1. 8-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Project Management Chapter 8

2. 8-2 ■The Elements of Project Management ■CPM/PERT Networks ■Probabilistic Activity Times ■Formulating the CPM/PERT Network as a Linear Programming Model Chapter Topics Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3. 8-3 ■Network representation is useful for project analysis. ■Project examples: Construction of a building, organization of a conference, installation of a computer system etc. ■Networks show how project activities are organized and are used to determine time duration of projects. ■Network techniques used are: ▪ CPM (Critical Path Method) ▪ PERT (Project Evaluation and Review Technique) ■Developed independently during late 1950’s. Overview Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4. 8-4 Elements of Project Management ■Management is generally perceived as concerned with planning, organizing, and control of an ongoing process or activity. ■Project Management is concerned with control of an activity for a relatively short period of time after which management effort ends. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

5. 8-5 Project Planning ■Objectives ■Project Scope ■Contract Requirements ■Schedules ■Resources ■Control ■Risk and Problem Analysis Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

6. 8-6 Project Planning ■All activities (steps) of the project should be identified. ■The sequential relationships of the activities (which activity comes first, which follows, etc.) is identified by precedence relationships. ■Steps of project planning: ■Make time estimates for activities, determine project completion time. ■Compare project schedule objectives, determine resource requirements. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7. 8-7 Elements of Project Management Project Scheduling ■Project Schedule : Timely completion of project. ■Schedule development steps: 1. Define activities, 2. Sequence activities, 3. Estimate activity times, 4. Construct schedule. ■Gantt chart and CPM/PERT techniques can be useful. ■Computer software packages available, e.g. Microsoft Project. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

8. 8-8 Elements of Project Management Gantt Chart (1 of 2) ■Popular, traditional technique, also known as a bar chart -developed by Henry Gantt (1914). ■Used in CPM/PERT for monitoring work progress. ■A visual display of project schedule showing activity start and finish times and where extra time is available. ■Suitable for projects with few activities and precedence relationships. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

9. 8-9 Elements of Project Management Gantt Chart (2 of 2) Figure 8.4 A Gantt chart Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

10. 8-10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ■A branch reflects an activity of a project. ■A node represents the beginning and end of activities, referred to as events. ■Branches in the network indicate precedence relationships. ■When an activity is completed at a node, it has been realized. The Project Network CPM/PERT Activity-on-Arc (AOA) Network

11. 8-11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ■Time duration of activities shown on branches. ■Activities can occur at the same time (concurrently). ■A dummy activity shows a precedence relationship but reflects no passage of time. ■Two or more activities cannot share the same start and end nodes. The Project Network Concurrent Activities Figure 8. 7 A Dummy Activity

12. 8-12 The Project Network House Building Project Data No. ActivityActivity Predecessor Duration (Months) 1. Design house and - 3 obtain financing 2. Lay foundation 1 2 3. Order Materials 1 1 4. Build house 2, 3 3 5. Select paint 2, 3 1 6. Select carpet 5 1 7. Finish work 4, 6 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

13. 8-13 The Project Network AOA Network for House Building Project Figure 8.6 Expanded Network for Building a House Showing Concurrent Activities Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

14. 8-14 The Project Network AON Network for House Building Project Activity-on-Node (AON) Network  A node represents an activity, with its label and time shown on the node  The branches show the precedence relationships  Convention used in Microsoft Project software Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Figure 8.8 Label Duration

15. 8-15 The Project Network Paths Through a Network Table 8.1 Paths Through the House-Building Network Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall PathEvents A12471247 B1256712567 C13471347 D1356713567

16. 8-16 The critical path is the longest path through the network; the minimum time the network can be completed. From Figure 8.8: Path A: 1  2  4  73 + 2 + 3 + 1 = 9 months Path B: 1  2  5  6  73 + 2 + 1 + 1 + 1= 8 months Path C: 1  3  4  73 + 1 + 3 + 1 = 8 months Path D: 1  3  5  6  73 + 1 + 1 + 1 + 1 = 7 months The Project Network The Critical Path Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

17. 8-17 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The Project Network Activity Start Times Figure 8.9 Activity start time

18. 8-18 The Project Network Activity-on-Node Configuration Figure 8.10 Activity-on-Node Configuration Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

19. 8-19 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ■ES is the earliest time an activity can start: ES = Maximum (EF) ■EF is the earliest start time plus the activity time: EF = ES + t The Project Network Activity Scheduling : Earliest Times Figure 8.11 Earliest activity start and finish times

20. 8-20 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ■LS is the latest time an activity can start without delaying critical path time: LS = LF - t ■LF is the latest finish time. LF = Minimum (LS) The Project Network Activity Scheduling : Latest Times Figure 8.12 Latest activity start and finish times

21. 8-21 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Slack is the amount of time an activity can be delayed without delaying the project: S = LS – ES = LF - EF Slack Time exists for those activities not on the critical path for which the earliest and latest start times are not equal. The Project Network Activity Slack Time (1 of 2) Table 8.2 *Critical path ActivityLSESLFEFSlack, S *100330 *233550 343541 *455880 565761 676871 *788990

22. 8-22 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The Project Network Activity Slack Time (2 of 2) Figure 8.13 Activity slack

23. 8-23 ■Activity time estimates usually cannot be made with certainty. ■PERT used for probabilistic activity times. ■In PERT, three time estimates are used: most likely time (m), the optimistic time (a), and the pessimistic time (b). ■These provide an estimate of the mean and variance of a beta distribution: variance: mean (expected time): Probabilistic Activity Times Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

24. 8-24 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Probabilistic Activity Times Example (1 of 3) Figure 8.14 Network for Installation Order Processing System

25. 8-25 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Probabilistic Activity Times Example (2 of 3) Table 8.3 Activity Time Estimates for Figure 8.14

26. 8-26 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Probabilistic Activity Times Example (3 of 3) Figure 8.15 Earliest and Latest Activity Times

27. 8-27 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall ■Expected project time is the sum of the expected times of the critical path activities. ■Project variance is the sum of the critical path activities’ variances ■The expected project time is assumed to be normally distributed. ■Epected project time (t p ) and variance (v p ) interpreted as the mean (  ) and variance (  2 ) of a normal distribution:  = 25 weeks  2 = 62/9 = 6.9 (weeks) 2 Probabilistic Activity Times Expected Project Time and Variance

28. 8-28 ■Using the normal distribution, probabilities are determined by computing the number of standard deviations (Z) a value is from the mean. ■The Z value is used to find corresponding probability in Table A.1, Appendix A. Probability Analysis of a Project Network (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

29. 8-29 Probability Analysis of a Project Network (2 of 2) Figure 8.16 Normal Distribution of Network Duration Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

30. 8-30 What is the probability that the new order processing system will be ready by 30 weeks? µ = 25 weeks  2 = 6.9  = 2.63 weeks Z = (x-  )/  = (30 -25)/2.63 = 1.90 Z value of 1.90 corresponds to probability of.4713 in Table A.1, Appendix A. Probability of completing project in 30 weeks or less: (.5000 +.4713) =.9713. Probability Analysis of a Project Network Example 1 (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

31. 8-31 Probability Analysis of a Project Network Example 1 (2 of 2) Figure 8.17 Probability the Network Will Be Completed in 30 Weeks or Less Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

32. 8-32 ■A customer will trade elsewhere if the new ordering system is not working within 22 weeks. What is the probability that she will be retained? Z = (22 - 25)/2.63 = -1.14 ■Z value of 1.14 (ignore negative) corresponds to probability of.3729 in Table A.1, appendix A. ■Probability that customer will be retained is (0.5 -.3729) =.1271 Probability Analysis of a Project Network Example 2 (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

33. 8-33 Probability Analysis of a Project Network Example 2 (2 of 2) Figure 8.18 Probability the Network Will Be Completed in 22 Weeks or Less Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

34. 8-34 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Given this network and the data on the following slide, determine the expected project completion time and variance, and the probability that the project will be completed in 28 days or less. Example Problem Problem Statement and Data (1 of 2)

35. 8-35 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Example Problem Problem Statement and Data (2 of 2)

36. 8-36 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Example Problem Solution (1 of 4) Step 1: Compute the expected activity times and variances.

37. 8-37 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Example Problem Solution (2 of 4) Step 2: Determine the earliest and latest activity times & slacks

38. 8-38 Example Problem Solution (3 of 4) Step 3: Identify the critical path and compute expected completion time and variance.  Critical path (activities with no slack): 1  3  5  7  Expected project completion time: t p = 9+5+6+4 = 24 days  Variance: v p = 4 + 4/9 + 4/9 + 1/9 = 5 (days) 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

39. 8-39 Example Problem Solution (4 of 4) Step 4: Determine the Probability That the Project Will be Completed in 28 days or less (µ = 24,  =  5) Z = (x -  )/  = (28 -24)/  5 = 1.79 Corresponding probability from Table A.1, Appendix A, is.4633 and P(x  28) =.4633 +.5 =.9633. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

40. 8-40 ■Project duration can be reduced by assigning more resources to project activities. ■However, doing this increases project cost. ■Decision is based on analysis of trade-off between time and cost. ■Project crashing is a method for shortening project duration by reducing one or more critical activities to a time less than normal activity time. Project Crashing and Time-Cost Trade-Off Overview Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

41. 8-41 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Project Crashing and Time-Cost Trade-Off Example Problem (1 of 5) Figure 8.19 The Project Network for Building a House

42. 8-42 Project Crashing and Time-Cost Trade-Off Example Problem (2 of 5) Figure 8.20 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Crash cost & crash time have a linear relationship:

43. 8-43 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Table 8.4 Project Crashing and Time-Cost Trade-Off Example Problem (3 of 5)

44. 8-44 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Figure 8.21 Network with Normal Activity Times and Weekly Crashing Costs Project Crashing and Time-Cost Trade-Off Example Problem (4 of 5)

45. 8-45 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Figure 8.22 Revised Network with Activity 1 Crashed Project Crashing and Time-Cost Trade-Off Example Problem (5 of 5) As activities are crashed, the critical path may change and several paths may become critical.

46. 8-46 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 8.16 Project Crashing and Time-Cost Trade-Off Project Crashing with QM for Windows

47. 8-47 Project Crashing and Time-Cost Trade-Off General Relationship of Time and Cost (1 of 2) ■Project crashing costs and indirect costs have an inverse relationship. ■Crashing costs are highest when the project is shortened. ■Indirect costs increase as the project duration increases. ■Optimal project time is at minimum point on the total cost curve. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

48. 8-48 Project Crashing and Time-Cost Trade-Off General Relationship of Time and Cost (2 of 2) Figure 8.23 The Time-Cost Trade-Off Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

49. 8-49 General linear programming model with AOA convention: Minimize Z =  x i subject to: x j - x i  t ij for all activities i  j x i, x j  0 Where: x i = earliest event time of node i x j = earliest event time of node j t ij = time of activity i  j The objective is to minimize the project duration (critical path time). The CPM/PERT Network Formulating as a Linear Programming Model i Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

50. 8-50 The CPM/PERT Network Example Problem Formulation and Data (1 of 2) Figure 8.24 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

51. 8-51 Minimize Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 subject to: x 2 - x 1  12 x 3 - x 2  8 x 4 - x 2  4 x 4 - x 3  0 x 5 - x 4  4 x 6 - x 4  12 x 6 - x 5  4 x 7 - x 6  4 x i, x j  0 The CPM/PERT Network Example Problem Formulation and Data (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

52. 8-52 Minimize Z = $400y 12 + 500y 23 + 3000y 24 + 200y 45 + 7000y 46 + 200y 56 + 7000y 67 subject to: y 12  5y 12 + x 2 - x 1  12 x 7  36 y 23  3y 23 + x 3 - x 2  8 x i, y ij ≥ 0 y 24  1y 24 + x 4 - x 2  4 y 34  0y 34 + x 4 - x 3  0 y 45  3y 45 + x 5 - x 4  4 y 46  3y 46 + x 6 - x 4  12 y 56  3y 56 + x 6 - x 5  4 y 67  1x 67 + x 7 - x 6  4 x i = earliest event time of node I x j = earliest event time of node j y ij = amount of time by which activity i  j is crashed Project Crashing with Linear Programming Example Problem – Model Formulation Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Objective is to minimize the cost of crashing

53. 8-53 Minimize Z = $400y 12 + 500y 23 + 3000y 24 + 200y 45 + 7000y 46 + 200y 56 + 7000y 67 subject to: y 12  5 x 2 - x 1  12-y 12 x 7  36 y 23  3x 3 - x 2  8-y 23 x i, y ij ≥ 0 y 24  1 x 4 - x 2  4-y 24 y 34  0x 4 - x 3  0-y 34 y 45  3x 5 - x 4  4-y 45 y 46  3x 6 - x 4  12-y 46 y 56  3x 6 - x 5  4- y 56 y 67  1x 7 - x 6  4-x 67 x i = earliest event time of node I x j = earliest event time of node j y ij = amount of time by which activity i  j is crashed Project Crashing with Linear Programming Example Problem – Model Formulation Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Objective is to minimize the cost of crashing

54. 8-54 Activity onActivityActivity on Node (AON)MeaningArc (AOA) A comes before B, which comes before C (a) A B C BAC A and B must both be completed before C can start (b) A C C B A B B and C cannot begin until A is completed (c) B A C A B C A Comparison of AON and AOA Network Conventions

55. 8-55 Activity onActivityActivity on Node (AON)MeaningArc (AOA) C and D cannot begin until both A and B are completed (d) A B C D B AC D C cannot begin until both A and B are completed; D cannot begin until B is completed. A dummy activity is introduced in AOA (e) CA BD Dummy activity A B C D A Comparison of AON and AOA Network Conventions

56. 8-56 Activity onActivityActivity on Node (AON)MeaningArc (AOA) B and C cannot begin until A is completed. D cannot begin until both B and C are completed. A dummy activity is again introduced in AOA. (f) A C DB AB C D Dummy activity A Comparison of AON and AOA Network Conventions