1. 1 5. M ONTE C ARLO B ASED CPM Objective: To understand how to apply the Monte Carlo based CPM method to planning construction projects that are subject to uncertainty: Summary: Part I: General Principles 5.1 Introduction 5.2 Monte Carlo Sampling 5.3 Application to a Construction Problem

2. 2 Part II: P3 Monte Carlo Related Issues 5.4 The Use of Lead and Lag Times 5.5 Total Float Calculations 5.6 Probabilistic & Conditional Activity Branches 5.7 Duration Distribution Types 5.8 Correlated Activity Durations

3. 3 Part I: General Principles 5.1 I NTRODUCTION Monte Carlo is a statistical method that allows a sample of possible project outcomes to be made. The probability of sampling an outcome is equal to the probability of it occurring in the actual project. Some characteristics: –its accuracy increases with the number of samples made; –it is computationally expensive, but well within the capabilities of today’s desktop computers; –the method is very flexible, allowing many real-world factors to be included in the analysis.

4. 4 It is used in similar situations to PERT: –when there is a lot of uncertainty about activity durations; –when there is a lot to be lost by finishing late (or by missing out on large bonuses for early completion). Its advantages over PERT are: –much more accurate; –much more flexible: measures variance on floats (as well as project duration); measures probability of alternative paths becoming critical; can be extended to take account of correlation between the durations of different activities; can be extended to allow for exclusive activity branching.

5. 5 Fig. 5-1: Observed Frequency Distribution Observed Frequency on Site Activity duration 20212022232425262728293031323334 0 1 2 3 4 5 6 7 5.2 M ONTE C ARLO S AMPLING For each cycle through a Monte Carlo analysis, we need to randomly select a duration for every activity. The probability of selecting a duration should match the likelihood of its occurrence on site: most likely duration T = 27 days Observations from similar past activities can be plotted on a frequency chart Observations from similar past activities can be plotted on a frequency chart

6. 6 Fig. 5-2: Converting a Frequency Distribution to a PDF Frequency Distribution Activity duration 20212022232425262728293031323334 0 1 2 3 4 5 6 7 The frequency distribution can be converted (automatically by the computer) into a probability density function (to give a continuous distribution): –not necessary, but can give more accurate results for larger numbers of observations. For example, if the distribution approximates a Normal distribution, then can calculate: sample mean: sample standard deviation: For example, if the distribution approximates a Normal distribution, then can calculate: sample mean: sample standard deviation: Probability Density Function (pdf)

7. 7 Fig. 5-3: Converting to the Cumulative Distribution (a) frequency to cumulative frequency distribution A good method of sampling from a frequency distribution, or a probability density function, is the inverse transform method (ITM) (automatic by computer). STEP 1: convert the frequency distribution to its cumulative frequency distribution: Frequency Distribution Activity duration 23242526272829 0 1 2 3 4 5 6 7 8 9 Activity duration Cumulative Frequency Distribution 23242526272829 0 1 2 3 4 5 6 7 8 9 Total = 8 observations

8. 8 Fig. 5-3: Converting to the Cumulative Distribution (b) probability density to cumulative probability function Activity duration Probability Density Function 23242526272829 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability of completing within 26 days = shaded area = 0.65 If dealing with a probability density function then convert it to its cumulative probability function: Activity duration Cumulative Probability Function 23242526272829 0 0.5 1.0 Probability of completing within 26 days = height = 0.65

9. 9 STEP 2: generate a uniformly distributed random number between 0.0 and 1.0; Fig. 5-4: Selecting a Duration from the Cumulative Distribution Activity duration Cumulative Frequency Distribution 23242526272829 0 1 2 3 4 5 6 7 8 9 0.01.0 PDF Uniform Distribution * r = 0.81 multiply it by the number of observations; go to the multiplied number on the vertical axis; read across until you hit a cumulative distribution bar and select the corresponding duration R = 0.81 * 8 = 6.48 Selected Activity Duration = 27 days

10. 10 It can be seen that the probability of selecting a duration in this way is proportional to the length of the left-side exposed face of its bar. Eg: Fig. 5-5: Probability of Selecting an Activity Duration Activity duration Cumulative Frequency Distribution 23242526272829 0 1 2 3 4 5 6 7 8 9 2 exposed blocks for 25 days 3 exposed blocks for 26 days 0 exposed blocks for 28 days –the bar for a duration of 26 days has 3 blocks exposed; –the bar for 28 days has 0 blocks exposed; and the number of exposed blocks corresponds to the number of site observations made at that duration (see the left side of Fig 5-3 (a)). – the bar for a duration of 25 days has 2 blocks exposed;

11. 11 5.3 A PPLICATION TO A C ONSTRUCTION P ROBLEM The following will perform 6 Monte Carlo cycles on a very simple project:

12. 12 Fig. 5-6: Example of Monte Carlo CPM Process 2 1 cf d act ‘A’ 58 2 1 cf d act ‘B’ 91113 3 4 2 1 cf d act ‘C’ 51015 3 4 9 8 19 18 11 1 10 act ‘C’ 8 8 19 00 0 11 act ‘B’ 0 0 8 8 00 0 8 act ‘A’dummy 19 Cycle 1: ESEFLSLFTFFFIFd cycle n ESEFLSLFTFFFIFdESEFLSLFTFFFIFd Activity ‘A’Activity ‘B’Activity ‘C’ Mean TF = 0Mean FF = 0 Mean IF = 0Critical Index = 1.0 Mean TF = 1.33Mean FF = 1.33 Mean IF = 1.33Critical Index = 0.67 Mean TF = 3.17Mean FF = 3.17 Mean IF = 3.17Critical Index = 0.33 810808000118198 00010818919111 52050500013518720222155205 000 530505000115165 00055101116666 84080800098171423666158238 000 550505000115165 00055101116666 860808000118198 00058131419666

13. 13 Fig. 5-7: Analysis of Results from Monte Carlo CPM Project duration Sampled Frequency Distribution 17181920212223 0 1 2 16 Mean Project Duration = = 18.8 days Standard Deviation on Duration = = 2.64 days Need More Samples (cycles) Variance on Duration = 6.98 days 2. What is the probability of completing within the deterministic duration (17.75 days)? 3. What is the probability of activity ‘B’ having <=5 days float? A: 6.50 (act A) + 11.25 (act C) = 17.75 days (note, this is less than the mean project duration according to the Monte Carlo analysis) A:  p = 34.5 % (much less than 50%) Questions: 1. What is the deterministic project duration? (use the observed frequencies as not a Normal distribution) A: p = 5/6 = 83.3%

14. 14 Part II: P3 Monte Carlo Related Issues 5.4 T HE U SE OF L EAD AND L AG T IMES P3 Monte Carlo interprets negative delays on links (negative lags in P3 terminology) as zero lag: –so how can we represent situations requiring negative delays? –first of all, we will review what we mean by negative delays, then we will explore possible solutions.

15. 15 Fig. 5-8: Dealing with Delays (b) negative delay (a) simple delay PRECEDENCE DIAGRAMBAR CHART act A dAdA act B dBdB +5 days act B dBdB act A dAdA 5 days act A dAdA act B dBdB - 5 days act B dBdB act A dAdA -5 days Earliest B can start is 5 days after end of A Note: negative delay Earliest B can start is 5 days before end of A Example: delay is for cure concrete. Example: Until the last 5 days of A, both activities need same space.

16. 16 Can we get around the problem by: –reversing the direction of the activity link, and then using a positive delay?

17. 17 Fig. 5-9: Reversing the Direction of the Activity Link PRECEDENCE DIAGRAMBAR CHART act A dAdA act B dBdB - 5 days act B dBdB act A dAdA -5 days Earliest B can start is 5 days before end of A act A dAdA act B dBdB + 5 days Are these equivalent ? act B dBdB act A dAdA -5 days Latest B can start is 5 days before end of A Example: A is dry walling and B is inspection of to be hidden columns. NO !

18. 18 Can we get around the problem by: –using a positive delay from the start of the activity?

19. 19 Fig. 5-10: Measuring the Delay from the Start of the Activity Link PRECEDENCE DIAGRAMBAR CHART act A dAdA act B dBdB - 5 days act B dBdB act A dAdA -5 days Earliest B can start is 5 days before end of A Are these equivalent ? Only if d A is fixed ! act B dBdB Earliest B can start is 5 days before end of A act A dAdA act B dBdB (d A -5) days Using Monte Carlo d A changes ! act A dAdA (d A -5) days

20. 20 For example, if d A is expected to last 14 days: –then the delay from start of activity A to start of B = 14 - 5 = 9 days; –this must be specified before all the Monte Carlo cycles start (it cannot change from cycle to cycle). but in Monte Carlo, the duration is variable from cycle to cycle: –thus, if in one Monte Carlo analysis, activity A was 12 days, then the 9 day delay would allow B to start just 3 days before the end of A !!! act A dAdA act B dBdB (d A -5) days

21. 21 Fig. 5-11: Introducing a Dummy Activity with F-F and S-S Links PRECEDENCE DIAGRAMBAR CHART act A dAdA act B dBdB - 5 days act B dBdB act A dAdA -5 days Earliest B can start is 5 days before end of A Are these equivalent ? act B dBdB Earliest B can start is 5 days before end of A act A dAdA dAdA act B dBdB dummy 5 Yes! as long as dummy cannot be delayed dummy: 5

22. 22 5.5 T OTAL F LOAT C ALCULATIONS P3 Monte Carlo provides several choices for calculating Total Float: –Finish Total Float; –Start Total Float; –Critical Total Float; and –Interruptible Total Float. For most activities, these different types of Total Float will have the same value. Differences occur when there are links from the start of an activity, or to the finish of an activity. What do the different types of Total Float represent, and when should we use each one?

23. 23 Fig. 5-12: Graphical Interpretation of Different Types of Total Float BAR CHART 0 0 22 10 act ‘A’ 0 0 22 10 5 act ‘B’ 20 30 10 act ‘C’ 0 0 8 20 PRECEDENCE DIAGRAM act ‘A’ act ‘C’ P3: Start TF = 0 - 0 = 0 P3: Start TF = 0 - 0 = 0 P3: Finish TF = 22 - 10 = 12 P3: Finish TF = 22 - 10 = 12 Note, activity B must be interrupted 5 days. It cannot start after 0, and cannot finish before 10, yet it only has 5 days of work Note, activity B must be interrupted 5 days. It cannot start after 0, and cannot finish before 10, yet it only has 5 days of work P3: Start TF = 0 P3: Finish TF = 12 Standard TF includes necessary interruptions to the activity (and Finish TF) = 17 Standard TF includes necessary interruptions to the activity (and Finish TF) = 17 act ‘B’ Standard TF = 22 - 0 - 5 = 17 Standard TF = 22 - 0 - 5 = 17

24. 24 Critical Total Float is defined as: smallest of Start Total Float and Finish Total Float –use Start TF when concerned about allowable delays to the start of the activity; –use Finish TF when concerned about allowable delays to the finish of the activity; –use Critical TF when concerned about the worst case for the activity; Interruptible Total Float is defined as: Finish Total Float with: late start time = early start time + Total Float. The Standard Total Float is not available in P3 MC, but it tells us: –the sum of all allowable delays on the activity, including those resulting from forced interruptions (necessary to ensure the logic of the network).

25. 25 5.6 P ROBABILISTIC AND C ONDITIONAL A CTIVITY B RANCHES P3 Monte Carlo provides decision branches: –points in a network where alternative sequences of activities can be performed; –two types: probabilistic - where the choice of branch is random; conditional - where the choice of branch is dependent on whether a task has started (or finished).

26. 26 Fig. 5-13: Decision Branches in a Network (a) probabilistic (random) (b) conditional fail or pass ? inspect HVAC equipment pass: p = 95% next task… make-good installation fail: p = 5% act ‘X’ finished ? position steel in found exc. yes use crane to convey conc to found use concrete pumps no activity ‘X’ (uses crane)

27. 27 Fig. 5-14: Decision Branch Rules (a) decision node must be only predecessor (b) no actual start times for branch activities (b) no SS and SF links from branch activities act ‘B’ x not allowed decision node outcome ‘b’ outcome ‘a’ act ‘A’ act ‘B’ decision node outcome ‘b’ outcome ‘a’ act ‘A’ act ‘B’ A specified actual start time. A specified actual start time. x not allowed x decision node outcome ‘b’ outcome ‘a’ act ‘A’ act ‘B’

28. 28 Fig. 5-15: Multiple Decisions (Trees) pass /fail ? inspect HVAC equipment act ‘B’ pass: p = 80% fail: p = 20% major /minor ? act ‘B’ minor problems: p = 75% act ‘B’ major problems: p = 25% combined probability of major installation problems is: 20% x 25% = 5% combined probability of major installation problems is: 20% x 25% = 5%

29. 29 5.7 D ISTRIBUTION T YPES P3 Monte Carlo provides 4 basic types of activity duration distribution: –Triangular; –Modified Triangular; –Poisson; –Custom.

30. 30 Fig. 5-16: Activity Distribution Types in P3 MC (a) Triangular Distribution act duration probability pessimistic most likely optimistic (b) Modified Triangular Distribution act duration probability pessimistic most likely optimistic Anything sampled within the end X & Y percentiles are read as the limit (thus creating a spike at each end). Anything sampled within the end X & Y percentiles are read as the limit (thus creating a spike at each end). Y% (c) Poisson Distribution act duration probability S = ? P = ? (d) Custom act duration probability A discrete distribution A discrete distribution Define the durations that can occur and their respective probabilities Define the durations that can occur and their respective probabilities X%

31. 31 5.8 C ORRELATED A CTIVITY D URATIONS P3 Monte Carlo allows activity durations to be either completely correlated or completely uncorrelated: –Uncorrelated: the duration of activity ’A’ has no relationship to the duration of activity ‘B’; –Correlated: the duration of activity ’A’ is either dependent on the duration of activity ‘B’ or is dependent on something else that ‘B’ is also dependent on.

32. 32 Fig. 5-17: Types of Correlation Between Two Activity Durations (a) Linear Perfect Correlation duration A duration B x y duration of A is a linear function of duration of B duration of A is a linear function of duration of B (b) Non-Linear Perfect Correlation duration A duration B x y duration of A is a nonlinear function of duration of B duration of A is a nonlinear function of duration of B (c) Linear Perfect Negative Correlation duration A duration B x y as duration of A increases, duration of B decreases as duration of A increases, duration of B decreases (d) Partially Correlated duration A duration B x duration of A is partially correlated to duration of B duration of A is partially correlated to duration of B range of possible values of y for a given x

33. 33 Fig. 5-18: Types of Correlation Between Two Activity Durations cumulative pdf activity ‘A’ duration P3 Monte Carlo correlates two activity durations by using the same random number to generate their durations: Same random number for both activities Same random number for both activities cumulative pdf activity ‘B’ duration

34. 34 Fig. 5-19: Impact of Correlated Activity Durations on Project Duration (a) sequential, correlated activities act Aact B 10+20=30 prob 1020 duration A Correlated 20+30=50 (b) sequential, uncorrelated activities act Aact B 10+20=30 prob 1020 duration A Un- Correlated 10+30=4020+20=40 prob 2030 duration B prob 2030 duration B 20+30=50 30 project duration prob 30 50 project duration prob p=0.5 5040