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Irwin/McGraw-Hill © The McGraw-Hill Companies, Inc., 2003 8.1 PERT/CPM Models for Project Management

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.2 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 PERTProject Evaluation and Review Technique CPMCritical Path Method –A graphical or network approach for planning and coordinating large-scale projects.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.3 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.4 Gantt Chart

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.5 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.6 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.7 PERT/CPM Project Network

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.8 Calculation of ES, EF, LF, LS, and Slack GOING FORWARD ES = Maximum of EFs for all predecessors EF = ES + t GOING BACKWARD LF = Minimum of LS for all successors LS = LF – t Slack = LS – ES = LF – EF

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.9 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.10 PERT/CPM Project Network

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.11 Example #2: ES, EF, LS, LF, Slack ActivityESEFLSLFSlack a04040 b04151 c47581 d48480 e4125131 f4116132 g8 8 0 h81110132 i 1813180 j111613182

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.12 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?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.13 Activity List for Reliable Construction ActivityActivity Description Immediate Predecessors Estimated Duration (Weeks) AExcavate2 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.14 Reliable Construction Project Network

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.15 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.)

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.16 The Paths for Reliables 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.17 ES and EF Values for Reliable Construction for Activities that have only a Single Predecessor

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.18 ES and EF Times for Reliable Construction

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.19 LS and LF Times for Reliables Project

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.20 The Complete Project Network

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.21 Slack for Reliables Activities ActivitySlack (LF–EF)On Critical Path? A0Yes B0 C0 D4No E0Yes F0 G4No H4 I2 J0Yes K1No L0Yes M4No N0Yes

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.22 Spreadsheet to Calculate ES, EF, LS, LF, Slack

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.23 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.24 Beta Distribution Assumption: The variability of the time estimates follows the beta distribution.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.25 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.26 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:

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.27 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.28 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.29 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)

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.30 Example Question: What is the probability that the project will be finished by day 12?

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.31 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.32 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.33 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.34 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.35 Reliable Construction Project Network

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.36 Reliable Problem: Time Estimates for Reliables 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.37 Pessimistic Path Lengths for Reliables 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.38 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.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.39 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

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 8.40 Spreadsheet for PERT Three-Estimate Approach

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