sum the variance along the critical path.) McGraw-Hill/Irwin 2 = 41 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 15 p(t < D) t TE = 54 D=53 Z = D - TE 2 cp 53 - 54 = = -.156 41 p(Z < -.156) = .438, or 43.8 % There is a 43.8% probability that this project will be completed in less than 53 weeks. (from the z table) McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Ex 1. Additional Probability Exercise 16 What is the probability that the project duration will exceed 56 weeks? McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Example 1. Additional Exercise Solution 17 p(t < D) TE = 54 Z = D - TE 2 cp t D=56 56 - 54 = = .312 41 p(Z > .312) = .378, or 37.8 % McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 18 Activity List for Example Problem McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 19 Example 2 Activity A B C D E F G H I J a 1 3 2 2 4 1 2.5 1 4 1.5 m 3 4.5 3 4 7 1.5 3.5 2 5 3 b 5 9 4 6 16 5 7.5 3 6 4.5 ET 2 ET = (a + 4m + b)/6 σ 2 = (b - a)2/36 = [(b - a)/6]2 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 20 Example 2 Activity A B C D E F G H I J a 1 3 2 2 4 1 2.5 1 4 1.5 m 3 4.5 3 4 7 1.5 3.5 2 5 3 b 5 9 4 6 16 5 7.5 3 6 4.5 ET 3 5 3 4 8 2 4 2 5 3 2 0.44 1 0.11 0.44 4 0.44 0.69 0.11 0.11 0.25 T = (a + 4m + b)/6 σ 2 = (b - a)2/36 = [(b - a)/6]2 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 21 Example 2 Activity A B C D E F G H I J a 1 3 2 2 4 1 2.5 1 4 1.5 m 3 4.5 3 4 7 1.5 3.5 2 5 3 b 5 9 4 6 16 5 7.5 3 6 4.5 ET 3 5 3 4 8 2 4 2 5 3 2 0.44 1 0.11 0.44 4 0.44 0.69 0.11 0.11 0.25 T = (a + 4m + b)/6 σ 2 = (b - a)2/36 = [(b - a)/6]2 McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 22 Calculations •ET = 23 weeks •Σσ2CP = σ 2B + σ 2C + σ 2D + σ 2E + σ 2J = 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! McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 23 22 is 0.42 SD below Mean 0.4 16% chance between 22 and 23 (from table) 0.3 0.2 0.1 0 34% chance < 22 22 McGraw-Hill/Irwin 50% chance > 23 ET = 23 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Variability of Completion Time for Noncritical Paths 24 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. McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.