Upload
obada-alsaqqa
View
42
Download
0
Embed Size (px)
Citation preview
Fuzzy Time-Delay Model in Fault-Tree Analysis
for Critical Path Method
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Obada Alsaqqa
Graduate Program in Civil Engineering
The Ohio State University
2015
Master's Examination Committee:
Dr. Fabian Tan, Advisor
Dr. Tarunjit Butalia
Dr. Rachel Kajfez
Copyright by
Obada Alsaqqa
2015
ii
Abstract
Construction projects are always expected to be delayed, but the likelihood of a
delay varies between projects because of the particular circumstances and schedule for
that project. It is usually left to the scheduler to estimate these future circumstances of the
project when preparing the schedule and determining the duration of the project.
However, the schedule of the Critical Path Method (CPM) does not indicate the factors
that are assumed to participate in determining the likelihood for delay. These
deterministic durations, apart from the relationship between the activities, are the
dominant contributor to the critical path in the CPM calculation. Risk management
focuses on the processes that are considered critical, although delay may emerge from
non-critical paths.
In this study, a new fuzzy model is proposed to provide a subjective assessment of
the likelihood of delay for activities in different periods. Using this model, the
scheduler’s assessment of the likelihood of delay for each activity can be combined to
determine the likelihood of a project delay. This process is done utilizing fuzzy logic and
fault-tree analysis and is then combined with the CPM schedule of the project. The result
is a fuzzy fault-tree that shows the potential delay of the project and its contributing
paths.
iii
Applying this method on a sample project, the results show that risk of delay
comes not only from critical paths but also from non-critical paths. Consequently, the
CPM schedule duration can be reevaluated such that the project can be rescheduled to
account for the new findings and, at the very least, the risk of delay can be accounted for.
iv
To Mom, Dad, Noor and Saed.
v
Acknowledgments
I would like to thank Dr. Fabian Tan for his support and advisory efforts, as well
as the committee members: Dr. Tarunjit Butalia and Dr. Rachel Kajfez. I also would like
to thank Turner Construction Company - Columbus, Ohio, for providing me with the
sample project, especially Adam Baker, Nigel Carter and Stephen Howell. I thank
American Journal Experts for editing this thesis. Last, but not least, I would not have
been able to do this work without the Department of State, Fulbright scholarship,
AMIDEAST, and the Jordan Binational Committee who believed in me and provided the
support and funding for my studies in the US.
vi
Vita
March 15, 1987 ...................Born – Amman, Jordan
Jun 2009 to Jul 2009 ..........Intern. Site Engineer, Wyre Borough Council – UK
Oct 2009 to Jan 2010 .........Water and Wastewater Systems Designer, Associated
Consulting Engineers International – Jordan
Jan 2010 ..............................B.S. Civil Engineering, University of Jordan
Feb 2010 to Jul 2010 ..........Resident Engineer, Al-Ufuq Engineering Office – Jordan
Aug 2010 to Dec 2010 .......Resident Engineer, Faris and Faris Architects – Jordan
Jan 2011 to Jul 2013 ..........Project Manager, Al-Ufuq Engineering Office – Jordan
Aug 2013 to present ............Graduate Student, The Ohio State University
May 2014 to Oct 2014 .......Co-op Assistant Superintendent, Turner Construction – USA
Fields of Study
Major Field: Civil Engineering
Specialization: Construction Engineering and Management
vii
Table of Contents
Abstract ............................................................................................................................... ii
Acknowledgments............................................................................................................... v
Vita ..................................................................................................................................... vi
Fields of Study ................................................................................................................... vi
Table of Contents .............................................................................................................. vii
List of Tables ..................................................................................................................... xi
List of Figures ................................................................................................................... xii
List of Abbreviations ........................................................................................................ xv
List of Notations .............................................................................................................. xvi
Chapter 1: Introduction ....................................................................................................... 1
1.1 Background ............................................................................................................... 1
1.2 Significance ............................................................................................................... 3
1.3 Scope and Limitations ............................................................................................... 4
1.4 Potential Benefit ........................................................................................................ 5
1.5 Tasks.......................................................................................................................... 6
Chapter 2: Literature Research ........................................................................................... 7
viii
2.1 Weighing Risk and Opportunity ............................................................................... 7
2.1.1 Classical Probability ........................................................................................... 7
2.1.2 Fuzzy Logic ...................................................................................................... 11
2.2 Pure Risk ................................................................................................................. 12
2.3 Delay Fuzzy Models................................................................................................ 15
Chapter 3: Fuzzy Time-Delay Model ............................................................................... 19
3.1 Introduction to Fuzzy Logic .................................................................................... 19
3.2 Modeling Process .................................................................................................... 23
3.2.1 The Fuzzy Member ........................................................................................... 23
3.2.2 Two-Parameters Model .................................................................................... 23
3.2.3 “Absolutely Unlikely” ...................................................................................... 25
3.3.4 “Absolutely Likely” .......................................................................................... 26
3.3.5 Between the Two “Absolutes” ......................................................................... 28
3.3.6 Modification Function ...................................................................................... 30
3.3.7 The Criteria ....................................................................................................... 32
3.3.8 The Final Step ................................................................................................... 35
3.2 Fuzzy Time-Delay Model (FTDM)......................................................................... 37
3.5 Defuzzification ........................................................................................................ 48
Chapter 4: Converting a CPM Schedule into a Fuzzy Fault-Tree .................................... 52
ix
4.1 Introduction ............................................................................................................. 52
4.2 Critical Path Method ............................................................................................... 52
4.3 Fuzzy Fault-tree....................................................................................................... 53
4.4 The Conversion Analysis ........................................................................................ 56
4.4.1 Activities in Series ............................................................................................ 56
4.4.2 Float .................................................................................................................. 59
4.4.3 Activities in Parallel ......................................................................................... 60
4.5 Methods of Conversion ........................................................................................... 62
4.5.1 Paths Method .................................................................................................... 62
4.5.2 Basic Method .................................................................................................... 67
Chapter 5: Sample Project ................................................................................................ 72
5.1 Introduction ............................................................................................................. 72
5.2 The Project Schedule ............................................................................................... 72
5.3 Scheduler Assessment ............................................................................................. 77
5.4 Computer Application ............................................................................................. 80
5.4.1 Using Coordinates ............................................................................................ 80
5.4.2 Fuzzy Operations .............................................................................................. 83
5.4.3 Rearranging FFT............................................................................................... 83
5.4.4 Plotting Membership Functions........................................................................ 84
x
5.5 Analysis Results ...................................................................................................... 85
5.5.1 FFT ................................................................................................................... 85
5.5.2 Likelihood of Project Delay ............................................................................. 87
5.5.3 Criticality .......................................................................................................... 89
Chapter 6: Summary, Conclusions and Recommendations .............................................. 94
6.1 Summary ................................................................................................................. 94
6.2 Conclusions ............................................................................................................. 96
6.3 Recommendations ................................................................................................... 98
Bibliography ................................................................................................................... 101
Appendix A. Numerical Solutions .................................................................................. 107
Appendix B. Email Correspondence with Turner Construction Company .................... 110
Appendix C. Sample Project Gantt/bar Chart – Partial Screenshots .............................. 112
Appendix D. FFT Analysis – Command Events Sub-Trees ........................................... 116
xi
List of Tables
Table 1. Trials in Numerical Solution for Finding 𝑝𝑈𝑁 .................................................... 36
Table 2. Final Fuzzy Sets for 1-Day Delay Likelihood and Some of their Properties ..... 36
Table 3. Trials of Numerical Solution for Calculating 𝜇(3) of the Fuzzy Set “Fairly
Likely” to Be Delayed for 2 days ......................................................................... 40
Table 4. Membership of Fuzzy Member 2 Days in 1-Day Delay Assessment ................. 43
Table 5. Activity Example – Scheduler Assessment ........................................................ 46
Table 6. Defuzzification Process – Example .................................................................... 49
Table 7. Defuzzified Result – Example ............................................................................ 51
Table 8. Delay Likelihood Assessment – “Fuzzy Sum” Example.................................... 58
Table 9. CPM Schedule Table for Sample Project ........................................................... 73
Table 10. Scheduler Delay Subjective Assessment for the Sample Project ..................... 79
Table 11. Paths and Project Delay Fuzzy Members Values – Highlighting Contributing
Paths in Project Delay Likelihood ........................................................................ 90
xii
List of Figures
Figure 1. Normal Distribution - Probability Density Function ........................................... 9
Figure 2. Beta Distributions – Probability Density Function ........................................... 14
Figure 3. Baldwin’s Fuzzy Rotational Model ................................................................... 16
Figure 4. Failure General Fuzzy Set – After Shiraishi & Furuta (1983) .......................... 17
Figure 5. Fuzzy Triangular Translation Model ................................................................. 18
Figure 6. Venn Diagram of Ordinary Sets ........................................................................ 20
Figure 7. Venn Diagram of Fuzzy Sets ............................................................................. 20
Figure 8. “Absolutely Unlikely” Fuzzy Set is Represented by Horizontal Line at the
Abscissa from Zero to Infinity and a Vertical Line from Zero to One at the
Ordinate................................................................................................................. 25
Figure 9. Primitive Model for 1-Day Delay Likelihood Using Power Function .............. 30
Figure 10. Unmodified vs. Exponentially Modified “Absolutely Likely” Fuzzy Set ...... 32
Figure 11. Proposed Fuzzy Model for t-Day Delay Likelihood ....................................... 37
Figure 12. Fuzzy Set "Fairly Unlikely" for 2-Days Delay ................................................ 40
Figure 13. Fuzzy Sets for 1-Day Delay Likelihood .......................................................... 42
Figure 14. Likelihood of Change for FTDM at 𝑥 = 𝑡 ...................................................... 44
Figure 15. Membership functions of “Likely” Assessment for One, Two, and Three-Day
Delay ..................................................................................................................... 45
Figure 16. Example of Creating Activity Delay Likelihood Using “Fuzzy Or” .............. 47
xiii
Figure 17. Types of Fault Events ...................................................................................... 55
Figure 18. Logic Gates in Fault-Trees .............................................................................. 55
Figure 19. Schedule Predecessor Diagram – “Fuzzy Sum” Example .............................. 57
Figure 20. Fuzzy Fault-Tree – “Fuzzy Sum” Example ..................................................... 58
Figure 21. “Fuzzy Sum” Operation Demonstration .......................................................... 59
Figure 22. Schedule Predecessor Diagram – “Fuzzy Or” Example .................................. 61
Figure 23. Fuzzy Fault-Tree – “Fuzzy Or” Example ........................................................ 61
Figure 24. Generic FFT using Paths Method .................................................................... 63
Figure 25. Algorithm of the Paths Method for Converting a CPM Schedule Table into a
Fuzzy Fault-Tree ................................................................................................... 65
Figure 26. Algorithm of the Basic Method for Converting a CPM Schedule Table into
Fuzzy Fault-Tree ................................................................................................... 69
Figure 27. Gantt Chart – Sample Project .......................................................................... 76
Figure 28. Representation of Fuzzy Set "Neutral" for 1 Day with 100 Points of Linear
Ordinate Relationship ........................................................................................... 81
Figure 29. Representation of Fuzzy Set "Neutral" for 1 Day with 100 Points of Quadratic
Ordinate Relationship ........................................................................................... 82
Figure 30. Representation of the Fuzzy Membership Function in the Computer Program
............................................................................................................................... 84
Figure 31. FFT Layout - Sample Project .......................................................................... 86
Figure 32. Defuzzified Likelihood of Project Delay – Sample Project ............................ 88
Figure 33. Path 12 FFT ..................................................................................................... 91
xiv
Figure 34. Path 8 FFT ....................................................................................................... 92
Figure 35. Path 9 FFT ....................................................................................................... 93
xv
List of Abbreviations
AL Absolutely Likely
AU Absolutely Unlikely
CPM Critical Path Method
FFT Fuzzy Fault-Tree
FL Fairly Likely
FTA Fault-Tree Analysis
FTDM Fuzzy Time-Delay Model
FU Fairly Unlikely
LI Likely
MCS Monte Carlo Simulation
ML Most likely duration in PERT
NE Neutral
O Optimistic duration in PERT
P Pessimistic duration in PERT
PERT Program and Evaluation Review Technique
UN Unlikely
VL Very Likely
VU Very Unlikely
xvi
List of Notations
𝜎 Standard Deviation
𝑐 Model Parameter
𝜇 Mean
𝑥 Time-Delay Fuzzy Member
𝜇(𝑥) Membership Function
𝑡 Delay Period Parameter
𝑝 Fuzzy Set Power Parameter
𝑒 Mathematical Constant ≈ 2.71828
𝑙𝑛 Natural Logarithm
∫ Integral
𝜋 Mathematical Constant ≈ 3.14159
1
Chapter 1: Introduction
1.1 Background
Projects can be delayed for various reasons, such as weather or a shortage of
resources, and these delays are expected to occur during a project’s duration. During the
scheduling process, planners know most of the reasons for a possible delay and can thus
attempt to forecast the project duration using their best knowledge and previous
experience. This forecasting must include possibility for the worst to happen. In other
words, the more accurate the risk is estimated, and so accounted for, the better.
Scheduling means setting up a plan for the future. This plan includes breaking the
project down into activities. Each activity can be identified, and the duration of the
activity, from start to finish, can be estimated. It is necessary that each activity start for
the project to be finished. Moreover, the completion of all activities should lead to the
completion of the project.
One of the major scheduling methods in construction is the Critical Path Method
(CPM). Another common scheduling method is the Program and Evaluation Review
Technique (PERT). CPM is a deterministic method in that each activity is given only one
duration, whereas PERT assigns three different durations for each activity. However, the
construction industry still favors CPM over PERT because CPM is simpler and easier to
implement.
2
Though it can be said that project delays are inevitable, with enough additional
duration included in the project schedule after considering the possible risks, many
projects will be completed on time. Goldratt (1997) tried applying his Theory of
Constrains (TOC) to the field of project management to illustrate the Critical Chain
Method, which emphasizes the resources required to perform the activities. According to
Goldratt, one major reason for a high percentage of delayed projects is the method used
to determine a safe duration for the project. The scheduler usually estimates an optimum
duration and then adds some extra duration, a safety duration, to the activity to account
for the risk of delay. Goldratt believes that due to the “student syndrome,” i.e.,
procrastination, essentially all of the extra duration included in each activity is used. In
this case, the project is inevitably delayed without crashing, which is reducing the
activities duration requiring more resources. Both crashing and delay mean extra cost
regardless of who is responsible for it. Goldratt suggests estimating each activity’s
optimum duration without adding any safety duration and instead adding the safety
duration for the project at the end of the schedule.
However, Goldratt’s suggestion of placing the safety duration at the end of the
project schedule can be difficult to implement in the construction industry because CPM
has become such a part of the contract that the schedule may hold high significance in
court. As a result, placing extra time at the end of the project would not be acceptable
because this would be designated a float, which is a responsibility that is considered to be
owned by the project owner. Float is as excess time an activity has without affecting the
project finish point. Float is explained further in Chapter 4. Using a float requires the
3
owner’s approval, but according to Goldratt’s suggestion, the contractor is entitled to this
float. A further issue with this method is that it is not possible to prove delay claims.
Therefore, many activities will be delayed as part of the plan. Nonetheless, it would be
worth trying this method because many projects are delayed anyway.
Many improvements to CPM have been introduced since it was first proposed.
Nevertheless, most of these probabilistic and fuzzy studies attempted to weigh the
opportunity and risk together. Doing so would mean providing a possible duration for the
activities that can be either more or less than the expected duration. However, the author
of this study believes that the focus should be on risk only, so this study focuses on just
the risk.
1.2 Significance
In this study, the author proposes a new fuzzy model to represent the likelihood of
delay for some duration. This model, termed the Fuzzy Time-Delay Model (FTDM),
makes it possible to refine the likelihood of delay for an activity in a direct, practical way.
The refinement is implemented by asking the scheduler to use a linguistic, subjective
assessment of the likelihood of a one-day delay, two-day delay, and so on, for each
activity. Fuzzy logic is then used to combine these likelihoods and thereby determine the
likelihood of delay for that activity. Then, using Fault-Tree Analysis (FTA) and fuzzy
logic an overall likelihood of delay for the whole project is computed.
FTDM creates a way to make it possible for schedulers to qualitatively determine
the risk of quantitative delay periods for a scheduled project, which is a unique feature of
this method. This method not only consolidates all of the scheduler’s decisions in
4
choosing the duration of each activity but also considers all of the factors that can cause
the delay.
1.3 Scope and Limitations
This study applies to CPM scheduled projects. It focuses on the risk of the delay
and disregards the opportunity for the activity to finish earlier than its planned duration.
All paths, both critical and non-critical, can be included in the analysis. Moreover, it is
not limited to any network complexity.
This study assumes that the schedule is already calculated according to the CPM,
considering both early and late start and finish times as well as float. However, it is
limited to the finish-to-start relationships between activities. A finish-to-start relationship
is the most common relationship used and can be sufficient in some simple projects.
Other relationships, such as start-to-start and finish-to-finish, require further research
concerning the way in which the activities are structured in the fault-tree. Although the
start-to-start relationship can be transformed into a finish-to-start relationship and vice
versa by changing the lag value, this process is not used in this study because it is
assumed that the lag between all activities is zero.
It is assumed that no constraints on the activities dates are set in the schedule,
such as dates of material deliveries or state road-access permits. Furthermore, all
activities share the same calendar.
In addition, it is important to note that the result relies primarily on the subjective
assessment of the scheduler and the schedule network. Because the scheduler is the one
5
who assigns the durations of the activities in the schedule, he/she controls the total risk of
delay for the project.
1.4 Potential Benefit
FTDM may be applicable to other problems related to time and delay and can be
adopted in areas beyond construction management.
Every planning engineer wants to have insight into how the schedule is expected
to run and what the potential for delay may be. Combining real rather than theoretical
input allows one to arrive at results more conveniently. In addition, using linguistic terms
instead of numerical probability is not only much easier and more accurate but also more
appropriate for this application.
Another important property of this study is that the method is able to show the
paths that have the highest potential for project delay. Even a non-critical path may
contribute to the potential for delaying the project despite any possible float that it may
have. Therefore, the important property in determining a critical path shifts from finding
the deterministic longest path to finding the group of paths that contribute in the potential
delay likelihood of the project.
This study can be an advancement in solving the impracticality of Goldratt’s
suggestion. By calculating the potential delay of a tightly scheduled project, the result can
be used as the sum of safety durations added to the end of the schedule. However, this
also requires more research to prove its practicality.
6
1.5 Tasks
Firstly, the fuzzy member was defined with its delay duration parameter. Then, a
model was created to resemble its membership value. The modeling process defined the
following:
The linguistic terms
The nature of the membership function in relation to the fuzzy member
The boundary fuzzy sets
The poles and pivot of the model
The model parameters and constants
Criteria
Defuzzification
Following the creation of the model, the different combinations of relations in the CPM
were assigned to appropriate gates in the fault-tree.
In summary, it is necessary to create a method for converting the CPM to the
Fuzzy Fault-Tree (FFT). Two methods were created one general and another limited one.
Finally, a sample project was analyzed using the FFT, and the results were studied.
7
Chapter 2: Literature Research
Risk management plays a vital role in project management. Although the project
management is concerned with time, budget and quality, the focus of this study is on
time, and more specifically, the risk of project delay.
2.1 Weighing Risk and Opportunity
Risk can refer to either the combination of risk and opportunity or the risk only.
Several references on project delay, regardless of whether they are quantitative or
qualitative studies, have included opportunity in the risk. More specifically, the variable
that is most often studied has been the duration of the project and its activities rather than
the delay itself. The methods used to study the duration of activities within the project
included providing a statistical distributions for those durations. These distributions
include the possibility for an activity to finish before the end of the original duration
assigned by the scheduler. This is where the opportunity in the expected total duration of
the project originates. However, by studying the delay only, the influence of the optimism
is removed, and only the risk is considered.
2.1.1 Classical Probability
Project duration has been extensively studied probabilistically to account for the
deterministic nature of the Critical Path Method (CPM) since the method was first
introduced by Kelley and Walker in 1959. This method assigns each activity a
8
deterministic duration. Accordingly, the whole project has a deterministic total duration.
In CPM, the paths in the project with the longest duration in the project are called critical
paths. All other paths should have a float, which makes them non-critical.
In the same year that CPM was introduced, the US Navy developed the Program
Evaluation and Technique Review (PERT), which gives each activity three durations:
optimistic time, most likely time, and pessimistic time, which makes this method on the
non-deterministic side (Fazar, 1959; Malcolm et al. 1959). The expected duration (TE) of
each activity is called the expected time or the best estimate. The expected duration is
calculated using a weighted average in which the most likely (ML) duration is weighted
four times more than the optimistic (O) or pessimistic (P) durations. The formula is
simply 𝑇𝐸 = (𝑂 + 4 ∗ 𝑀𝐿 + 𝑃)/6. The rest of the calculations used to find the critical
path are identical to the CPM forward and backward passes but use the expected time.
PERT gives results that include a slight underestimation of the total duration of the
project compared to other methods that were introduced later, such as Monte Carlo
Simulation (MCS) (Diaz & Hadipriono, 1993, p. 65).
Many studies have been conducted to account for the uncertainty of the project
durationMCS is one method that assigns distributions to the duration of an activity (Van
Slyke, 1963). Monte Carlo methods are an iterative computational algorithm with random
sampling from a preselected distribution. As a result, it generates different results after
every run. Nonetheless, if the number of trials is sufficiently large, the results will
converge. MCS produces results that can be considered a reference to compare with. One
9
of the most commonly used distributions is the normal distribution. Figure 1 shows
different variations of a normal distribution.
Figure 1. Normal Distribution - Probability Density Function
Other successful trials to account for the uncertainty in activities duration were
introduced later and included the Probabilistic Network Evaluation Technique (PNET)
(Ang et al. 1975). PNET improves PERT by changing the parameters of the mean
duration and the correlations among the network paths. It also provides results that agree
with MCS with even less time for run processing. Unlike PERT, PNET covers all
schedule paths. Ranasinghe's (1990; 1994) work applied PNET by assigning a Pearson
family of distributions for the activities duration.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-4 -2 0 2 4
f(x)
x
Normal Distribution
μ =0, σ =1
μ =0, σ =2
μ =0, σ =0.5
μ =1, σ =1
μ =-1, σ =0.5
𝑓 𝑥 =1
𝜎 2𝜋𝑒
−𝑥−𝜇 2
2𝜎2
10
The Narrow Reliability Bounds (NRB) “method was developed for structural
reliability analysis by Ditlevsen (1979), and was earlier applied for scheduling by
Laferriere (1981). Like PNET, the NRB model is based on the probability of failure of
each path. Failure occurs when the network duration is longer than a predetermined target
duration. A failure mode is equivalent to a network path. Each path is considered to be
normally distributed with expected duration … and standard deviation”. “NRB finds two
probabilities of failure for the combination of all existing paths: lower bound probability
(PL) and upper bound probability (PU)” (Ditlevsen, 1979) in (Diaz & Hadipriono, 1993,
p. 43).
A Simplified MCS (SMCS) developed by Diaz-Suarez (1989) was as effective as
MCS, took less processing time and produced more conservative results (Diaz &
Hadipriono, 1993, p. 55). This method simplifies the network discarding activities on
paths with less than a certain minimum duration. The minimum duration is chosen by the
scheduler. Similarly, Fast and Accurate Risk Evaluation (FARE) by Jun & El-Rayes
(2011) is an approximation method that identifies and removes insignificant paths, which
also makes it faster than MCS.
It should be noted that there are other significant methods for finding the total
project durations probabilistically. One is the Modified Stochastic Assignment Mode
(MSAM), as introduced by Guo (2001), which features adding project economics to the
equation of project duration.
11
2.1.2 Fuzzy Logic
Zadeh (1965), using fuzzy set theory, made it possible to use linguistic terms
instead of numerical probabilities to obtain results. Fuzzy logic is explained in more
details in Chapter 3.
Fuzzy logic has been used instead of classical probabilities to closely estimate
projects durations a couple of decades later than normal probabilities. However, it has
also been used to allow the risk to include both the opportunity and the safety. Ayyub &
Haldar (1984) and AbouRizk & Sawhney (1993) both applied fuzzy logic to problems in
the scheduling process, such as assessing the impact of some factors on the duration of
activities.
Fuzzy Networking Evaluation Technique (FNET) introduced by Lorterapong &
Moselhi (1996) obtains results that are close to MCS. In FNET, the expected duration is
considered a fuzzy number rather than a deterministic one. The results of FNET would be
identical to those of PNET in the case that the membership function of the fuzzy element
in FNET was proportional to the distribution of the activity duration in PNET. However,
FNET used the fuzzy trapezoidal model. Nonetheless, the FNET results are close to the
MCS results because of the central limit theory, which states that the distribution of a
large number of random variables with a well-defined variance will be approximately a
normal distribution, regardless of the distribution of the variable.
Boussabaine (2001) used a fuzzy inference system to estimate the project duration
from its contributing factors. The fuzzy inference system has a predefined set of rules in
addition to fuzzy models. After inputting the contributing factors into the system, the
12
fuzzy logic operations are calculated according to the given rules, and then the result is
determined. Then, the system defuzzifies the result to a resulting estimated duration.
Wu & Hadipriono (1994) used an angular fuzzy model, whereas Oliveros &
Fayek (2005) used triangular and pi-shape fuzzy models to create the fuzzy duration of an
activity by subjectively assessing some of its contributing factors. Long & Ohsato (2008)
took an extra step by combining a fuzzy trapezoidal model for the duration of the
activities with resource constraints loaded on the schedule.
Sebt, Rajaei, & Pakseresht (2007) used a combination of frequency of
occurrences and adverse consequences as a fuzzy model, similar to Ayyub & Haldar's
(1984) fuzzy translational model, to represent weather delays based on a time impact
analysis.
2.2 Pure Risk
There have been few studies that examined the risk of delay while disregarding
the opportunity, compared with studies that included the opportunity. However, these
varied from normal probabilistic models to fuzzy approaches.
In 1986, fault-tree analysis was applied to CPM scheduling using normal
probability statistics while focusing on the risk of delay and considering the delay as a
failure (Hadipriono, 1988a; Hadipriono, Larew, & Lin, 1987; Tirtotjonro, 1986).
Nevertheless, it is not easy to give a percentage of failure for an activity accurately,
though it is easier and more realistic to give a subjective opinion. In addition, when using
normal probabilities, the issue of statistical independency required calculating the
probability of the different combinations of delays. However, this issue is not found in
13
fuzzy logic because the statistical dependency or independency is implied in the
subjective assessment.
Many studies have tried to determine risk factors to modify the original duration,
such as the Probabilistic Risk Factors by Dawood (1998), Construction Schedule Risk
System by Mulholland & Christian (1999) and Evaluating Risk in Construction–Schedule
Model (ERIC-S) by (Nasir, McCabe, & Hartono, 2003). In the first study, Dawood
(1998) tried to find the impact of several factors on the activity duration, such as weather,
labor productivity and equipment availability. In contrast, Mulholland & Christian (1999)
calculated the risk factors through a “system that provides a structured approach to
identify the sources of risk in a project and based on these risks determine the range of
schedule outcomes”. Nasir et al. (2003) identified the construction schedule risks and
their cause and effect relationships. Moreover, through field experts’ review, they were
able to develop their belief network model. Although their work can be modeled using
fuzzy logic, they chose to use normal probabilities. Furthermore, the Correlated Schedule
Risk Analysis Model (CSRAM) by Ökmen & Öztaş (2008) combines providing a
distribution of the activity duration with employing the correlated risk factors method.
Similar to the probabilistic approach, fuzzy logic was utilized to produce risk
factors that modify the activity duration (Kim et al. 2006; Zeng et al. 2004). Each of these
studies uses different ways of identifying the risk factors to modify the duration of the
activities or the project by utilizing fuzzy logic. However, AbouRizk & Sawhney (1993)
used a system that best fits the custom Beta distribution for each activity with the ability
to modify the input through linguistic variables for both an optimistic and pessimistic
14
case by following PERT methodology. They found in their study that Beta distributions
could represent construction activities durations better than any other distribution could,
as long as the parameters of the Beta distribution are chosen correctly. Figure 2 illustrates
different cases of the Beta distribution.
Figure 2. Beta Distributions – Probability Density Function
Al-Humaidi & Hadipriono Tan (2010a) considered the delay by accounting purely
for risk while excluding safety, but the delay was assessed qualitatively and as a whole.
They identified a number of the contributing factors to the project delay as a whole. They
classified these factors into enabling, triggering and procedural factors. From a subjective
0.0
0.5
1.0
1.5
2.0
2.5
0 0.2 0.4 0.6 0.8 1
f(x)
x
Beta Distributions
α =0.5, β =0.5
α =5, β =1
α =1, β =3
α =2, β =2
α =2, β =5
𝑓 𝑥 =1
Β 𝛼, 𝛽𝑥𝛼−1(1 − 𝑥)𝛽−1
where B is the Beta function
15
assessment and by applying fuzzy fault-tree analysis, they found the delay likelihood
qualitatively for the project as a whole but not through the schedule activities separately.
They used fuzzy translational and rotational models (Al-Humaidi & Hadipriono Tan,
2010a, 2010b). In a different manner and under the name of Safety Management System
(SMS), Ingle, Atique, & Dahad (2011) produced a similar work in which the different
factors were classified upon stages of the project and used a fuzzy trapezoidal model.
2.3 Delay Fuzzy Models
Many fuzzy models have been created since Zadeh (1965) introduced fuzzy logic.
Because the model introduced in this study contains both power and exponential terms,
only the models that have similar forms will be mentioned.
Baldwin's (1979) fuzzy rotational model uses the power function to differentiate
between levels of truth-values in the membership function of opposite notions or
concepts. The area under the curve plays a major role in the value of the index of the
fuzzy set. Al-Humaidi & Hadipriono Tan (2010a) used this model, shown in Figure 3, to
determine the likelihood of delay for the project qualitatively.
16
Figure 3. Baldwin’s Fuzzy Rotational Model
Translational models can represent fuzzy members with an unlimited maximum
value, similar to delay being represented as a period. Many researchers have used
translational models. Shiraishi & Furuta (1983) used a failure fuzzy set in their fuzzy
reliability analysis, and its fuzzy membership is shown in Figure 4. The function itself
was not specified in the study so a general one was used.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mem
ber
ship
Fuzzy Member
Rotational Model
Absolutely True
Very True
True
Fairly True
Fairly False
False
Very False
Absolutely False
17
Figure 4. Failure General Fuzzy Set – After Shiraishi & Furuta (1983)
Hadipriono (1995) used the translational model as well as other fuzzy models in
his structural mechanics study. Part or all of a triangular, trapezoidal or a bell shaped
distributions; such as normal distributions, π-distributions, t-distributions and Cauchy
distributions, are all possible membership functions that can be set as fuzzy translational
models. Figure 5 shows a fuzzy triangular translational model.
1
Failure Region
M(z): Membership
z: Fuzzy Member
18
Figure 5. Fuzzy Triangular Translation Model
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mem
ber
ship
Fuzzy Member
Likelihood Distribution
Absolutely Unlikely
Vey Unlikely
Unlikely
Fairly Unlikely
Fairly Likely
Likely
Very Likely
Absolutely Likely
19
Chapter 3: Fuzzy Time-Delay Model
3.1 Introduction to Fuzzy Logic
Even today, in the minds of nonprofessionals, the word “logic” seems to imply
only binary logic, although fuzzy logic is implicitly being used in everyday dialogue. For
example, the process or reasoning requires logic to be completed. With fuzzy logic,
reasoning becomes an approximate logic scheme. The term “fuzzy logic” was introduced
to the world by Zadeh (1965), although it has been studied since the 1920s as infinite-
valued logic by Lukasiewicz (Pelletier, 2000). The applications of fuzzy logic are
numerous and include linguistics, decision making and clustering.
Set theory is traditionally handled with binary logic, which is why the boundaries
its sets are abrupt, as shown in the Venn diagram in Figure 6, where the sample space 𝑆 is
the power set containing all possible outcomes of the variable being studied. Conversely,
fuzzy logic is an extension of set theory. The boundaries of fuzzy sets are imprecise, and
the transition is gradual rather than sudden, as illustrated in Figure 7.
20
Figure 6. Venn Diagram of Ordinary Sets
Figure 7. Venn Diagram of Fuzzy Sets
In fuzzy logic, the sample space is called the universe of discourse and is denoted
as 𝑆’ in Figure 7. By looking at Figure 6, it can be said that the member 𝑥 belongs to set
𝐵 and does not belong to set 𝐴. In contrast, in Figure 7, it cannot be said that the fuzzy
𝐵 𝐴
𝑆
𝑥
𝐴’ 𝑆’
𝑥′
21
member 𝑥′ does not belong to the fuzzy set 𝐴’. The membership of 𝑥′ in 𝐴’ is, in fact,
partial.
This membership of the fuzzy member can be represented with a value between
zero and one, which represents the degree of membership. A value of zero indicates that
𝑥′ is absolutely not a member, and a value of one indicates that 𝑥′ is absolutely a member.
The representation of membership of a fuzzy variable is denoted by 𝜇(𝑥) where 𝑥 is the
fuzzy member.
Binary logic or boolean logic is concerned with mapping true and false to an
indicator function with two contradicting values. The indicator function denotes the
variable under study. In simpler terms, a statement can be either true or false and its
opposite would be false or true respectively. Instead, fuzzy logic maps continuous truth-
values to a continuous indicator function. Mathematically, if, similar to a function, the
mapping is represented by a domain and a range, then the following shows then the
difference in the mapping for binary and fuzzy logic:
Binary:{0,1} → {0,1}
The braces {} indicates listing included members. For example, the domain can be 0 or 1.
Fuzzy: [0,1] → [0,1]
The square brackets [] indicates a range with a start and finish inclusive. Consequently,
the range can be any value between 0 and 1 including 0 and 1.
Similar to binary logic, fuzzy logic has its own algebraic operations. For example,
the synonym for “Or”, i.e., the union of sets, is “Fuzzy Or”. Furthermore, “Fuzzy And” is
a synonym for “And”, i.e., the intersection of sets. However, these operations are
22
calculated in a different manner. For example, the membership function of the union of
two fuzzy sets is calculated from Equation (1)
𝜇𝐴⋃𝐵(𝑥) = max {𝜇𝐴(𝑥), 𝜇𝐵(𝑥)} ......................................... (1)
where 𝑥 is the fuzzy member, 𝜇𝐴⋃𝐵(𝑥) is the membership function of the union of A and
B, 𝜇𝐴(𝑥) is the membership function of fuzzy set A, and 𝜇𝐵(𝑥) is the membership
function of fuzzy set B. The intersection of two fuzzy sets A and B denoted by 𝐴 ∩ 𝐵 is
calculated from Equation (2):
𝜇𝐴∩𝐵(𝑥) = min {𝜇𝐴(𝑥), 𝜇𝐵(𝑥)}.......................................... (2)
The membership of the complement of 𝐴 denoted by �̅� would simply be found as in
Equation (3):
𝜇�̅�(𝑥) = 1 − 𝜇𝐴(𝑥) .................................................. (3)
Some fuzzy operations, such as the use of hedges like “very” and “fairly”, and
some of the relations, such as implication, vary, depending on the fuzzy model.
23
3.2 Modeling Process
Delay has never been modeled using fuzzy logic in a way in which the delay is
calculated quantifiably while simultaneously assessing the delay subjectively. In an effort
to make this calculation possible, the author introduces a new model. The process of
modeling is explained and described in this section.
3.2.1 The Fuzzy Member
The fuzzy member represents the duration of the delay starting from the moment
an activity is planned to finish to its actual finish. In this study, this duration is denoted
with the letter 𝑥. For any activity, this fuzzy member 𝑥 will equal zero at the point where
the activity under assessment is supposed to finish. Moreover, the fuzzy member 𝑥 can
never be negative because this model focuses on the risk and excludes the opportunity for
an activity to finish before its scheduled finish.
3.2.2 Two-Parameters Model
Each fuzzy set in this model is described using two parameters, which are denoted
with the letters 𝑝 and 𝑡. The first parameter 𝑝 represents the linguistic variable. The
linguistic variable will be describing the likelihood of delay. Between “Absolutely
Likely” set and “Absolutely Unlikely” set, there is a medium likelihood or a “Neutral”
fuzzy set. The high likelihood is represented by “Likely” fuzzy set, and the low
likelihood is labeled as “Unlikely”. To make the model more precise, two fuzzy sets are
added to the “Likely” and “Unlikely” by adding the modifier “Very” and “Fairly” to
them.
24
The linguistic variables are shown in the following list, along with their respective
meaning:
1. Absolutely Unlikely (AU)..................No Likelihood
2. Very Unlikely (VU) ...........................Very Low Likelihood
3. Unlikely (UN) ....................................Low Likelihood
4. Fairly Unlikely (FU) ..........................Fairly Low Likelihood
5. Neutral (NE).......................................Medium Likelihood
6. Fairly Likely (FL) ..............................Fairly High Likelihood
7. Likely (LI) ..........................................High Likelihood
8. Very Likely (VL) ...............................Very High Likelihood
9. Absolutely Likely (AL)......................Extremely High Likelihood
The value of the parameter 𝑝 that corresponds to each of these linguistic variables will be
determined later in this section.
The other parameter in this model 𝑡 is the number of days under assessment. The
values that this parameter can take are only positive integers from one to infinity. Each
fuzzy set will be a combination of a linguistic variable and a duration under assessment.
So typically, a fuzzy set in this model describes an activity likelihood of delay for a
certain period of time-delay. For example, let an activity be assessed as “Unlikely” to be
delayed for 1 day. In this example, the first parameter is a value of 𝑝 corresponding to the
assessment “Unlikely” and the other parameter 𝑡 is the duration under assessment, which
equals one.
25
3.2.3 “Absolutely Unlikely”
The first basic fuzzy set is the set “Absolutely Unlikely”. This assessment means
that there is no likelihood for that activity to be delayed; therefore, this fuzzy set can be
represented as 𝜇(𝑥) = 0, where 𝜇(𝑥) is the membership function of the fuzzy member 𝑥,
except when 𝑥 = 0 in which the case the likelihood becomes one. This means that the set
is a fuzzy singleton at 𝑥 = 0, as shown in Figure 8. The final membership function of
“Absolutely Unlikely” would simply be Equation (4) that is not dependent on the
parameter 𝑡, which is the number of days under assessment.
𝜇(𝑥) = {1 , 𝑥 = 00 , 𝑥 > 0
................................................. (4)
Figure 8. “Absolutely Unlikely” Fuzzy Set is Represented by Horizontal Line at the
Abscissa from Zero to Infinity and a Vertical Line from Zero to One at the
Ordinate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
Absolutely Unlikely
26
3.3.4 “Absolutely Likely”
The opposite of “Absolutely Unlikely” is the “Absolutely Likely” set. Because it
is absolute, from time 0 until time 𝑡, the likelihood of delay must follow the membership
function 𝜇(𝑥) = 1.
For any activity and up to any particular period of expected delay, the likelihood
of delay must be higher before that particular time and must be lower after that time. For
example, if an activity delay is possible for 3 days, then the likelihood of a 2-day delay is
higher and the likelihood of a 4-day delay is lower. In other words, as time increases, the
likelihood of delay decreases. Therefore, the membership function of time-delay for any
fuzzy set must be decreasing.
The “Absolutely Likely” membership function equals one from 𝑥 = 0 to 𝑥 = 𝑡,
and when 𝑥 > 𝑡 the likelihood should start decreasing below one. Because there will
always be some likelihood of delay no matter how much time increases, the likelihood of
delay will become zero only at infinite delay duration. Therefore, following the first point
represented as ( 𝜇(𝑥) | 𝑥 ) = ( 1 | 0 ), the second pole of this model is the point ( 0 | ∞ ).
Upon searching through different functions that can represent the distribution of
delay likelihood over time, bell-shaped functions were the best representation of the
distribution in terms of converging to infinity on the tails and being concentrated at the
mean. Among bell functions, the exponential is used as a distribution function for many
time-related problems. The author used the basic exponential function as the first trial in
the modeling process, but it failed to match the criteria the author set, which will be
27
described later in this section. In contrast, the normal distribution, which is another
exponential function, is able to accommodate those criteria.
Another condition is that the transition of the function must be smooth. Because at
𝑥 = 𝑡, the slope is zero from the left, the slope must be zero from the right. That is true
for the normal distribution but not for the simple exponential function.
Although the Beta distributions can better represent the duration of construction
activities according to AbouRizk & Sawhney (1993), the normal distribution is a
practical assumption for the distribution of the likelihood of delay. In contrast, by using
Beta distributions, each activity would have to be modeled individually. Although the use
of a Beta distribution is precise, it is not a practical option. In addition, the likelihood of
delay for an activity will be refined in this study through an assessment of the scheduler
and so the likelihood will be customized. Anyway, the normal distribution has been used
in MCS to represent the activities duration and has been proved to be valid. Moreover,
Beta distributions are limited to be in the range of two values, whereas a distribution that
converges to infinity is needed for this model.
The normal distribution function is shown in Equation (5), where 𝜇 is the mean, 𝑧
is the variable and 𝜎 is the standard deviation:
𝑓(𝑧) =1
𝜎√2𝜋𝑒
−(𝑧−𝜇)2
2𝜎2 ................................................. (5)
28
Some variables were defined and substituted into Equation (5) to make the equation
transition into the previous function 𝜇(𝑥) = 1 from 𝑥 = 𝑡. Furthermore, only the right
side of the bell curve will used. The membership function is given by Equation (6):
𝜇(𝑥) = {𝑐
𝜎√2𝜋∗ 𝑒
−(𝑥𝑡
−1)2
2𝜎2 , 𝑥 > 𝑡
1 , 𝑥 ≤ 𝑡
........................................ (6)
The constant c is used as a modifier for the function to make the maximum value
of the membership function equal to one because the highest point in ordinate changes
with the change of the standard deviation 𝜎. At first, the mean 𝜇 is substituted with the
value of one into Equation (5) because this is the point of transition from the constant
function associated with “Absolutely Likely” to the exponential function. Then, the
parameter 𝑡 is inserted as a multiplication factor in the inverse function of the likelihood.
In the membership function, the parameter 𝑡 becomes translated as a division factor to the
fuzzy member. This determines how the likelihood of an activity is constructed by asking
the scheduler about his/her opinion regarding a 1-day delay, 2-day delay, and so on. It is
possible to make the model applicable to more than one day by considering 𝑡 to be a scale
factor that can be translated by either multiplication or division.
3.3.5 Between the Two “Absolutes”
The transition between the fuzzy sets must be modeled using a function with the
ability to connect the two poles of the model ( 1 | 0 ) and ( 0 | ∞ ). Baldwin (1979) was
able to use a power functions in his fuzzy rotational model to represent the change of
29
membership between the fuzzy sets. However, his model handles two opposite notions or
concepts that are bounded in value between zero and one with two poles for each concept
(Figure 3; Page 16). This study’s time-delay model focuses on one concept, which is the
risk of delay, and excludes the safety or the opportunity emerging from the possibility of
finishing an activity earlier than the originally expected duration.
Inspired by Baldwin’s model, the author chooses a power function of the form
𝜇(𝑥) = (1 − 𝑥)1
𝑝 to describe the change of membership between the fuzzy sets. The
power parameter 𝑝 is set in the denominator in the membership function, so that it would
be in the nominator in its inverse because the inverse of the function will be the form
used later.
Baldwin’s model used the area as an index to differentiate between the fuzzy sets
and as a defuzzification method. In this study, the same concept can be used, but it is not
necessary. The defuzzification process will be explained later in this section. However,
when assessing a certain duration, the area under the curve of likelihood of delay
represents the total likelihood of delay, so a linear change in the area for the fuzzy sets for
the same number of days under assessment should be maintained.
The linear change can be ensured by solving the function for the power
parameter 𝑝 for the fuzzy sets according to the criteria chosen. Figure 9 shows all the
fuzzy sets using the current modeling stage assuming that the “Absolutely Likely” fuzzy
set is 𝑥 = 𝑡. For the purpose of this study, this model in Figure 9 is labeled as a primitive
model because the model is not in its final form. This primitive model is not acceptable
for the simple reason that the membership of delay at 𝑥 = 𝑡 for all of the linguistic
30
variables cannot be zero, except for “Absolutely Unlikely.” Furthermore, none of the
fuzzy sets extends to infinity in this primitive model.
Figure 9. Primitive Model for t-Day Delay Likelihood Using Power Function
3.3.6 Modification Function
The inverse membership function of “Absolutely Likely” fuzzy set transforms
from 𝑥 = 𝑡 to the inverse of the function in Equation (6) according to Equation (7):
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0t 0.1t 0.2t 0.3t 0.4t 0.5t 0.6t 0.7t 0.8t 0.9t 1.0t
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
Primitive Model for t-Day Delay Likelihood
Absolutely Likely
Absolutely Unlikely
Very Likely
Likely
Fairly Likely
Neutral
Fairly Unlikely
Unlikely
Vey Unlikely
31
𝑥 = 𝑡 (1 + √−2𝜎2 ln (𝜎√2𝜋
c∗ 𝜇(𝑥))) ................................... (7)
The exponential function of “Absolutely Likely” can be considered as a modifying
function for all of the fuzzy sets functions because the previous function of “Absolutely
Likely” was given by 𝑥 = 1 and because the values of the fuzzy element 𝑥 are less than
one for all of the inverses of the power functions. This modification is shown in contrast
to the unmodified “Absolutely Likely” fuzzy set for any 𝑡 in Figure 10. Alternatively, the
power functions can be termed “slicing functions” because they slice the “Absolutely
Likely” membership function, as they do in the primitive model in Figure 9.
The combination can be represented by multiplying the inverse functions the
power function and inverse function of the exponential function. The resulting
combination of the power function and the natural logarithm as membership function
relation can be shown in Equation (8)
𝑥 = 𝑡(1 − 𝜇(𝑥))𝑝 ∗ (1 + √−2𝜎2 ln (𝜎√2𝜋
c∗ 𝜇(𝑥))) ........................ (8)
32
Figure 10. Unmodified vs. Exponentially Modified Membership Function of “Absolutely
Likely” Fuzzy Set
3.3.7 The Criteria
The next step would be to find the parameter 𝑝 and the constants of the model. To
find 𝑐 and 𝜎 along with 𝑝𝑓𝑢𝑧𝑧𝑦 𝑠𝑒𝑡, some criteria must be set. Two criteria are concerned
with the fuzzy set “Neutral,” which is the one between “Absolutely Likely” and
“Absolutely Unlikely.” The first is that the area of the fuzzy set “Neutral” must be half
the area of the set “Absolutely Likely” to cover the criteria of the linear change of area.
The area for any fuzzy set can be calculated by integrating Equation (8), as shown in
Equation (9):
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0t 1t 2t 3t 4t 5t
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
Membership Function of "Absolutely Likely" Fuzzy Set
Unmodified Exponentially Modified
33
𝐴 = ∫ 𝑡(1 − 𝜇(𝑥))𝑝 ∗ (1 + √−2𝜎2 ln (𝜎√2𝜋
c∗ 𝜇(𝑥))) 𝑑(𝜇(𝑥))
1
0 .............. (9)
The likelihood of the delay at 𝑥 = 𝑡 being “Neutral” means that it is exactly
between zero and one. Therefore, the other criterion is that the likelihood for the fuzzy set
“Neutral” at 𝑥 = 𝑡 should be 50%. This can be implemented by simply substituting the
fuzzy set “Neutral” membership function with the point (𝜇(𝑥) | 𝑥) = (0.5 | 𝑡). This point
can be considered a center or pivot point for the model.
The two equations emerging from these two criteria for the fuzzy set “Neutral”
are Equations (10) and (11) respectively.
𝐴𝑁𝐸 = 0.5𝐴𝐴𝐿 = ∫ 𝑡(1 − 𝜇(𝑥))𝑝𝑁𝐸 ∗ (1 + √−2𝜎2 ln (𝜎√2𝜋
c∗ 𝜇(𝑥))) 𝑑(𝜇(𝑥))
1
0 (10)
1 = (1 − 0.5)𝑝𝑁𝐸 ∗ (1 + √−2𝜎2 ln (𝜎√2𝜋
c∗ 0.5)) ........................ (11)
Because the maximum value of the normal distribution changes with its standard
deviation, the constant 𝑐 was introduced to adjust that point to a maximum value of one.
This can be represented by Equation (12) by substituting 𝑥 with 𝑡 and 𝜇(𝑥) with one in
Equation (6).
34
1 =𝑐
𝜎√2𝜋 ......................................................... (12)
To make it possible to solve these equations, the area of “Absolutely Likely” set
must be formulated using Equation (9). The result is calculated with the help of Equation
(12) and is shown in Equation (13):
𝐴𝐴𝐿 = ∫ 𝑡 (1 + √−2𝜎2 ln(𝜇(𝑥))) 𝑑(𝜇(𝑥)) = 𝑡 (1 +c
2)
1
0 .................. (13)
Then, combining Equations (10) and (13) results in Equation (14):
0.5 (1 +c
2) = ∫ (1 − 𝜇(𝑥))𝑝𝑁𝐸 ∗ (1 + √−2𝜎2 ln(𝜇(𝑥))) 𝑑(𝜇(𝑥))
1
0 .......... (14)
In Equation (14), the variable 𝑡 is omitted. The system of equations consisting of
Equations (11), (12) and (14) can be solved by substituting c value from Equation (12) in
Equations (11) and (14) and then numerically solving for 𝜎 and 𝑝𝑁𝐸. The numerical
solution is provided in Appendix A. The solution yields the result of 𝜎 = 1.572 ,
𝑐 = 1.572√2𝜋 ≈ 3.9404 and the parameter 𝑝𝑁𝐸 = 1.5114.
Consequently, Equation (8) can be rewritten as Equation (15) after substituting
the resulting constants:
𝑥 = 𝑡(1 − 𝜇(𝑥))𝑝 ∗ (1 + √−4.942368 ln(𝜇(𝑥))) ....................... (15)
35
3.3.8 The Final Step
The final step is solving the functions of the rest of the fuzzy sets for the
parameter 𝑝 while maintaining the linear change in the area by integrating Equation (15)
as in Equation (16).
𝐴 = ∫ 𝑡(1 − 𝜇(𝑥))𝑝
∗ (1 + √−4.942368 ln(𝜇(𝑥)))1
0𝑑(𝜇(𝑥)) .............. (16)
The area of the “Absolutely Likely” must be calculated first because one criterion
is keeping the linear change considering “Absolutely Likely” the extreme likelihood. By
substituting in Equation (16) with 𝑝 = 0, the resulting area is 𝐴𝐴𝐿 = 2.970210𝑡.
Taking the final step, there are nine linguistic variables with eight area changes.
For example, to find the parameter 𝑝 that corresponds to the linguistic variable
“Unlikely,” the area of this fuzzy set must be 2/8 the area of “Absolutely likely” set as it
is the second change of likelihood between the eight changes. Using Equation (16), the
parameter 𝑡 is omitted, and the the result is shown in Equation (17). Solving this equation
numerically for 𝑝𝑈𝑁, the trials are listed in Table 1.
0.742552 = ∫ (1 − 𝜇(𝑥))𝑝𝑈𝑁
∗ (1 + √−4.942368 ln(𝜇(𝑥)))1
0𝑑(𝜇(𝑥)) ...... (17)
36
Trial 𝒑 Error
1 5 -1.7%
2 4.9 -0.30%
3 4.88 -0.014%
4 4.8790 -0.00020%
Table 1. Trials in Numerical Solution for Finding 𝑝𝑈𝑁
Table 2 lists the fuzzy sets with their respective parameter 𝑝, as well as the area
under the curve of the membership function and the membership value at 𝑥 = 𝑡. The
numerical solutions for finding the value of the parameter 𝑝 for all linguistic variables are
provided in Appendix A. In addition, Figure 11 shows the final proposed fuzzy model
with all of its fuzzy sets at 𝑥 = 𝑡. When 𝑡 changes the area under the membership
function changes proportionally, whereas the membership value at 𝑥 = 𝑡 stays the same.
The parameter 𝑝 is not used for the fuzzy set “Absolutely Unlikely” because the
membership function of this fuzzy set does not follow the general formula but rather
Equation (4).
Fuzzy Set 𝒑 Likelihood
𝝁(𝒕)
Area
Absolutely Unlikely - 0% 0
Very Unlikely 12.2068 11.2% 1/8t*AAL = 0.371276t
Unlikely 4.8790 23.4% 2/8t*AAL = 0.742552t
Fairly Unlikely 2.5980 36.3% 3/8t*AAL = 1.113829t
Neutral 1.5114 50.0% 4/8t*AAL = 1.485105t
Fairly Likely 0.8849 64.2% 5/8t*AAL = 1.856381t
Likely 0.4816 78.4% 6/8t*AAL = 2.227657t
Very Likely 0.2028 91.7% 7/8t*AAL = 2.598934t
Absolutely Likely 0.0000 100.0% t*AAL = 2.970210t
Table 2. Final Fuzzy Sets for 1-Day Delay Likelihood and Some of their Properties
37
Figure 11. Proposed Fuzzy Model for t-Day Delay Likelihood
3.2 Fuzzy Time-Delay Model (FTDM)
In this model, the representing value of 𝑝 to the assessment for any duration of
delay 𝑡 days is set as in Table 2. This value will be substituted as the power parameter in
the membership function of the fuzzy set.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
t-Day Delay Fuzzy Model
Absolutely Unlikely Very Unlikely Unlikely
Fairly Unlikely Neutral Fairly Likely
Likely Very Likely Absolutely Likely
38
The Fuzzy Time-Delay Model (FTDM) membership function is shown as a
relation in Equation (18). It is in an inverse form, which means that it is in terms of 𝑥
instead of 𝜇(𝑥).
𝑥 = {
0 , 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑙𝑦 𝑈𝑛𝑙𝑖𝑘𝑒𝑙𝑦
𝑡(1 − 𝜇(𝑥))𝑝 ∗ (1 + √−4.942368 ln(𝜇(𝑥))) , 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (18)
This model’s membership function consists of the fuzzy member variable 𝑥, the two
parameters 𝑝 and 𝑡, and constants. In addition, Equation (18) is a multiplication of the
inverse of a power function with the inverse of an exponential function, which is another
power function and a natural logarithm function, respectively.
In its general form shown in Equation (18), the membership function is set in an
inverse form. It is not defined in the standard form, that is, in terms of 𝜇(𝑥) because of
the mathematical complexity of having 𝜇(𝑥) appears twice in different multiplied
functions. That is not the case for the fuzzy sets “Absolutely Likely” and “Absolutely
Unlikely”. For the “Absolutely Likely” fuzzy set, when substituting 𝑝 with its
corresponding value zero the power function becomes one and 𝜇(𝑥) ends up appearing
once as in Equation (19).
𝑥 = 𝑡 (1 + √−4.942368 ln(𝜇(𝑥))) ................................... (19)
39
The membership function of the “Absolutely Likely” set in terms of the membership
value can be reformulated as in Equation (20).
𝜇(𝑥)𝐴𝐿,𝑡 = {0.8004 𝑒(𝑥
𝑡−1)
2
, 𝑥 > 𝑡1 , 𝑥 ≤ 𝑡
................................... (20)
In contrast, the membership function of the “Absolutely Unlikely” in terms of the
membership is defined according to Equation (4).
As an illustration of a sample fuzzy set, if the assessment of an activity is “The
activity is fairly likely to be delayed for two days”, then the linguistic variable is “Fairly
Likely” and the duration of assessment is two days. Based on Table 2, the result means
that the corresponding value of 𝑝 = 0.8849 and that the value of 𝑡 = 2. The membership
function of this fuzzy set “Fairly Likely” for 2-day delay is shown in Equation (21).
𝑥 = 2(1 − 𝜇(𝑥)𝐹𝐿,2)0.8849
∗ (1 + √−4.942368 ln(𝜇(𝑥)𝐹𝐿,2)) .............. (21)
This membership function can be plotted in a Cartesian coordinate system. The abscissa
would be the fuzzy member, and the membership value would be the ordinate. Figure 12
shows the plot of the membership function of “Fairly Likely” for a 2-day delay.
40
Figure 12. Fuzzy Set "Fairly Unlikely" to Be Delayed for 2-Days
For instance, the membership of the fuzzy member three days (𝑥 = 3) in the
fuzzy set “Fairly Likely” to be delayed for 2 days can be found by either tracing a plot
such as in Figure 12, or by substituting and numerically solving Equation (21). A
numerical solution is shown in Table 3.
Trial 𝜇(3) Error
1 0.5 +2.9%
2 0.51 +0.15%
3 0.511 -0.12%
4 0.5105 +0.016%
5 0.51055 +0.0026%
Table 3. Trials of Numerical Solution for Calculating 𝜇(3) of the Fuzzy Set “Fairly
Likely” to Be Delayed for 2 days
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10 11
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
"Fairly Likely" to be Delayed for 2-Days
0.5106
41
This membership value 𝜇(3) is approximately equal to 0.5106. The value 51.06%
indicates how much the fuzzy member 3 days belongs to the fuzzy set “Fairly Likely” for
a 2-day delay.
Additionally, the fuzzy set can be described by the common practice of showing
the fuzzy sets as pairs of numbers where each pair is the membership value followed by
the corresponding fuzzy member, with a vertical line as a delimiter separating them. The
following is the representation for the fuzzy set “Fairly Likely” (FL) to be delayed for 2
days:
𝐹𝐿, 2 = [ 1|0 , 0.9|0.4488, 0.8|0.9870, 0.7|1.6042, 0.6|2.3015, 0.5|3.0877,
0.4|3.9810, 0.3|5.0169, 0.2|6.2716, 0.1|7.9683, 0|∞ ]
The pairs can selected in a different manner depending on the variable that
requires focus. The previous representation focused on the membership value, so a linear
change in that value shows how much time of delay is expected as the membership
decreases. Another way would be showing the membership values to a constant change
of the time in days for the same example of the fuzzy set “Fairly Likely” (FL) to be
delayed for 2 days as follows:
𝐹𝐿, 2 = [ 1|0 , 0.7977|1, 0.6418|2, 0.5106|3, 0.3980|4, 0.3015|5,
0.2197|6, 0.1522|7, 0.0985|8, 0.0584|9, 0.0311|10, … , 0|∞ ]
42
Not all fuzzy sets can be shown in one graph because the 𝑡 values are infinite
range. However, upon setting a single value for 𝑡, the fuzzy sets for variation in the
parameter 𝑝 produces the transition between the linguistic variable for one duration of
delay. For example, if an assessment for one-day delay is needed, then the value of 𝑡 = 1.
Hence, plotting all of the fuzzy sets, as in Figure 13, shows the variation of the likelihood
of delay and the change in the memberships of the fuzzy members.
Figure 13. Fuzzy Sets for 1-Day Delay Likelihood
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
1-Day Delay Fuzzy Sets in FTDM
Absolutely Unlikely Very Unlikely Unlikely
Fairly Unlikely Neutral Fairly Likely
Likely Very Likely Absolutely Likely
0.67
0.138
0.215
0.301
0.398
0.511
0.644
0.817
43
For instance, Table 4 shows the different memberships for the fuzzy member 𝑥 =
2 for each fuzzy set of the possible assessments for a 1-day delay, which is also shown in
Figure 13.
Fuzzy Set 𝝁(𝒙 = 𝟐)
Absolutely Unlikely (AU) 0.0%
Very Unlikely (VU) 6.7%
Unlikely (UN) 13.8%
Fairly Unlikely (FU) 21.5%
Neutral (NE) 30.1%
Fairly Likely (FL) 39.8%
Likely (LI) 51.1%
Very Likely (VL) 64.4%
Absolutely Likely (AL) 81.7%
Table 4. Membership of Fuzzy Member 2 Days in 1-Day Delay Assessment
Using the same example, when 𝑥 is multiplied by 𝑡, the result represents the
FTDM in general. The change in the membership values at 𝑥 = 𝑡 throughout the different
linguistic variables are shown in Table 2. These values are also plotted in Figure 14. With
an R2 value, which is the coefficient of determination of how well a model fits some line
or curve, of 0.998 and a 2% standard deviation from the perfect linear relation, 𝑦 =
0.125𝑥 , it can be said that the change in the membership at 𝑥 = 𝑡 in the model is almost
linear.
44
Figure 14. Change of Membership at 𝑥 = 𝑡 in FTDM
Because the parameter 𝑡 is a multiplication factor in the general form of the
membership function formula in Equation (15), the fuzzy sets of the same linguistic
variable and the parameter p are similar, and the scalar factor is the parameter 𝑡.
For example, the fuzzy sets of the linguistic variable “Likely,” when assessing the
delay duration of a one-, two- or three-day delay, can be plotted as in Figure 15. For
instance, for the membership value 0.7, the corresponding fuzzy element for a one-, two-
y = 0.1299x - 0.1437R² = 0.998
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1μ
(t)
Linguistic Variable of FTDM
Likelihood Change of Membership at x=t in FTDM
Fuzzy Time Delay Model Perfect Linear Relation Linear (Fuzzy Time Delay Model)
45
and three-day delay assessed as “Likely” are 1.3035, 2.607 and 3.9105 days, respectively.
The results can be calculated by substitution into the general form of the membership
function in Equation (15), or by tracing the plot of the membership in Figure 15.
Figure 15. Membership functions of “Likely” Assessment for One, Two, and Three-Day
Delay
Based on the previous results, it can be deducted that the ratio between the
parameter 𝑡 and the corresponding fuzzy member 𝑥 at the same membership value 𝜇(𝑥)
stays the same. This can be proved by looking at the general form of the membership
function in Equation (18), where 𝑡 plays a multiplicative role in the inverse function.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
"Likely" for One, Two, and Three-Day Delay
Likely for 1 Days Likely for 2 Days Likely for 3 Days
1.3035 2.607 3.9105
46
3.4 Activity Delay Likelihood
One way of creating an activity’s delay likelihood is by directly assessing it. To
do that, the scheduler is repeatedly asked about his/her assessment for the likelihood of
delay for a number of days, starting with one and stopping when the assessment input is
“Absolutely Unlikely.” Then, each input is represented with a fuzzy set from the FTDM.
After that, the gate “Fuzzy Or” is applied on these fuzzy sets to determine the final result
of the activity likelihood. This refinement creates a better representation to the
distribution of delay likelihood for the activity.
In the following example, this process of assessment is demonstrated. If an expert
is asked for his/her opinion on the likelihood of delay for scheduled future activity to
assess, a table with the possible number of days, increasing from one in numerical order,
can be provided. On this survey table, each day must be filled with one of the linguistic
variables of this model. Table 5 shows a survey table filled with an illustrative example.
Delay (Days)
Likelihood of Delay
AL VL L FL N FU U VU AU
1 X
2 X
3 X
4 X
5 X
Table 5. Activity Example – Scheduler Assessment
The scheduler should keep assessing more days of delay until the assessment from
a given number of days is “Absolutely Unlikely.” The assessment should be inversely
proportional with time. This means for more days of delay, the likelihood of delay must
47
decrease. Otherwise, the assessment is not logical. For example, if an activity is “Likely”
to be delayed for 2 days, then it cannot be “Unlikely” to be delayed for 1 day.
The membership of likelihood of delay for the activity would be the result of a
“Fuzzy Or” operation between all the inputs of the scheduler assessment. For each input,
the fuzzy set selected is substituted with the respective 𝑡. Then, the maximum
membership function is selected at each fuzzy member. The “Fuzzy Or” operation is
illustrated in Figure 16.
Figure 16. Example of Creating Activity Delay Likelihood Using “Fuzzy Or”
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 2 4 6 8 10 12 14 16 18
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
Activity Dealy Lieklihood Example
Absolutely Unlikely for 5 Days Very Unlikely for 4 Days
Fairly Unlikely for 3 Days Fairly Likely for 2 Days
Very Likely for 1 Days Fuzzy Or (Max)
(8,0.16)
(4,0.4)
48
This operation combines the highest likelihood in every input to come up with a
better and less conservative representation for the delay likelihood of the activity. In this
way, the representation of the likelihood of delay for an activity is a more accurate
estimation and is a custom calibration. For instance, in Figure 16 and at 𝑥 = 4, the
membership value is approximately 0.4, which is the highest likelihood between the five
fuzzy sets being the “Fairly Likely” for 2 days. Additionally, at 𝑥 = 8, the membership
value is approximately 0.16, which came from the membership of “Fairly Unlikely” for 3
days.
3.5 Defuzzification
Although the model is constructed in a way that maintains a linear change of area
between the fuzzy sets, the area alone is not sufficient for defuzzification. Defuzzification
here means translating the membership function back to the most suitable linguistic
variable and delay duration parameter. The result of an area can have more than one
defuzzification because the model has two parameters. So for each value of the
parameter 𝑡, the area can be read as one of the linguistic variables. That is why each of
the linguistic variables will have a range of defuzzified 𝑡 values.
Upon defuzzification, one of the parameter must be assumed to find the other,
which means that result can be represented with unlimited number of fuzzy sets because
the model has two parameters. For each linguistic variable, the result of defuzzification is
a range of values of 𝑡. The most conservative delay duration in each fuzzy set is the
maximum number of days of delay 𝑡. The maximum number of delay days for the
“Absolutely Unlikely” fuzzy set is always infinity.
49
At each value of 𝑡, the defuzzification process involves simply finding the best-
fitting curve from the fuzzy sets to choose the closest linguistic variable.
The steps of defuzzification are as follows:
1. First, the resulting values are compared with each membership function of the
fuzzy sets at 𝑡 = 1 using the least squares error.
2. This process is repeated by adding one day to the variable 𝑡 at each time, while
recording the value of 𝑡 until the closest fuzzy set is “Absolutely Unlikely.”
3. Then, the maximum 𝑡 recorded for each fuzzy set is set as the maximum number
of delay days expected for that fuzzy set.
Using the same example of the activity assessment filled in by the scheduler in Table 5,
the result of that assessment would be defuzzified as in Table 6.
𝒕
(Days)
Likelihood of Delay
AL VL LI FL NE FU UN VU AU
1 X
2 X
3 X
4 X
5 X
6 X
7 X
8 X
9 X
X: Selected, X: Selected then discarded
Table 6. Defuzzification Process – Example
In Table 6, the cells with the nearest fuzzy set at each 𝑡 are marked with an “X”
according to Steps 1 and 2 in the defuzzification process. For each fuzzy set, if a range of
50
days is recorded, the one with the highest likelihood is selected. The discarded results
have a strikethrough the “X” like this “X.” The remaining results mark the maximum
number of days for each fuzzy set, except for “Absolutely Unlikely,” that would be the
minimum duration of delay for the activity. If for any number of delay days an
assessment is asked, the fuzzy set that includes that range could be selected. For instance,
and for this activity, a delay of 6 days is “Very Unlikely” and a delay of 11 days is
“Absolutely Unlikely.”
For the linguistic variables that have no durations assigned, the number of delay
days of a higher likelihood is assigned to be conservative. For example, if asked how
many days of delay are “Fairly Likely” to occur, it is not 3 days but rather 2 days.
Although it could be useless as the fuzzy set with higher likelihood and the same number
of days prevails in likelihood. That is way such fuzzy sets should not be mention in the
defuzzified result. The defuzzification can be summarized in differently as in Table 7.
51
Fuzzy Set Expected Delay (Range of 𝒕)
[days]
Absolutely Unlikely (AU) 9 ~ infinity
Very Unlikely (VU) 5 ~ 8
Unlikely (UN) 4
Fairly Unlikely (FU) 3
Neutral (NE) 3
Fairly Likely (FL) 2
Likely (LI) 2
Very Likely (VL) 1
Absolutely Likely (AL) 1
Table 7. Defuzzified Result – Example
The results are notable and require explanation. The basic assessment of the
scheduler for a 1-day delay was “Very Likely”, but the defuzzification indicates that it is
“Absolutely Likely” to be delayed for one day. This discrepancy can be explained by the
fact that the scheduler had given the likelihood of delay for more days, which contributes
to the likelihood of a 1-day delay increasing to be “Absolutely Likely.” In contrast, the
assessment was “Absolutely Unlikely” for 5 days, but the defuzzification minimum
duration of delay corresponding to “Absolutely Unlikely” is 9 days. This is due to the
likelihood provided in the assessment of durations of delay less than 5 days, which hold
higher likelihood. The result shows how the fuzzy operation created a different
assessment for the durations from the input assessment done on the same durations
individually.
52
Chapter 4: Converting a CPM Schedule into a Fuzzy Fault-Tree
4.1 Introduction
This chapter includes an introduction to the Critical Path Method, Fuzzy Fault-
Trees, the analysis needed for FTDM application, and methods for converting the CPM
schedule into a FFT.
4.2 Critical Path Method
CPM is one of the best and most wide spread scheduling methods in construction
industry (O’Brien, 2010). It is based on giving each activity one value for its duration,
unlike PERT, for example, which gives three values. Relations such as the Finish-to-Start
relationships are used to link activities. Lag can be used to show the period of time
overlap or duration between the activities.
After the durations of the activities are set and following simple arithmetic
calculations, the earliest start and finish of each activity as well as the latest start and
finish are found. These calculations are called forward and backward passes. Moreover,
the difference between how late and early an activity can start or finish is called the total
float. If the total float is zero or negative, then the activity is critical and lies on a critical
path. The activity is termed critical because if that activity duration is increased, it will
affect the whole project duration.
53
There are many types of float. One type of float is the total float, which was
explained earlier. Another popular and easy to understand type of float is the free float.
The free float is a part of the total float and is the period of time an activity can be
delayed without affecting any other activity in the schedule.
There are many ways to illustrate a schedule graphically. One method is to use a
Gantt/bar chart. In this method, each activity is represented as a bar on a separate line or
row and is plotted no a time scale. Arrows can be used to show logical relationships and
colors can be used to show critical and non-critical activities. Another is an arrow
diagram where the activity is represented by an arrow. In some cases, the logical
relationships require the use of a dummy dashed arrow to indicate the representation
logical. In addition, a predecessor diagram (activity on node) method can be used. In this
method, a rectangular node represents each activity and arrows are used to show the
logical relationships.
Nonetheless, all of the information of a CPM schedule can be summarized in a
table, which contains all of the activities’ information with each row representing an
activity. This information includes but is not limited to the following: Activity Name,
Duration, Early Start, Early Finish, Late Start, Late Finish, Predecessors, Successors,
Total Float and Free Float.
4.3 Fuzzy Fault-tree
Fault-Tree Analysis (FTA) is a type of deductive failure analysis because it
answers the question of how a system can fail or be unavailable. Fault-trees are top down
trees. The top event is called the top undesired event, which represents the failure of the
54
whole system under study; below this top undesired event, lower-level events occur and
are analyzed using binary logic.
Fuzzy Fault-Trees (FFT) are modified fault-trees. They are used to accommodate
the failure events that do not have crisp sets of causes. Therefore, fuzzy sets are used
instead of normal probabilities.
Because the analysis of interest in this study is the project delay, the author chose
a deductive FTA because the analysis varies from consequences to causes. In other
words, all the possible causes of the project delay need to be identified, which is an FTA
method.
Al-Humaidi & Hadipriono Tan (2010a, 2010b) classified the causes of
construction projects delays into procedural, triggering and enabling causes. Most of the
causes of delay are variant, that is, due to human nature, in contrast to machines and due
to the complexity of the projects. The causes of delays are also inconsistent in time,
similar to weather and natural disasters being more inconsistent than controlled and
closed environments. This uncertainty makes it difficult to use well-defined probabilities
of failure to finish an activity in the time scheduled. Thus, the use of fuzzy sets for delay
likelihood can be easier and more accurate.
Fault-trees focus on the failure, and the events are classified according to the type
of failure as primary, secondary and command events. Primary events are used when the
component failure is within its design envelop while secondary events are used when the
failure is outside its design envelope. The combinations of events failures utilize
55
command events. Different shapes are used to represent different types of events, as
shown in Figure 17.
Figure 17. Types of Fault Events
Combining two or more events in a command event requires defining the logic
governing the relation between the events. In fault-trees, these relations are called gates.
“And” and “Or” are examples of common gates and are illustrated in Figure 18.
Figure 18. Logic Gates in Fault-Trees
In addition, the gates can be represented as fuzzy gates to account for the fuzzy
logic calculations, and depend on the fuzzy model chosen. As a result, the “Or” gate
Primary Secondary Command
+ .
Or And
56
becomes the “Fuzzy Or” gate. Another fuzzy gate that is used in this study is the “Fuzzy
Sum” gate, which will be described later in this chapter.
Each activity’s delay likelihood can be created either through direct assessment or
by expanding the deduction to determine the causes of each delay. The first method is
used here because the proposed time-delay model is new and will need to be
demonstrated first.
4.4 The Conversion Analysis
When applying the concept of FFT to this study, delay is considered a failure
event, and each activity’s delay is a primary event. The top undesired event is the delay
of the project. The FFT will require fuzzy gates, such as the “Fuzzy Sum” and “Fuzzy
Or” gate, depending on the method of conversion, which is explained in the next section
in addition to the network interrelations.
4.4.1 Activities in Series
If two activities are related by a finish-to-start relationship, then the likelihood of
delay for the activity is the sum of delay at each level of likelihood. The “Fuzzy Sum”
gate is used here as a simple summation of the fuzzy member in each event, which is
governed by the gate at every level of confidence or by the same membership value, in
the same way as the fuzzy translational model (Hadipriono, 1988b). The order of the
activities in this model does not matter. For each chain of activities that has finish-to-start
relationship and no other relationships, the events are added together using The “Fuzzy
Sum” gate.
57
Just for distinction, the “Fuzzy Sum” applied on Baldwin’s model that was
introduced by Fujino and Hadipriono (1994) is not the same “Fuzzy Sum” used here
because the fuzzy models are different.
To demonstrate the “Fuzzy Sum” operation, an example of a schedule with four
activities is shown in Figure 19. In this figure, a predecessor diagram shows the relations
between the activities A, B, C, and D, as finish-to-start. As a result, the operation that can
combine the failure or delay of these activities is the “Fuzzy Sum”.
Figure 19. Schedule Predecessor Diagram – “Fuzzy Sum” Example
The resulting FFT is shown in Figure 20, where a new command event is created
with the gate “Fuzzy Sum,” which is labeled as “Sum,” and the primary events are
created beneath each activity. The names of the events are the same as the names of the
activities but with an apostrophe added to indicate the failure of the activity to finish on
time, otherwise known as the delay. Note that the “Fuzzy Sum” gate is represented by
inscribing the letter “S” inside a circle in a general gate shape.
B C D A
58
Figure 20. Fuzzy Fault-Tree – “Fuzzy Sum” Example
For each activity in Table 8, an assessment of the likelihood of delay is provided.
For instance, activity B is “Unlikely” to be delayed for 2 days.
Activity Likelihood of Delay Days of Delay
A Very Unlikely 2
B Unlikely 2
C Fairly Unlikely 1
D Neutral 1
Table 8. Delay Likelihood Assessment – “Fuzzy Sum” Example
“Fuzzy Sum” operation adds up the fuzzy members from each event as 𝑥𝑆𝑈𝑀 =
𝑥𝐴 + 𝑥𝐵 + 𝑥𝐶 + 𝑥𝐷 to have the same membership value such that 𝜇(𝑥𝐴) = 𝜇(𝑥𝐵) =
𝜇(𝑥𝐶) = 𝜇(𝑥𝐷) = 𝜇(𝑥𝑆𝑈𝑀). Figure 21 shows the membership functions of the four
events A’, B’, C’, D’, and the result of the operation “Fuzzy Sum.” For instance, as noted
on the plot in Figure 21, at the membership value of 0.3025 the fuzzy member values of
A’, B’, C’, and D’ are 0.0844, 1.1833, 1.3457, 1.9904 days, respectively. The “Fuzzy
Sum” member at the membership 0.3025 is then the summation, which equals 4.6 days.
B’ C’ D’ A’
Sum
S
59
Figure 21. “Fuzzy Sum” Operation Demonstration
4.4.2 Float
If a chain of activities that are related by finish-to-start relationships is not critical,
then the total float for all of the activities will be one value greater than zero. This total
float is shown as the free float for the last activity in the chain. Because the order of the
activities in the chain does not matter, it is sufficient to use the free float. Either way, this
float is modeled is by defining it as 𝑥 = −𝑓𝑙𝑜𝑎𝑡. Then, the float is simply added to the
chain of events under the “Fuzzy Sum” gate. This becomes a modification of the regular
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
"Fuzzy Sum" Operation
A: Very Unlikely for 2 Days B: Unlikely for 2 Days
C: Fairly Unlikely for 1 Days D: Neutral for 1 Days
T: (Fuzzy Sum)
𝑥𝑆𝑈𝑀 = 𝑥𝐴 + 𝑥𝐵 + 𝑥𝐶 + 𝑥𝐷
𝑥𝐶
𝑥𝐵
𝑥𝐷
𝑥𝐴
At the Same Level of Membership 0.3025
4.6
60
fault-trees because the usage of the float should be a condition not an event. Hadipriono
et al. (1987) used the inhibit gate in their fault-tree to account for the free float as a
condition to abide by the fault-tree general rules, which is also a good practice to follow
to keeping the FFT unmodified. In either case, the calculations remain the same.
If, for the previous example (see Figure 21), the chain of activities had a total
float of one day then the membership function of the fuzzy sum would include
subtracting one from each summation operation. For instance, at 𝜇(𝑥) = 0.3025, the
fuzzy member will be 3.6 days instead of 4.6 days.
4.4.3 Activities in Parallel
For two activities that are scheduled to happen at the same time, in which their
successors are some common activity and their predecessor is some other common
activity, see Figure 22, then the likelihood of delay will come from the highest likelihood
of delay for them altogether. Thus, the “Fuzzy Or” operation is used to represent the
relationship between two chains of activities having the same predecessor and the same
successor. “Fuzzy Or” considers the maximum likelihood for each value, of the variable
as shown in Equation (1), page 22.
The “Fuzzy Or” operation has been demonstrated previously in Figure 16, page
47. However, in Figure 22, an example of a CPM schedule is turned into the FFT in
Figure 23. In Figure 22, activities B and C have the same predecessor and the same
successor, so they are scheduled to be executed concurrently. Thus, if any of them is
delayed, the successor will be delayed. As a result, “Fuzzy Or” governs their overall
delay likelihood. To utilizes the “Fuzzy Or” gate, a new command event O is created for
61
the result from the operation with that gate (see Figure 23) and this event O shares a
“Fuzzy Sum” gate with A’ and D’.
Figure 22. Schedule Predecessor Diagram – “Fuzzy Or” Example
Figure 23. Fuzzy Fault-Tree – “Fuzzy Or” Example
B
C
D A
O D’ A’
Sum
S
B’ C’
62
4.5 Methods of Conversion
In this section, the author introduces two methods of converting a CPM schedule
to a FFT. Each is different in its limitations and procedure.
4.5.1 Paths Method
Any complex schedule network can be analyzed with this method. However, all
relations should be finish-to-start, and no lag should be assigned. In this method, every
path is considered to be an entire chain, and all of the paths are considered to be different
possible scenarios. For that reason, it is termed the Paths Method.
In the Path Method fuzzy-fault tree, the top undesired event will have a “Fuzzy
Or” gate, and each path as an event will be below it. For each path, there will be a
command event with a “Fuzzy Sum” gate. Each path will include all of the activities in
its chain. Moreover, the maximum total float for each path is added as a secondary event.
Float is then included in the calculation within the “Fuzzy Sum” gate for that path. Figure
24 shows a generic FFT using this method.
63
Figure 24. Generic FFT using Paths Method
The procedure for converting the CPM table of activities using this method is
shown in Figure 25 as an algorithm. In general, the algorithm will read the table and
follow the activities along each path, in addition to the float, to create the FFT. The
required data from a CPM schedule table are as follows:
Activity ID
Total Float or Free Float
Predecessors and Successors
Duration, Activity Name, Early and Late Start and Finish are all information that help the
scheduler identify the activity during assessment.
Path 2 Path 1
Project Delay
S
S
…
S
.
Activity’ Activity’ Float’
Dashed lines represent an extension
for more possible branching
64
The steps of this algorithm, shown in Figure 25, are described as follows:
First, a “Fuzzy Or” gate is added to the top undesired event, i.e., “Project Delay.”.
A new command event is added under the top undesired event whenever a new
path is started.
The schedule is read to find the starting activity, which is the activity with no
predecessors.
For each activity being read:
o The activity is checked whether it has been read before or not.
If the activity has not been checked, then the likelihood of delay is
retrieved from the scheduler or from a prefilled survey and the
“Fuzzy Or” operation is applied to the input fuzzy sets.
If the activity has been read before, then previous assessment of
the activity is used.
o The maximum total float is recorded for the path being processed.
o A primary event is added with the name given by the Activity ID assigned
during the assessment.
o Then, if the number successors to the activity
is only one, then read that activity
is more than one,
the activity is added to a list of branching activities with a
counter assigned to that activity on the list starting with 1
65
Figure 25. Algorithm of the Paths Method for Converting a CPM Schedule Table into a Fuzzy Fault-Tree
65
66
Whether the activity is new on the list or existing, the
activity with the order of its counter on its successors is
read. For instance, if the activity being read has three
successors and its counter is two, then the second successor
is read.
If there are no successors to the activity, this means it is the last in
the path; thus, if the maximum total float recorded is not zero, then
another event is added with that total float.
Because the algorithm has a list of branching activities, which are the ones with
more than one successors, this list and its counters are manipulated to shift the
path. This is done by the checking the last activity on the list:
o If the activity’s counter equals its number of successors, then the activity
is deleted and the new last activity on the list is checked. This means that
all branches of this activity have been passed and that the previous
branching activity must be checked.
o If not, then its counter is less than its number of successors. In this case,
One is added to its counter. This secures a new path that is passed
in the next run.
A new command event is created under the top event, which
accounts for a new path.
The predefined starting activity is read, but all events are now
added under the new command event for the new path.
67
If the list is empty, meaning that all paths have been passed, then the procedure is
done.
4.5.2 Basic Method
Because the Paths Method requires a great deal of processing because each
activity delay likelihood is calculated for every path it lies on, an alternative but limited
method is proposed to minimize the computations and the size of the fault-tree by
ensuring that each activity is only represented by one event. This is similar in concept to
the minimal cut sets method Hadipriono (1988a) used but involves a different procedure.
This method is called basic because it is limited. The limitation lies in the
schedule network. The network must be restricted to contain either parallel or series
chains without relationships between the chains. Furthermore, the method contains the
limitations of the previous general method.
The steps of this method are shown in Figure 26. This method treats the whole
project as one chain and substitutes small chains and parallel events with equivalent
chains. For this reason, the top undesired event will have a “Fuzzy Sum” gate with the
activities beneath.
This method requires the use of the free float instead of the total float because the
activities of every small continuous chain are combined under the “Fuzzy Sum” gate.
This means that only the float found locally in that chain should be evaluated, not the
total float on the path.
The procedure will actually process one whole path of activities from beginning
to end. Then, by going back to where it had branched earlier, the procedure will continue
68
the process on the different path until arriving back to a preprocessed activity. Then, the
activity will be moved to its correct location on the fault-tree.
The required data fields from the CPM schedule table for this alternative method
are the same as the Paths Method.
The steps of this algorithm, shown in Figure 26, are described as follows:
A variable, termed loop for example, is created with a starting value of zero.
The top undesired event is created for the project delay with a “Fuzzy Sum” gate.
The schedule table is read and the starting activity is found to be the one with no
predecessors.
Reading an activity is performed as follows:
o If it is read for the first time, then
A primary event is created for the activity, which is added under
the current level. Then, assessment of the likelihood of delay is
carried out with “Fuzzy Or”
If there exists a free float assigned to the activity, then a secondary
event is created next to that activity and is associated with the
activity failure event for any further relocation.
The number of successors to the activity can be:
Only one successor: In this case, the successor activity is
read.
69
Figure 26. Algorithm of the Basic Method for Converting a CPM Schedule Table into Fuzzy Fault-Tree
69
70
More than one successor:
o One is added to the variable loop.
o A command event is created with a “Fuzzy Or” gate
as a stakeholder for the result of the combination of
events beneath it.
o A list of successors is created. For the first
successor, the following process is run:
A new command event is created with a
“Fuzzy Sum” gate, and the next events are
created beneath it.
If the successor being processed is the last
successor, the variable loop is decreased by
one.
The successor being processed is read as an
activity.
No successors. In this case, the processing of the activity is
finished, and if the value of the variable loop is
o Zero, then all nested branches have been taken care
of. The procedure is done.
o Otherwise, the level where events are created must
be elevated as filling activities under the current
71
command is done. Then, the next successor on the
list is processed as described earlier.
o If it is the second time reading the activity, then it is raised up two levels
along with any secondary events associated to it. This means that it is
relocated to its parent’s parent activity.
o If this is the third or more time that the activity is being read, no action is
required, and the next step depends on the value of the variable loop:
If the variable loop is zero, then the activity successors’ case is
studied as previously mentioned.
If the variable loop is not zero, then the next successors on the list
is processed as described earlier.
72
Chapter 5: Sample Project
5.1 Introduction
In this chapter, an actual project schedule is analyzed as a demonstration. The
scheduler is asked for an assessment of delay for the activities. Using the Paths Method, a
FFT is created, and the result is defuzzified.
The whole process, except for the CPM calculations, was programmed using the
C# language. Therefore, some aspects of the method need more clarification in a separate
section to explain how the computer program handled the calculations and the
presentation.
The sample project is a renovation project for Boehringer Ingelheim Roxane, Inc.
(BIRI) Spirit Services Building Renovation, Columbus, Ohio. The contractor, who is
responsible for the project and its schedule is Turner Construction Company, Columbus,
Ohio. Appendix B includes the correspondence with Turner Construction Company in
which includes the approval for the project information usage.
5.2 The Project Schedule
The project schedule consists of 36 activities. One of the activities has zero
duration, which means it is a milestone. The start of the project is February 9, 2015.
Table 9 shows a list of these activities and their respective durations and relations.
73
Activity ID
Activity Name Duration Predecessors Successors Early Start Early Finish Late Start Late Finish Free Float
Total Float
A1000 CM RFP Proposal Due 0 A1030, A1040,
A1050 9-Feb-15 9-Feb-15 0 0
A1030 CM Award 2 A1000 A1060 9-Feb-15 10-Feb-15 29-Apr-15 30-Apr-15 14 57
A1040 DD- Estimate 5 A1000 A1060 9-Feb-15 13-Feb-15 24-Apr-15 30-Apr-15 11 54
A1050 CD- Construction Documents 16 A1000 A1060, A1070 9-Feb-15 2-Mar-15 9-Feb-15 2-Mar-15 0 0
A1060 City of Columbus Permit
Process 20
A1050, A1040, A1030
A1370 3-Mar-15 30-Mar-15 1-May-15 28-May-15 43 43
A1070 Bid Period 10 A1050 A1080, A1020, A1120, A1140
3-Mar-15 16-Mar-15 3-Mar-15 16-Mar-15 0 0
A1020 Doors/Frames/Hardware 40 A1070 A1280 17-Mar-15 11-May-15 24-Mar-15 18-May-15 0 5
A1080 GMP Creation/Developed 5 A1070 A1090, A1130, A1110, A1250
17-Mar-15 23-Mar-15 17-Mar-15 23-Mar-15 0 0
A1120 Light Fixtures- Florescent 4wks 20 A1070 A1240 17-Mar-15 13-Apr-15 27-Mar-15 23-Apr-15 0 8
A1140 Fire Alarm Permit 20 A1070 A1190 17-Mar-15 13-Apr-15 1-Apr-15 28-Apr-15 5 11
A1090 GMP Review/Sign-off 5 A1080 A1110, A1160 24-Mar-15 30-Mar-15 1-Apr-15 7-Apr-15 0 6
A1130 Flooring Lead Time 20 A1080 A1270 24-Mar-15 20-Apr-15 24-Mar-15 20-Apr-15 0 0
A1250 Millwork Lead Time 30 A1080 A1300 24-Mar-15 4-May-15 9-Apr-15 20-May-15 0 12
A1110 Demolition Floors, Walls, misc.
ceilings 10 A1080, A1090 A1180 31-Mar-15 13-Apr-15 8-Apr-15 21-Apr-15 0 6
A1160 Wall Covering Removal Existing
Walls 5 A1090 A1170, A1150 31-Mar-15 6-Apr-15 9-Apr-15 15-Apr-15 0 7
A1150 Frame New Walls 4 A1160 A1180, A1260 7-Apr-15 10-Apr-15 16-Apr-15 21-Apr-15 0 7
A1170 Prep Existing walls for Paint 10 A1160 A1230 7-Apr-15 20-Apr-15 1-May-15 14-May-15 12 18
A1260 HVAC Modifications 8 A1150 A1330 13-Apr-15 22-Apr-15 5-May-15 14-May-15 10 16
A1180 In-Wall Electric 5 A1150, A1110 A1190, A1320 14-Apr-15 20-Apr-15 22-Apr-15 28-Apr-15 0 6
Continued
Table 9. CPM Schedule Table for Sample Project
.
73
74
Table 9 Continue
Activity ID
Activity Name Duration Predecessors Successors Early Start Early Finish Late Start Late Finish Free Float
Total Float
A1240 Light Fixture
Replacement/Install A1120 A1330 14-Apr-15 4-May-15 24-Apr-15 14-May-15 2 8
A1190 Rough Wall Inspections A1180, A1140 A1200 21-Apr-15 22-Apr-15 29-Apr-15 30-Apr-15 0 6
A1270 Flooring Installation 15 A1130 A1290, A1310 21-Apr-15 11-May-15 21-Apr-15 11-May-15 0 0
A1320 Technology Cabling 10 A1180 A1330 21-Apr-15 4-May-15 1-May-15 14-May-15 2 8
A1200 Drywall Installation/Patching 5 A1190 A1210, A1220 23-Apr-15 29-Apr-15 1-May-15 7-May-15 0 6
A1210 Drywall Finishing 5 A1200 A1230 30-Apr-15 6-May-15 8-May-15 14-May-15 0 6
A1220 ACT Ceiling Install/Patching 5 A1200 A1330 30-Apr-15 6-May-15 8-May-15 14-May-15 0 6
A1300 Millwork Install 3 A1250 A1360 5-May-15 7-May-15 21-May-15 25-May-15 12 12
A1230 Prime/Paint 10 A1170, A1210 A1370 7-May-15 20-May-15 15-May-15 28-May-15 6 6
A1330 Above Ceiling Inspections 2 A1320, A1220,
A1240, A1260 A1340 7-May-15 8-May-15 15-May-15 18-May-15 0 6
A1340 New/Old ACT Replacement 5 A1330 A1350 11-May-15 15-May-15 19-May-15 25-May-15 0 6
A1280 Doors/Frames/Hardware
Install 5 A1020 A1360 12-May-15 18-May-15 19-May-15 25-May-15 5 5
A1290 Final Painting 10 A1270 A1360 12-May-15 25-May-15 12-May-15 25-May-15 0 0
A1310 Owner System Furniture
Installation 6 A1270 A1370 12-May-15 19-May-15 21-May-15 28-May-15 7 7
A1350 Glass and Glazing 3 A1340 A1370 18-May-15 20-May-15 26-May-15 28-May-15 6 6
A1360 Punch List 3 A1290,
A1280, A1300 A1370 26-May-15 28-May-15 26-May-15 28-May-15 0 0
A1370 Final Inspections 3
A1350, A1060, A1230,
A1310, A1360
29-May-15 2-Jun-15 29-May-15 2-Jun-15 0 0
.
74
75
As an example, activity A1150 is “Frame New Walls” with a duration of 4
working days. Its predecessor A1160 is “Wall Covering Removal Existing Walls”, and its
successors are A1180 and A1260.
The calendar of the project is a standard 5-day calendar, which means that of the
seven days in the week, five are working days. All holidays were omitted for the purpose
of this study.
Some of the relationships were start-to-start with lag. With the help of the
scheduler, these relationships were reconfigured in a way that maintains the progress and
logic of the schedule while abiding by the limitation of this thesis, which requires the
relationships to be finish-to-start. This required splitting them in some cases.
The CPM calculations were performed using a computer program, and the results
are shown in Figure 27, which is a Gantt/bar chart of the project. Due to space
limitations, the entire schedule is shown in Figure 27, while partial screen shots give a
closer look in Appendix C. The red bars represent the critical activities. The green bars
are non-critical activities, which mean that they have a total float associated with them.
Arrows show the relationships between the activities.
In addition, the CPM results are populated in Table 9 and then sorted according to
the early start column. The critical activities can also be identified in this table as the ones
with zero days of total float.
76
Figure 27. Gantt Chart – Sample Project
76
77
According to the CPM passes, the projected finish is Jun 2, 2015 with a total
duration of 82 working days. The project schedule has nine critical activities. The value
of the total values of the activities has a maximum of 57 days. Additionally, the free float
values are observed for 13 activities, with a maximum value of 43 days.
The activity A1150 can start as early as April 7, 2015, and as late as April 16,
2015. It has no free float, but because it has a total float of 7 days, it is not critical.
Following the relationships from the starting activity A1000 to the finishing one
A1370, the schedule consists of 21 paths. Only one critical path exists in this schedule.
This path includes all of the critical activities in the schedule. The critical path is A1000,
A1050, A1070, A1080, A1130, A1270, A1290, A1360, and then A1370.
Some paths can be considered near critical, depending on the value of the total
float. The value below which the path is considered near critical varies from project to
project and depends on the scheduler. For this project, the scheduler considers paths with
five working days of float or less to be near critical. Only one path follows this criterion
with five critical activities of the seven total activities. This near critical path is A1000,
A1050, A1070, A1020, A1280, A1360, and then A1370.
5.3 Scheduler Assessment
An assessment is filled out by the scheduler. As previously explained, an answer
to the question “What is the likelihood of delay for 1 day for this activity?” is sought. For
the same activity, the question is repeated again for a greater number of days until the
assessment is “Absolutely Unlikely”. Subsequently, this questionnaire is repeated for all
of the activities.
78
For this project, Table 10 shows that assessment summary with the activities
sorted by early start, similar to that in Table 9. Critical activities are noted with an
asterisk next to its ID.
Upon observing the scheduler assessment, it is noted that the schedule has little
likelihood of delay because it is believed that not a single activity has any likelihood of
delay for more than three days. This is especially true given that the minimum near-
critical path has 5 days of total float. In addition, no single assessment has the high
likelihood band of “Fairly Likely” or more. Therefore, for the whole project, the expected
number of days of delay for the low likelihood band should be moderate.
In contrast, some critical activities have some likelihood of delay. This will
accumulate some likelihood of delay for the whole project, and might be sufficient to
appear in the high likelihood of delay. Moreover, activity A1270 is a critical activity and
has a delay likelihood assessment of “Very Unlikely” for 2 days. This will cause a major
addition to the high likelihood of delay of the whole the project.
79
Activity ID 1-Day delay 2-Day delay 3-Day delay 4-Day delay
A1000* AU
A1030 AU
A1040 VU AU
A1050* AU
A1060 VU AU
A1070* AU
A1020 FU UN VU AU
A1080* VU AU
A1120 VU AU
A1140 VU VU VU AU
A1090 VU AU
A1130* VU AU
A1250 VU AU
A1110 UN VU AU
A1160 VU AU
A1150 UN AU
A1170 VU AU
A1260 FU VU AU
A1180 VU AU
A1240 UN AU
A1190 VU AU
A1270* VU VU AU
A1320 UN VU AU
A1200 VU AU
A1210 VU AU
A1220 VU AU
A1300 UN VU AU
A1230 UN UN VU AU
A1330 VU AU
A1340 VU AU
A1280 UN VU AU
A1290* UN AU
A1310 NE UN VU AU
A1350 VU AU
A1360* VU AU
A1370* VU AU
NE: Neutral, FU: Fairly Unlikely, UN: Unlikely, VU: Very Unlikely, AU: AU
* Critical Activity in CPM
Table 10. Scheduler Delay Subjective Assessment for the Sample Project
80
5.4 Computer Application
In this section, the computer output and the issues that may face any user of the
FTDM utilizing a computer are discussed.
5.4.1 Using Coordinates
After applying the different fuzzy operations, the membership functions
occasionally lose their smooth transition, and it can thus be more difficult to represent the
trend in a function form. In addition, in the computer program, it is easier to use points or
coordinates rather than mathematical functions. This causes an issue of how well the
membership function is represented.
For example, if a constant relation of ordinate distribution were used to plot a
hundred points; {1,0.99,0.98,…,0.02,0.01,0} , then the minimum value of interest in the
ordinate would be 0.01 as the value 0 is already known to have ∞ as a coordinate. This
means that the fuzzy members of membership less than 1% are not recorded.
Furthermore, substituting this value in any of the fuzzy sets would come up with a small
range of fuzzy member values, although the membership functions have tails extending
to infinity. For demonstration, the fuzzy set “Neutral” to be delayed for 1 day is plotted
using 100 points with linear relationship in Figure 28. In this Figure, the plotted points
are dense at the lower values of the fuzzy member and then they are spaced as the fuzzy
member value increases.
81
Figure 28. Representation of Fuzzy Set "Neutral" for 1 Day with 100 Points of Linear
Ordinate Relationship
To overcome this issue, the author suggests using a quadratic relationship as a
distribution for the ordinate to improve the representation of the membership function
tail. This relationship can be shown in Equation (21), where 𝑟 is the rank out of a hundred
and 𝑟’ is the ordinate to be used.
𝑟′ = (𝑟
100)
2
....................................................... (22)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
"Neutral" for 1 Day
82
To show this modification, the point ordinate mentioned earlier transforms from
0.01 to 0.0001. For example, more than 99.99% of the area of the fuzzy set “Neutral” is
covered with is modification instead of 99.6% without. The improvement can be
visualized in Figure 29 where the same “Neutral” to be delayed for 1 day fuzzy set is
plotted in the same range of fuzzy member 0-8 days in comparison with the unmodified
points in Figure 28. In Figure 29, although the points are spaced at the tail, they have
reached a very low membership values and they have covered almost all of the summed
likelihood of delay.
Figure 29. Representation of Fuzzy Set "Neutral" for 1 Day with 100 Points of Quadratic
Ordinate Relationship
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
Mem
ber
ship
μ(x
)
Fuzzy Member x (Days)
"Neutral" for 1 Day
83
Further, in the defuzzification process and after the point where 99.99% of the
area is covered, the tail will have less impact on the result because all of the fuzzy sets
converge to the same zero point at infinity. As a result, the tails become very close to the
abscissa and become very similar.
5.4.2 Fuzzy Operations
In this study analysis, there exist only two operations: “Fuzzy Sum” and “Fuzzy
Or.” The operation “Fuzzy Sum” can be calculated by adding the fuzzy element for each
value of the membership. While the operation “Fuzzy Or” is the maximum membership
function at each fuzzy membership. However, the maximum fuzzy member value at the
same membership value have the same result because all the membership functions are
connected between the model’s two poles or points (1,0) and (0, ∞) and are always
decreasing. Thus, the same number of points for each membership function is sufficient
as the both operations can be done on the same value of membership.
5.4.3 Rearranging FFT
Both by observing visually and by using the Paths Method, the entire FFT would
be very wide, considering that all of the events would have to be next to each other for all
paths at the same horizontal level. In an attempt to reduce the effect of this problem, the
software is programmed to reorder the FFT in a way that stacks all the paths’ chain of
events over each other to allow the layout to be more compact and practical. This makes
it easier to navigate by scrolling through on a computer screen but still does not solve the
84
need for large sized paper for printing out. Moreover, the paths are numbered to make
them distinguishable.
5.4.4 Plotting Membership Functions
Due to the limited space, the membership function for each event in the FFT is
plotted in a square. However, the timescale is changed for each event to show to what
extent the function covers almost 99% of its area. The time scale is shown by drawing red
vertical lines as grid lines at integer numbers of days, and a green line is drawn at zero as
illustrated in Figure 30.
Additionally, using coordinates to plot the membership function will mean having
a limited number of points. Therefore, the maximum fuzzy member value can be the limit
at which the plot will stop and that is how the timescale can be decided.
Figure 30. Representation of the Fuzzy Membership Function in the Computer Program
Fuzzy Member (x) in Days
Membership Value μ(x)
Membership Function (Blue)
Gridlines (Red) at
Integer Values of 𝑥
Line (Green):
𝑥 = 0
85
From Figure 30, the likelihood of delay can be read for this event following the
membership function in blue. While the range of the plot in time is 11 days, which is the
count of the red lines, and the plot starts from zero on the far left. It can be concluded that
the likelihood of an 11-day delay is very low. On the other hand, as the membership
value of a 1-day delay is exceeding one-third the plot height and so the likelihood of 1-
day delay would be significant. However, the whole membership function must be
defuzzified to give results that compile all the likelihood of delay.
5.5 Analysis Results
With Intel® Core™ i5-4200U at 1.6GHz processor, computer application written
on C# took 55 seconds for processing the whole analysis of this sample project using the
Paths Method and utilizing 100 points for each membership function including activities’
refinement, fuzzy operations, drawing FFT and defuzzifying the top undesired event.
5.5.1 FFT
Due to the limited space, as explained earlier, the entire FFT is too large to
display on a paper print out, so it is shown in Figure 31, as compiled from screenshots.
However, partial shots of the tree are displayed in larger segments during the discussion
of the FFT analysis, and Appendix D contains all of the paths’ FFTs separately. In
Appendix D, whenever the path has more than ten events beneath it, it issplit into two
rows.
86
Figure 31. FFT Layout - Sample Project
87
5.5.2 Likelihood of Project Delay
First, the most important result is the top undesired event, which is the delay of
the project. The associated membership function is as follows:
𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐷𝑒𝑙𝑎𝑦 = [ 1|0, 0.9|0, 0.8|0.0008, 0.7|0.0060, 0.6|0.0297, 0.5|0.0989,
0.4|0.2806, 0.3|0.8627, 0.2|2.843, 0.1|11.68, 0|∞ ]
Alternatively, it can be presented as:
𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐷𝑒𝑙𝑎𝑦 = [ 1|0, 0.2884|1, 0.2296|2, 0.1924|3, 0.1705|4, 0.1538|5,
0.1381|6, 0.1270|7, 0.1192|8, 0.1129|9, 0|∞ ]
These numbers are pairs of membership and fuzzy member. For instance, the pair
(0.2|2.843) means that with a membership of 20% a delay of 2.84 days is possible, while
the pair (0.1381|6) means that a delay of 6 days is believed to be possible with 13.8% of
membership. It is hard to deduct the likelihood of delay by reading these values
separately nonetheless, the defuzzification compiles all of them into more meaningful
delay likelihood.
According to the FTDM, the total likelihood of delay is defuzzified and illustrated
as in Figure 32. For clarification, the red background behind the membership function is
actually a very dense set of vertical red gridlines that represent the fuzzy member 𝑥 at
integer values, as explained earlier.
88
Figure 32. Defuzzified Likelihood of Project Delay – Sample Project
The results shown in Figure 32 support the previous logical analysis of the
scheduler assessment, with a several number of days in the low likelihood range and a
few of days in the high likelihood. For example, the results shows that the project is
“Very Unlikely” to be delayed for 21 days and “Likely” to be delayed for 1 day. In
addition, the project delay is not “Absolutely Likely” nor “Very Likely” to be delayed.
The scheduler can add further duration to the project to account for the expected
risk of delay. For this project, a duration of 1 day is suggested to be added to the schedule
in its critical path because a 1-day delay is “Likely” to occur according to this analysis
and the FTDM.
The paths that contribute in the likelihood of delay for the whole project can be
deduced by comparing the membership functions of each path with the membership
function of the project delay. The paths at which the membership range has the maximum
89
values of the fuzzy members will be the ones that contribute to the project delay because
the delay is governed by the “Fuzzy Or” gate using the Paths Method.
5.5.3 Criticality
The critical path is no longer the one governed by CPM but rather the ones that
contribute in the delay likelihood of the whole project. However, in this sample project,
the paths that contributed to the delay likelihood of the whole project are paths number 8,
9 and 12, as shown by Table 11, with the contributing values highlighted. Path 12 is the
critical path for this project CPM, and its FFT is shown in Figure 33.
It is notable that non-critical paths contribute in the project delay likelihood. The
total float of both paths 8 and 9 is 6 days, which is shown in their FFT in Figure 34 and
Figure 35. The critical path is dominant at a membership value of approximately 13%
and above, with a corresponding fuzzy member of nearly 𝑥 = 7 days or less. The
likelihood of delay for 𝑥 > 7 days is governed by path 8 until around 𝑥 = 24 days of
delay, and it is then shared with path 9 afterwards, that is, with a delay likelihood less
than approximately 5.8%.
In this sample project, the critical path from CPM (path 12) is still the major path
that contributes in the high likelihood of delay. However, it is not the only contributor.
90
μ(x) 0.0441 0.0484 0.0529 0.0576 0.0625 0.0676 0.0729 0.0784 0.0841 0.09 0.0961 0.1024 0.1089 0.1156 0.1225 0.1296 0.1369 0.1444 Event Name
Fuzzy Member x
Project Delay
29.21 27.05 24.95 23.12 21.40 19.74 18.13 16.59 15.10 13.68 12.33 11.04 9.81 8.64 7.55 6.73 6.13 5.57
Path 1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Path 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Path 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Path 4 28.10 25.89 23.74 21.87 20.11 18.41 16.78 15.21 13.71 12.28 11.05 9.87 8.75 7.69 6.69 5.74 4.85 4.01
Path 5 28.10 25.89 23.74 21.66 19.66 17.74 15.90 14.14 12.47 10.89 9.51 8.22 7.01 5.86 4.80 3.80 2.87 2.01
Path 6 25.26 23.23 21.26 19.36 17.52 15.76 14.07 12.46 10.93 9.48 8.36 7.30 6.30 5.35 4.46 3.62 2.83 2.09
Path 7 10.73 9.26 7.83 6.65 5.56 4.50 3.48 2.50 1.56 0.65 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Path 8 29.21 27.05 24.95 23.12 21.40 19.74 18.13 16.59 15.10 13.68 12.33 11.04 9.81 8.64 7.55 6.51 5.53 4.62
Path 9 29.21 27.05 24.95 22.92 20.96 19.07 17.25 15.52 13.86 12.29 10.80 9.39 8.06 6.82 5.65 4.57 3.56 2.62
Path 10 26.37 24.39 22.47 20.61 18.82 17.09 15.43 13.84 12.32 10.88 9.64 8.46 7.35 6.30 5.31 4.39 3.52 2.70
Path 11 19.53 17.74 16.00 14.31 12.68 11.11 9.73 8.48 7.28 6.14 5.05 4.02 3.04 2.11 1.23 0.41 0.00 0.00
Path 12 21.00 19.76 18.56 17.39 16.26 15.17 14.13 13.12 12.17 11.25 10.38 9.56 8.79 8.06 7.37 6.73 6.13 5.57
Path 13 15.73 14.26 12.83 11.65 10.56 9.50 8.48 7.50 6.56 5.65 4.79 3.97 3.19 2.45 1.74 1.08 0.46 0.00
Path 14 25.26 23.23 21.26 19.56 17.97 16.44 14.96 13.53 12.17 10.88 9.76 8.71 7.70 6.74 5.83 4.97 4.16 3.40
Path 15 25.26 23.23 21.26 19.36 17.52 15.76 14.07 12.46 10.93 9.48 8.23 7.06 5.95 4.91 3.94 3.03 2.19 1.40
Path 16 22.41 20.57 18.78 17.05 15.39 13.78 12.25 10.79 9.39 8.07 7.07 6.13 5.24 4.40 3.60 2.85 2.14 1.48
Path 17 5.05 3.94 2.87 1.83 0.83 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Path 18 14.89 13.60 12.35 11.34 10.42 9.52 8.66 7.82 7.02 6.25 5.63 5.04 4.48 3.93 3.41 2.90 2.42 1.96
Path 19 12.16 11.11 10.08 9.09 8.12 7.20 6.30 5.45 4.63 3.84 3.10 2.40 1.73 1.10 0.51 0.00 0.00 0.00
Path 20 22.41 20.57 18.78 17.26 15.83 14.46 13.13 11.86 10.63 9.47 8.36 7.30 6.30 5.35 4.46 3.62 2.83 2.09
Path 21 22.41 20.57 18.78 17.05 15.39 13.78 12.25 10.79 9.39 8.07 6.83 5.65 4.55 3.52 2.56 1.68 0.85 0.10
Table 11. Paths and Project Delay Fuzzy Members Values – Highlighting Contributing Paths in Project Delay Likelihood
90
91
Figure 33. Path 12 FFT
91
92
Figure 34. Path 8 FFT
92
93
Figure 35. Path 9 FFT
93
94
Chapter 6: Summary, Conclusions and Recommendations
6.1 Summary
The author proposes a new method of assessing the risk of project delay. The risk
of the project is defined in this study by the risk, excluding the opportunity or safety. The
assessment, however, is both quantitative and qualitative at the same time, and the
durations of delay are subjectively described with linguistic variables, such as “Very
Likely” to be delayed for 1 day.
In this study, the assessment of the delay likelihood is obtained from the
scheduler. The delay is thus not dependent on the causes of the delay because the
subjective assessment assumes that these causes are considered during assessment.
The methods in this study are not a substitute for the CPM but instead transform
its deterministic nature into a more realistic one. All CPM calculations must be carried
out before applying the proposed method because the method depends on the CPM float
values. In contrast to CPM, which only shows a unique critical path, this method shows
the paths that contribute to the potential likelihood of delay. Furthermore, all of the paths
can be assessed for likelihood of delay.
The entire analysis depends on the FTDM that the author proposes. This model
relies on fuzzy logic as a platform and is unique because it is the first to consider the
delay likelihood using a quantitative measure. The model describes the likelihood of
95
delay using colloquial words for a certain period of delay, such as “Very Likely” to be
delayed for one day and “Unlikely” to be delayed for two days.
FTDM is based upon several assumptions and criteria. First, it utilizes the normal
distribution as the maximum likelihood of delay for any period of time after that period.
The normal distribution is an exponential function, which fits the indefinite nature of the
time variable. In addition, the model is embedded with a power function to distinguish
between the different levels of likelihood, while maintaining a linear change of area
because the area represents the total likelihood of delay.
The FTDM can be used to refine the delay likelihood of an activity by acquiring
several input from the scheduler assessing the likelihood of delay for more than one delay
period. That is done by combining all the input and considering the maximum delay
likelihood for all of the activities.
The author proposes two methods to convert the CPM schedule to a FFT. The
first method is called the Paths Method as it follows each path in the schedule and treats
each path as one chain of events, each of which is an activity delay likelihood that is
combined using the “Fuzzy Sum” gate. Then, the top undesired event (Project Delay)
combines the highest likelihood of all the paths using the “Fuzzy Or” gate.
The other method is intended for a simple network in which no relationships
between activities in parallel exist. This method is called the basic method and minimizes
the calculations by allowing each activity delay event to be mentioned just once in the
whole FFT. In this method, the whole project is treated as one chain of events (in series),
96
which are combined by a “Fuzzy Sum” gate. Activities in parallel are treated with a
command event with a “Fuzzy Or” gate.
The defuzzification, which translates the resulting delay likelihood, uses a simple
process of finding the best fitting curve with the least square errors value. This is done to
the top undesired event, i.e. the project delay, and can be done to any event in the FFT in
the same way as the paths’ command events.
The results of the FFT analysis are used to assess the risk of delay for the project
and to make the appropriate adjustment to the activities durations to dampen that risk.
The author wrote and used a computer program to carry out the fuzzy
calculations, draw the FFT and defuzzify the results. Some modifications were made such
as rearranging the layout of the FFT and configuring the ordinate in a quadratic
relationship for a better representation.
6.2 Conclusions
The FTDM is a promising fuzzy model that solves a problem that other fuzzy
models could not solve, which is combining the linguistic variables with a parameter of
interest, i.e., the delay period under assessment. The result is a linguistic assessment of a
quantifiable delay, such as the statement that the project is “Likely” to be delayed for 2
days. Such a complex model will need further research and development.
The area under the membership function in many fuzzy models, including
Baldwin’s model, is used as index for the fuzzy sets and for defuzzification. However,
the area in the FTDM represents the total expectation of the delay likelihood, but it is not
the defuzzification criterion because this model has two parameters.
97
For all fuzzy sets in the FTDM, the delay likelihood diminishes after 7t days.
Therefore, for a 1-day delay assessment, the likelihood of delay is believed to be almost
negligible for more than a week regardless of the linguistic variable chosen. This can be
logical in a sense where construction industry usually monitors a week progress per
minimum for its schedule updates. Nonetheless, the refinement of the likelihood of delay
for the activities covers that issue if more than 1-day delay is expected.
Although each activity delay likelihood is refined by multiple inputs, this method
of determining the delay of an activity has the issue that the assessment relies primarily
on the scheduler assessment. There would be no way to consider whether the scheduler
assessment is right or wrong. However, the experience of the scheduler can be an
indicator of accuracy. Because the schedule is produced by the same scheduler, the
assessment can be assumed to be consistent. Moreover, that is not the only possible way
of using the FTDM to refine the delay likelihood of the activity.
The FFT resulting from the Paths Method of analysis is huge, but by using
today’s super-fast processors the computational time is still acceptable. Though the basic
method requires less computational time, it is limited and can be impractical for most
construction schedules, which are often complex.
The defuzzification of the top undesired event splits the time scale into durations
of possible delay and gives them linguistic terms that describe that likelihood of delay for
these durations. This allows the scheduler to understand the potential for delay better and
so readjust the durations of the activities. At the very least, the risk of delay is better
assessed.
98
FFT analysis creates a new concept of criticality from deterministic critical path
from CPM to including all the paths that contribute in the likelihood of delay for the
project. Consequently, all of the activities in those paths, particularly those with float,
should have their durations readjusted or at least monitor their progress upon updating the
schedule.
In the analysis of a sample project scheduled using CPM the FFT shows logical
results. Because the schedule was complex, the path method was used for analysis and
creating the FFT. The critical path of the CPM was not the only path that holds likelihood
of delay. Nevertheless, a group of three paths contributed in the likelihood of delay for
the project even though two of them had 6 days of float.
The assessment by the scheduler on the likelihood of delay for the activities did
not exceed a likelihood of delay of more than 3 days. In addition, none of the critical
activities had an assessment of more that “Unlikely” to be delayed for 1 day. However,
the project was “Likely” to be delayed for 1 day according to the FFT analysis using the
FTDM. This shows how the analysis compiled different likelihoods of delay into one
resulting risk of delay, which should be brought to the scheduler.
The need for computers to carry out the calculations is unescapable, due to the
mathematics of the fuzzy model, the calculations of the fuzzy operations and the
complexity of the schedules’ networks. However, the use of computers prevails other
issues that need closer look and customized solutions.
6.3 Recommendations
99
In the FTDM, depending on the type of the activity, it may be more appropriate to
use functions other than normal distributions for the membership function of the fuzzy set
“Absolutely Likely.” There are many other bell shaped functions to be tried and tested,
such as the Beta distribution. In addition, a simplification for the membership function
may be sought to make it easier and faster to compute the fuzzy logic calculations.
Finding the best fitting curve was the most practical way of defuzzification.
Nevertheless, there could be other ways.
The float fuzzy set model was simple and direct, but there could be a reason to
have a specific model for it.
In this study, the assessment required from the scheduler on the likelihood of
delay for an activity is for all of its progress from its starting point to its finishing point. If
the assessment of an activity delay is split into two assessments, one for the activity to
start and another for it to finish, then this may help overcome the limitation of the
relationships between the activities. However, it can be more difficult for the scheduler to
make the assessment in this case.
Alternatively, a fuzzy inference system can be developed applying the FTDM,
where rules are created to transform causes of delay for each activity to likelihood of
delay. Then, by collecting historical data or a survey of experts’ opinions, the rules can be
determined. In this way, the scheduler assessment can be a calibrated.
Although the FTDM is applied in this study on a CPM schedule and thus falls into
the area of construction management, other areas may find this model applicable to
problems related to time and delay.
100
Some of the limitations of this study that are related to the schedule, such as
constraints on dates and lag, will need more research to determine how to incorporate
them and how they are translated into the FFT. The notion of splitting the events for each
activity into a delay of start and a delay of finish can be helpful in overcoming these
restrictions.
This method of analysis with a new fuzzy model requires testing in more projects.
The assessment of the scheduler before the project started should be compared with the
actual delay that occurred. This will require a consistent schedule because if updating the
schedule includes adding or removing activities or relationships, it will be difficult to
make a valid comparison.
101
Bibliography
AbouRizk, S. M., & Sawhney, A. (1993). Subjective and interactive duration estimation.
Canadian Journal of Civil Engineering, 20(3), 457–470.
Al-Humaidi, H. M., & Hadipriono Tan, F. (2010a). A fuzzy logic approach to model
delays in construction projects using rotational fuzzy fault tree models. Civil
Engineering and Environmental Systems, 27(4), 329–351.
http://doi.org/10.1080/10286600903150721
Al-Humaidi, H. M., & Hadipriono Tan, F. (2010b). A fuzzy logic approach to model
delays in construction projects using translational models. Civil Engineering and
Environmental Systems, 27(4), 353–364.
http://doi.org/10.1080/10286600903362797
Ang, A. H.-S., Chaker, A. A., & Abdelnour, J. (1975). Analysis of Activity Networks
under Uncertainty. Journal of the Engineering Mechanics Division, 101(4), 373–
387.
Ayyub, B. M., & Haldar, A. (1984). Project scheduling using fuzzy set concepts. Journal
of Construction Engineering and Management, 110(2), 189–204.
Baldwin, J. F. (1979). A new approach to approximate reasoning using a fuzzy logic.
Fuzzy Sets and Systems, 2(4), 309–325.
102
Boussabaine, A. H. (2001). Neurofuzzy modelling of construction projects’ duration I:
principles. Engineering Construction and Architectural Management, 8(2), 104–
113.
Dawood, N. (1998). Estimating project and activity duration: a risk management
approach using network analysis. Construction Management and Economics,
16(1), 41–48. http://doi.org/10.1080/014461998372574
Diaz, C. F., & Hadipriono, F. C. (1993). Nondeterministic networking methods. Journal
of Construction Engineering and Management, 119(1), 40–57.
Diaz-Suarez, C. F. (1989). Probabilistic network analyses for construction projects.
Ditlevsen, O. (1979). Narrow Reliability Bounds for Structural Systems. Journal of
Structural Mechanics, 7(4), 453–472. http://doi.org/10.1080/03601217908905329
Fazar, W. (1959). Program evaluation and review technique. The American Statistician,
13(2), 10.
Fujino, T., & Hadipriono, F. C. (1994). New gate operations of fuzzy fault tree analysis.
In Fuzzy Systems, 1994. IEEE World Congress on Computational Intelligence.,
Proceedings of the Third IEEE Conference on (pp. 1246–1251). IEEE. Retrieved
from http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=343643
Goldratt, E. M. (1997). Critical chain. North River Press Great Barrington, MA.
Retrieved from http://www.chaine-
critique.com/medias/fichiers/resume_2_du_livre_critical_chain.pdf
Guo, Q. L. (2001). Development of risk analysis models for decision-making in project
management.
103
Hadipriono, F. C. (1988a). Fault tree network analysis for construction. Civil Engineering
Systems, 5(1), 42–48. http://doi.org/10.1080/02630258808970501
Hadipriono, F. C. (1988b). Fuzzy set concepts for evaluating performance of constructed
facilities. Journal of Performance of Constructed Facilities, 2(4), 209–225.
Hadipriono, F. C. (1995). Fuzzy sets in probabilistic structural mechanics. In
Probabilistic Structural Mechanics Handbook (pp. 280–316). Springer.
Hadipriono, F. C., Larew, R. E., & Lin, C. C. S. (1987). A New Networking Method For
Construction Education. In ASC Proceedings of the 23rd Annual Conference (pp.
62–67). Purdue University - West Lafayette, Indiana. Retrieved from
http://ascpro0.ascweb.org/archives/cd/1987/Hadipriono87.htm
Ingle, M. M., Atique, M., & Dahad, S. O. (2011). RISK ANALYSIS USING FUZZY
LOGIC, 96–99.
Jun, D. H., & El-Rayes, K. (2011). Fast and Accurate Risk Evaluation for Scheduling
Large-Scale Construction Projects. Journal of Computing in Civil Engineering,
25(5), 407–417. http://doi.org/10.1061/(ASCE)CP.1943-5487.0000106
Kelley, J. E., Jr., & Walker, M. R. (1959). Critical-path planning and scheduling. In
Papers presented at the December 1-3, 1959, eastern joint IRE-AIEE-ACM
computer conference (pp. 160–173). ACM. Retrieved from
http://dl.acm.org/citation.cfm?id=1460318
Kim, J., Lee, S., Hong, T., & Han, S. (2006). Activity vulnerability index for delay risk
forecasting. Canadian Journal of Civil Engineering, 33(10), 1261–1270.
http://doi.org/10.1139/l06-075
104
Long, L. D., & Ohsato, A. (2008). Fuzzy critical chain method for project scheduling
under resource constraints and uncertainty. International Journal of Project
Management, 26(6), 688–698. http://doi.org/10.1016/j.ijproman.2007.09.012
Lorterapong, P., & Moselhi, O. (1996). Project-network analysis using fuzzy sets theory.
Journal of Construction Engineering and Management, 122(4), 308–318.
Malcolm, D. G., Roseboom, J. H., Clark, C. E., & Fazar, W. (1959). Application of a
Technique for Research and Development Program Evaluation. Operations
Research, 7(5), 646–669. http://doi.org/10.1287/opre.7.5.646
Mulholland, B., & Christian, J. (1999). Risk assessment in construction schedules.
Journal of Construction Engineering and Management, 125(1), 8–15.
Nasir, D., McCabe, B., & Hartono, L. (2003). Evaluating Risk in Construction–Schedule
Model (ERIC–S): Construction Schedule Risk Model. Journal of Construction
Engineering and Management, 129(5), 518–527.
http://doi.org/10.1061/(ASCE)0733-9364(2003)129:5(518)
O’Brien, J. J. (2010). CPM in construction management (7th ed). New York: McGraw-
Hill.
Ökmen, Ö., & Öztaş, A. (2008). Construction project network evaluation with correlated
schedule risk analysis model. Journal of Construction Engineering and
Management, 134(1), 49–63.
Oliveros, A. V. O., & Fayek, A. R. (2005). Fuzzy Logic Approach for Activity Delay
Analysis and Schedule Updating. Journal of Construction Engineering and
105
Management, 131(1), 42–51. http://doi.org/10.1061/(ASCE)0733-
9364(2005)131:1(42)
Pelletier, F. J. (2000). Hájek Petr. Metamathematics of fuzzy logic. Trends in logic, vol.
4. Kluwer Academic Publishers, Dordrecht, Boston, and London, 1998, viii + 297
pp. Bulletin of Symbolic Logic, 6(03), 342–346. http://doi.org/10.2307/421060
Ranasinghe, K. A. M. K. (1990). Analytical method for quantification of economic risks
during feasibility analysis for large engineering projects.
Ranasinghe, M. (1994). Quantification and management of uncertainty in activity
duration networks. Construction Management and Economics, 12(1), 15–29.
http://doi.org/10.1080/01446199400000003
Sebt, M. H., Rajaei, H., & Pakseresht, M. M. (2007). A Fuzzy Modeling Approach to
Weather Delays Analysis in Construction Projects. INTERNATIONAL JOURNAL
OF CIVIL ENGINEERING, 5(3), 169–181.
Shiraishi, N., & Furuta, H. (1983). Reliability analysis based on fuzzy probability.
Journal of Engineering Mechanics, 109(6), 1445–1459.
Tirtotjonro, H. S. (Harry S. (1986). Construction network scheduling using a modified
fault tree concept / Harry S. Tirtotjondro.
Van Slyke, R. M. (1963). MONTE CARLO METHODS AND THE PERT PROBLEM.
Operations Research, 11(5), 839–860.
Wu, R. W.-K., & Hadipriono, F. C. (1994). Fuzzy modus ponens deduction technique for
construction scheduling. Journal of Construction Engineering and Management,
120(1), 162–179.
106
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.
http://doi.org/10.1016/S0019-9958(65)90241-X
Zeng, J., An, M., Chan, A. H. C., & Lin, Y. (2004). A methodology for assessing risks in
the construction process. In Proceedings of the twentieth annual conference.
Association of Researchers in Construction Management (ARCOM). Edinburgh,
UK (pp. 1165–74). Retrieved from http://www.arcom.ac.uk/-
docs/proceedings/ar2004-1165-1174_Zeng_et_al.pdf
107
Appendix A. Numerical Solutions
108
Numerical Solution for 𝝈 and𝒑𝑵𝑬.
The equations after substitution of Equation (12):
0.5 (1 + 𝜎√𝜋
2) = ∫ (1 − 𝑦)𝑝𝑁𝐸 (1 + √−2𝜎2 ln(𝑦)) 𝑑𝑦
1
0
1 = (1 − 0.5)𝑝𝑁𝐸 (1 + √−2𝜎2 ln(0.5))
Trial 𝝈 𝒑𝑵𝑬 Error
1 1.0 1.1226 25.0%
2 2.0 1.7461 -11.0%
3 1.5 1.4675 2.5%
4 1.6 1.5279 -0.83%
5 1.57 1.5111 0.058%
6 1.573 1.512008 -0.027%
7 1.572 1.511412 -0.0017%
Numerical Solution for Parameter 𝒑:
Trial 𝒑 Error
“Very Unlikely”
1 12 +1.4%
2 12.5 -1.9%
3 12.2 +0.044%
4 12.21 -0.022%
5 12.206 +0.0048%
6 12.2068 +0.00021%
“Unlikely”
1 5 -1.7%
2 4.9 -0.30%
3 4.88 -0.014%
4 4.8790 -0.00020%
“Fairly Unlikely”
1 3 -8.3%
2 2.5 +2.3%
3 2.6 -0.046%
4 2.599 -0.023%
5 2.598 -0.00078%
109
Trial 𝒑 Error
“Fairly Likely”
1 0.8 +3.6%
2 0.85 +1.4%
3 0.90 -0.61%
4 0.88 +0.20%
5 0.885 -0.0055%
6 0.8849 -0.0011%
“Likely”
1 0.5 -0.92%
2 0.49 -0.42%
3 0.48 +0.081%
4 0.481 +0.030%
5 0.4815 +0.0051%
6 0.4816 +0.00012%
“Very Likely”
1 0.2 +0.17%
2 0.21 -0.44%
3 0.202 +0.046%
4 0.2025 +0.011%
5 0.2027 +0.0033%
6 0.2028 -0.0028%
110
Appendix B. Email Correspondence with Turner Construction Company
111
Obada Alsaqqa <[email protected]>
The Sample Project
2 messages
Obada Saqqa <[email protected]> 4 March 2015 at 10:48
To: Adam Baker <[email protected]>
Adam-
I want to ask you if it's OK to use the project you gave me along with mentioning the name
of the project and Turner as my resource..
Is it OK if I acknowledge Turner and the personnel who've helped me out too?
I was also wondering if I could get more info about the project, like location, area .. etc.
Thank you,
Obada Alsaqqa
Baker, Adam - (Ohio) <[email protected]> 4 March 2015 at 10:49
To: Obada Saqqa <[email protected]>
Sure go ahead.
Adam Baker, LEED AP │ Ohio Region Scheduling Manager
Turner Construction Company │ 262 Hanover Street<x-apple-data-detectors://0> │
Columbus, OH 43215<x-apple-data-detectors://1/0>
direct 614.984.3000<tel:614.984.3000> │
mobile 614.506.8575<tel:614.506.8575>│ [email protected]<mailto:[email protected]>
website<http://www.turnerconstruction.com/> │
linkedin<http://www.linkedin.com/company/turner-
construction/careers?trk=tabs_biz_career> │
facebook<http://www.facebook.com/TurnerConstructionOhio?ref=hl> │
twitter<https://twitter.com/Turner_Ohio>
│youtube<http://www.youtube.com/user/TurnerOhio?feature=guide> │
pinterest<http://pinterest.com/turnerprojects>
112
Appendix C. Sample Project Gantt/bar Chart – Partial Screenshots
113
114
115
116
Appendix D. FFT Analysis – Command Events Sub-Trees
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137