1 5. m onte c arlo b ased cpm objective: to understand how to apply the monte carlo based cpm method...
TRANSCRIPT
1
5. MONTE CARLO BASED 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 Introduction5.2 Monte Carlo Sampling5.3 Application to a Construction Problem
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
Part I: General Principles
5.1 INTRODUCTION• 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
• 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.
5Fig. 5-1: Observed Frequency Distribution
Observed Frequencyon Site
Activity duration20 2120 22 23 24 25 26 27 28 29 30 31 32 33 3401234567
5.2 MONTE CARLO SAMPLING• 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 likelyduration T = 27 days
Observations from similarpast activities can be plotted
on a frequency chart
Observations from similarpast activities can be plotted
on a frequency chart
6Fig. 5-2: Converting a Frequency Distribution to a PDF
FrequencyDistribution
Activity duration20 2120 22 23 24 25 26 27 28 29 30 31 32 33 3401234567
• 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 distributionapproximates a Normal distribution,then can calculate:
sample mean: sample standarddeviation:
For example, if the distributionapproximates a Normal distribution,then can calculate:
sample mean: sample standarddeviation:
n
dX
1or
2
nn
dXS
Probability DensityFunction (pdf)
7Fig. 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:
FrequencyDistribution
Activity duration23 24 25 26 27 28 29
01234567
89
Activity duration
Cumulative FrequencyDistribution
23 24 25 26 27 28 290123456789 Total = 8 observations
FDCFD
8
Fig. 5-3: Converting to the Cumulative Distribution(b) probability density to cumulative probability function
Activity duration
ProbabilityDensity Function
23 24 25 26 27 28 290.00.10.20.30.40.50.60.70.80.9
Probability ofcompleting within26 days = shadedarea = 0.65
If dealing with a probability density function then convert it to its cumulative probability function:
PDFCPF
Activity duration
Cumulative ProbabilityFunction
23 24 25 26 27 28 290
0.5
1.0
Probability ofcompleting within26 days = height= 0.65
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 FrequencyDistribution
23 24 25 26 27 28 290123456789
0.0 1.0
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 = 6.48Selected ActivityDuration = 27 days
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 DurationActivity duration
Cumulative FrequencyDistribution
23 24 25 26 27 28 290123456789
2 exposed blocksfor 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
5.3 APPLICATION TO A CONSTRUCTION PROBLEM
The following will perform 6 Monte Carlo cycles on a very simple project:
12Fig. 5-6: Example of Monte Carlo CPM Process
21
cf
d
act ‘A’
5 8
21
cf
d
act ‘B’
9 11 13
34
21
cf
d
act ‘C’
5 10 15
34
9
8
19
181 1
1
10
act ‘C’
8
8
19
190 0
0
11
act ‘B’
0
0
8
80 0
0
8 act ‘A’ dummy
19
19
Cycle 1:
ES EF LS LF TF FF IFd
cycle n ES EF LS LF TF FF IFd ES EF LS LF TF FF IFd
Activity ‘A’ Activity ‘B’ Activity ‘C’
Mean TF = 0 Mean FF = 0 Mean IF = 0 Critical Index = 1.0
Mean TF = 1.33 Mean FF = 1.33Mean IF = 1.33 Critical Index = 0.67
Mean TF = 3.17 Mean FF = 3.17 Mean IF = 3.17 Critical Index = 0.33
81 0 8 0 8 0 0 0 11 8 19 8 19 0 0 0 10 8 18 9 19 1 1 1
52 0 5 0 5 0 0 0 13 5 18 7 20 2 2 2 15 5 20 5 20 0 0 0
53 0 5 0 5 0 0 0 11 5 16 5 16 0 0 0 5 5 10 11 16 6 6 6
84 0 8 0 8 0 0 0 9 8 17 14 23 6 6 6 15 8 23 8 23 0 0 0
55 0 5 0 5 0 0 0 11 5 16 5 16 0 0 0 5 5 10 11 16 6 6 6
86 0 8 0 8 0 0 0 11 8 19 8 19 0 0 0 5 8 13 14 19 6 6 6
13
Fig. 5-7: Analysis of Results from Monte Carlo CPM
Project duration
Sampled FrequencyDistribution
17 18 19 20 21 22 23012
16
n
dX Mean Project Duration = = 18.8 days
1
2
n
dXSStandard 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)
4.0
98.6
8.1875.17
z 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
Part II: P3 Monte Carlo Related Issues
5.4 THE USE OF LEAD AND LAG TIMES
• 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.
15Fig. 5-8: Dealing with Delays
(b) negative delay
(a) simple delay
PRECEDENCE DIAGRAM BAR CHART
act AdA
act BdB
+5 days
act BdB
act A
dA
5 days
act AdA
act BdB
-5 days act BdB
act A
dA
-5 days
Earliest B can start is5 days after end of A
Note: negative delay
Earliest B can start is5 days before end of A
Example:delay is for
cure concrete.
Example:Until the last 5 days of
A, both activitiesneed same space.
16
• Can we get around the problem by:– reversing the direction of the activity link, and then
using a positive delay?
17Fig. 5-9: Reversing the Direction of the Activity Link
PRECEDENCE DIAGRAM BAR CHART
act AdA
act BdB
- 5 daysact B
dB
act A
dA
-5 days
Earliest B can start is5 days before end of A
act AdA
act BdB
+ 5 days
Are these equivalent ?
act BdB
act A
dA
-5 days
Latest B can start is5 days before end of A
Example:A is dry walling and B
is inspection ofto be hidden columns.NO !
18
• Can we get around the problem by:– using a positive delay from the start of the
activity?
19Fig. 5-10: Measuring the Delay from the Start of the Activity Link
PRECEDENCE DIAGRAM BAR CHART
act AdA
act BdB
- 5 daysact B
dB
act A
dA
-5 days
Earliest B can start is5 days before end of A
Are these equivalent ?
Only ifdA is fixed !
act BdB
Earliest B can start is5 days before end of A
act AdA
act BdB
(dA-5) days
UsingMonte CarlodA changes !
act A
dA
(dA-5) days
20
• For example, if dA 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 AdA
act BdB
(dA-5) days
21Fig. 5-11: Introducing a Dummy Activity with F-F and S-S Links
PRECEDENCE DIAGRAM BAR CHART
act AdA
act BdB
- 5 daysact B
dB
act A
dA
-5 days
Earliest B can start is5 days before end of A
Are these equivalent ?
act BdB
Earliest B can start is5 days before end of A
act A
dA
act AdA
act BdB
dummy 5
Yes! as long asdummy cannot be
delayeddummy: 5
22
5.5 TOTAL FLOAT CALCULATIONS• 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?
23Fig. 5-12: Graphical Interpretation of Different Types of Total Float
BAR CHART
0
0
22
10
10 act ‘A’
0
0
22
10
5 act ‘B’
20
20
30
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 mustbe interrupted 5 days.
It cannot start after 0, andcannot finish before 10,
yet it only has 5 days of work
Note, activity B mustbe interrupted 5 days.
It cannot start after 0, andcannot finish before 10,
yet it only has 5 days of work
P3: Start TF = 0P3: Start TF = 0
P3: Finish TF = 12P3: Finish TF = 12
Standard TF includesnecessary interruptions to
the activity (and Finish TF) = 17
Standard TF includesnecessary interruptions to
the activity (and Finish TF) = 17
act ‘B’
Standard TF =22 - 0 - 5 = 17
Standard TF =22 - 0 - 5 = 17
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
5.6 PROBABILISTIC AND CONDITIONAL
ACTIVITY BRANCHES
• 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).
26Fig. 5-13: Decision Branches in a Network
(a) probabilistic (random)
(b) conditional
failor pass
?
inspect HVACequipment
pass: p = 95%next task…
make-good installation
fail: p = 5%
act ‘X’finished
?
position steelin found exc.
yes
use craneto convey
conc to found
use concretepumps
no
activity ‘X’ (uses crane)
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’ xnot allowed
decisionnode
outcome ‘b’
outcome ‘a’act ‘A’
act ‘B’
decisionnode
outcome ‘b’
outcome ‘a’act ‘A’
act ‘B’
A specified actualstart time.
A specified actualstart time.xnot allowed
xnot allowed
decisionnode
outcome ‘b’
outcome ‘a’act ‘A’
act ‘B’
28
Fig. 5-15: Multiple Decisions (Trees)
pass/fail
?
inspect HVACequipment
act ‘B’pass: p = 80%
fail: p = 20%
major/minor
?
act ‘B’minor problems: p = 75%
act ‘B’major problems: p = 25%
combined probabilityof major installation
problems is:20% x 25% = 5%
combined probabilityof major installation
problems is:20% x 25% = 5%
29
5.7 DISTRIBUTION TYPES
• P3 Monte Carlo provides 4 basic types of activity duration distribution:– Triangular;– Modified Triangular;– Poisson;– Custom.
30Fig. 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 withinthe end X & Y percentiles are
read as the limit (thuscreating a spike at each end).
Anything sampled withinthe end X & Y percentiles are
read as the limit (thuscreating a spike at each end).
Y%
(c) Poisson Distribution
act duration
probability
S = ?P = ?
(d) Custom
act duration
probabilityA discretedistribution
A discretedistribution
Define the durations thatcan occur and their
respective probabilities
Define the durations thatcan occur and their
respective probabilities
X%
31
5.8 CORRELATED ACTIVITY DURATIONS
• 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.
32Fig. 5-17: Types of Correlation Between Two Activity Durations
(a) Linear Perfect Correlation
duration A
duration B
x
y duration ofA is a linearfunction of
duration of B
duration ofA is a linearfunction of
duration of B
(b) Non-Linear Perfect Correlation
duration A
duration B
x
y
duration ofA is a nonlinear
function ofduration of B
duration ofA is a nonlinear
function ofduration of B
(c) Linear Perfect Negative Correlation
duration A
duration B
x
y
as duration ofA increases,duration of B
decreases
as duration ofA increases,duration of B
decreases
(d) Partially Correlated
duration A
duration B
x
duration ofA is partiallycorrelated toduration of B
duration ofA is partiallycorrelated toduration of B
rangeof possiblevalues of y
for a given x
33Fig. 5-18: Types of Correlation Between Two Activity Durations
cum
ulat
ive
activity ‘A’ d
uration
• P3 Monte Carlo correlates two activity durations by using the same random number to generate their durations:
Same randomnumber for
both activities
Same randomnumber for
both activities
cum
ulat
ive
activity ‘B’ d
uration
34Fig. 5-19: Impact of Correlated Activity Durations on Project Duration
(a) sequential, correlated activities
act A act B
10+20=30
prob
10 20duration A
Correlated
20+30=50
(b) sequential, uncorrelated activities
act A act B
10+20=30
prob
10 20duration A
Un-Correlated
10+30=40
20+20=40
prob
20 30duration B
prob
20 30duration B
20+30=50
30project duration
prob
30 50project duration
prob
p=0.5 p=0.5
50 40