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.

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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.

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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

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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 <obada.saqqa@gmail.com>

The Sample Project

2 messages

Obada Saqqa <obada.saqqa@gmail.com> 4 March 2015 at 10:48

To: Adam Baker <abaker@tcco.com>

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) <abaker@tcco.com> 4 March 2015 at 10:49

To: Obada Saqqa <obada.saqqa@gmail.com>

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>│ abaker@tcco.com<mailto:abaker@tcco.com>

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>

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Appendix C. Sample Project Gantt/bar Chart – Partial Screenshots

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Appendix D. FFT Analysis – Command Events Sub-Trees

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