MCPA#4

MGT 6304, Managing Complex Projects 1

Course Learning Outcomes for Unit IV Upon completion of this unit, students should be able to:

5. Employ program evaluation and review technique (PERT) methodology to evaluate project schedule completion probability. 5.1 Develop a PERT schedule. 5.2 Identify the project critical path. 5.3 Perform a Monte Carlo analysis on a project schedule.

Course/Unit Learning Outcomes

Learning Activity

5.1 Unit Lesson Chapter 7 Unit IV Assignment

5.2

Unit Lesson Chapter 7 Chapter 7 Supplement Unit IV Assignment

5.3 Unit Lesson Chapter 7 Supplement Unit IV Assignment

Required Unit Resources Chapter 7: Scheduling Stochastic Projects Chapter 7 Supplement: Monte Carlo Simulation

Unit Lesson

Complex Schedule Analysis We observed in the breakfast schedule example from the previous unit that a schedule may be both apparently simple as well as complex at the same time. In the breakfast example, the complex element of the schedule was the fact that the critical path—and overall project duration—could not be determined by inspection. It took critical path analysis in the form of the network diagram and the forward and backward pass algorithm to determine this. The schedule for breakfast was inherently simplified, however, by assuming that the duration of each activity in the breakfast-making process is fixed and known with certainty. However, it is almost never the case that each activity duration in a schedule is known with certainty. When highly specific duration estimates are assigned to each activity, the schedule that results is not likely to be accurate. Fortunately, there are techniques that account for the natural duration variability associated with activity durations. Such techniques also open the door to making probabilistic estimates of the overall schedule. Probabilistic schedules are an essential component of risk planning for the project scope and budget.

Program Evaluation and Review Technique (PERT) and the Three-Point Estimate The starting point for incorporating duration variability into activities is by means of the three-point estimate. Instead of assigning a fixed duration to each activity, three duration estimates are used. These are the best- case estimate, the worst-case estimate, and the most-likely estimate. This can be demonstrated by returning

UNIT IV STUDY GUIDE

Complex Schedule Analysis

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to the breakfast schedule, using the previous duration estimate as the most-likely estimate and then adding a best-case and worst-case estimate as follows:

Number Activity Predecessor Best-Case Likely Worst-Case

1 Retrieve ingredients from refrigerator.

--- .25 .5 1.5

2 Place egg and bacon pan on stove.

1 .15 .5 1.0

3 Place hash brown pan on stove.

2 .1 .5 .75

4 Place toast in toaster.

7 .5 1 1.5

5 Cook eggs. 7 3 5 8

6 Cook bacon. 7 2 5 9

7 Cook hash browns.

3 4 10 15

8 Brew coffee. 1 8 10 17

9 Make toast. 4 3 4 7

10 Pour orange juice.

7,12 .5 1 1.5

11 Pour coffee. 7 .15 1 1.5

12 Plate breakfast food.

7 1 2 3

13 Serve coffee. 12, 8 .25 1 1.5

14 Serve breakfast. 13 .5 1 2.5

Once the three-point estimates are in place, there are two fundamental approaches to using them to estimate the overall project duration. The first, known as PERT (program evaluation and review technique) uses a form of weighted average of the three-point estimates to create a single duration estimate for each activity that incorporates each of the three points. The weighted average uses the beta distribution and uses the following formula:

Worst-case + (4 times Most likely) + Best-case 6

The theory behind the use of the beta distribution is not important for this analysis. All that is required is the ability to use this simple formula for all project activities. Each is calculated as follows:

Number Activity Predecessor Best-Case Likely Worst-Case Beta Result

1 Retrieve ingredients from refrigerator.

--- .25 .5 1.5 0.63

2 Place egg and bacon pan on stove.

1 .15 .5 1.0 .53

3 Place hash brown pan on stove.

2 .1 .5 .75 .48

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4 Place toast in toaster.

7 .5 1 1.5 1.00

5 Cook eggs. 7 3 5 8 5.17

6 Cook bacon. 7 2 5 9 5.17

7 Cook hash browns.

3 4 10 15 9.83

8 Brew coffee. 1 8 10 17 10.83

9 Make toast. 4 3 4 7 4.33

10 Pour orange juice.

7,12 .5 1 1.5 1

11 Pour coffee. 7 .15 1 1.5 .94

12 Plate breakfast food.

7 1 2 3 2

13 Serve coffee.

12, 8 .25 1 1.5 .96

14 Serve breakfast.

13 .5 1 2.5 1.17

Once the activity durations are calculated using the beta distribution, they are then used to populate the network diagram with the revised durations, and the forward and backward pass is carried out once again. Note that the overall project duration slightly changes, and the critical path shifts as observed in the revised network diagram.

In this example, the changes to the schedule appear to be slight. It begs the question, “What’s the point?” Given that making breakfast is not normally thought of as involving a high-precision schedule. First, while this schedule changes only slightly, an actual project schedule may change quite a bit more. There is a second major benefit to using weighted averages for activity durations, and this is because the use of averages provides a way to calculate the probability of achieving a project target date.

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The Mean, the Standard Deviation, and Probability The schedule end time is 20.93 minutes. Since this duration is now the sum of multiple weighted averages, it is also an average or mean. In statistics, the mean is the number located at the center of the normal curve.

This position is the 50% point, halfway between the beginning and the end of the curve. The normal curve is used to calculate probability in a way that will become clear in the following example. However, because the duration as calculated is the result of a sum of weighted averages, or means and is a mean itself, its position within the normal curve is at the 50% probability mark. This is an important observation: any schedule calculated using the PERT methodology will initially have a 50% probability of achieving the PERT schedule date because of its position as a mean on the normal curve. Most project sponsors and clients will not be pleased with the prospect of a project schedule that has only a 50% chance of meeting the due date. Most would find the 95% date to be a better option. But, what would be the 95% due date? This is a duration found at approximately 1.65 standard deviations above the mean. So then, all we need to do is add 1.65 standard deviation’s worth of time to the PERT schedule to have a schedule with a 95% probability of achievement. This requires calculating the project standard deviation, and once the PERT duration (mean) and standard deviation are known then it is a simple matter of calculating the 95% probability schedule. The good news is that calculating the project standard deviation is a simple matter using the following steps:

1. Find the variance of each task on the critical path (Subtract the best-case duration from the worst- case duration, divide by 6, and square it:

〖((𝑊𝐶−𝐵𝐶)/6))]^2 2. Add the variances of each activity on the critical path: (𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒1+𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒2+….) 3. Take the square root of the sum of the variances: √((𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒1+𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒2…))

Number Activity Predecessor Best Case Likely Worst- Case

Variance Number

1 Retrieve ingredients from refrigerator.

--- .25 .5 1.5 0.04 1

2 Place egg and bacon pan on stove.

1 .15 .5 1.0 0.02 2

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3 Place hash brown pan on stove.

2 .1 .5 .75 0.01 3

4 Place toast in toaster.

7 .5 1 1.5 0.03 4

5 Cook eggs. 7 3 5 8 0.69 5

6 Cook bacon.

7 2 5 9 1.36 6

7 Cook hash browns.

3 4 10 15 3.36 7

8 Brew coffee.

1 8 10 17 2.25 8

9 Make toast. 4 3 4 7 0.44 9

10 Pour orange juice.

7,12 .5 1 1.5 0.03 10

11 Pour coffee.

7 .15 1 1.5 0.05 11

12 Plate breakfast food.

7 1 2 3 0.11 12

13 Serve coffee.

12, 8 .25 1 1.5 0.04 13

14 Serve breakfast.

13 .5 1 2.5 0.11 14

The sum of the variances is 8.56. The square root of the variances is 2.93. Since the PERT project duration is 20.93 minutes and the project standard deviation is 2.93 minutes, then the 95% probability schedule is: 20.93 + (1.65*2.93) = 25.76 minutes. What does this mean in practice? It means that it is far more likely that breakfast will be ready in about 26 minutes rather than 21 minutes!

Another Approach: Monte Carlo Analysis Instead of using a special weighted average of three-point estimates using PERT, computers make it possible today to quickly and systematically run thousands of project trials with each trial run picking up one of the three-point estimates from each activity. The diagram illustrates a simple sequence of seven activities, each with three- point activity estimates. In one run, the computer picks the best-case for activity 1, the most-likely case for activity 2, and continues through the sequence picking three-point estimates at random. After thousands of runs, the most likely schedule duration will become obvious by observing the resulting statistical distribution.

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Breakfast Schedule Monte Carlo Simulation Chapter 7 and the Chapter 7 Supplement of the textbook explains how to create a Monte Carlo simulation using Excel. The breakfast schedule was modeled following the steps described in the supplement as follows: Schedule With Dependencies

Task Duration ES EF LS LF TS Critical Best Likely Worst

Start 0

1

Retrieve ingredients from refrigerator. 0.48 0 0.48 0.00 0.48 0.00

0.25 0.5 1.5

2

Place egg and bacon pan on stove. 0.56 0.48 1.04 0.48 1.04 0.00 TRUE

0.15 0.5 1

3

Place hash brown pan on stove. 0.42 1.04 1.46 1.04 1.46 0.00 TRUE

0.1 0.5 0.75

4 Place toast in toaster. 1.10 11.01 12.11 11.01 12.11 0.00 TRUE

0.5 1 1.5

5 Cook eggs. 4.76 11.01 15.77 11.60 16.36 0.59 FALSE 3 5 8

6 Cook bacon. 4.10 11.01 15.11 12.25 16.36 1.25 FALSE

2 5 9

7 Cook hash browns. 9.55 1.46 11.01 1.46 11.01 0.00 TRUE

4 10 15

8 Brew coffee. 9.85 0.48 10.33 8.20 18.05 7.72 FALSE

8 10 17

9 Make toast. 4.25 12.11 16.36 12.11 16.36 0.00 TRUE 3 4 7

10

Pour orange juice. 0.83 11.01 11.84 17.22 18.05 6.22 FALSE

0.5 1 1.5

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11 Pour coffee. 0.51 11.01 11.52 17.54 18.05 6.53 FALSE

0.15 1 1.5

12

Plate breakfast food. 1.69 16.36 18.05 16.36 18.05 0.00 TRUE

1 2 3

13 Serve coffee. 1.25 18.05 19.31 18.05 19.31 0.00 TRUE

0.25 1 1.5

14 Serve breakfast. 1.03 19.31 20.33 19.31 20.33 0.00 TRUE

0.5 1 2.5

Beta Distribution Duration Generator

Duration Variance Alpha Beta Random

0 0 0 0

0.63 0.04 1.97 4.59 0.479324

0.53 0.02 3.47 4.40 0.556425

0.48 0.01 4.49 3.29 0.42139

1.00 0.03 4.00 4.00 1.104018

5.17 0.69 3.40 4.44 4.762679

5.17 1.36 3.58 4.34 4.103697

9.83 3.36 4.22 3.74 9.551932

10.83 2.25 2.13 4.64 9.847528

4.33 0.44 2.33 4.67 4.245059

1.00 0.03 4.00 4.00 0.826443

0.94 0.05 4.53 3.20 0.51191

2.00 0.11 4.00 4.00 1.692862

0.96 0.04 4.44 3.40 1.254745

1.17 0.11 2.33 4.67 1.027765

Critical Path Identifier

Path Length

Start-1-2-3-7-4-9-12-13- 14 20.33

Start-1-2-3-7-5-12-13-14 19.75

Start-1-8-13-14 12.61

Start-1-2-3-7-6-12,13,14 19.09

Start-1-2-3-7-10-13-14 14.12

Start-1-2-3-7-11-13-14 13.80

Critical Path # 1

Critical Path Start-1-2-3-7-4-9-12-13- 14

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Trial Runs

Trial Makespan Critical Path

20.33 1

1 23.33879 4

2 21.12358 4

3 23.58932 4

4 19.86953 2

5 25.00882 4

6 20.32944 2

7 21.21638 4

8 26.99103 4

9 21.31321 4

10 23.83108 4

Graph of Trial Run Durations

For the purpose of this example, only 10 runs were performed, but a Monte Carlo simulation often uses thousands of runs so that the high probability schedule duration may be easily observed. The spreadsheet was created according to the textbook in Chapter 7 and will be made available in this unit’s assignment so that it may be used as a guide.

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10

Schedule Duration