# PERT/CPM Models for Project Management

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

Project Management Characteristics of Projects Examples
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. Slides 8.47–8.57 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). It covers an introduction to project management, calculating the ES, EF, LS, LF, and slack, and determining the critical path. © The McGraw-Hill Companies, Inc., 2003

Example: Building a House
Activity Time (Days) Immediate Predecessor Foundation 4 Framing 10 Plumbing 9 Electrical 6 Wall Board 8 Plumbing, Electrical Siding 16 Paint Interior 5 Paint Exterior Fixtures Int. Paint, Ext. Paint © The McGraw-Hill Companies, Inc., 2003

Gantt Chart Gantt Chart—a tool for planning and scheduling simple projects. © The McGraw-Hill Companies, Inc., 2003

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

Notation t Duration of an activity
ES The earliest time an activity can start EF The earliest time an activity can finish (EF = ES + t) LS The latest time an activity can start and not delay the project LF The latest time an activity can finish and not delay the project Slack The extra time that could be made available to an activity without delaying the project (Slack = LS – ES) Critical Path The sequence(s) of activities with no slack © The McGraw-Hill Companies, Inc., 2003

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

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

Building a House: ES, EF, LS, LF, Slack
Activity ES EF LS LF Slack (a) Foundation 4 (b) Framing 14 (c) Plumbing 23 17 26 3 (d) Electrical 20 6 (e) Wall Board 31 34 (f) Siding 30 (g) Paint Interior 36 39 (h) Paint Exterior (i) Fixtures 45 © The McGraw-Hill Companies, Inc., 2003

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

Example #2: ES, EF, LS, LF, Slack
Activity ES EF LS LF Slack a 4 b 1 5 c 7 8 d e 12 13 f 11 6 2 g h 10 i 18 j 16 © The McGraw-Hill Companies, Inc., 2003

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: How can the project be displayed graphically to better visualize the activities? What is the total time required to complete the project if no delays occur? When do the individual activities need to start and finish? What are the critical bottleneck activities? For other activities, how much delay can be tolerated? What is the probability the project can be completed in 47 weeks? What is the least expensive way to complete the project within 40 weeks? How should ongoing costs be monitored to try to keep the project within budget? © The McGraw-Hill Companies, Inc., 2003

Activity List for Reliable Construction
Activity Description Immediate Predecessors Estimated Duration (Weeks) A Excavate 2 B Lay the foundation 4 C Put up the rough wall 10 D Put up the roof 6 E Install the exterior plumbing F Install the interior plumbing 5 G Put up the exterior siding 7 H Do the exterior painting E, G 9 I Do the electrical work J Put up the wallboard F, I 8 K Install the flooring L Do the interior painting M Install the exterior fixtures N Install the interior fixtures K, L Table 8.1 Activity list for the Reliable Construction Co. project. © The McGraw-Hill Companies, Inc., 2003

Reliable Construction Project Network
Figure 8.1 The project network for the Reliable Construction Co. project. © The McGraw-Hill Companies, Inc., 2003

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.) © The McGraw-Hill Companies, Inc., 2003

The Paths for Reliable’s Project Network
Length (Weeks) StartA B C D G H M Finish = 40 Start A B C E H M Finish = 31 Start A B C E F J K N Finish = 43 Start A B C E F J L N Finish = 44 Start A B C I J K N Finish = 41 Start A B C I J L N Finish = 42 Table 8.2 The paths and path lengths through Reliable’s Project Network. © The McGraw-Hill Companies, Inc., 2003

ES and EF Values for Reliable Construction for Activities that have only a Single Predecessor
Figure 8.4 Earliest start time (ES) and earliest finish time (EF) values for the initial activities in Figure 8.1 that have only a single immediate predecessor. © The McGraw-Hill Companies, Inc., 2003

ES and EF Times for Reliable Construction
Figure 8.5 Earliest start time (ES) and earliest finish time (EF) values for all the activities (plus the start and finish nodes) of the Reliable Construction Co. project. © The McGraw-Hill Companies, Inc., 2003

LS and LF Times for Reliable’s Project
Figure 8.6 Latest start time (LS) and latest finish time (LF) for all the activities (plus the start and finish nodes) of the Reliable Construction Co. project. © The McGraw-Hill Companies, Inc., 2003

The Complete Project Network
Figure 8.7 The complete project network showing ES and LS (in the upper parentheses next to the node) and EF and LF (in the lower parentheses next to the node) for each activity of the Reliable Construction Co. project. The darker arrows show the critical path through the project network. © The McGraw-Hill Companies, Inc., 2003

Slack for Reliable’s Activities
Activity Slack (LF–EF) On Critical Path? A Yes B C D 4 No E F G H I 2 J K 1 L M N Table 8.3 Slack for Reliable’s activities. © The McGraw-Hill Companies, Inc., 2003

Spreadsheet to Calculate ES, EF, LS, LF, Slack
Figure 8.8 A spreadsheet to calculate the ES, EF, LS, LF, slack, and whether or not it is critical, for each activity in Reliable Construction Co.’s project network. © The McGraw-Hill Companies, Inc., 2003

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 Slides 8.58–8.69 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). It covers PERT with uncertain activity durations. © The McGraw-Hill Companies, Inc., 2003

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

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

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: © The McGraw-Hill Companies, Inc., 2003

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. © The McGraw-Hill Companies, Inc., 2003

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. s2path = ∑ all activities on path s2 The standard deviation of the length of the path is the square root of the variance. © The McGraw-Hill Companies, Inc., 2003

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

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

Expected Duration and Variance of Activities (Step #1)
Activity o m p a 2 3 4 3.00 1/9 b 5 3.83 1/4 c 1 7 3.33 d 6 4.17 e 8 3.67 © The McGraw-Hill Companies, Inc., 2003

Expected Length of Each Path (Step #2)
Expected Length of Path a - b - d = 11 a - c - e = 10 The mean-critical path is a - b - d. © The McGraw-Hill Companies, Inc., 2003

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. s2path = ∑ all activities on path s2 = 1/9 + 1/4 + 1/4 = 0.61 The standard deviation of the length of the path is the square root of the variance. © The McGraw-Hill Companies, Inc., 2003

Probability Mean-Critical Path Completes by t=12 (Step #4)
The probability that the mean critical path (with expected length 11 and standard deviation 0.71) has duration ≤ 12? Then, from Normal Table: Prob(Project ≤ 12) = Prob(z ≤ 1.41) = 0.92 © The McGraw-Hill Companies, Inc., 2003

Reliable Construction Project Network
Figure 8.1 The project network for the Reliable Construction Co. project. © The McGraw-Hill Companies, Inc., 2003

Reliable Problem: Time Estimates for Reliable’s Project
Activity o m p Mean Variance A 1 2 3 1/9 B 3.5 8 4 C 6 9 18 10 D 5.5 E 4.5 5 4/9 F G 6.5 11 7 H 17 I 7.5 J K L M N Table 8.4 Expected value and variance of the duration of each activity for Reliable’s project © The McGraw-Hill Companies, Inc., 2003

Pessimistic Path Lengths for Reliable’s Project
Pessimistic Length (Weeks) StartA B C D G H M Finish = 70 Start A B C E H M Finish = 54 Start A B C E F J K N Finish = 66 Start A B C E F J L N Finish = 69 Start A B C I J K N Finish = 60 Start A B C I J L N Finish = 63 Table 8.5 The paths and path lengths through Reliable’s project network when the duration of each activity equals its pessimistic estimate. © The McGraw-Hill Companies, Inc., 2003

Three Simplifying Approximations of PERT/CPM
The mean critical path will turn out to be the longest path through the project network. 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. 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. © The McGraw-Hill Companies, Inc., 2003

Calculation of Project Mean and Variance
Activities on Mean Critical Path Mean Variance A 2 1/9 B 4 1 C 10 E 4/9 F 5 J 8 L N 6 Project duration mp = 44 s2p = 9 Table 8.6 Calculation of mp and s2p for Reliable’s project. © The McGraw-Hill Companies, Inc., 2003

Spreadsheet for PERT Three-Estimate Approach
Figure This Excel template in your MS Courseware enables efficient application of the PERT three-estimate approach, as illustrated here for Reliable’s project. © The McGraw-Hill Companies, Inc., 2003

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