1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.1 Table of Contents Chapter 8 (PERT/CPM Models for Project Management) A Case Study: The Reliable Construction Co. Project (Section 8.1)8.2–8.3 Using a Network to Visually Display a Project (Section 8.2)8.4–8.7 Scheduling a Project with PERT/CPM (Section 8.3)8.8–8.17 Dealing with Uncertain Activity Durations (Section 8.4)8.18–8.26 Considering Time-Cost Tradeoffs (Section 8.5)8.27–8.38 Scheduling and Controlling Project Costs (Section 8.6)8.39–8.46 Introduction to Project Management (UW Lecture)8.47–8.57 These slides are based upon a lecture from the course “Principles of Operations Management”, (an undergraduate business class) at the University of Washington (as taught by one of the authors). Project Management with Uncertain Activity Durations (UW Lecture)8.58–8.69 These slides are based upon a lecture from the course “Principles of Operations Management” (an undergraduate business class) at the University of Washington (as taught by one of the authors). Time-Cost Tradeoffs (UW Lecture)8.70–8.81 These slides are based upon a lecture from the course “Principles of Operations Management” (an undergraduate business class) at the University of Washington (as taught by one of the authors).

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.2 Reliable Construction Company Project The Reliable Construction Company has just made the winning bid of $5.4 million to construct a new plant for a major manufacturer. The contract includes the following provisions: –A penalty of $300,000 if Reliable has not completed construction within 47 weeks. –A bonus of $150,000 if Reliable has completed the plant within 40 weeks. Questions: 1.How can the project be displayed graphically to better visualize the activities? 2.What is the total time required to complete the project if no delays occur? 3.When do the individual activities need to start and finish? 4.What are the critical bottleneck activities? 5.For other activities, how much delay can be tolerated? 6.What is the probability the project can be completed in 47 weeks? 7.What is the least expensive way to complete the project within 40 weeks? 8.How should ongoing costs be monitored to try to keep the project within budget?

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.3 Activity List for Reliable Construction ActivityActivity Description Immediate Predecessors Estimated Duration (Weeks) AExcavate—2 BLay the foundationA4 CPut up the rough wallB10 DPut up the roofC6 EInstall the exterior plumbingC4 FInstall the interior plumbingE5 GPut up the exterior sidingD7 HDo the exterior paintingE, G9 IDo the electrical workC7 JPut up the wallboardF, I8 KInstall the flooringJ4 LDo the interior paintingJ5 MInstall the exterior fixturesH2 NInstall the interior fixturesK, L6

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.4 Project Networks A network used to represent a project is called a project network. A project network consists of a number of nodes and a number of arcs. Two types of project networks: –Activity-on-arc (AOA): each activity is represented by an arc. A node is used to separate an activity from its predecessors. The sequencing of the arcs shows the precedence relationships. –Activity-on-node (AON): each activity is represented by a node. The arcs are used to show the precedence relationships. Advantages of AON (used in this textbook): –considerably easier to construct –easier to understand –easier to revise when there are changes

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.5 Reliable Construction Project Network

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.6 Microsoft Project Gantt Chart

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.7 Microsoft Project—Project Network

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.8 The Critical Path A path through a network is one of the routes following the arrows (arcs) from the start node to the finish node. The length of a path is the sum of the (estimated) durations of the activities on the path. The (estimated) project duration equals the length of the longest path through the project network. This longest path is called the critical path. (If more than one path tie for the longest, they all are critical paths.)

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.9 The Paths for Reliable’s Project Network PathLength (Weeks) Start  A  B  C  D  G  H  M  Finish 2 + 4 + 10 + 6 + 7 + 9 + 2 = 40 Start  A  B  C  E  H  M  Finish 2 + 4 + 10 + 4 + 9 + 2 = 31 Start  A  B  C  E  F  J  K  N  Finish 2 + 4 + 10 + 4 + 5 + 8 + 4 + 6 = 43 Start  A  B  C  E  F  J  L  N  Finish 2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 = 44 Start  A  B  C  I  J  K  N  Finish 2 + 4 + 10 + 7 + 8 + 4 + 6 = 41 Start  A  B  C  I  J  L  N  Finish 2 + 4 + 10 + 7 + 8 + 5 + 6 = 42

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.10 Earliest Start and Earliest Finish Times The starting and finishing times of each activity if no delays occur anywhere in the project are called the earliest start time and the earliest finish time. –ES = Earliest start time for a particular activity –EF = Earliest finish time for a particular activity Earliest Start Time Rule: ES = Largest EF of the immediate predecessors. Procedure for obtaining earliest times for all activities: 1.For each activity that starts the project (including the start node), set its ES = 0. 2.For each activity whose ES has just been obtained, calculate EF = ES + duration. 3.For each new activity whose immediate predecessors now have EF values, obtain its ES by applying the earliest start time rule. Apply step 2 to calculate EF. 4.Repeat step 3 until ES and EF have been obtained for all activities.

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.11 ES and EF Values for Reliable Construction for Activities that have only a Single Predecessor

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.12 ES and EF Times for Reliable Construction

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.13 Latest Start and Latest Finish Times The latest start time for an activity is the latest possible time that it can start without delaying the completion of the project (so the finish node still is reached at its earliest finish time). The latest finish time has the corresponding definition with respect to finishing the activity. –LS = Latest start time for a particular activity –LF = Latest finish time for a particular activity Latest Finish Time Rule: LF = Smallest LS of the immediate successors. Procedure for obtaining latest times for all activities: 1.For each of the activities that together complete the project (including the finish node), set LF equal to EF of the finish node. 2.For each activity whose LF value has just been obtained, calculate LS = LF – duration. 3.For each new activity whose immediate successors now have LS values, obtain its LF by applying the latest finish time rule. Apply step 2 to calculate its LS. 4.Repeat step 3 until LF and LS have been obtained for all activities.

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.14 LS and LF Times for Reliable’s Project

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.15 The Complete Project Network

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.16 Slack for Reliable’s Activities ActivitySlack (LF–EF)On Critical Path? A0Yes B0 C0 D4No E0Yes F0 G4No H4 I2 J0Yes K1No L0Yes M4No N0Yes

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.17 Spreadsheet to Calculate ES, EF, LS, LF, Slack

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.18 The PERT Three Estimate Approach Most likely estimate (m) = Estimate of most likely value of the duration Optimistic estimate (o) = Estimate of duration under most favorable conditions. Pessimistic estimate (p) = Estimate of duration under most unfavorable conditions.

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.19 Mean and Standard Deviation An approximate formula for the variance (  2 ) of an activity is An approximate formula for the mean (  ) of an activity is

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.20 Time Estimates for Reliable’s Project ActivityompMeanVariance A1232 1/91/9 B23.5841 C6918104 D45.51061 E14.554 4/94/9 F441051 G56.51171 H581794 I37.5971 J39981 K44440 L15.5751 M1232 1/91/9 N5 96 4/94/9

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.21 Pessimistic Path Lengths for Reliable’s Project PathPessimistic Length (Weeks) Start  A  B  C  D  G  H  M  Finish 3 + 8 + 18 + 10 + 11 + 17 + 3 = 70 Start  A  B  C  E  H  M  Finish 3 + 8 + 18 + 5 + 17 + 3 = 54 Start  A  B  C  E  F  J  K  N  Finish 3 + 8 + 18 + 5 + 10 + 9 + 4 + 9 = 66 Start  A  B  C  E  F  J  L  N  Finish 3 + 8 + 18 + 5 + 10 + 9 + 7 + 9 = 69 Start  A  B  C  I  J  K  N  Finish 3 + 8 + 18 + 9 + 9 + 4 + 9 = 60 Start  A  B  C  I  J  L  N  Finish 3 + 8 + 18 + 9 + 9 + 7 + 9 = 63

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.22 Three Simplifying Approximations of PERT/CPM 1.The mean critical path will turn out to be the longest path through the project network. 2.The durations of the activities on the mean critical path are statistically independent. Thus, the three estimates of the duration of an activity would never change after learning the durations of some of the other activities. 3.The form of the probability distribution of project duration is the normal distribution. By using simplifying approximations 1 and 2, there is some statistical theory (one version of the central limit theorem) that justifies this as being a reasonable approximation if the number of activities on the mean critical path is not too small.

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.23 Calculation of Project Mean and Variance Activities on Mean Critical PathMeanVariance A2 1/91/9 B41 C104 E4 4/94/9 F51 J81 L51 N6 4/94/9 Project duration  p = 44  2 p = 9

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.24 Probability of Meeting Deadline

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.25 Probability of Meeting a Deadline P(T ≤ d) –3.00.001400.50 –2.50.00620.250.60 –2.00.0230.50.69 –1.750.0400.750.77 –1.50.0671.00.84 –1.250.111.250.89 –1.00.161.50.933 –0.750.231.750.960 –0.50.312.00.977 –0.250.402.50.9938 00.503.00.9986

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.26 Spreadsheet for PERT Three-Estimate Approach

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.27 Considering Time-Cost Trade-Offs Question: If extra money is spent to expedite the project, what is the least expensive way of attempting to meet the target completion time (40 weeks)? CPM Method of Time-Cost Trade-Offs: –Crashing an activity refers to taking special costly measures to reduce the duration of an activity below its normal value. Special measures might include overtime, hiring additional temporary help, using special time-saving materials, obtaining special equipment, etc. –Crashing the project refers to crashing a number of activities to reduce the duration of the project below its normal value.

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.28 Time-Cost Graph for an Activity

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.29 Time-Cost Trade-Off Data for Reliable’s Project Time (weeks) Cost Maximum Reduction in Time (weeks) Crash Cost per Week Saved ActivityNormalCrashNormalCrash A21$180,000$280,0001$100,000 B42320,000420,000250,000 C107620,000860,000380,000 D64260,000340,000240,000 E43410,000570,0001160,000 F53180,000260,000240,000 G74900,0001,020,000340,000 H96200,000380,000360,000 I75210,000270,000230,000 J86430,000490,000230,000 K43160,000200,000140,000 L53250,000350,000250,000 M21100,000200,0001100,000 N63330,000510,000360,000

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.30 Marginal Cost Analysis for Reliable’s Project Initial Table Length of Path Activity to Crash Crash Cost ABCDGHM ABCEHM ABCEFJKN ABCEFJLN ABCIJKN ABCIJLN 403143444142

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.31 Marginal Cost Analysis for Reliable’s Project Table After Crashing One Week Length of Path Activity to Crash Crash Cost ABCDGHM ABCEHM ABCEFJKN ABCEFJLN ABCIJKN ABCIJLN 403143444142 J$30,000403142434041

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.32 Marginal Cost Analysis for Reliable’s Project Table After Crashing Two Weeks Length of Path Activity to Crash Crash Cost ABCDGHM ABCEHM ABCEFJKN ABCEFJLN ABCIJKN ABCIJLN 403143444142 J$30,000403142434041 J$30,000403141423940

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.33 Marginal Cost Analysis for Reliable’s Project Table After Crashing Three Weeks Length of Path Activity to Crash Crash Cost ABCDGHM ABCEHM ABCEFJKN ABCEFJLN ABCIJKN ABCIJLN 403143444142 J$30,000403142434041 J$30,000403141423940 F$40,000403140413940

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.34 Marginal Cost Analysis for Reliable’s Project Final Table After Crashing Four Weeks Length of Path Activity to Crash Crash Cost ABCDGHM ABCEHM ABCEFJKN ABCEFJLN ABCIJKN ABCIJLN 403143444142 J$30,000403142434041 J$30,000403141423940 F$40,000403140413940 F$40,000403139403940

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.35 Project Network After Crashing

36. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.36 Using LP to Make Crashing Decisions Restatment of the problem: Consider the total cost of the project, including the extra cost of crashing activities. The problem then is to minimize this total cost, subject to the constraint that project duration must be less than or equal to the time desired by the project manager. The decisions to be made are the following: 1.The start time of each activity. 2.The reduction in the duration of each activity due to crashing. 3.The finish time of the project (must not exceed 40 weeks). The constraints are: 1.Time Reduction ≤ Max Reduction (for each activity). 2.Project Finish Time ≤ Desired Finish Time. 3.Activity Start Time ≥ Activity Finish Time of all predecessors (for each activity). 4.Project Finish Time ≥ Finish Time of all immediate predecessors of finish node.

37. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.37 Spreadsheet Model

38. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.38 Mr. Perty’s Conclusions The plan for crashing the project only provides a 50 percent chance of actually finishing the project within 40 weeks, so the extra cost of the plan ($140,000) is not justified. Therefore, Mr. Perty rejects any crashing at this stage. The extra cost of the crashing plan can be justified if it almost certainly would earn the bonus of $150,000 for finishing the project within 40 weeks. Therefore, Mr. Perty will hold the plan in reserve to be implemented if the project is running well ahead of schedule before reaching activity F. The extra cost of part or all of the crashing plan can be easily justified if it likely would make the difference in avoiding the penalty of $300,000 for not finishing the project within 47 weeks. Therefore, Mr. Perty will hold the crashing plan in reserve to be partially or wholly implemented if the project is running far behind schedule before reaching activity F or activity J.

39. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.39 Scheduling and Controlling Project Costs PERT/Cost is a systematic procedure (normally computerized) to help the project manager plan, schedule, and control costs. Assumption: A common assumption when using PERT/Cost is that the costs of performing an activity are incurred at a constant rate throughout its duration.

40. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.40 Budget for Reliable’s Project Activity Estimated Duration (weeks) Estimated Cost Cost per Week of Its Duration A2$180,000$90,000 B4320,00080,000 C10620,00062,000 D6260,00043,333 E4410,000102,500 F5180,00036,000 G7900,000128,571 H9200,00022,222 I7210,00030,000 J8430,00053,750 K4160,00040,000 L5250,00050,000 M2100,00050,000 N6330,00055,000

41. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.41 PERT/Cost Spreadsheet (Earliest Start Times)

42. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.42 PERT/Cost Spreadsheet (Earliest Start Times)

43. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.43 PERT/Cost Spreadsheet (Latest Start Times)

44. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.44 PERT/Cost Spreadsheet (Latest Start Times)

45. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.45 Cumulative Project Costs

46. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.46 PERT/Cost Report after Week 22 Activity Budgeted Cost Percent Completed Value Completed Actual Cost to Date Cost Overrun to Date A$180,000100%$180,000$200,000$20,000 B320,000100320,000330,00010,000 C620,000100620,000600,000–20,000 D260,00075195,000200,0005,000 E410,000100410,000400,000–10,000 F180,0002545,00060,00015,000 I210,00050105,000130,00025,000 Total$2,180,000$1,875,000$1,920,000$45,000

47. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.47 Project Management Characteristics of Projects –Unique, one-time operations –Involve a large number of activities that must be planned and coordinated –Long time-horizon –Goals of meeting completion deadlines and budgets Examples –Building a house –Planning a meeting –Introducing a new product PERT—Project Evaluation and Review Technique CPM—Critical Path Method –A graphical or network approach for planning and coordinating large-scale projects.

48. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.48 Example: Building a House ActivityTime (Days) Immediate Predecessor Foundation4— Framing10Foundation Plumbing9Framing Electrical6Framing Wall Board8Plumbing, Electrical Siding16Framing Paint Interior5Wall Board Paint Exterior9Siding Fixtures6Int. Paint, Ext. Paint

49. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.49 Gantt Chart

50. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.50 Gantt Chart

51. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.51 PERT and CPM Procedure 1.Determine the sequence of activities. 2.Construct the network or precedence diagram. 3.Starting from the left, compute the Early Start (ES) and Early Finish (EF) time for each activity. 4.Starting from the right, compute the Late Finish (LF) and Late Start (LS) time for each activity. 5.Find the slack for each activity. 6.Identify the Critical Path.

52. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.52 Notation tDuration of an activity ESThe earliest time an activity can start EFThe earliest time an activity can finish (EF = ES + t) LSThe latest time an activity can start and not delay the project LFThe latest time an activity can finish and not delay the project SlackThe extra time that could be made available to an activity without delaying the project (Slack = LS – ES) Critical PathThe sequence(s) of activities with no slack

53. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.53 PERT/CPM Project Network

54. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.54 Calculation of ES, EF, LF, LS, and Slack ES = Maximum of EF’s for all predecessors EF = ES + t LF = Minimum of ES for all successors LS = LF – t Slack = LS – ES = LF – EF

55. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.55 Building a House: ES, EF, LS, LF, Slack ActivityESEFLSLFSlack (a) Foundation04040 (b) Framing4144 0 (c) Plumbing142317263 (d) Electrical1420 266 (e) Wall Board233126343 (f) Siding143014300 (g) Paint Interior313634393 (h) Paint Exterior303930390 (i) Fixtures394539450

56. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.56 PERT/CPM Project Network

57. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.57 Example #2: ES, EF, LS, LF, Slack ActivityESEFLSLFSlack a04040 b04151 c47581 d48480 e4125131 f4116132 g8 8 0 h81110132 i 1813180 j111613182

58. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.58 PERT with Uncertain Activity Durations If the activity times are not known with certainty, PERT/CPM can be used to calculate the probability that the project will complete by time t. For each activity, make three time estimates: –Optimistic time: o –Pessimistic time: p –Most-likely time: m

59. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.59 Beta Distribution Assumption: The variability of the time estimates follows the beta distribution.

60. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.60 PERT with Uncertain Activity Durations Goal: Calculate the probability that the project is completed by time t. Procedure: 1.Calculate the expected duration and variance for each activity. 2.Calculate the expected length of each path. Determine which path is the mean critical path. 3.Calculate the standard deviation of the mean critical path. 4.Find the probability that the mean critical path completes by time t.

61. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.61 Expected Duration and Variance for Activities (Step #1) The expected duration of each activity can be approximated as follows: The variance of the duration for each activity can be approximated as follows:

62. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.62 Expected Length of Each Path (Step #2) The expected length of each path is equal to the sum of the expected durations of all the activities on each path. The mean critical path is the path with the longest expected length.

63. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.63 Standard Deviation of Mean Critical Path (Step #3) The variance of the length of the path is the sum of the variances of all the activities on the path.  2 path = ∑ all activities on path  2 The standard deviation of the length of the path is the square root of the variance.

64. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.64 Probability Mean-Critical Path Completes by t (Step #4) What is the probability that the mean critical path (with expected length t path and standard deviation  path ) has duration ≤ t? Use Normal Tables (Appendix A)

65. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.65 Example Question: What is the probability that the project will be finished by day 12?

66. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.66 Expected Duration and Variance of Activities (Step #1) Activityomp a2343.00 1/91/9 b2453.83 1/41/4 c1373.331 d3464.17 1/41/4 e2383.671

67. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.67 Expected Length of Each Path (Step #2) PathExpected Length of Path a - b - d3.00 + 3.83 + 4.17 = 11 a - c - e3.00 + 3.33 + 3.67 = 10 The mean-critical path is a - b - d.

68. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.68 Standard Deviation of Mean-Critical Path (Step #3) The variance of the length of the path is the sum of the variances of all the activities on the path.  2 path = ∑ all activities on path  2 = 1 / 9 + 1 / 4 + 1 / 4 = 0.61 The standard deviation of the length of the path is the square root of the variance.

69. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.69 Probability Mean-Critical Path Completes by t=12 (Step #4) The probability that the mean critical path (with expected length 11 and standard deviation  has duration ≤ 12? Then, from Normal Table: Prob(Project ≤ 12) = Prob(z ≤ 1.41) = 0.92

70. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.70 Time-Cost Tradeoffs It is often possible to expedite activities at extra cost (called “crashing”). Benefits of completing a project earlier: –Monetary incentives for timely or early completion of project –Beating the competition to the market –Reduce indirect cost Procedure: 1.Obtain estimates of regular and crash times and costs for each activity. 2.Determine the lengths of all paths using regular time for each activity. 3.Identify the critical path(s) 4.Crash activities on the critical path(s) in order of increasing cost as long as crashing costs do not exceed benefits.

71. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.71 Time-Cost Trade-Off Data for House Project Time (days) Cost Maximum Reduction in Time (days) Crash Cost per Day Saved ActivityNormalCrashNormalCrash (a) Foundation44$8,000 0— (b) Framing1089,0009,8002$400 (c) Plumbing975,6006,0002200 (d) Electrical648,4008,8002200 (e) Wall Board8611,50012,7002600 (f) Siding161213,40015,6004300 (g) Paint Interior523,0005,7003900 (h) Paint Exterior975,0006,4002700 (i) Fixtures756,5008,90021,200

72. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.72 Building a House Project Network Question: Suppose $1,000 in indirect costs can be saved for each day the project is shortened. Which activities should be crashed?

73. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.73 Marginal Cost Analysis for House Project Initial Table Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945

74. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.74 Marginal Cost Analysis for House Project After Crashing 1 Day Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944

75. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.75 Marginal Cost Analysis for House Project After Crashing 2 Days Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944 (f) Siding$300423943

76. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.76 Marginal Cost Analysis for House Project After Crashing 3 Days Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944 (f) Siding$300423943 (f) Siding$300423942

77. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.77 Marginal Cost Analysis for House Project After Crashing 4 Days Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944 (f) Siding$300423943 (f) Siding$300423942 (b) Framing$400413841

78. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.78 Marginal Cost Analysis for House Project After Crashing 5 Days Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944 (f) Siding$300423943 (f) Siding$300423942 (b) Framing$400413841 (b) Framing$400403740

79. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.79 Marginal Cost Analysis for House Project After Crashing 6 Days Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944 (f) Siding$300423943 (f) Siding$300423942 (b) Framing$400413841 (b) Framing$400403740 (c&f) Plumbing & Siding$500393739

80. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.80 Marginal Cost Analysis for House Project Final Table (After Crashing 7 Days) Length of Path Activity to Crash Crash Costabcegiabdegiabfhi 423945 (f) Siding$300423944 (f) Siding$300423943 (f) Siding$300423942 (b) Framing$400413841 (b) Framing$400403740 (c&f) Plumbing & Siding$500393739 (c&h) Plumbing & Paint Ext.$900383738

81. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.81 Marginal Cost Analysis for House Project Seven days have been saved off the length of the project (38 days rather than 45). –$7,000 saved in indirect costs. The total crashing costs were –$300 + $300 + $300 + $400 + $500 + $900 = $2,700 Total savings = $7,000 – $2,700 = $5,300.