Incorporating Uncertainty in the Life Cycle Cost Analysis of Pavements
by
MASSACHUSE
Omar Abdullah Swei
OFTEC
B.S. Civil and Environmental Engineering
University of Massachusetts - Amherst, 2010
LiBR,
Submitted to the Department of Civil & Environmental Engineering and EngineeringSystems
Division in Partial Fulfillment of the Requirements for the Degrees of
Master of Science in Civil and Environmental Engineering
Master of Science in Engineering Systems
at the
Massachusetts Institute of Technology
September 2012
@ 2012 Massachusetts Institute of Technology.
All rights reserved.
Signature of author
Department of Civil and Environmental Engineering
Engineering Systems Division
Certified by
Randgiph E. Kirchain
Principal Research Scientist of Engineering Systems Division
Certified by
Jonn ucnsenuon
1
Associate Professor of Building Technology and Civil and Environmental Engineering
.
DThesi- Retnder
Certified by
-3_eremy uregty./
Research Associate of Engineering Systems Division
Thesis Reader
Accepted by
Olivier de Weck
Chair,
D Education/
7
licy Comn ttee
Accepted by
e i
epf
Chair, Departmental Committee for Graduate Stuents
Incorporating Uncertainty in the Life Cycle Cost Analysis of Pavements
by
Omar Swei
Submitted to the Department of Civil and Environmental Engineering and
Engineering Systems Division on August 10, 2012 in Partial Fulfillment of the
Requirement for the Degrees of Master of Science in Civil Engineering and Master
of Science in Engineering Systems at the Massachusetts Institute of Technology
ABSTRACT
Life Cycle Cost Analysis (LCCA) is an important tool to evaluate the economic performance of
alternative investments for a given project. It considers the total cost to construct, maintain, and
operate a pavement over its expected life-time. Inevitably, input parameters in an LCCA are
subject to a high level of uncertainty, both in the short-term and long-term. Under its current
implementation in the field, however, LCCA inputs are treated as static, deterministic values.
Conducting such an analysis, although computationally simpler, hides the underlying uncertainty
of the inputs by only considering a few possible permutations. This suggests that although
computationally simpler, the answer from the analysis may not necessarily be the correct one.
One methodology to account for uncertainty is to treat input parameters as probabilistic values,
allowing the analysis to consider a range of possible outcomes. There are two major reasons as
to why probabilistic LCCAs, although recommended, have not been streamlined into practice.
First, the LCCA of construction projects is a large-scale problem with many input parameters
with a high-level of uncertainty. Second, there is a significant gap in research that statistically
quantifies uncertainty for input values. This research addresses the latter point by statistically
quantifying four types of uncertainty: the unit cost of construction, quantity of material inputs,
occurrence of maintenance activities, and a particular emphasis is placed upon characterizing the
evolution of material prices over time. Having statistically characterized uncertainty in the
LCCA analysis, the application of the probabilistically derived inputs is illustrated in three
scenarios. Pavement alternative designs are derived for a set of traffic conditions in a given
location. The results of the analysis indicate the integration of probabilistic input parameters in
the LCCA process allows for more robust conclusions when evaluating alternative pavement
designs. Additionally, the case study shows treating input parameters probabilistically could
potentially alter the pavement selection, and one parameter that greatly influences this is
material-specific price projections.
Thesis Supervisor: Randolph E. Kirchain
Principal Research Scientist, Engineering Systems Division
3
ACKNOWLEDGEMENTS
I am incredibly grateful to so many people who were able to make the past two years as great as
they have been. First and foremost, thank you John Ochsendorf. When I came to your office for
the first time two years ago, I came to you with the dream of one day being a structural engineer.
Although my ambitions and aspirations evolved the past two years, you have nevertheless always
been incredibly supportive of every decision I have made. Your continual support of me does
not go unnoticed.
I am also truly thankful to Franz Ulm for giving me the opportunity to be a member of the
Concrete Sustainability Hub, which has given me the opportunity to work on exciting, real-world
problems. A special thank you is due to James Mack. You've really helped me understand the
major issues facing the paving industry and gave me direction for my work.
Thank you, Kris Kipp, for always being helpful and responsive to all of my questions, Elizabeth
Milnes for making me feel a part of ESD, and Kathleen for always being receptive whenever I
would sneak into the BT lab. You have all helped me along this journey.
This past year has been an amazing opportunity to work in the Material Systems Laboratory.
Thank you, Terra Choflin, for all the things you do that makes MSL as great as it is. A heartfelt
thank you is owed to Frank Field for those times you've asked me thought provoking questions,
Elsa Olivetti for joining our weekly subgroup meetings, and Rich Roth for being such a great
person to talk to both in and out of the lab. Also, thank you Arash and Maggie for your ideas
and thoughts during our weekly subgroup meetings.
I am especially grateful to Jeremy Gregory and Randy Kirchain. Jeremy has led our research
project with unwavering leadership and guidance. I could not have done as much as I have
without his timely and thoughtful input each and every week. Randy has been an amazing
mentor and resource to deal with many of problems I've come across on this journey. Both of
you, with your great sense of humor, made me a believer that Wednesday mornings can be
enjoyable.
Last but not least, I am forever indebted to my family for all they have done for me. Thank you
Ali, Aliya, and Anisa for being a part of my life. Thank you, dad, for all of your support and
giving me the opportunity to accomplish my goals. Most of all, thank you mom for giving me
guidance and being supportive of me for 24 years and counting. I love you all very much.
4
CONTENTS
Abstract........................................................................................................
3
Acknowledgem ents......................................................................................
4
List of Figures..............................................................................................
7
List of Tables ...............................................................................................
9
1 Introduction ...........................................................................................
11
1.1
Thesis Outline ...................................................................................................
2 Literature Review .................................................................................
2.1
2.2
General LCCA Methodology ..........................................................................
Pavement LCCAs............................................................................................
2.2.1
12
13
13
15
Scope and boundary ....................................................................................
15
2.2.2 Previous LCCA Studies.............................................................................
17
2.3
Forecasting Future Prices ..................................................................................
18
2.4
Gap Analysis ...................................................................................................
20
3 M ethodology .........................................................................................
22
4
3.1
General Five-Step Probabilistic LCCA Approach...........................................
22
3.2
Structure and Scope.........................................................................................
23
3.3
Quantify Uncertainty.......................................................................................
3.3.1 Unit-cost of construction activities............................................................
3.3.2 Inputs with no empirical data....................................................................
3.3.3 Future material costs ..................................................................................
24
24
25
27
3.4
M onte Carlo Simulations .................................................................................
35
3.5
Using Results to Evaluate Alternatives ............................................................
35
3.6
Understanding Which Types of Uncertainty Impacts an LCCA.....................
36
3.7
M ethodology Summary..................................................................................
37
Quantification of Uncertainty in LCCA Input Data ................
4.1
38
Analysis of Historical Bid Data .....................................................................
38
4.2
Projection of Future M aterial Prices ..............................................................
4.2.1 Data collection ...........................................................................................
4.2.2 Step 1: Assessing the historical behavior of pavement materials ..............
4.2.3 Step 2: Testing the price-link between paving materials and energy prices..
4.2.4 Step 3: Testing if long-term forecasts are plausible ................
40
41
42
42
50
4.2.5 Step 4: Probabilistically projecting the future ...........................................
4.3 Summary ..........................................................................................................
54
57
5 Case Study.............................................................................................59
5.1
Scope of Analysis............................................................................................
59
5.2
Deterministic Results ......................................................................................
61
5
5.3 Results Incorporating Uncertainty ...................................................................
5.3.1 High-Volum e scenario................................................................................
5.3.2 M edium -Volume scenario ........................................................................
5.3.3 Low-Volum e scenario.................................................................................
5.4 Case Study Discussion .....................................................................................
5.4.1 Influence of size of project .........................................................................
5.4.2 Sensitivity analysis ....................................................................................
63
64
65
66
68
68
69
...................................................................................................................................
72
...................................................................................................................................
72
5.4.3
5.5
Param eters which im pact pavement selection ...........................................
Summ ary ..........................................................................................................
6 Conclusions ..........................................................................................
73
75
77
6.1
Key Findings ...................................................................................................
77
6.2
Areas of future work .......................................................................................
78
References ............................................................................................
81
Appendix.....................................................................................................
88
7
6
LIST OF FIGURES
Figure 2-1: Sample cash flow with respect to time....................................................................
13
Figure 2-2: Historical real interest rates published by OMB (OMB 2011)..............................
Figure 3-1: Simplified scope and boundary of current LCCA study........................................
14
23
Figure 3-2: Plot for a hypothetical time-series which follows an a) real-price inflation rate model
b) Geometric Brownian Motion (GBM) model and c) a mean-reverting model. The dotted red
line represents the time-series data and the smooth black line represents the forecast ............ 33
Figure 3-3: Sample CDF output from Monte Carlo simulations. Dashed lines represent a riskaverse (75t percentile) and highly risk-averse ( 9 5 th percentile) perspective............................. 36
Figure 4-1: Histogram of collected bid data for JPCP pavements over a 12-month span in CA
(C alTrans 20 12a) ..........................................................................................................................
39
Figure 4-2: Regression analysis of unit-price of JPCP winning pavement bids with respect to bid
v olum e...........................................................................................................................................
39
Figure 4-3: Monthly real-price data of asphalt, concrete, oil, natural gas, and coal used in
cointegration analysis (BP 2012; BLS 2012a; EIA 2012a; BLS 2012b; EIA 2012b; BLS 2012c;
B LS 20 12d)...................................................................................................................................
41
Figure 4-4: Annualized Ready-Mix Concrete and Asphalt Paving time-series (2011 = 100 for
both data sets) (BLS 2012a; BLS 2012b; BLS 2012d).............................................................
42
Figure 4-5: Real price of Construction Sand & Gravel time-series (Kelly and Matos 2012a)..... 51
Figure 4-6: Backcasting results from the Construction Sand & Gravel time-series................. 52
Figure 4-7: Backcasting results for USGS Sand & Gravel time-series using a GBM process to
project the future. Projections are made using at most 35 and 55 years of available historical data
.......................................................................................................................................................
52
Figure 4-8: Real price of Cement time-series (Kelly and Matos 2012b)...................................
53
Figure 4-9: Backcasting results from the Cement time-series ..................................................
53
Figure 4-10: Backcasting results for USGS Cement time-series using an Ornstein-Uhlenbeck
mean-reverting process. Projections compared use at most 35-years of historical data versus as
much historical data readily available for a particular forecast. ...............................................
54
Figure 4-11: Long-term price projection of concrete based upon time-series analysis. The dashed
lines represent the 5th and 95th percentiles of the probabilistic distribution of results............ 56
Figure 4-12: Long-term price projection of asphalt based upon time-series analysis. The dashed
lines represent the 5 th and 95t percentiles of the probabilistic distribution of results.............. 57
Figure 5-1: Deterministic initial cost for three scenarios.........................................................
62
Figure 5-2: Discounted life-cycle cost for design scenarios....................................................
62
Figure 5-3: Initial and life-cycle cost of asphalt pavement designs relative to concrete designs for
high volume, medium volume, and low volume scenarios.......................................................
63
Figure 5-4: High-volume roadway cumulative distribution results not including material-specific
projections.....................................................................................................................................64
Figure 5-5: Medium-volume roadway cumulative distribution results not including materialspecific projections .......................................................................................................................
65
7
Figure 5-6: Medium-volume roadway cumulative distribution results including material-specific
projections.....................................................................................................................................
66
Figure 5-7: Low-volume roadway cumulative distribution results not including material-specific
projections.....................................................................................................................................67
Figure 5-8: Low-volume roadway cumulative distribution results including material-specific
projections.....................................................................................................................................68
Figure 5-9: Percent contribution of initial cost to total life-cycle cost using a discount rate of 4%
.......................................................................................................................................................
69
Figure 5-10: Percent contribution of initial cost to total life-cycle costs using a discount rate of
2 % .................................................................................................................................................
69
Figure 5-11: Percent contribution of initial cost to total life-cycle costs using a discount rate of
5 % .................................................................................................................................................
70
Figure 5-12: Relative life-cycle cost savings of asphalt designs for medium-volume roadway for
different GBM growth rate parameters.........................................................................................72
Figure 5-13: Relative life-cycle cost savings of asphalt designs for medium-volume roadway for
different GBM growth rate parameters.........................................................................................72
Figure 6-1: Historical real-price of oil (BP 2012) ....................................................................
8
79
LIST OF TABLES
26
Table 3-1: Pedigree matrix quality indicator criterion (Weidema et al. 2011).........................
Table 3-2: Uncertainty factors (variances of the underlying normal distribution) for the pedigree
matrix quality indicators (Weidema et al. 2011) .....................................................................
27
37
Table 3-3: Structure of case study analysis...............................................................................
Values
in
Table 4-1: Quantification of unit-cost uncertainty for significant input parameters.
40
parenthesis represent the standard error of the regression coefficients .....................................
Table 4-2: Historical Growth rate and standard deviation of growth rate for Asphalt and Concrete
.......................................................................................................................................................
42
Table 4-3: Goodness of Fit performance of paving material time-series fitted to ADF regression.
Bold values represent number of lags with lowest AIC or BIC value for all lags considered..... 43
Table 4-4: Goodness of Fit performance of energy price time-series fitted to ADF regression.
43
Bold values represent number of lags with lowest AIC or BIC value for all lags considered .
Table 4-5: ADF test results for lag orders selected by BIC and AIC criteria. Bold values are less
44
than 5% critical values..................................................................................................................
Table 4-6: Concrete-Oil VAR Goodness of Fit performance for different number of lags.
Italicized numbers are best-fitting for each "goodness of fit" measure and bolded italics is best-
fitting for A IC criteria...................................................................................................................45
Table 4-7: Asphalt-Oil VAR Goodness of Fit performance for different number of lags.
Italicized numbers are best-fitting for each "goodness of fit" measure and bolded italics is best-
fitting for A IC criteria...................................................................................................................
45
Table 4-8: Johansen test results for Concrete-Oil and Asphalt-Oil models. Bold values are less
46
than the 5% critical values ............................................................................................................
Table 4-9: Concrete-Natural Gas VAR "goodness of fit" performance for different number of
lags. Italicized numbers are best-fitting for each "goodness of fit" measure and bolded italics is
the best-fitting A IC .......................................................................................................................
47
Table 4-10: Asphalt-Natural Gas VAR "goodness of fit" performance for different number of
lags. Italicized numbers are best-fitting for each "goodness of fit" measure and bolded italics is
the best-fitting A IC .......................................................................................................................
47
Table 4-11: Johansen test results for concrete-natural gas and asphalt-natural gas models. Bold
values indicate when it can be confirmed no more than r cointegration relationships exist ........ 48
Table 4-12: concrete-coal VAR "goodness of fit" performance for different number of lags.
Italicized numbers are best-fitting for each "goodness of fit" measure and bolded italics is the
best-fitting A IC .............................................................................................................................
48
Table 4-13: asphalt-coal VAR "goodness of fit" performance for different number of lags.
Italicized numbers are best-fitting for each "goodness of fit" measure and bolded italics is the
best-fitting A IC .............................................................................................................................
49
Table 4-14: Johansen test results for concrete-coal and asphalt-coal models. Bold values indicate
49
when it can be confirmed no more than r cointegration relationships exist ..............................
55
Table 4-15: ADF test results for asphalt paving time-series (Source: BLS)............................
9
Table 4-16: Logarithmic growth rate and volatility for various sample sizes ...........................
Table 5-1: Three roadway scenarios in Southern CA considered............................................
Table 5-2: JPCP initial pavement designs for three scenarios..................................................
Table 5-3: HMA initial pavement designs for three scenarios ..................................................
Table 5-4: Maintenance schedule for high-volume roadway alternatives ................................
Table 5-5: Maintenance schedule for medium-volume roadway alternatives ..........................
Table 5-6: Maintenance schedule for low-volume roadway alternatives ..................................
Table 5-7: Significant unit-cost inputs for the deterministic analysis .......................................
55
59
60
60
60
60
61
61
Table 5-8: Cost of asphalt pavement relative to concrete alternatives for all three scenarios...... 63
Table 5-9: Probabilistic cost of high-volume scenario alternatives not including material-specific
price projections............................................................................................................................
64
Table 5-10: Probabilistic cost of high-volume scenario alternatives including material-specific
price projections............................................................................................................................
65
Table 5-11: Probabilistic cost of medium-volume scenario alternatives not including materialspecific price projections ..............................................................................................................
65
Table 5-12: Probabilistic cost of medium-volume scenario alternatives including materialspecific price projections ..............................................................................................................
66
Table 5-13: Probabilistic cost of low-volume scenario alternatives not including material-specific
price projections............................................................................................................................
67
Table 5-14: Probabilistic cost of low-volume scenario alternatives including material-specific
price projections............................................................................................................................
67
Table 5-15: Life-cycle cost of pavement designs using a discount rate of 2%......................... 70
Table 5-16: Life-cycle cost of pavement designs using a discount rate of 4%......................... 71
Table 5-17: Life-cycle cost of pavement designs using a discount rate of 5%......................... 71
Table 5-18: Pavement selection for probabilistic analyses a) excluding material-specific price
projections b) including material-specific price projections and the lowest life-cycle cost
pavement is selected c) including material-specific price projections and selecting an initially
more expensive pavement if it offers at least 10% life-cycle cost savings and d) the same rule as
(c) but for a threshold value of 20%. A discount rate of 4% is used in all analyses................ 73
Table 5-19: Pavement selection for probabilistic LCCA including material-specific price
projections with a discount rate of 2%. Pavements are selected by a) lowest life-cycle cost b)
initially more expensive pavement selected if its life-cycle cost savings are at least 10% and c)
the same decision rule as (b) but the threshold value is 20% ..................................................
74
Table 5-20: Pavement selection for probabilistic LCCA including material-specific price
projections with a discount rate of 5%. Pavements are selected by a) lowest life-cycle cost b)
initially more expensive pavement selected if its life-cycle cost savings are at least 10% and c)
the same decision rule as (b) but the threshold value is 20% ....................................................
75
Table A-1: Pedigree matrix uncertainty indicator scores for parameters with no empirical data 88
Table A-2: Variances calculated by pedigree matrix approach ................................................
89
10
1
INTRODUCTION
There is an undeniable need to upgrade infrastructure across the United States, which has been
recognized by the media, general public, politicians, and decision-makers. It is estimated that a
total investment of $2 trillion is needed to bring America's infrastructure network to a
satisfactory level (Miller 2011). Making intelligent investment decisions will play a crucial role
in revitalizing the infrastructure network across the United States. One component of the
infrastructure network is pavements, the focus of this thesis.
As demand for the United States pavement network continues to grow, state Departments of
Transportation (DOTs) are increasingly under pressure to satisfy consumers with limited
resources.
short-term
work-zone
Clearly a
mentioned
This means that decision-makers and pavement engineers are not just concerned with
costs, but also long-term costs resulting from future maintenance activities, excessive
closures, and overall unsatisfactory performance of paving materials over time.
robust framework for analysis is needed to economically weight the previously
concerns of decision-makers. One such tool is Life Cycle Cost Analysis (LCCA).
Life Cycle Cost Analysis (LCCA) is an analytical method to assess the value of alternative
investments. It includes the total cost of ownership, operation, and maintenance for a given
project (Smith 1998). For pavement projects, this encompasses costs associated with initial
construction, maintenance and rehabilitation, and user costs (i.e. vehicle operation costs). A
major benefit of LCCA is it allows for the fair comparison of pavement alternatives with
different cash flows by converting costs across all designs into one time-perspective, frequently
the present.
Despite its merits, a recent survey showed that highway officials place a greater emphasis upon
initial costs rather than life-cycle costs (Rangaraju et al. 2008). One likely reason for this is it is
generally more plausible to predict costs with a much higher level of precision in the near-term
than 20, 30, 50, or even 100 years into the future. It is likely that decision-makers will only
weigh life-cycle costs more heavily in decision-making if more advanced analytical models are
constructed to account for such uncertainties (Frangopol et al. 2001). One methodology to
account for the uncertainty in an LCCA is to conduct a probabilistic LCCA. In a probabilistic
LCCA, uncertain parameters are characterized as probabilistic values, which allows the analysis
to account for a range of possible outcomes (Tighe 2001). In fact, the concept of conducting a
probabilistic LCCA is not necessarily novel; the use of a probabilistic LCCA has been
recommended by the Federal Highway Administration (FHWA) for almost fifteen years (Smith
1998).
However, despite the recognized uncertainty when constructing an LCCA, and the advent of a
methodology which can account for them, practitioners have implemented LCCA in the field by
treating inputs as static, deterministic values (Chan et al. 2008). Modeling an LCCA using static
values hides the underlying uncertainty of the inputs, but makes the pavement selection process
11
simple: one chooses the pavement alternative with the lowest total life-cycle cost. The issue with
this deterministic analysis is a pavement alternative is selected based upon only a few considered
scenarios. It is highly likely, therefore, that the values being compared are not what the actual
life-cycle costs will be. This suggests that although computationally simpler, the answer from
the deterministic analysis may lead the decision-maker to select a pavement design with a higher
life-cycle cost.
There are two major reasons why probabilistic LCCAs, although recommended, have rarely been
implemented in practice. First, the LCCA of pavement alternatives is a large-scale problem with
many input parameters with a high-level of uncertainty, which makes implementation of the
methodology challenging. Second, there is currently a gap in the literature to statistically
quantify uncertainty for input values (Tighe 2001). If a decision-maker incorrectly quantifies the
underlying uncertainty of input parameters, the results of the analysis are rendered useless.
This research aims to build upon previous work to move the probabilistic LCCA methodology
into practice by statistically quantifying relevant input parameters and applying them to a case
study. More broadly speaking, this work aims at statistically quantifying two types of
uncertainties: statistically characterized inputs, which are things that cannot always be known but
can be characterized, and known unknowns, which are recognized uncertainties but with no
readily available data (McManus and Hastings 2005).
1.1
Thesis Outline
This thesis is organized as follows:
Chapter 2 presents an overview of LCCA literature in the domain of pavements. It presents
which uncertainties have been characterized and provides background for how previous
researchers have quantified uncertainty.
Chapter 3 provides information regarding the methodology implemented in this thesis to
characterize uncertainty in pavement LCCAs.
Chapter 4 presents the data collected and probabilistic characterization of the data. A particular
emphasis is placed upon quantifying material-specific price projections by testing historical price
data for relevant materials in the LCCA of pavements
Chapter 5 illustrates how the inclusion of probabilistic values impacts the results of an LCCA by
implementing the described methodology for three different scenarios.
Chapter 6 includes final remarks from the analysis and discusses future areas of work that should
be done to build upon this thesis.
12
2
LITERATURE REVIEW
This section outlines relevant literature used as the basis for this thesis. An overview of the
general LCCA methodology, including key terminology and important considerations, is
provided. Subsequently, previous pavement LCCA studies, both deterministic and probabilistic
researchers have conducted, are presented. A particular focus is placed upon evaluating previous
work which has characterized future construction costs, which will be a major point of emphasis
in this thesis. Lastly, the major gaps from the literature are presented, which this thesis aims to
resolve.
2.1
General LCCA Methodology
LCCA provides a framework to compare the total cost of ownership of multiple alternatives (Lee
2002). Costs in an LCCA are defined by costs in the present (i.e. initial costs) and costs to
maintain and operate the system (i.e. future costs). Figure 2-1 presents a sample cash flow
diagram in which costs accrue over time. To compare the economic performance of alternative
designs, an array of performance metrics is at the disposal of a decision-maker. Some of the
more common performance metrics include the net present value (NPV), present annual worth
(PAW), internal rate of return (IRR), and cost/benefit ratio (Smith 1998). Pavement LCCA
literature suggests that the more commonly used metrics in pavement LCCAs are the NPV,
followed by the PAW (Rangaraju et al. 2008).
Figure 2-1: Sample cash flow with respect to time
Initial Costs
Future Costs
Time
The present value (PV) represents the value at time 0 of a series of incoming or outgoing cash
flows into the future (Riggs 1986). The summation of all incoming and outgoing PVs, therefore,
13
represents what is known as the net present value (NPV). To account for the time-value of
money, future cash flows are discounted at a prescribed discount rate.
Clearly, the selection of discount rate is an important parameter when converting future cash
flows into a present value. One study from the 1980s suggests that the time-value of money (i.e.
discount rate) to be used in an analysis is 5% (Rangaraju et al. 2008). The author found that for
LCCAs where the discrepancy in NPV between the pavement alternatives is more than 20%, the
results of the LCCA are insensitive to the selected discount rate; that is, the superior alternative
will tend to be the case for any reasonable discount rate selected. In the case of a final result
where the discrepancy between the alternatives is less than 20%, a sensitivity analysis is
FHWA's Life Cycle Cost Analysis in Pavement Design, perhaps the most cited
suggested.
source for conducting pavement LCCAs, analyzes historical real-interest rate data provided by
the Office of Management and Budget (OMB) in order to suggest a real-discount rate of 4%
(OMB 1992; Walls and Smith 1998). The 4% value suggested by FHWA has been integrated
into the LCCA process for many state Department of Transportations (DOTs) (MoDOT 2004;
WSDOT 2005). A likely issue with using the 4% discount rate value, however, is it is not
representative of today's real interest rate values. Figure 2-2 presents the annual real interest
rates published by OMB since 1979 for 30-year analyses (OMB 2011). Real-interest rates have
continued to drop since FHWA published its report in 1998, and as such, a lower discount rate is
potentially more realistic for what should be used in an LCCA.
Figure 2-2: Historical real interest rates published by OMB (OMB 2011)
8%
6%
tilIIiiii
4%
2%
1979
1984
1989
1999
1994
2004
2009
Year
Clearly the selection of an appropriate discount rate is an important consideration. Therefore,
although this thesis does not statistically analyze historical discount rates, a sensitivity analysis
for the case study is conducted in Section 5.4.2 for different real discount rates.
14
2.2
Pavement LCCAs
This section now presents the general scope and boundary of pavement LCCA analyses, as well
as the type of analyses that have previously been conducted.
2.2.1
Scope and boundary
In practice, an LCCA can be conducted at two different levels: either project-specific or the
entire network. In a project-specific analysis, a set of structurally equivalent pavement designs
for a proposed project are compared and the lowest cost roadway is selected. This type of
analysis, although more common in the field, selects the lowest cost option while ignoring
possible funding allocation issues that a state DOT may face (Tighe 2001). A network-level
analysis considers all funds and policies that a DOT has at their disposal in selecting how to
divide resources for a series of projects (Ravirala and Grivas 2002; Zhang et al. 2012). Although
the latter type of analysis has become an increasingly important topic for researchers, most
LCCAs currently conducted are at the project-specific level. This, in part, can be attributed to
the number of software tools currently available to conduct such an LCCA (Frangopol et al.
2001).
For the former type of LCCA, FHWA has defined the steps and scope when conducting an
analysis (Walls and Smith 1998). The defined procedure is as follows:
1) Establish alternative pavement design strategies for a pre-determined analysis period
2) Determine rehabilitation activities and their timings
3) Estimate agency costs (i.e. the DOT)
4) Estimate user costs
5) Determine the expected cash flow of the alternatives
6) Compute each alternative's life-cycle cost
7) Analyze the results
8) Evaluate design alternatives
For steps 1 and 2, it is important that alternative pavement designs are functionally equivalent.
This means that roadway design parameters, traffic conditions, and any other project-specific
parameters are equivalent between the alternatives considered. The selection of rehabilitation
schedules, and specifically their years of occurrence, has been an evolving process. For most
DOTs, a pre-determined rehabilitation schedule is applied for all pavement alternatives
considered (MoDOT 2004; WSDOT 2005; CalTrans 2012c).
A growing body of work,
however, has evolved to better understand the mechanistic performance of pavements. The
FHWA Long-Term Pavement Performance (LTPP) program has collected data regarding
pavement performance for hundreds of pavements across the United States (Wang et al. 2011).
The collection of data has led to the recently developed Mechanistic Empirical Pavement Design
15
Guide (MEPDG) and Darwin-ME software program, which incorporates empirical field data to
predict the life-time performance of a pavement design (Wang et al. 2011). The inclusion of
MEPDG in the LCCA process is currently being evaluated and gradually introduced to state
DOTs across the United States (El-Badawy et al. 2011).
Steps 3 and 4 present the general concept that pavement costs are bifurcated into two broad
categories: agency cost and user cost. Agency cost is defined as the cost to build, maintain, and
operate a roadway, which can be attributed to the DOT agencies that finance the roadway. The
user cost is the external costs associated with a roadway that the driver (i.e. user) incurs (BenAkiva and Gopinath 1995). Current software programs that allow for the consideration of user
cost tend to only incorporate the user cost of time (FHWA 2005). The user cost of time is the
value of time that a user loses when a roadway is designed inappropriately or when traffic delay
results in a roadway to not fulfill its mission. An increasingly recognized user cost, however,
that is not incorporated in software is the differential impact of pavement alternatives on fuel
consumption, known as pavement vehicle interaction (Santero and Horvath 2009). Research has
shown that different pavement materials, due to roughness and deflection, may perform
differently over time, impacting the performance of the vehicles that drive over the pavement.
Although the fuel difference between pavement materials is very small when considering the
impact on a single vehicle, the impact can become quite significant on a high-volume roadway.
One study suggests that the maximum fuel consumption difference between flexible (i.e. asphalt)
and rigid (i.e. concrete) pavements is 0.007 liters per vehicle-kilometer (Taylor and Pattern
2006). If the average annual daily traffic (AADT) for a one kilometer stretch was 100,000
vehicles, typical of a high volume roadway, this would amount to a maximum differential fuel
consumption of 700 liters per day. Although there is a general consensus that different materials
induce differential fuel consumption, there is enough conflicting evidence that this effect has
generally been omitted in LCCAs (Santero et al. 2011).
When performing steps 5 and 6, where the cash flow and life-cycle cost of the alternatives are
computed, it is important that an appropriate time-frame of the analysis is selected. Since
roadways can be constructed using different construction materials with different mechanical
properties, the occurrence of future maintenance activities can vary significantly between
alternative designs. By selecting too short of a time-horizon a decision-maker would heavily
weigh initial cost of the alternatives, while considering too long of a time-horizon may introduce
too many uncertainties that are unrealistic to capture.
The 1993 AASHTO design guide, which has been incorporated into the design process for
multiple DOTs, has a set of recommendations for the analysis period for the LCCA of pavements
(AASHTO 1993). For high volume interstate roads, it suggests that an analysis period between
20 and 50 years be used, and for low volume roads a range between 10 to 25 years. More
recently, FHWA has suggested that a minimum time-horizon of 35 years be used for any LCCA
to be conducted for new or reconstruction projects, and it is generally considered good practice
the analysis period accounts for at least one pavement rehabilitation (Walls and Smith 1998).
16
Given that both sources have only specified a range of years to use in an analysis period, there is
clearly no definitive answer for how long of an analysis period an LCCA should use. However,
what is evident is that increasingly DOTs are pushing towards longer time-horizons. In the past
20 years, the number of state DOTs that use a time-horizon of 50 years increased from only 7%
to 20%, and 2005 was the first time that any state DOT used a time-horizon of 65 years in their
LCCA procedure (Rangaraju et al. 2008). Much of this, the authors suggest, can be attributed to
the increased performance of materials and construction.
2.2.2
Previous LCCA Studies
Following the National Highway System Designation Act of 1995, which required states to
conduct an LCCA for project costing over $25 million, state DOTs and researchers have focused
their efforts on improving the overall LCCA process (1995). In particular, the Federal Highway
Administration (FHWA) has been the major body in both promoting and funding LCCA
research, which has led to significant advancements the past 15 years (Chan et al. 2008).
Early research amongst the pavement LCCA community focused on comparative assessments
for a range of different applications. Embacher and Snyder (2001) compared the life-cycle cost
of asphalt and concrete pavements for low-volume roadways. Huang et al. (2004) created a
decision support system for identifying optimal concrete bridge deck repairs. Fagen and Phares
(2000) compared the life-cycle costs of steel beam precast, concrete beam precast, and
continuous concrete slab bridge deck for a low-volume roadway. Zimmerman and Peshkin
(2003) used LCCA to identify optimal timings for preventative maintenance procedures.
Although these are major contributions, a drawback associated with all of these studies is that
input parameters were treated as deterministic values. Recognizing this, research in the past
decade has focused on developing a probabilistic approach to deal with uncertainty.
The majority of probabilistic LCCAs have focused on statistically characterizing a select few
input parameters with historical data using variety of best-fit probability density functions.
Tighe (2001) collected empirical data to characterize cost and pavement thickness variation. In
general, running a chi-square best-fit test, the types of uncertainties considered were best
explained using a log-normal distribution. Osman (2005) developed a risk-based methodology
by considering uncertainty only with respect to pavement performance over time, which was
described with a Weibull distribution. Li and Madanu (2009) created a life-cycle cost/benefit
model by characterizing cost uncertainty. Historical bid data was collected from the state of
Indiana over an 11-year span and was used to characterize the unit-cost of construction and
maintenance activities using a Beta distribution. Salem et al. (2003) characterized uncertainty
related to year of pavement failure with a Weibull distribution. Although all of the above LCCA
studies have contributed significantly to the probabilistic LCCA domain, there is a lack of
studies which account for a range of input uncertainties.
For all of the aforementioned studies, a probabilistic LCCA is conducted through Monte Carlo
simulations rather than more elaborate analytical methods. The use of Monte Carlo simulations
17
to conduct a probabilistic LCCA has not only been utilized by researchers, but has been highly
encouraged by FHWA (Walls and Smith 1998). The usefulness in conducting a probabilistic
LCCA is that a final probability distribution can be assimilated from all iterations to describe the
probability of an event occurring. This is typically conducted so researchers can understand the
risk of alternative investments (Reigle and Zaniewski 2002; Salem et al. 2003; Osman 2005).
Risk, which is discussed extensively throughout the finance literature, implies that there is a
potential down-side associated with an investment that can lead to financial losses (de Neufville
and Scholtes 2011). For pavements, uncertainty in short-term and long-term costs suggests that
the actual life-cycle costs for an alternative will be potentially higher (or lower) than expected.
Two important financial metrics to measure the potential downside (or upside) of an investment
is the value at risk (VaR) and value at gain (VaG). The VaR represents the threshold value that
an investment cost for a given probability (de Neufville and Scholtes 2011). As an example, if a
roadway had a 10% chance of costing more than $1 million, it can be inferred that the "value at
risk" for the investment at the 10% confidence level is $1 million.
Despite the progress, deterministic LCCAs are still mostly conducted by state DOTs (Chan et al.
2008). Likely, this is in part due to the above studies characterizing uncertainty for specific
materials, data for a specific region, etc. which are not applicable to every project or state DOT.
Therefore, a systematic methodology is needed more than anything else so decision-makers can
characterize uncertainty for their particular project in a robust manner.
2.3 Forecasting Future Prices
One set of LCCA parameters with a high level of uncertainty, and will be a particular point of
emphasis in this study, is the evolution of material prices over time. Estimates of cost for a given
project may occur months or years before construction begins, and the completion of
construction can take multiple years. Infrastructure projects are also expected to have long lifecycles with maintenance activities occurring decades into the future. If prices remained static
over time, planners could accurately project costs for construction projects. Due to the volatile
nature of material prices, however, projecting short-term and long-term construction costs has
shown to be historically difficult, resulting in inaccurate cost projections (Gransberg and Rierner
2009). This raises two basic questions: can the uncertainty when projecting future construction
costs be statistically characterized, and, if so, would it improve the final decision.
Forecasting future costs in terms of a deterministic value implicitly hides the underlying
uncertainty of the forecast model. Despite this, the current literature is generally split between
studies that have conducted deterministic and probabilistic analyses. Attala and Hegazy (2003)
comparatively assess the deterministic performance of a regression model versus an artificial
neural networks model to predict the cost overrun for a given project. Trost and Oberlender
(2003) used factor analysis and multivariate regression to make early prediction, single point
forecasts of likely construction costs. On the other hand, Touran (2003) incorporates uncertainty
18
through defined probabilistic distributions to calculate the probability of cost overrun for a
project. This research follows the latter approach, recognizing that no single point value will
perfectly predict the future. In fact, this is particularly pertinent to the paving materials due to the
expected volatility of construction materials based on historical experience.
The methodology to forecast future construction costs has generally been conducted in two
separate fashions. First, studies have derived a relationship between construction costs and
factors that drive those costs to predict future costs (Wilmot and Cheng 2003). The second, and
far more common, employs univariate time-series models, which ignores the underlying drivers
of change, in order to project future costs (Wilmot and Cheng 2003; Hwang 2009). Although
these types of forecasting models are simpler, they have generally performed well when
projecting future costs.
The majority of these studies are based off of the univariate
Autoregressive Integrated Moving Average (ARIMA) model. Hwang (2011) projected a future
Construction Cost Index (CCI) value based off of the aforementioned ARIMA model. Ashuri
and Lu (2010) used an ARIMA model that accounted for seasonality to project future values
from the Engineering News Record (ENR) cost index. Ng et al. (2000) used a similar type of
model to project future costs for the Hong Kong construction industry. The general conclusion
from above is a majority of construction cost forecasting studies use the basic ARIMA model,
and tend to project future construction costs using their own construction cost index (CCI). The
index is a composite measure of labor, materials, and equipment (Wilmot and Cheng 2003).
If material costs have behaved differently over time, however, a flaw induced by forecasting
future prices using a CCI is it assumes all construction inputs increase at the same rate (Hwang et
al. 2012). Since different projects will use different construction materials, different quantities,
and require different amounts of labor, it is clearly inappropriate to treat each alternative the
same. This is particularly important within the pavement LCCA community; it is likely that
future costs of asphalt and concrete paving will grow at different rates over time, and even more
likely, have different price volatilities, the latter of which is a particular focus of this research.
To validate the historical differential price movement between concrete and asphalt, it is
pertinent to quantify the historical price spillover with other volatile commodities. In particular,
it is likely a price-link exists between energy prices and construction costs. This assumed pricelink, although not statistically characterized previously, has been assumed in recent news articles
to explain the recent price hike in construction prices. A potentially robust analysis to assess if
paving materials are tied to the price of energy prices, and if so, which ones, is to test if the timeseries are "cointegrated". Prior to the advent of cointegration, economists evaluated the longterm price relationship through other analytical methods, such as regression analysis. It was
discovered, however, that many commodities that were clearly independent of one another could
inappropriately be quantified as correlated (Franses and Dijk 2009). Cointegration allows for the
analysis of time-series that, although they may seem to drift randomly, tend to not drift far away
from other closely linked time-series (Franses and Dijk 2009). Testing for cointegration between
commodities, although never conducted for paving materials, has been examined in other
19
industries to understand price and volatility spillovers between markets. Xiarchos (2006) tested
the price spillovers between primary and secondary scrap metals. Ewing et al. (2002) and
Ramberg (2010) quantified the price-link between natural gas and crude oil. The price-link
between oil and refined products has also been quantified (Gjoldberg and Johansen 1999; Lanza
et al. 2005). It is pertinent for the paving industry to understand price spillovers between
commodities in order to quantify the appropriate volatility measure to be used in projecting
future maintenance costs, a serious topic of discussion within the asset management community
who have evaluated the cost-benefit associated with future maintenance events (Herabat et al.
2002; Poovadol Sirirangsi 2003).
2.4
Gap Analysis
The previously discussed literature has led to the following general conclusions:
*
The inclusion of LCCA in the decision-making process for paving materials has grown
significantly the past 15-20 years.
*
Despite this, a greater emphasis is still placed on initial costs, in part due to the
significant uncertainties in trying to project the future.
*
In the past ten years there has been a steady growth in LCCA studies which
probabilistically characterize uncertainty in forecasting construction costs.
" Probabilistic LCCA studies which have characterized uncertainty do so by fitting
historical empirical data to best-fit probability distributions. Of these studies, they have
either focused on probabilistically characterizing historical cost data or pavement
performance over time.
*
Projections of future construction costs tend to employ univariate time-series models
which only model price with respect to time. These projections are typically made for a
construction cost index which assumes all construction activities will grow at the same
rate.
" Previous non-paving related studies have quantified price volatility of a commodity by
quantifying if a price spillover exists with other commodities.
Based upon the preceding conclusions, this thesis now aims to resolve some of the gaps which
exist in the literature.
Quantify cost uncertainty by characterizingpossible relationships
The previously mentioned studies characterize uncertainty and variability in an LCCA but may
overestimate it by ignoring likely relationships. For example, Herbsman (1986) showed there is
a direct correlation between bid volume and unit-cost of bid items. One potential way to reduce
the uncertainty and make the results of the analysis more realistic is to account for such
relationships. Therefore, this thesis explores if such relationships exist, and if so, do they explain
much of the variability in the data.
20
Quantify uncertainty for input parameters with no empirical data
All of the previous studies make use of empirical data in statistically characterizing input
parameters. It is likely, however, that many input parameters exist with no readily available
data. A methodology should be introduced in the LCCA literature which is able to account for
such uncertainties.
Predicting future material prices based upon historical data
Projecting future prices with a construction cost index assumes all costs will behave the same.
Making such an assumption, however, is likely invalid if historically different paving materials
have moved differently. Therefore, it is pertinent to assess if historically different construction
materials have behaved non-static and differently, and if so, how can an LCCA account for both
of these aspects?
Does conducting a probabilistic analysis change the pavement selection, and if so, what is
driving that change?
Lastly, the aforementioned probabilistic LCCA studies have typically focused on only
characterizing one type of uncertainty, and applying the characterized uncertainty in a case
study. Two pertinent questions to ask are a) does the inclusion of probabilistic values lead to a
different pavement selection than a deterministic analysis and b) if so, which probabilistic
parameters are causing that change?
21
3
METHODOLOGY
This chapter presents the general methodology used to characterize three types of uncertainties
that exist in the pavement LCCA literature. First, historical bid data is collected and
characterized by considering a possible relationship between cost and bid volume. Second, an
accepted methodology in the environmental life-cycle assessment (LCA) community is presented
to quantify uncertainty in the absence of empirical data. Third, material price forecasting is
conducted by addressing a) have materials historically behaved differently, and if so, is this
expected to continue b) are long-term projections decades into the future plausible, and c) if the
above are true, how should one forecast future material prices. Having characterized the above
uncertainties, a case study analysis can be conducted to assess if a probabilistic LCCAs could
potentially alter the likely pavement selection.
3.1
General Five-Step Probabilistic LCCA Approach
The most common reference for practitioners in conducting an LCCA is FHWA's 1998 Life
Cycle Cost in Pavement Design report. FHWA suggests that four major steps are needed to
conduct an LCCA (Walls and Smith 1998):
Identify the Structure of the Problem
An LCCA is used to evaluate a range of alternatives for a proposed pavement project. The first
step is to define the contextual conditions that will define the project, including: location, soil
conditions, traffic flow, expected design life, and more. Based upon the contextual conditions, a
designer will derive a range of satisfactory pavement designs with an expected design life and
expected maintenance schedule.
Quantify Uncertainty
Choosing a single, best-estimate value for an input hides the underlying uncertainty. In
developing probabilistic input values, a modeler is capturing the range of possible values for
different parameters. This can be done either by statistically analyzing data, if available, or
exercising expert opinion (Smith 1998; McManus and Hastings 2005). When data is available,
there are multiple ways to characterize the data. This includes fitting data to a best-fit
distribution, using goodness of fit indicators to identify how well a distribution fits a dataset,
conducting regression analysis to model a trend, and others. In choosing the best way to
characterize a dataset, it is best to first visualize the data before conducting statistical tests.
Perform Simulation
After constructing an LCCA that is linked with characterized uncertain parameters, the next step
is to derive the probabilistic total cost of the alternatives considered. One way to accomplish this
is through Monte Carlo simulation, where random numbers are generated to compute something
that is not random. For each iteration, a random value from each probabilistically defined input
22
is selected and the model calculates the output value from the iteration. This process is
conducted hundreds, or thousands, of times and each iteration is tracked. Once the simulation is
complete, the results can be used to develop a probabilistic distribution describing the likelihood
of an outcome. The advantage of Monte Carlo simulation is that a model with hundreds of
probabilistically characterized input values can perform thousands of iterations in seconds.
Interpret Monte Carlo Simulation Results and Make a Consensus Decision
After performing a Monte Carlo simulation, a decision-maker must try to make sense of the
results. There are multiple financial metrics to compare the performance of alternatives for a
project. Alternatives can be compared based upon the mean present value, also known as the
expected value. The expected value is the weighted sum of possible outcomes from a Monte
Carlo simulation, and in financial theory, represents the preferred cost from a risk-neutral
perspective. Another metric is the Value at Risk (VaR), which is the given probability value that
an asset can fall below a certain threshold.
3.2
Structure and Scope
This thesis quantifies the probabilistic economic cost to build and maintain a new roadway. It
only focuses on the cost to finance a project and ignores user costs associated with traffic delays
previous studies have explored (Vadakpat et al. 2000; Lee 2002; Temple et al. 2004). It also
assumes that a decision has already been made to build a new roadway, ignoring the underlying
policies and impacts a roadway has on existing infrastructure (Stamatiadis et al. 2010). Although
these are important considerations, they are ignored to reduce the complexity of the problem at
hand. Figure 3-1 is a simplified flow chart of all phases considered within the system boundary
of this research.
The four general life-cycle phases include: materials, construction,
maintenance, and end-of-life. It is important to emphasize that this study is a comparative
assessment of pavement alternatives, and as such, costs incurred irrespective of pavement
selection are ignored. If land had to be cleared for a new roadway, for example, costs to clear
land are independent of the pavement selection, and are subsequently outside the scope of this
work. To allow for the fair comparison between pavement alternatives with likely different cash
flows, all costs are converted into a net present value (NPV) to allow for an equivalent time
perspective.
Figure 3-1: Simplified scope and boundary of current LCCA study
- Pavement Removal
- Extraction and
production
- Transportation
- Onsite equipment
- Labor and overhead
- Materials
- Construction
23
- Landfilling
- Recycling
- Transportation
3.3
Quantify Uncertainty
The LCCA analysis in this research is structured to allow for the incorporation of uncertainty
related to the unit-cost of construction, future material prices, bid volume, and occurrence of
maintenance activities (with the last two categories not based on empirical data). The following
section describes the process of statistically characterizing the uncertain parameters considered.
3.3.1
Unit-cost of construction activities
When producing a product, not all costs are the same. Some costs change with respect to
In more formal economic
production levels, while some are constant in the short-run.
terminology, the total cost is composed of both fixed and variable costs. Fixed costs are defined
as costs that are independent of output and can only be eliminated by changing business activity
(Pindyck and Rubinfield 2012). A variable cost is a cost whose magnitude varies depending
upon level of activity. Fixed costs in the short-term may include rent, insurance, and energy costs
to operate the facility. Variable costs, such as raw materials, number of employees (beyond a
minimal level), and distribution costs increase as output increases.
By understanding the differences between fixed and variable costs, the discussion now shifts to
marginal and average costs. Marginal cost, also known as incremental cost, is the cost to
produce one more unit of output (Pindyck and Rubinfield 2012). It is expected that the firm will
stop producing when its marginal revenue, or incremental revenue per unit, is less than marginal
cost. Average total cost, or average economic cost, is the total cost over total production. It
describes the average cost to produce one unit and has two components: average fixed cost and
average variable cost. Since fixed costs remain constant irrespective of output levels, the
average fixed costs should decrease as production levels increase.
Understanding average total cost, it is evident that the average cost of producing an output will
decrease as production increases, to a certain level. Obviously as one increases production levels
the average fixed cost component will decrease, decreasing the average total cost. Average
variable costs would also be expected to decrease for less obvious reasons, such as lower costs to
obtain raw materials by buying in large quantities, employing specialized workers who can work
more efficiently, etc. This concept of decreasing average costs with production levels is also
known as economies of scale. It should be noted that economies of scale can only occur up to a
certain level of production, at which point average costs will start to increase if production is too
large.
Based upon this basic economic theory, this research characterizes uncertainty in the LCCA of
pavements by modeling cost as a function of quantity. This phenomena, as previously assessed
by Herbsman (1986), has only been probabilistically characterized by Tighe (2001), where
historical bid price data is segmented by orders of magnitude and statistically characterized by a
best-fit log-normal distribution. This work differs slightly, in that the relationship between cost
and quantity is treated as a continuous, rather than step, function.
24
To characterize unit-cost
uncertainty, substantial historical bid data is made publically available by state DOTs which
includes bid volume and total bid cost. The total bid cost provided by DOTs typically convolves
manufacturing of materials, transportation to site, and labor and overhead into one number. This
makes it difficult to differentiate the relative contribution of variable and fixed costs to the total
bid price. In spite of this, one way to quantify the relationship between cost and quantity is to
evaluate the average unit-cost (total cost divided by quantity) with respect to bid volume. It is
likely for most relevant construction activities, a statistically significant relationship will exist
To test if the relationship is in fact statistically significant, a univariate
regression analysis can be conducted between average unit-cost and bid volume. In conducting a
regression analysis, both average unit-cost and bid volume should undergo a log-transformation
to allow the data to meet the assumptions of the linear regression analysis. Once estimated, an
between the two.
output from any statistical software package (e.g. Microsoft EXCEL) will report the p-value of
the estimated coefficients. The p-value represents a statistical measure, which ranges from zero
to one, to evaluate if a parameter can be classified as statistically significant (Selke et al. 1999).
If the p-value is less than a threshold value, a, one is 1-a confident the parameter is statistically
significant. Typical p-values used in statistics can range between 0.1 and 0.01, meaning one is
90% to 99% confident that the dependent variable, bid volume in this case, is statistically
significant. For this particular research, a threshold p-value of 0.05 is used. For all construction
activities that show a statistically significant relationship between average unit-cost and bid
volume, the regression analysis is used to project initial costs. Other factors not captured by this
relationship, such as construction site variability, day versus night construction, and
transportation costs from the plant to the project are modeled by using the standard error of the
univariate regression equation when conducting a Monte Carlo simulation. For construction
activities where a statistically insignificant relationship exists between average unit-cost and bid
volume, a chi-square best-fit log-normal distribution is fitted to the data, consistent with the
approach of Tighe (2001).
3.3.2
Inputs with no empirical data
Given that uncertainty is an important factor in risk analysis, it is important to have a general
methodology to characterize inputs with no data available. Many inputs in an LCCA, such as
the thickness of a pavement layer or density of a mixture, have variability but no historical data
required for statistical characterization. The area of characterizing uncertainty using expert
judgment has been an evolving field. One general method is to evaluate the subjective risk of a
given input (Morgan and Henrion 1990). Although such a methodology has not been formalized
in the pavement LCCA literature, this work will adopt risk factors quantified in the life-cycle
analysis (LCA) community, known as the pedigree matrix approach (Weidema et al. 2011).
The ecoinvent pedigree matrix recognizes two extremely general types of uncertainty: basic
uncertainty and additional uncertainty (Weidema et al. 2011). Basic uncertainty is defined as
values with variation and stochastic error resulting from process, geographic, or temporal
variation. For example, the prescribed pavement thickness by a pavement engineer will not be
25
perfectly met in the field.
Additional uncertainty is defined as uncertainty resulting from
temporal, geographic, or technological correlation, as well as completeness and reliability in the
underlying data. If a construction activity is estimated based upon data from five years ago, for
example, there is temporal uncertainty related to the data in use.
The power of the pedigree matrix approach is it allows for the quantification of uncertainty by
qualitatively evaluating the input parameters (Weidema and Wesmaes 1996). Quality indicator
scores are used to assess the input values being used in the LCCA model and are transformed
into variances which are applied to the input parameters in the form of a log-normal distribution.
The power of the pedigree matrix approach is that it allows for the estimation of uncertainty by
simply qualitatively evaluating the input values. For additional uncertainty, quality indicators
are applied to assess the input values being used in the LCCA model. The basis for qualitatively
assessing input values is shown in Table 3-1.
Table 3-1: Pedigree matrix quality indicator criterion (Weidema et al. 2011)
Reliability
1
2
Verified data
Verified data
based on
measurement
partly based on
assumptions or
non-verified
data based on
measurements
Completeness
Representative
data from all
sites relevant for
the market
Representative
data from >50%
of the sites
relevant for the
considredaover
considered, over
a rk
onied
thpeens
market
an adequate
period, to even
period to even
out normalout normal
fluctuations
out normal
fluctuations
Temporal
Correlation
Geographic
Correlation
3
4
5
Non-verified
data partly
based on
qualified
estimates
Quantified
estimate
Non-quantifiable
estimate
Representative
data from only
some sites
(<50%) relevant
Representative
data from only
one site relevant
Representativeness
unknown or data
considered or
some sites but
from shorter
number of sites
and from shorter
periods
temreat
(<o0%
for
d
>50% of sites
0of site
butperiods
Less than 6
years of
difference to the
time period of
Less than 10
years of
difference to the
time period of
the dataset
the dataset
the dataset
Averaged data
Data from area
from larger area
area nder sudy
with similar
p
production
c
araudrsuy
conditions
is included
Further
Technological
Correlation
from a small
periods
peidpeisAgofat
Less than 3
years of
difference to the
time period of
Data from area
under study
for the market
Less than 15
years of
difference to the
time period of
the dataset
Age of data
unknown or more
than 15 years of
difference to the
time pdaasetof the
Data from area
with slightly
proucio
Data from
unknown or
distinctly different
prdcinarea
conditions
Data from
Data from
enterprises,
processes and
materials under
processes, and
materials under
study but from
materials under
study
btfrom
studystudy
different
study
~technology
26
Data on related
processes or
materials
Data on related
processes on
laboratory scale or
from different
technology
tcnlg
The quality indicator scores are then transformed into variances for an underlying normal
distribution. The associated variances for each indicator score are show in Table 3-2. It should
be noted that the basic uncertainty variance for all input parameters uses the uncertainty factor
for primary resources of 0.006, the most closely related category to this thesis. The total variance
calculation, presented in Equation 3-1, is applied as a log-normal distribution:
6
.2
02
(3-1)
02 = basic uncertainty (default value of 0.006 applied)
'22 = uncertainty variance of reliabilitydistribution
'32 = uncertainty variance of completeness distribution
2
= uncertainty variance of temporal correlationdistribution
52 = uncertainty variance of geographicalcorrelationdistribution
-62 = uncertaintyvariance of technological correlationdistribution
Table 3-2: Uncertainty factors (variances of the underlying normal distribution) for the pedigree matrix
quality indicators (Weidema et al. 2011)
1
2
3
4
5
Reliability
0.000
0.0006
0.002
0.008
0.04
Completeness
0.000
0.0001
0.0006
0.0002
0.0008
Temporal Correlation
0.000
0.0002
0.002
0.008
0.04
Geographic Correlation
0.000
0.000025
0.0001
0.0006
0.002
Further technological Correlation
0.000
0.0006
0.0008
0.05
0.12
Indicator Score
The pedigree matrix approach is used for all inputs with uncertainty but no empirical data. For
this study, that includes pavement design inputs as well as years of maintenance activities.
Future material costs
Cash flow estimates (and the various project parameters from which they derive) for an
LCCA are inherently uncertain, both in the short-term and long-term. One set of LCCA
parameters with a high level of uncertainty is the evolution of material prices over time.
Estimates of cost for a given project may occur months or years before construction begins, and
the completion of construction can take multiple years. Infrastructure projects are also expected
to have long life-cycles with maintenance activities occurring decades into the future. If prices
3.3.3
27
remained static over time, planners could accurately project costs for construction projects. Due
to the volatile nature of material prices, however, projecting short-term and long-term
construction costs has shown to be historically difficult, resulting in inaccurate cost projections.
For example, Gransberg and Rierner (2009) discuss the difficulties in projecting construction
prices and the contingencies DOTs implement to account for these risks. This raises two basic
questions: can the uncertainty when projecting future construction costs be statistically
characterized, and, if so, would it improve the final decision around selecting a pavement? The
general methodology of this work, and the more detailed questions which are asked, are outlined
in these four steps:
1) Evaluating historical price behavior: It is fundamental to first make the assertion that
material prices have historically behaved non-static. If data shows both concrete and
asphalt have non-zero growth-rate and volatility measures, and additionally they are
different between the two, it can be confirmed both materials are non-static and have
historically behaved differently. The latter statement is likely true given that the two
materials require different proportions of different inputs, but is fundamental to validate
nevertheless.
2) Quantifying how energy prices drive paving costs: If historical prices of asphalt and
concrete have behaved non-statically and differently, a question to ask is why. This
question has likely several answers, but one of the most likely is that a price spillover
exists between energy prices and construction materials. Energy prices play a pivotal
role in manufacturing and transportation costs for construction materials.
This is
amplified in the case of asphalt, a petroleum product. Understanding this, and knowing
that energy prices, in particular oil and natural gas, have been historically volatile, one
likely reason paving material prices would have behaved non-linearly over time is that a
significant price-link exists between paving and energy prices. If asphalt and concrete
have historically had differential price-links to energy prices (i.e. the price of asphalt is
either driven by different energy sources or the same sources but at different rates) it
would likely explain much of the differential price behavior. Therefore, this research
tests if a price-link exists between paving materials and three major non-renewable
energy sources (oil, natural gas, and coal) through cointegration.
By testing for
cointegration, it can be statistically quantified if a) different commodities are linked to
different energy prices and b) does a differential price-link exist for energy prices that
both paving materials can be "cointegrated" with.
This step is only used to explain potentially why materials have historically behaved
differently and not actually used in steps 3 and 4 to forecast future prices. The results in
this analysis, however, could be expanded upon in future studies if a statistically
significant relationship exists between a paving material and an energy commodity by
projecting the future price of an energy commodity as a proxy.
28
3)
Testing if forecasting long time-horizons is plausible: If the previous steps show
concrete and asphalt behave quite differently, it is important to validate if projecting
future prices decades into the future is possible. This research tests whether projecting
the future is plausible and effective by backcasting historical price data.
4) Lastly, if the three previous steps conclude asphalt and concrete historically have
behaved non-static and different and projecting them is possible, the next step is to
quantify probabilistic price projections which can be implemented in an LCCA. This
work will only consider univariate processes which forecast price as a function of time,
but as mentioned previously, future studies could potentially use the long-run price
relationships derived in Step 2 to project future prices.
3.3.3.1 Step 1: Evaluating historical price behavior
The purpose of this first step is to validate historically the real-price of asphalt and concrete have
behaved both non-statically and differently. This will be accomplished by collecting historical
real-price data and calculating the historical growth-rate and volatility of asphalt and concrete. If
the growth-rate and volatility between the two commodities is both non-zero and different, it
would suggest an LCCA would account for such behavior
3.3.3.2 Step 2: Quantifying how energy prices drive paving costs
As discussed, it is expected the cost of all construction materials are related to the price energy.
If the price behavior of asphalt and concrete has historically been different, one likely
explanation is a differential price spillover (i.e. price-link) exists between the pavement materials
and different energy commodities (i.e. oil, natural gas, coal). One relatively recent method to
test the relationship of different commodities is cointegration (Franses and Dijk 2009).
Cointegration allows for one to test if a statistically significant price-link exists between multiple
time-series, and if so, what is the price-link. The latter is done by calculating a stationary linear
combination which describes the long-run price equilibrium between the variables and
characterizes two important metrics. First, in this particular analysis, the long-run price
equilibrium explains how a unit change in the price of an energy source impacts the price of the
two paving materials. Second, the coefficient of determination, commonly denoted as R 2,and
represents the goodness of fit of a given model, can be used to measure how much of the
volatility the two paving materials experience is explained by the price-link with a given energy
source. If either asphalt and concrete are cointegrated with different energy sources or both
metrics are significantly different between shared, cointegrated energy sources, it would likely
explain the historically differential price behavior of different paving materials.
Testing if it is possible the time-series are "cointegrated"
Commodities are considered cointegrated if they shift stochastically through time but never far
apart from one another. Before testing for cointegration, or in fact, conducting any econometric
analysis, it is important to understand if the time-series are "non-stationary" or "stationary". If a
29
time-series is non-stationary, it implies the average value shifts over time, whereas stationary
indicates the time-series tends to shift randomly around a single mean value (Chatfield 2003).
One method to assess if a time-series is stationary or non-stationary is to test if a unit-root exists.
A unit-root represents an autoregressive process, which will be expanded upon briefly, whose
autoregressive coefficient equals one. If a time-series does, in fact, have a unit root, it suggests
that the data exhibits autocorrelation. That is, the price in the future is dependent upon previous
data, and as such, future values are not completely random (Ramberg 2010). A time-series that
is autocorrelated (i.e. exhibits a trend) can be classified as non-stationary. It is possible that two
non-stationary time-series are cointegrated, however, only if they are integrated to the same
order. To test the order of integration for a non-stationary time-series, a unit root test can be
conducted for each difference of the time-series until it exhibits no unit-roots and can therefore
be classified as stationary. As an example, if a time-series must be differenced twice until it is
stationary, it can be said that the data is integrated to the second order. Two of the more
common tests are the Augmented Dickey-Fuller and Phillips-Perron tests (Leybourne and
Newbold 1999). For this research, the Augmented Dickey-Fuller (ADF) test is used to test if the
relevant data sets are stationary or not.
The basis of the ADF test is an ARIMA (p, d, q) model, where p is the number of autoregressive
terms, d is the number of non-seasonal differences, and q is the number of moving average
terms. More generally, a time-series, Y, is defined by:
q
P
Yt
where a and
1-
a, Li (1 - L)d =
Et
1 -
fli Li
(3-2)
fl are the coefficients of the autoregressive and moving average terms, and L is the
lag operator. The ADF tests the null hypothesis of a non-stationary, autoregressive process with
a trend against a stationary, autoregressive model that is constant over time. In other words, an
ARIMA (p, 1, 0) is the null-hypothesis, and is compared to an ARIMA (p+1, 0, 0) process
(Cheung and Lai 1995). The regression equation of the ADF test is as follows:
p-1
AYt = I
+
yt
+
aYt-
lAYt-i + Et
(3-3)
i=1
where A Y, is the first difference operator, p is a constant, y is the time trend, and E represents the
white noise of the process. For the given regression equation, there are three types of ADF tests
that may be conducted. For simplification, if it is assumed that a given time-series follows an
ARIMA (1, 0, 0), the three types of ADF tests that may be conducted are:
1) Unit-root test with no constants: AYt =
#AYt-
30
1
+ Et
2) Unit-root test with drift: AYt = /I + PAYt-
1
+ Et
3) Unit-root test with drift and time-trend: AYt = p + yt + flAYt-
1
+ Et
It should be noted that the first two tests are simply the third test with pi and/or y set equal to
zero. The selection of which of the three tests to use is not necessarily intuitive, and it can
significantly influence the results of an ADF test. For example, if a time-series has a timerelated trend but is tested using only the drift term, it will likely result in the ADF test suggesting
a unit-root exists since the regression can only account for the trend by including the
autoregressive term. One simple method to account for this is to conduct a unit-root test with
both a drift and time trend, and if a unit-root cannot be rejected, one can say with general
confidence that a unit-root does exist (Elder and Kennedy 2001). If a unit-root does not exist,
however, further testing is required.
An important consideration in running an ADF test is selecting a number of lags that is neither
too high nor too low. To select an appropriate number of autoregressive terms, p, empirical
research suggests running the ADF regression equation for a series of lag orders and selecting
the lag order which minimizes one of two possible "goodness of fit" measures: Akaike
Information Criterion (AIC) and Bayesian Information Criterion (BIC) (Cheung and Lai 1995).
Unlike the coefficient of determination, both the AIC and BIC penalize the regression for
selecting too many lag orders.
Testing if a statistically significant relationship exists between paving materials and energy
prices, and if so, is it different
If two time-series are integrated to the same order it is possible that they are cointegrated.
The next step is to see if a stationary linear combination exists between the multiple time-series.
If a stationary linear combination does exist, it can be confirmed that the time-series are
cointegrated, and the stationary linear combination quantifies the long-run price relationship
between the time-series of interest. The most popular cointegration test is the Johansen test,
which tests the restrictions imposed by cointegration on an unrestricted VAR model (Xiarchos
2006). A VAR model captures the interdependencies between multiple time-series, and is
described by the following equation:
Yt = C + A1Y_1 + A 2 Yt-2 +
+A Y
+ Et
(3-4)
where C is a k x 1 vector of constants, Y, is a k x 1 vector of the considered variables, A is a k x k
matrix that describes the price transmission between the variables considered, e is a k x 1 vector
of error terms, and p represents the number of lag terms in the VAR model. Given data prior to
t-1, the model can project the price for different commodities in year t. One important
consideration when constructing a VAR model is to select the appropriate number of lags, p, for
the model. Unfortunately, the selection of the correct lag order is not necessarily intuitive. This
issue in the VAR literature has been considered, and to date, a large body of empirical research
31
has explored the topic (Killian 2001). For the type of large data sets (greater than 240 data
points) available for this research, empirical data suggests that selecting a lag order resulting in
the lowest Akaike Information Criterion (AIC), one statistical goodness of fit measure, best
selects the true lag order of the model (Killian 2001).
As mentioned, the Johansen cointegration test is used to test the restrictions imposed by
cointegration on the above, unrestricted VAR model. This is accomplished by transforming the
VAR into the following Vector Error Correction Model (VECM):
p-1
APt = H Pt_1 + >IAPi
+
(3-5)
Et
i=1
p
H=I
p
Ai-I
and
i=1
fi=
A;
(3-6)
j=i+1
where I is the identity matrix while all other terms remain the same as in the VAR model. If the
coefficient matrix, H, reduces in rank order, r, than it can be said that there are r cointegration
relationships amongst the variables of interest. From the VECM, a stationary linear combination
of the variables that are cointegrated can be derived which describes the long-run price
equilibrium.
3.3.3.3 Step 3: Testing ifforecasting long time-horizons is plausible
The basis of time-series forecasting is that past historical data are a good predictor for future
events. Of course, it is important to validate this assumption given the extended time-horizon of
infrastructure projects. One method to accomplish this is backcasting, where a future outcome is
known, and the forecast is compared to the known future (McDowall and Eames 2004).
Employing backcasting techniques, there are two parallel questions to answer. First, how do
different forecasting techniques compare to the currently practiced "no change" forecast, and
second, how much data is needed to outperform the "no change" forecast? This second question
is particularly pertinent for asphalt, where there are significantly less historical data available
than its counterpart in the paving industry, concrete. If, in fact, it can be inferred that there is not
enough data to project the future, than perhaps more advanced analytical techniques must be
employed in this analysis.
To answer these questions, data sets provided by the United States Geological Service (USGS)
for cement and construction sand and gravel, dating back to 1900, are statistically analyzed using
a real-price inflation rate, Geometric Brownian Motion (GBM), and mean reverting model (Kelly
and Matos 2012a; Kelly and Matos 2012b). The real-price inflation rate and GBM models
assumes future prices will follow a trend. The directionality of the two models is the same, but
the degree of trend will vary. The mean-reverting model, however, assumes future prices revert
32
back to a mean value over time. Figure 3-2 presents sample plots of what these types of models
typically look like.
Figure 3-2: Plot for a hypothetical time-series which follows an a) real-price inflation rate model b)
Geometric Brownian Motion (GBM) model and c) a mean-reverting model. The dotted red line represents
the time-series data and the smooth black line represents the forecast
Time
Time
Time
The real-price inflation rate model is equivalent to a geometric random walk model i.e. ARIMA
(0, 1, 0). The equation for the real-price inflation model is as follows:
Pt = Pts * (1 + r)
(3-7)
where r is the real-price inflation rate and P is the price in a given year. Research at the MIT
Concrete Sustainability Hub stochastically simulated the above model thousands of times to
probabilistically project future real-prices of major construction materials over the next 50 years
(Lindsey et al. 2011). Estimations of the real-price inflation rate, r, were based on fitting
historical real-price change data to a chi-square best-fit distribution, and as such, this thesis will
utilize the same methodology for estimating the real-price inflation rate parameter.
The GBM process is based upon the following stochastic differential equation (Gerber and Shlu
2000):
dPt = ysedt + astdWt
(3-8)
where P, is the stochastic process, p and u represent the logarithmic growth rate and standard
deviation (volatility measure), and W, is a Wiener process. The expected issue with the two
previously mentioned models is they are autoregressive processes, which are only appropriate for
time-series that are non-stationary. Since it is possible some construction time-series are
stationary, meaning random but tending to be random around one mean value, a more
33
appropriate time-series model may be a mean-reverting process. One such random process is the
Ornstein-Uhlenbeck process, which follows the stochastic differential equation (Pindyck 1999):
dPt = K(y - Pt)dt + c-dWt
(3-9)
where K is the speed of mean-reversion, p is the mean value the process tends to revert to, and a
represents the volatility. The above differential equation has the flexibility to allow for modeling
of price which reverts back to a changing mean (i.e. linear trend, quadratic trend) if the data has a
constantly reverting mean. This, however, is not used in this particular study. The implication
of the above process is if one is forecasting while at a point above the mean, the forecast will
revert downward asymptotically towards the mean, and of course, vice versa. In the financial
literature, mean reverting processes, although recognized, are oftentimes ignored since the time
required to revert back to a mean value is longer than the time-horizon of the analysis (Pindyck
1999). Given that infrastructure projects, however, have maintenance activities decades into the
future, such a process may be appropriate.
Each of the three processes mentioned above are applied to the Cement and Construction Sand
and Gravel time-series to answer the questions of how much data is needed to project the future
and does projecting the future outperform the "no change" forecast. To answer both questions,
forecasts are made between 1940 and 1980 under two scenarios: the first using as much data
available up until the year of the forecast, and the second limiting the forecast by using no more
than 35 years of data. The forecasted values are then compared to what actually occurred, and the
accuracy of the forecast is measured by Mean Absolute Percent Error (MAPE), one of the more
common performance metrics (Armstrong 2001). The MAPE is defined
_
MAPE =
100% i
n
iActual - Predicted(
|
Actual
i=1
(
where n is the number of forecasts. The MAPE for each of the three mentioned processes is
calculated and compared to the MAPE of the "no change" forecast, with the hypothesis the
forecasting model will outperform it. If it is possible to outperform the "no change" forecast, a
time-series model is constructed to probabilistically project the price of asphalt and concrete.
3.3.3.4 Step 4: Derive probabilisticforecasts
If the analysis generally concludes that concrete and asphalt are two commodities with two
different volatility measures, and it is plausible to project the future, the next step is to actually
derive probabilistic forecast values for concrete and asphalt. First, to assess which model is most
appropriate, both time-series are subject to a unit-root test to conclude whether each series is
stationary or non-stationary. It should be noted that unless the time-series reverted back to a
mean value quickly, it is likely the test results will not be able to reject the null hypothesis of a
unit-root existing due the limited sample size in both cases. If, as anticipated, a unit-root cannot
be rejected, the model is assumed to be non-stationary. Under this scenario, a GBM model is
most likely the most appropriate model if the growth rate and volatility of the model is relatively
34
constant over a range of sample sizes (Pindyck 1999). This is because if the commodity actually
follows a slowly mean-reverting process that the unit-root test cannot recognize, the assumption
of a GBM model is appropriate if volatility is relatively constant for any time-period.
3.4
Monte Carlo Simulations
When performing a Monte Carlo simulation, random values are sampled from probabilistic
distributions thousands of times to form a distribution of possible outcomes. An important
consideration when conducting a Monte Carlo simulation is that the values selected, although
random, structurally make sense. Therefore, it is paramount that the simulations account for
both correlation and dependencies between input parameters. Dependencies represent statistical
relationships between multiple variables. For example, the year of occurrence for a second
maintenance is clearly dependent upon the first maintenance activity. Correlation is the common
inputs each alternative shares. For instance, the source of materials for two asphalt designs is
expected to be the same. Considering dependencies and correlation allows for a Monte Carlo
simulation to select values that are reasonable and not completely random and unrealistic.
To account for correlation, the LCCA analysis is structured such that during an iteration, the
randomly drawn value for correlated inputs is the same. Dependent random variables are
accounted for by linking them together to ensure sensible outputs occur during a Monte Carlo
simulation.
3.5
Using Results to Evaluate Alternatives
One way to visualize the results from a Monte Carlo simulation is to examine the results in the
form of a cumulative distribution function (CDF); this is the integral of the more commonly
known probability distribution function (PDF). The usefulness of a CDF is a decision-maker can
select an alternative based upon his or her risk-perspective. It is expected that a decision-maker
is not necessarily interested in the expected cost for a project, but potentially more interested in
limiting his or her losses (risk-averse). From a risk-averse perspective, a decision-maker would
use a CDF to select a pavement that has a high probability of costing less than a certain
threshold. Figure 3-3 presents a sample CDF for two alternative designs. In this particular
example, Design B is significantly superior to Design A for a low cumulative probability. As the
cumulative probability increases, however, the superiority of Design B steadily declines until
around the 90'h percentile when Design A becomes superior. This means if a decision-maker had
a risk-seeking or risk-neutral perspective, the selection of Design B is obvious. However, as the
decision-maker becomes more and more risk-averse, Design A gradually becomes the superior
selection.
35
Figure 3-3: Sample CDF output from Monte Carlo simulations. Dashed lines represent a risk-averse (7 5 *h
percentile) and highly risk-averse (9 5th percentile) perspective
100%
95* percentile
80%
75* percentile
60%
-5
---- Design A
CU40%
-Design
B
20%
0%
Cost of Roadway
Three particular values are pulled from the CDF and compared between the alternatives for each
scenario: the mean value (risk-neutral perspective), 7 5th percentile (risk-averse perspective), and
95t percentile (highly risk-averse perspective). The 75*h and 95th percentile values are more
commonly referred to as the Value at Risk (VaR) for a given probability. For example, a VaR of
25% means there is a 25% probability that the cost of a design will exceed a certain value,
representing the 75th percentile from the CDF. By considering different points along the CDF
curve, this thesis aims to understand how the selection changes based upon the risk-profile of the
decision-maker.
3.6
Understanding Which Types of Uncertainty Impacts an LCCA
This thesis focuses on quantifying four types of uncertainty: the unit-cost of construction
activities, the quantity of inputs, the occurrence of maintenance activities, and future material
costs. As discussed in Chapter 2, most previous probabilistic LCCAs have focused on only one
type of uncertain parameter. One effort of this thesis is to determine which uncertainties
significantly impact the LCCA analysis to steer future research efforts towards those areas.
For the case study, four types of analyses are conducted: a deterministic analysis (including and
excluding material-price forecasts) and a probabilistic analysis (including and excluding
material-price forecasts). If a probabilistic analysis does not lead to a different pavement
selection excluding material-price projections, but does including, it suggests accounting for
36
differential material price behavior in an LCCA may be significant. Table 3-3 presents the
conceptual framework of this case study analysis.
Table 3-3: Structure of case study analysis
Probabilistic Analysis
Including
Excluding
Material-Price
Forecasts
Material-Price
Forecasts
C6
Including
Material-Price
Forecasts
Excluding
Material-Price
Forecasts
3.7
Methodology Summary
This chapter has presented a methodology to characterize different types of uncertainties when
conducting a probabilistic LCCA
e
Historical bid data will be characterized by considering the likely relationship between
cost and bid volume. In the case where a construction activity shows no statistically
significant relationship between the two, a best-fit probability distribution is fitted to the
data.
*
Uncertain input parameters with no empirical data available (i.e. pavement thickness,
pavement mixture) will be characterized using the LCA pedigree matrix approach, which
transforms quality indicator scores into probabilistic uncertainties.
*
*
Differential probabilistic material price projections will be made if it can be validated:
o
Asphalt and concrete have historically behaved non-statically and at different
rates.
o
Empirical analyses show projecting decades into the future is plausible.
A case study analysis will be conducted to assess if probabilistic change the pavement
preference in a deterministic analysis, and if so, what input parameter(s) are driving the
change.
The following two chapters will implement the above methodology to answer the existing gaps
in the literature described in Section 2.4.
37
4
QUANTIFICATION OF UNCERTAINTY IN LCCA INPUT DATA
As discussed in Chapter 2, little guidance has been provided for practitioners on how to
probabilistically quantify uncertainty. Therefore, one of the key contributions of this thesis is the
probabilistic quantification of relevant input parameters for a pavement LCCA. This particular
chapter focuses on two types of uncertainty that are quantified: the unit-price of bid items and the
forecasting of future material prices.
The values presented in this chapter are specific to paving construction and limited to those only
used in the case study analysis. Further detailed information of quantification of uncertainty by
use of the pedigree-matrix approach is presented in the Appendix.
4.1 Analysis of Historical Bid Data
For many construction processes there is significant variability in cost due to several likely
factors such as construction site variability, day versus night construction, transportation
distances for materials, etc. Unfortunately, historical bid data made publically available by
DOTs only provide total cost and quantity information, convolving manufacturing,
transportation, labor, and other project specific information into one number. Due to this
limitation, one is unable to model many of the likely factors driving uncertainty and variability in
the data. For example, uncertainty in cost data likely varies between cities or counties, but
unfortunately must be ignored given that cost data is only provided at a state level. As a result,
this thesis only models cost as a function of quantity, both due to this limitation and the cited
literature which has shown a relationship between the two (Herbsman 1986; Tighe 2001).
Although this may be a limitation of this work, it is likely a reasonable type of analysis to
conduct given that an LCCA typically occurs early on in the design process before many factors
are known.
This analysis is based off historical bid data provided by California's DOT (Caltrans) over a 12month span (CalTrans 2012a). For each relevant construction process, the average unit-cost is
calculated by simply dividing the total cost by total quantity for each bid. Data points undergo a
log transformation, and a univariate regression analysis is conducted where average unit-cost if a
function of quantity. If the p-value of the dependent variable (quantity) is less than 0.05, the
relationship between cost and quantity is considered statistically significant. Construction
processes that show no statistically significant relationship between cost and quantity are
quantified using a best-fit log-normal distribution.
Figure 4-land Figure 4-2 presents a sample analysis of the unit-price of Jointed Plain Concrete
Pavements (JPCP) over a 12-month span (CalTrans 2012a). Data collected, as shown in
Figure 4-1, is logarithmically transformed, and a univariate regression analysis of cost with
respect to quantity is conducted, as presented in Figure 4-2. For this particular dataset, the
coefficient of determination (R2) is 0.7, meaning 70% of the variation can be described by this
38
simple analysis. To account for other factors driving the variability, the standard error of the
regression equation coefficients are used when conducting Monte Carlo simulations.
Figure 4-1: Histogram of collected bid data for JPCP pavements over a 12-month span in CA (CalTrans
2012a)
30%
20%
10%
~IIIIII
0%
20
100
260
180
340
I
420
II
500
580
Unit Price of Concrete ($/CY)
Figure 4-2: Regression analysis of unit-price of JPCP winning pavement bids with respect to bid volume
LN (Unit-Price) = 6.70 - 0.18 * LN (Quantity)
7
6
5
~4-o
0
4
3
z
2
1
0
0
8
4
Natural Log of Quantity (Cubic Yards)
39
12
Table 4-1 presents the regression analysis results for the most statistically significant unit-cost
inputs for the model. As can be seen, the result in Figure 4-2 generally holds true for all unitcost parameters for the case study analysis. A p-value significantly less than 0.001 is found for
every regression analysis, and as such, a statistically significant relationship exists for all
parameters between cost and quantity. The coefficient of determination, representing the
"goodness of fit" of the regression equation, ranges between 50% and 70% for all input
parameters except for patching, where it is only 16%.
The standard error of the regression
equation coefficients ranges from 1% to 5% of the intercept values and 7% to 20% of the
dependent coefficient values.
Table 4-1: Quantification of unit-cost uncertainty for significant input parameters. Values in parenthesis
represent the standard error of the regression coefficients
Input
Units
Coefficient of
Determination
Regression Analysis
Ln(y)=a*Ln(x)+b
P-Value
Statistically
Significant?
Initial Concrete Design
Concrete
Cubic
a = -0.18 (0.019)
0.69
<<0.001
True
Lean Concrete
Base
Cubic
Yards
a = -0.13 (0.012)
b = 5.83 (0.093)
0.51
<<0.001
True
Aggregate Base
Cubic
a = -0.16 (0.033)
0.63
<<0.001
True
Yards
Yards
b = 6.70 (0.17)
b = 5.50 (0.21)
Initial Asphalt Design
Surface HMA
Tons
a = -0.20 (0.009)
b = 6.17 (0.06)
0.61
<<0.001
True
Intermediate
HMA
Tons
a = -0.26 (0.037)
b = 6.50 (0.24)
0.51
<<0.001
True
Aggregate Base
Cubic
a = -0.16 (0.033)
5.50 (0.2 1)
0.63
<<0.001
True
Aggregate Subbase
Cubic
Yards
a = -0.27 (0.038)
b = 5.48 (0.29)
0.62
<<0.001
True
Yards
b
=
Maintenance Specific Input Parameters
Diamond
Grinding
Square
Yards
a = -0.23 (0.016)
b = 4.31 (0.17)
0.56
<<0.001
True
Patch
Cubic
Yards
a = -0.38 (0.055)
b = 7.65 (0.16)
0.16
<<0.001
True
Mill
Square
Yards
a = -0.42 (0.030)
b = 5.55 (0.22)
0.62
<<0.001
True
4.2 Projection of Future Material Prices
The following presents the results of the pavement material-specific forecast analyses. The
purpose of this work is to characterize the historical behavior of asphalt and concrete in order to
project future prices of paving materials. In Step 1, the growth rate and standard deviation of
historical real-price data for asphalt and concrete are used to assess if the two construction
materials have historically behaved non-static and differently. From that simple analysis, Step 2
tests if the price of asphalt and concrete has shifted as a result of changes in energy prices. Step
40
3 tests whether it is plausible or not to forecast future prices through backcasting. Lastly, if the
expected price behavior of the paving materials is expected to change and empirical results
suggest it is both plausible and effective to project the future, probabilistic price projections are
made in Step 4.
4.2.1
Data collection
This analysis uses monthly nominal-price data from Bureau of Labor Statistics (BLS) for the
Ready-Mix Concrete and Asphalt Paving data sets (BLS 2012a; BLS 2012b). Energy prices are
collected from the U.S. Energy Information Administration (EIA) for natural gas and Brent Spot
oil prices and BLS for coal data (EIA 2012a; EIA 2012b; BLS 2012c). All time-series are
deflated with the Consumer Price Index (CPI) (BLS 2012d). Figure 4-3 presents the price
indexes analyzed in the cointegration analysis, with all data points normalized to a real-price
index of 100 in January of 1984, the earliest date where monthly price data was available for all
five time-series. The use of CPI has been used by various time-series models to forecast future
prices. For example, Alquist et al. (2011) deflate oil prices when attempting to forecast the realprice of oil. Despite this, studies have shown that bias may be introduced by selecting the CPI
(Peterson and Tomek 2000). Therefore, although this thesis only considers deflating time-series
with the CPI, future work should conduct a sensitivity analysis to see if the results hold true for
other time-series deflators.
Figure 4-3: Monthly real-price data of asphalt, concrete, oil, natural gas, and coal used in cointegration
analysis (BP 2012; BLS 2012a; EIA 2012a; BLS 2012b; EIA 2012b; BLS 2012c; BLS 2012d)
Natural Gas
200
150
Coal
Oil
1984
1989
1994
Date
1999
2004
2009
As can be seen above, the real-price of natural gas and oil have been quite volatile over the past
three decades relative to coal. Potentially, concrete and asphalt could be linked differently to the
three types of energy sources in this analysis, and as such, this could potentially drive the
volatility which both materials experience differentially.
41
Step 1: Assessing the historical behavior of pavement materials
4.2.2
Figure 4-4 presents the real-price index of concrete and asphalt as shown above but annualized
with 2011 as the base year. As can be seen, the real-price of concrete and asphalt has historically
behaved quite differently between the two paving materials. Table 4-2 presents the historical
real-price growth rate and volatility (i.e. standard deviation) of the concrete and asphalt timeseries. Although there is a slight difference between the two commodities in terms of growth
rate, there is a much more dramatic discrepancy when considering volatility.
Table 4-2: Historical Growth rate and standard deviation of growth rate for Asphalt and Concrete
Asphalt
1.3%
6.3%
Concrete
-0.2%
2.6%
Growth Rate
Standard Deviation
Figure 4-4: Annualized Ready-Mix Concrete and Asphalt Paving time-series (2011
(BLS 2012a; BLS 2012b; BLS 2012d)
=
100 for both data sets)
150
Concrete
120
90
.
-Concrete
60
-
-Asphalt
Asphalt
30
0
1958
1968
1978
1988
1998
2008
Year
Based upon the discrepancy in growth rate and volatility between concrete and asphalt, one
wonders why this has historically occurred, and if this is expected to continue, how should an
LCCA account for this. The former is the focus of the next step
4.2.3
Step 2: Testing the price-link between paving materials and energy prices
This section tests if asphalt and concrete are "cointegrated" with oil, natural gas, and/or coal to
assess which energy prices have historically driven the real-price of each paving material. This
is first done by seeing if it is plausible the different time-series could be considered
"cointegrated". It is only possible that time-series are cointegrated only if they are integrated to
the same order. To test the order of integration for all five time-series, an ADF test is conducted
for lag orders selected by both the BIC and AIC criteria in order to ensure the test results are
independent of the goodness of fit measure selected. Table 4-3 and Table 4-4 presents the results
running the regression analysis for the concrete, asphalt, and oil time-series. As can be seen
below, the AIC and BIC "goodness of fit" measures selected a different number of lags for both
the Ready Mixed Concrete and Asphalt Paving time-series.
42
Table 4-3: Goodness of Fit performance of paving material time-series fitted to ADF regression. Bold values
represent number of lags with lowest AIC or BIC value for all lags considered
Concrete
Asphalt
Lags
1
2
3
4
5
6
7
8
BIC
-21.94
-21.77
-16.27
-11.50
-6.09
-4.70
-0.70
3.58
AIC
1.610
1.601
1.609
1.615
1.623
1.618
1.622
1.626
BIC
-146.37
-142.22
-140.10
-148.83
-146.45
-145.29
-142.93
-143.38
AIC
2.973
2.976
2.969
2.935
2.932
2.925
2.922
2.910
9
8.99
1.634
-142.51
2.902
10
10.80
1.630
-141.01
2.896
11
9.13
1.615
-135.90
2.900
12
-3.19
1.563
-138.34
2.882
13
2.282
1.571
-132.18
2.890
14
7.03
1.577
-129.77
2.887
15
13.08
1.587
-125.20
2.891
Table 4-4: Goodness of Fit performance of energy price time-series fitted to ADF regression. Bold values
represent number of lags with lowest AIC or BIC value for all lags considered
Coal
Natural Gas
Oil
Lags
1
2
3
4
5
6
7
8
9
10
11
12
13
BIC
-45.65
-41.25
-35.51
-32.66
-28.97
-28.65
-0.7
3.58
8.99
10.8
9.13
-3.19
2.282
AIC
6.476
6.48
6.489
6.489
6.49
6.481
1.622
1.626
1.634
1.63
1.615
1.563
1.571
BIC
-1.01
-2.33
0.49
3.75
9.40
14.74
20.83
19.43
17.36
22.98
25.32
29.84
34.58
AIC
9.018
9.008
9.007
9.007
9.014
9.019
9.026
9.016
9.003
9.009
9.008
9.011
9.015
BIC
-7.28
-1.08
-1.59
3.81
10.06
15.85
20.23
25.64
31.61
34.15
37.22
36.52
42.02
AIC
2.648
2.656
2.646
2.651
2.658
2.664
2.667
2.673
2.679
2.678
2.676
2.667
2.672
14
15
7.03
13.08
1.577
1.587
40.59
46.64
9.023
9.030
42.97
47.38
2.667
2.670
43
The results from the ADF tests are presented in Table 4-5. For all five time-series, the test
statistic is greater than the 5% critical value when the time-series are not differenced, meaning
that the null hypothesis of a unit-root existing cannot be rejected at the 5% level. When the timeseries are differenced, however, the test statistic is less than the 1% critical value, indicating at
the 1% confidence level the null hypothesis of a unit-root existing can be rejected, and therefore
the first difference for all five time-series can be classified as stationary.
Table 4-5: ADF test results for lag orders selected by BIC and AIC criteria. Bold values are less than 5%
critical values
.
Criteria to
select lag order
Concrete
Concrete
Asphalt
Asphalt
Oil
Natural Gas
Natural Gas
Coal
Coal
BIC
AIC
BIC
AIC
BIC, AIC
BIC
AIC
BIC
AIC
Concrete
Concrete
Asphalt
Asphalt
Oil
Natural Gas
Natural Gas
Coal
Coal
BIC
AIC
BIC
AIC
BIC, AIC
BIC
AIC
BIC
AIC
No Difference
Test
statistic
1
-1.386
12
-2.299
4
-0.974
12
-0.645
1
-3.133
2
-3.027
9
-2.268
1
1.448
3
1.066
First Difference
1
-9.248
12
-2.964
4
-8.200
12
-4.990
1
-9.499
2
-11.617
9
-8.215
1
-14.062
3
-8.777
1% Critical
Value
5% Critical
Value
-3.987
-3.988
-3.987
-3.988
-3.987
-3.987
-3.988
-3.987
-3.988
-3.427
-3.428
-3.427
-3.428
-3.427
-3.427
-3.428
-3.427
-3.428
-2.58
-2.58
-2.58
-2.58
-2.58
-2.58
-2.58
-2.58
-2.58
-1.95
-1.95
-1.95
-1.95
-1.95
-1.95
-1.95
-1.95
-1.95
Since all five time-series are integrated to the same order, it is possible all three energy timeseries are cointegrated with the price of concrete and asphalt. The next step is to use a
constrained Vector Autoregression (VAR) model to compute if the time-series are cointegrated.
a) Testingfor a long-run price equilibrium between oil andpaving materials
As discussed in Section 3.3.3.2, the basis of cointegration is a VAR model. To select the lag
order of the model, a VAR model is fitted to the concrete-oil and asphalt-oil time-series up to 15
lag orders. Subsequently, four "goodness of fit" metrics are calculated for each VAR model: the
Final Prediction Error (FPE), Akaike Information Criterion (AIC), Hannan-Quinn Information
44
Criterion, and Schwartz Bayesian Information Criterion (SBIC) (Killian 2001). Given that the
sample size for this model is slightly greater than 400, empirical research suggests the best
metric to quantify the true number of lags, p, of the VAR model is the AIC (Killian 2001). The
results for the concrete-oil and asphalt-oil VAR models are presented in Table 4-6 and Table 4-7.
Based upon the AIC criteria, lag orders, p, of 13 and 11 are selected for the concrete-oil and
asphalt-oil VAR models.
Table 4-6: Concrete-Oil VAR Goodness of Fit performance for different number of lags. Italicized numbers
are best-fitting for each "goodness of fit" measure and bolded italics is best-fitting for AIC criteria
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
FPE
14.21
10.57
9.96
10.20
10.40
10.38
9.67
9.74
9.86
9.91
AIC
8.33
8.03
HQIC
8.36
8.08
SBIC
8.40
8.16
7.97
8.04
8.14
8.00
8.02
8.02
7.94
8.22
8.28
8.33
8.31
8.36
8.42
8.48
9.95
7.97
9.97
7.97
7.90
8.09
8.12
8.14
8.09
8.12
8.15
8.17
8.19
8.22
8.16
8.18
8.22
9.21
9.25
9.44
7.95
7.96
7.97
7.90
7.92
8.53
8.58
8.55
8.60
8.67
Table 4-7: Asphalt-Oil VAR Goodness of Fit performance for different number of lags. Italicized numbers
are best-fitting for each "goodness of fit" measure and bolded italics is best-fitting for AIC criteria
Lag
1
2
3
4
5
6
7
8
9
10
11
FPE
51.22
34.86
31.35
31.54
30.61
30.47
29.96
29.98
29.57
29.47
29.24
AIC
9.61
9.23
9.12
9.13
9.10
9.09
9.08
9.08
9.06
9.06
9.05
45
HQIC
9.64
9.28
9.19
9.21
9.20
9.22
9.22
9.24
SBIC
9.68
9.35
9.29
9.34
9.36
9.41
9.44
9.49
9.25
9.26
9.52
9.57
9.27
9.61
Lag
12
13
14
15
FPE
29.39
29.61
29.65
29.68
AIC
9.06
9.06
9.06
9.07
HQIC
9.30
9.32
9.34
9.36
SBIC
9.66
9.72
9.76
9.81
Having selected a lag order for each model, a Johansen cointegration test is subsequently
conducted as described in section 3.3.3.2. Two test statistics can be used to quantify the number
of cointegration relations that exist: the eigenvalue and trace statistics. The eigenvalue statistic
tests the null hypothesis that there are exactly r cointegration relationships, whereas the trace
statistic tests the null hypothesis of no more than r cointegration relationships (Franses and Dijk
2009). For this research, the trace statistic is used for both the concrete-oil and asphalt-oil
VECM. If the trace statistic is less than the critical value, the null hypothesis cannot be rejected.
Table 4-8 presents the Johansen cointegration test results, with the bold values representing when
the trace statistic is less than the 5% critical value. Both the concrete-oil and asphalt-oil have a
trace statistic less than the 5% critical value for a Rank of 1, meaning one is at least 95%
confident at a relationship does exist.
Table 4-8: Johansen test results for Concrete-Oil and Asphalt-Oil models. Bold values are less than the 5%
critical values
Concrete-Oil
5% critical value
trace statistic
15.41
18.70
0
3.76
2.05
1
Asphalt - Oil
5% critical value
trace statistic
Rank
15.41
17.76
0
3.76
0.55
1
Since both models can be considered cointegrated, a stationary linear combination exists
for both of the models. The stationary linear combinations, which represent the long-run price
equilibrium between the different commodities, are presented below:
Rank
Pconcrete,t = 75.6 + 0.171
*
PAsphait,t = 42.6 + 0.764
*
Pouit
(4-1)
Poi,t
(4-2)
That is, a unit change of 1 in the price index of oil impacts the price index of concrete and
oil by 17% and 76%, respectively. Additionally, the coefficient of determination, commonly
denoted as R2, is 0.42 for the concrete-oil relationship, whereas for the asphalt-oil relationship it
is 0.64.
b) Testing for a long-run price equilibrium between natural gas and paving materials
The same procedure is used to test if the price of paving materials is cointegrated with the price
of natural gas. Table 4-9 and Table 4-10 present the "goodness of fit" results depending up to 15
46
lags for the concrete-natural gas and asphalt-natural gas VAR models. Based upon AIC criteria,
lag orders of 13 and 10 are chosen for the concrete-natural gas and asphalt-natural gas models,
respectively.
Table 4-9: Concrete-Natural Gas VAR "goodness of fit" performance for different number of lags. Italicized
numbers are best-fitting for each "goodness of fit" measure and bolded italics is the best-fitting AIC
Lag
FPE
AIC
HQIC
SBIC
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
216.51
211.26
211.13
212.49
214.31
217.31
219.53
222.33
222.04
222.06
225.59
223.91
152.90
154.19
156.44
11.05
11.03
11.03
11.03
11.04
11.06
11.07
11.08
11.08
11.08
11.09
11.09
10.71
10.71
10.73
11.08
11.07
11.08
11.11
11.13
11.16
11.18
11.21
11.23
11.24
11.27
11.28
10.92
10.94
10.97
11.11
11.13
11.17
11.21
11.26
11.31
11.36
11.42
11.45
11.49
11.55
11.58
11.24
11.29
11.34
Table 4-10: Asphalt-Natural Gas VAR "goodness of fit" performance for different number of lags. Italicized
numbers are best-fitting for each "goodness of fit" measure and bolded italics is the best-fitting AIC
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
FPE
AIC
974.83
695.08
676.73
680.63
681.69
686.69
683.76
684.75
684.40
664.41
670.62
671.95
681.47
686.33
695.27
12.56
12.22
12.19
12.20
12.20
12.21
12.20
12.20
12.20
12.18
12.18
12.19
12.20
12.21
12.22
HQIC
12.58
12.26
12.25
12.27
12.29
12.31
12.32
12.34
12.35
12.34
12.36
12.38
12.41
12.43
12.46
47
SBIC
12.62
12.32
12.33
12.38
12.42
12.46
12.50
12.54
12.58
12.59
12.64
12.68
12.73
12.78
12.83
Table 4-11 presents the Johansen cointegration test results following the procedure in Section
3.3.3.1. Both the concrete-natural gas and asphalt-natural gas models have a trace statistic less
than the 5% critical value for Rank 0, as shown by the bold values. This means one is at least
95% confident no statistically significant relationship has historically existed between either
paving materials and natural gas. Since neither model is cointegrated, no stationary linear
combination is calculated for these models.
Table 4-11: Johansen test results for concrete-natural gas and asphalt-natural gas models.
indicate when it can be confirmed no more than r cointegration relationships exist
Bold values
Concrete-Natural Gas
Rank
trace statistic
5% critical value
0
1
12.14
3.78
15.41
3.76
Asphalt - Natural Gas
Rank
trace statistic
5% critical value
0
1
10.68
0.87
15.41
3.76
c) Testingfor a long-runprice equilibriumbetween coaland paving materials
The same analysis is lastly conducted between the two paving materials and coal. Table 4-12 and
Table 4-13 presents the goodness of fit performance of the concrete-coal and asphalt-coal VAR
models depending upon number of lags. Using the AIC criteria, 13 and 10 lags is selected for
each model, respectively.
Table 4-12: concrete-coal VAR "goodness of fit" performance for different number of lags. Italicized
numbers are best-fitting for each "goodness of fit" measure and bolded italics is the best-fitting AIC
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
FPE
0.374
0.367
0.373
0.367
0.373
0.376
0.378
0.380
0.377
0.364
0.366
0.367
0.257
AIC
4.69
4.67
4.69
4.67
4.69
4.70
4.70
4.71
4.70
4.67
4.67
4.67
4.32
HQIC
4.72
4.71
4.75
4.74
4.78
4.80
4.82
4.84
4.85
4.83
4.85
4.87
4.53
48
SBIC
4.75
4.77
4.83
4.85
4.91
4.95
5.00
5.04
5.08
5.08
5.13
5.17
4.85
Lag
14
15
FPE
0.262
0.263
AIC
4.33
4.34
HQIC
4.56
4.58
SBIC
4.91
4.95
Table 4-13: asphalt-coal VAR "goodness of fit" performance for different number of lags. Italicized numbers
are best-fitting for each "goodness of fit" measure and bolded italics is the best-fitting AIC
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
FPE
HQIC
AIC
1.88
1.28
1.29
1.25
1.21
1.22
1.24
1.23
1.23
1.20
1.20
1.21
1.21
1.21
1.22
6.31
5.92
5.93
5.89
5.87
5.87
5.89
5.88
5.88
6.33
5.96
5.98
5.99
5.96
5.97
6.00
6.02
6.03
6.02
6.04
6.06
6.08
6.09
6.11
5.86
5.86
5.87
5.86
5.87
5.87
SBIC
5.37
6.02
6.07
6.07
6.09
6.13
6.18
6.22
6.25
6.27
6.31
6.36
6.40
6.44
6.49
Table 4-14 presents the Johansen cointegration test results for the concrete-coal and asphalt-coal
models. For a rank of zero, the trace statistic is less than the 5% critical value for the asphaltcoal model, whereas for a rank of one the trace statistic is less than the 5% critical value for the
concrete-coal model, as shown by the bold values. This means the concrete-coal model shows
the two time-series can be classified as cointegrated, whereas the asphalt-coal model show no
statistically significant relationship.
Table 4-14: Johansen test results for concrete-coal and asphalt-coal models. Bold values indicate when it can
be confirmed no more than r cointegration relationships exist
Concrete-Coal
Rank
trace statistic
5% critical value
0
1
26.17
3.50
15.41
3.76
Asphalt - Coal
Rank
0
1
trace statistic
11.81
0.11
5% critical value
15.41
3.76
49
Since the concrete-coal model is cointegrated, a stationary linear combination exists which
describes the long-run price relationship between the commodities.
The long-run price
relationship is as follows:
Pconcrete,t = 55.9 + 0.553 * Pcoal,t
(4-3)
The coefficient of determination for the above equation is 0.36, meaning 36% of the price
volatility of concrete is explained by the volatility of coal.
d) Conclusions from cointegration work
By testing if the price of different paving materials and energy commodities are cointegrated (i.e.
stochastically shift together), this chapter has shown that:
*
A statistically significant price-link exists between asphalt and concrete with oil, although
the price-link is much stronger in the case of asphalt, as expected
*
The price of asphalt and concrete are not cointegrated with the price of natural gas
*
Only concrete is cointegrated with the price of coal
These results suggest the differential price-behavior between asphalt and concrete shown in Step
1 is directly linked to the above conclusions. Since asphalt is a product of oil production, it is
highly sensitive to price changes in oil, itself a historically volatile commodity. Concrete,
however, has a much weaker price-link with oil and has a long-run price equilibrium with coal,
which as shown in Figure 4-3, has historically been the most stable energy source of the three
considered. As such, one would suspect concrete prices to have been much more stable
historically than asphalt, which Step 1 illustrated has been the case.
Given such a strong price-link exists between asphalt and oil, one would suspect a forecasting
model would use this relationship to project future prices of asphalt since so much extensive data
is available for oil. This thesis, however, uses a time-series model which only forecasts asphalt
as a function of time, but future work could build upon this relationship.
4.2.4 Step 3: Testing if long-term forecasts are plausible
Steps 1 and 2 have shown that historically the price of concrete and asphalt have behaved
differently. In Step 3, this thesis now evaluates if projecting the future is plausible to account for
this. To understand if historical data is a good predictor for future performance and quantify
how much data, at a minimum, is needed, backcasting is conducted on data sets provided by the
USGS for Construction Sand & Gravel and Cement (Kelly and Matos 2012a; Kelly and Matos
2012b). These data sets are selected because they are construction related time-series which date
back to 1900 and, therefore, provide an adequate amount of data to backcast. For each timeseries, three models are considered: real-price inflation rate, Geometric Brownian Motion
(GBM), and an Ornstein-Uhlenbeck mean-reverting process (Pindyck 1999; Gerber and Shlu
2000).
50
USGS Sand & Gravel time-series
Figure 4-5 presents the real cost of the USGS Construction Sand & Gravel time-series in 1998
dollars (Kelly and Matos 2012a). It is evident that the real cost of Construction Sand & Gravel
has exhibited a small downward trend over time. This would suggest that the optimal model
would be one of the non-stationary models considered (i.e. either the real-price inflation model
or GBM process discussed in Section 3.3.3.3).
Figure 4-5: Real price of Construction Sand & Gravel time-series (Kelly and Matos 2012a)
8.00
17.00
S6.00
S5.00
, 4.00
$ 3.00
2.00
S1.00
0.00
1900
1920
1940
1960
Year
1980
2000
Figure 4-6 tracks the performance of the GBM process, which was the best performing nonstationary model, the Ornstein-Uhlenbeck mean-reverting process, and a "no-change" forecast
which assumes the real-price remains constant over time. Forecasts are made between 1940 and
1980 using the three different models, compared to what actually occurred in each forecasted
year, and used to measure the Mean Absolute Percent Error (MAPE) of each forecast as
described in Section 3.3.3.3. For example, the data point "10 years into the future" in Figure 4-6
represents the MAPE of all 10-year forecasts (i.e. the average error of a 1940 forecast projecting
1950, a 1941 forecast projecting 1951, and so on.) Using all historical data available for each
forecast, the results of the analysis, as hypothesized, suggest a non-stationary time-series model
has the lowest MAPE for all data points, and as such, is the "best performing" model. The "no
change" forecast outperforming the mean-reverting model is surprising but explainable.
Forecasts for the mean-reverting model tended to revert to a relatively high mean since it
weighted high prices in 1920 thru 1940, and as such, induced large errors in the forecasting
model. An important finding in this particular analysis, however, is the best-performing GBM
model outperforms the no-change forecast, implying that using historical data as a predictor for
the future is better than complete ignorance.
51
Figure 4-6: Backcasting results from the Construction Sand & Gravel time-series
20%
No Change
Reverting
16%Mean
W 16%
2
'0 12%
0%
8%
8%
GBM
S4%
0%*
I
0
10
20
30
40
Years into the Future
To quantify how much data are needed to outperform the "no change" forecast, the MAPE of the
GBM process is calculated using different amounts of historical data. Figure 4-7 presents the
performance of the GBM model using at most 35 and 55 years of available data from the
Construction Sand & Gravel time-series, representative of the amount of historical data for
concrete and asphalt via BLS, for price-projections up to 40 years. The results suggest that for a
non-stationary data set, both 35 years and 55 years of data outperform the "no change" forecast.
In fact, surprisingly, for projections 25 to 35 years into the future, using only 35-years of data
generally outperformed the model with more data. Further statistical analyses should be
conducted in future work to explain this phenomenon.
Figure 4-7: Backcasting results for USGS Sand & Gravel time-series using a GBM process to project the
future. Projections are made using at most 35 and 55 years of available historical data
20%
No change
16%
55 years of data
12%
-
8%
35 years of data
L4%
0%0
10
20
30
40
Years Into Future
USGS Cement Time-Series
Figure 4-8 presents real price data for the USGS Cement time-series in 1998 dollars (Kelly and
Matos 2012b). For the first 80 years of data, the time-series has generally shifted around a mean
52
value of $90 (in 1998 dollars). In the past thirty years, however, the real-price of cement has
exhibited a downward trend. Given this shift has recently occurred, it is not necessarily obvious
which forecasting model will perform best.
Figure 4-8: Real price of Cement time-series (Kelly and Matos 2012b)
$140
$120
-a
$100
$80
$60
AA
-
I
$40
$20
1900
1920
1940
1960
1980
2000
Year
Figure 4-9 illustrates the performance of the Ornstein-Uhlenbeck mean-reverting process, a "nochange" forecast, and the real-price inflation rate (best performing non-stationary model)
process, which is calculated following the same methodology as the previous case. For this
particular time-series, the Ornstein-Uhlenbeck process was indeed the best performed best
model. Figure 4-10 presents the MAPE of the Ornstein-Uhlenbeck mean-reverting process using
all available historical data for each forecast and limiting the amount of historical data used to
only 35 years of data. There is little change in performance between the two, again implying
enough empirical data might be available to capture the price behavior asphalt has exhibited.
Figure 4-9: Backcasting results from the Cement time-series
40%
I-
S30%
No change
20%
OrnsteinUhlenbeck
S10%
Real-price
inflation rate
0%
0
10
20
Years into Future
53
30
40
Figure 4-10: Backcasting results for USGS Cement time-series using an Ornstein-Uhlenbeck mean-reverting
process. Projections compared use at most 35-years of historical data versus as much historical data readily
available for a particular forecast.
40%
30%
All historical data
available used
20%
Limited to 35 years
of data
,E
10%
0
10
20
30
40
Years into Future
4.2.5 Step 4: Probabilistically projecting the future
The previous analyses have led to the following conclusions:
1) Asphalt and concrete are two commodities with historically different growth rates and
volatilities, and this is likely due to the types of energy sources which they are tied to
2) Empirical evidence in this thesis has shown modeling real-price as a function of time is
able to reasonably capture material price behavior with as little as 35-years of data. As such,
it is perhaps reasonable to project the real-price of concrete and asphalt using such a process.
This thesis now quantifies probabilistic price projections for the two commodities by:
1) Testing to see if a unit-root exists to conclude whether either time-series is stationary or
non-stationary. It should be noted that unless the time-series reverted back to a mean
value quickly, it is likely the test results will not be able to reject the null hypothesis of a
unit-root existing due the limited sample size in both cases.
2) If a unit-root cannot be rejected, the model is assumed to be non-stationary and a GBM
model is used to forecast future prices. To ensure this is appropriate, the growth rate and
volatility of the model is calculated for a range of samples. If the commodity actually
follows a slowly mean-reverting process that the unit-root test cannot recognize, the
assumption of a GBM model is appropriate if the growth rate volatility is relatively
constant for varying sample sizes (Pindyck 1999).
54
3) Probabilistically project the price of concrete over the next 50 years based upon the
selected model.
To project the future price of concrete and asphalt, a time-series model is constructed based off
of historical annual price index data for Asphalt Paving, which dates back to 1976, and ReadyMix Concrete, dating back to 1958, as was shown in Figure 4-4.
To understand if the time-series should be modeled as stationary or non-stationary, an ADF test
with trend is conducted for a lag order of one to maximize the amount of data available. The test
statistic from the ADF test is greater than the 5% critical values as shown in Table 4-15,
indicating that both time-series can be classified as non-stationary.
Table 4-15: ADF test results for asphalt paving time-series (Source: BLS)
Time-Series
Test statistic
1% Critical Value
5% Critical Value
10% Critical Value
Concrete
-2.599
-4.146
-3.498
-3.179
Asphalt
-0.119
-4.297
-3.564
-3.218
As discussed, it is likely that even if the two time-series are stationary, there may not be enough
data for a unit-root test to recognize this. Table 4-16 presents the growth rate and volatility of
the logarithmic time-series for different sample sets. Clearly the volatility and growth rate have
remained relatively constant over time, suggesting that even if the time-series were meanreverting, selecting a GBM model is unlikely to lead to large errors in the ultimate investment
decision (Pindyck 1999).
Table 4-16: Logarithmic growth rate and volatility for various sample sizes
Ready-Mix Concrete
Sample
Growth Rate
Volatility
1959 - 1988
-0.9%
2.4%
Asphalt Paving
Sample
Growth Rate
1977 - 1996
-0.8%
1964
-
1993
-0.6%
2.4%
1982
-
2001
-0.9%
2.7%
1969
-
1998
-0.5%
2.4%
1987
-
2006
0.6%
4.1%
1974-2003
0.0%
3.3%
1992-2011
2.2%
4.1%
1979-2008
-0.5%
3.4%
N/A
N/A
N/A
Volatility
2.9%
The discretized solution of the GBM process described in section 3.3.3.1 is as follows:
Pt
t-s
t
y+2
Atf)
(4-4)
where P, is the price in year t, P, is the price in year t-s, p is the drift (i.e. growth rate), o is the
volatility term, At is t-s, and W, is Weiner process normally distributed with a mean of zero and a
standard deviation of t-s. The above equation can be reduced in the case where t-s is equal to t-1
to the following:
55
Pt = Pt-1
(4-5)
2((y+)+N(0,1))
where N (0, 1) is a normal distribution with a mean of zero and a standard deviation of one.
Parameter estimation for the growth rate, p, and volatility, u, for the two time series are
calculated based upon the average and standard deviation values of In (P,) - In (P,.) for the
entire sample. The resulting GBM model for the discretized Concrete and Asphalt time-series is:
Concrete: Pt = Pt-1e(-0.002 8 4 +0.0
29 6
(4-6)
*N(0,1))
(4-7)
Asphalt: Pt = Pt-1e(o.00750+0.0400*N(0,1))
The real price of concrete and asphalt is stochastically simulated over the next 50 years using the
derived model. Figure 4-11 and Figure 4-12 present the real-price projection of concrete and
asphalt normalized around a real-price index value in 2011 of 100 by conducting 5,000
stochastic simulations. The material-specific price projections below have been conducted for
concrete and asphalt using only one simple time-series model, a GBM model, which accounts for
the historical growth rate and volatility of commodities. Although this research has shown it will
likely outperform a no-change forecast, future studies should consider if more advanced
analytical time-series models should be used to forecast future prices.
Figure 4-11: Long-term price projection of concrete based upon time-series analysis. The dashed
lines represent the 5th and 95th percentiles of the probabilistic distribution of results
150
125
100
50
25
0
2010
2020
2030
2040
Year
56
2050
2060
Figure 4-12: Long-term price projection of asphalt based upon time-series analysis. The dashed lines
represent the 5th and 9 5 h percentiles of the probabilistic distribution of results
250
200
-
a1
0
0
II
150
0
100
Li
50
0 412010
2020
2030
2040
2050
2060
Year
4.3 Summary
This chapter has quantified two types of uncertainties: the unit-price of bid items and the
forecasting of future material prices. The majority of construction activities evaluated, which
will be used in the case study, show a statistically significant relationship between cost and
quantity which characterizes much of the variability in the data. The material price forecasting
work showed historically the growth rate and volatility of asphalt and concrete has been quite
different. A likely reason is asphalt has been closely tied with the price of oil, whereas concrete
shows a much weaker tie to oil and is instead impacted by the price of coal, a much more stable
energy resource. Having identified that asphalt and concrete have historically behaved
differently, construction time-series which data back to 1900 are used to test if long-term price
projections are plausible. By backcasting historical data, results show long-term forecasts are
plausible, and even more importantly, for the two USGS time-series analyzed, empirical results
showed at least 35-years of data outperformed the no-change forecast up to 40 years into the
future.
Although further time-series work should confirm that 35-years of data is indeed
sufficient to project the future, the limited findings in this thesis are promising given that at least
35-years of data for both concrete and asphalt are available. Given that a reasonable amount of
data seems to be available to perform better than the "no change" forecast, probabilistic realprice forecasts are subsequently constructed, with the expected price of concrete slightly
57
decreasing over time, while the expected price of asphalt expected to increase by 50% in the next
fifty years.
The quantified uncertain parameters will now be used in Chapter 5 in a case study analysis to
assess how the probabilistic characterization of uncertainty in pavement LCCAs ultimately
affects the pavement selection of an analysis.
58
5
CASESTUDY
The purpose of this chapter is to implement the probabilistically characterized input parameters
in three different scenarios to assess how the inclusion of probabilistic values could potentially
affect the pavement selection process. For each scenario, three types of analyses are conducted:
a deterministic analysis and two probabilistic analyses, one that includes and one that excludes
material-specific price projections. The goal is twofold: to understand if considering uncertainty
changes the decision-maker's pavement selection, and if so, are material-specific projections the
driving parameter for that change.
5.1 Scope of Analysis
As discussed in Chapter 3, this thesis only includes the cost to construct, operate, and maintain a
new roadway. It assumes that the decision of whether or not a new roadway should be built has
already been decided and ignores the associated user cost with a new roadway. As such, the
costs shown in this case study are agency costs only.
All three projects are located in Southem California. Pavement designs, expected design life,
and rehabilitation schedules are derived from California Department of Transportation
(CalTrans) design manuals and are extrapolated over a 50-year analysis period (CalTrans 2012b;
CalTrans 2012c). Uncertainty related to the occurrence of maintenance activities, therefore, is
based upon the pedigree matrix approach rather than a more robust analysis by modeling
pavement performance with the Mechanistic Empirical Pavement Design Guide (MEPDG). A
real discount rate of 4% is used to stay consistent with the CalTrans LCCA process.
Table 5-1 provides information regarding the basic traffic and geometric design inputs for the
three scenarios. The input parameter which differentiates the three scenarios is traffic volume,
which impacts the pavement design and performance over time.
Table 5-1: Three roadway scenarios in Southern CA considered'
High Volume
Moderate Volume
Low Volume
Oxnard, CA
Ramona, CA
Ojai, CA
Speed (MPH)
65
35
40
Number of lanes (each direction)
3
2
1
2 inner/2 outer
2 inner/2 outer
2 unpaved shoulders
139,000
23,400
3,400
6,672
1,357
150
Location
Number of shoulders
AADT
AADTT
1 Pavement designs and maintenance schedules are derived with the help of James Mack from CEMEX
59
For each scenario, a hot mix asphalt (HMA) and Jointed Plain Concrete Pavement (JPCP) design
are compared. All JPCP designs are supported by a Lean Concrete and Aggregate base, whereas
the HMA designs bear on an Aggregate Base and Sub-base. Table 5-2 and Table 5-3 present the
thickness of the different layers for the JPCP and HMA designs. For each design, the CalTrans
LCCA manual provides guidance for the expected year and type of maintenance for each
alternative considered. Table 5-4, Table 5-5, and Table 5-6 provide the prescribed JPCP and
HMA maintenance schedule for the high, medium, and low volume roadways, respectively.
Table 5-2: JPCP initial pavement designs for three scenarios
High Volume
10.8 in
Moderate Volume
Low Volume
9.6 in
8.2 in
Lean Concrete Base Thickness
6.0 in
4.8 in
4.2 in
Aggregate Base Thickness
8.4 in
7.2 in
6.0 in
Concrete Thickness
Table 5-3: HMA initial pavement designs for three scenarios
High Volume
Moderate Volume
Low Volume
Asphalt Concrete Thickness
7.8 in
6.6 in
4.8 in
Aggregate Base Thickness
12.6 in
10.6 in
8.4 in
Aggregate Sub-base Thickness
9.6 in
8.4 in
6.0 in
Table 5-4: Maintenance schedule for high-volume roadway alternatives
Concrete Design
Maintenance
Number
1
Year of
Occurrence
45
2
Asphalt Design
Rehab Type
2% Patch & Diamond Grind
Year of
Occurrence
20
3" Asphalt Overlay
N/A
N/A
25
Mill and 4" Asphalt Overlay
3
N/A
N/A
35
Mill and 3" Asphalt Overlay
4
N/A
N/A
45
Mill and 4" Asphalt Overlay
Rehab Type
Table 5-5: Maintenance schedule for medium-volume roadway alternatives
Concrete Design
Maintenance
Number
1
Year of
Occurrence
25
2
Asphalt Design
Rehab Type
2% Patch & Diamond Grind
Year of
Occurrence
20
3" Asphalt Overlay
30
4% Patch & Diamond Grind
25
Mill and 4" Asphalt Overlay
3
40
6% Patch & Diamond Grind
35
Mill and 3" Asphalt Overlay
4
45
3" Asphalt Overlay
45
Mill and 4" Asphalt Overlay
60
Rehab Type
Table 5-6: Maintenance schedule for low-volume roadway alternatives
Concrete Design
Asphalt Design
Maintenance
Number
Year of
Occurrence
Rehab Type
Year of
Occurrence
Rehab Type
1
25
2% Patch & Diamond Grind
20
3" Asphalt Overlay
2
30
4% Patch & Diamond Grind
30
Mill and 3" Asphalt Overlay
3
40
6% Patch & Diamond Grind
40
Mill and 2.5" Asphalt Overlay
4
45
3" Asphalt Overlay
45
Mill and 3" Asphalt Overlay
Since the length of the project will influence the quantity of material required, and subsequently,
the unit-cost due to economies-of-scale, an appropriate length should be selected for the three
scenarios. In this particular analysis, cost is calculated over a five mile span for the high volume
roadway, while for the medium and low volume roads cost is only calculated over one mile. All
NPV calculations presented, however, are normalized into cost per mile.
5.2 Deterministic Results
This section assesses the likely pavement selection if a decision-maker used deterministic input
values by conducting an analysis in Microsoft Excel (Microsoft 2010). Given that currently in
the pavement selection process decision-makers tend to place more weight on initial rather than
life-cycle costs, both are calculated to understand if the life-cycle cost savings of an alternative
warrants its selection. Table 5-7 summarizes some of the more significant deterministic input
parameters in the analysis. The unit-cost data for the analysis is based on a simple weighted
average from historical bid data provided by CalTrans over a 12-month span. As mentioned in
the methodology section, only differential costs are considered in this analysis, and therefore,
costs such as engineering fees, traffic lighting, etc. are excluded from the scope of the analysis.
Table 5-7: Significant unit-cost inputs for the deterministic analysis
Input
Units
Cost($/unit)
JPCP
Cubic Yards
126.43
Lean Concrete Base
Cubic Yards
95.81
Surface HMA
Tons
82.60
Intermediate HMA
Tons
78.50
Aggregate Base
Cubic Yards
26.35
Aggregate Sub-base
Cubic Yards
18.62
Diamond Grinding
Square Yards
3.90
Patch
Cubic Yards
822.20
Mill
Square Yards
6.92
Initial Costs
61
Based upon the pavement designs and unit-cost inputs, Figure 5-1 presents the initial cost for all
three scenarios. For all three scenarios, the initial cost of the JPCP pavement is larger than the
HMA design, ranging from 15% larger in the case of the high-volume roadway to 28% larger for
the low volume roadway. The implication is a significant total life-cycle cost difference for the
JPCP designs would be needed to warrant its selection. As will be discussed in Section 5.4, the
initial cost difference, not just life-cycle cost difference, is an important consideration when
selecting a pavement, and as such, is considered when deciding between pavement designs.
Figure 5-1: Deterministic initial cost for three scenarios
5.0
1~
4.0
3.0
* Concrete
0
rj~
0
2.0
* Asphalt
1.0
0
Q
High Volume
Medium Volume
Low Volume
LCC not considering material-price forecasting
Accounting for both initial costs and future discounted costs while assuming future real-prices
will be static, Figure 5-2 presents the total life-cycle cost of all pavement alternatives in the
analysis. In the high-volume scenario the JPCP design presents a total life-cycle cost (LCC)
savings of 22%, indicating it would likely be the pavement of choice under the current analysis.
In all other scenarios, however, both the initial and total LCC of the HMA designs are less than
the concrete designs.
Figure 5-2: Discounted life-cycle cost for design scenarios
5.0
1
4.0
3.0
C-
" Concrete
2.0
" Asphalt
U
1.0
High Volume
Medium Volume
62
Low Volume
LCC accounting for material-price forecasting
Lastly, a deterministic analysis is conducted using the deterministic material-specific price
projections based upon Equation 4-6 and Equation 4-7 while keeping all remaining parameters
constant. Although the total LCC difference between the HMA and JPCP designs lessen for the
medium and low volume scenarios, it does not alter the pavement selection, as can be seen in
Figure 5-3.
Figure 5-3: Initial and life-cycle cost of asphalt pavement designs relative to concrete designs for high volume,
medium volume, and low volume scenarios
6.0
5.0
4.0
o 3.0
Concrete
-
2 Asphalt
2.0
1.0
High Volume
Medium Volume
Low Volume
Summary
Table 5-8 summarizes the final results from the deterministic analysis. For all three scenarios,
the HMA designs are initially less than the JPCP designs. Accounting for future maintenance
activities and discounting costs with a discount rate of 4%, the LCC of the JPCP pavement is
significantly less for the high-volume roadway, but higher in the case of the medium and lowvolume roadways. In general, accounting for the expected price projection of concrete and
asphalt decreases the relative LCC of the JPCP designs by 7-8%, but this does not influence the
final decision for these three scenarios.
Table 5-8: Cost of asphalt pavement relative to concrete alternatives for all three scenarios
High Volume
Medium Volume
Low Volume
Initial Cost
-15%
-16%
-28%
LCC with no price inflation
+22%
-7%
-19%
LCC with price inflation
+30%
0%
-12%
5.3 Results Incorporating Uncertainty
Having quantified probabilistic input parameters in Chapter 4, this section applies those values in
order to derive probabilistic life-cycle costs for each pavement alternative. For each scenario,
63
5,000 simulations are conducted using the Microsoft Excel add-in @Risk software program
(Palisade 2012). The results are presented in the form of a cumulative distribution that describes
the risk of each design. Within each scenario, a probabilistic analysis is conducted both with and
without material-specific price projections. Unit-cost uncertainty is based upon the values
presented in Section 4-1 and pedigree-matrix uncertainty values are presented in the Appendix.
5.3.1
High-Volume scenario
Probabilistic LCC not considering material-price forecasting
Table 5-9 presents the probabilistic life-cycle cost of the high-volume roadway assuming that
material prices grow at the rate of inflation (i.e. no real-price change). As was the case for the
deterministic analysis, the JPCP pavement outperforms the HMA alternative. Figure 5-4
presents the cumulative distribution results from the Monte Carlo simulations.
Table 5-9: Probabilistic cost of high-volume scenario alternatives not including material-specific price
projections
Asphalt Pavement
Concrete Pavement
Asphalt - Concrete
(millions of $'s)
(millions of $'s)
Mean Value
3.15
2.43
Asphalt
23%
7 5 *hpercentile
3.43
2.48
28%
percentile
4.08
2.6
36%
9 5th
Figure 5-4: High-volume roadway cumulative distribution results not including material-specific projections
100% 1
.W
.
.
I
80%
60%
40%
U 20%
0% .
2.25
2.75
3.25
3.75
4.25
Net Present Value (millions of $'s per mile)
Probabilistic LCC accounting for material-price projections
There is little change in the likely decision by including probabilistic material-price forecasts in
the analysis. Since the HMA pavement has four major rehabilitations rather than only one for
64
the JPCP alternative, the discrepancy in LCC increases in favor of the JPCP design for all riskprofiles, as can be seen in Table 5-10.
Table 5-10: Probabilistic cost of high-volume scenario alternatives including material-specific price
projections
Asphalt Pavement
Concrete Pavement
Asphalt - Concrete
(millions of $'s)
(millions of $'s)
Asphalt
3.36
2.43
28%
percentile
3.68
2.55
31%
* percentile
4.35
2.60
40%
Mean Value
7 5 th
95
5.3.2
Medium-Volume scenario
Probabilistic LCC not considering material-price forecasting
In the case of the medium-volume roadway, there is again little difference in the likely decision
when conducting a probabilistic analysis that does not account for material-price projections.
The HMA pavement is less than the JPCP alternative in terms of mean (ENPV), 75* percentile,
and 95th percentile values (albeit relatively small), as can be seen in Table 5-11. In fact, as can
be seen in Figure 5-5, risk associated with the alternative investment is nearly identical for all
risk-perspectives.
Table 5-11: Probabilistic cost of medium-volume scenario alternatives not including material-specific price
projections
Asphalt Pavement
(millions of $'s)
Mean Value
2.53
7 5 th
percentile
2.72
9 5 th
percentile
3.13
Concrete Pavement
(millions of $'s)
2.56
Asphalt - Concrete
Asphalt
-1%
2.91
3.16
-7%
-1%
Figure 5-5: Medium-volume roadway cumulative distribution results not including material-specific
projections
100% i
n
_-...-
80%
0
60%
40%
fQ
20%
0%
2
3.25
3
2.75
2.5
2.25
Net Present Value (millions of $'s per mile)
65
3.5
Probabilistic LCC accounting for material-price projections
The medium-volume roadway is the first scenario where the inclusion of probabilistic values,
and more specifically, the inclusion of material-specific price projections, could potentially
change the final pavement selection. The ENPV, 7 5th percentile, and 9 5 th percentile of the JPCP
designs are all less than the HMA designs, as presented in Table 5-12.
Additionally, the
discrepancy between the alternatives tends to increase as cumulative probability increases, as
shown in Figure 5-6. This indicates the JPCP design is more likely to be selected if the decisionmaker tends to be more risk-averse.
Table 5-12: Probabilistic cost of medium-volume scenario alternatives including material-specific price
projections
Asphalt Pavement
Concrete Pavement
Asphalt - Concrete
(millions of $'s)
(millions of $'s)
Asphalt
2.75
2.57
7%
* percentile
2.99
2.63
12%
percentile
3.54
3.19
10%
Mean Value
75
9 5 th
Figure 5-6: Medium-volume roadway cumulative distribution results including material-specific projections
100% 1
.
I
80%
60%
Cu
40%
0
Q
20%
0%
-F=
2.00
5.3.3
I
1
2.25
2.50
2.75
3.00
3.25
Net Present Value (millions of $'s per mile)
3.50
Low-Volume scenario
Probabilistic LCC not considering material-price forecasting
For the low-volume scenario, conducting a probabilistic LCCA while ignoring material-specific
price projections generally leads to the same conclusion from the deterministic analysis.
Figure 5-7 presents the cumulative distribution plot for the low-volume scenario. For all
percentiles, presented in Table 5-13, the HMA design is less than the JPCP pavement, and can
generally be concluded as superior in this particular analysis.
66
Figure 5-7: Low-volume roadway cumulative distribution results not including material-specific projections
100%
80% -
i
a
60% 05
i
Concrete
Asphalt
40% 20% I
I
I
I
0%
0.75
1.25
1.15
1.05
0.95
0.85
per
mile)
$'s
(millions
of
Value
Net Present
Table 5-13: Probabilistic cost of low-volume scenario alternatives not including material-specific price
projections
(millions of $'s)
Concrete Pavement
(millions of $'s)
0.89
0.94
Asphalt Pavement
Asphalt - Concrete
75*
percentile
0.94
0.97
Asphalt
-6%
-3%
95*
percentile
1.06
1.20
-13%
Mean Value
ProbabilisticLCC accountingfor material-priceprojections
As was the case for the medium-volume roadway, the inclusion of material-specific price
projections significantly impacts the conclusions from the analysis. For all risk perspectives the
JPCP pavement outperforms the HMA design, as shown in Figure 5-8. Additionally, as the
decision-maker is increasingly risk-averse, the concrete pavement increasingly outperforms the
asphalt design. As an example, the ENPV difference between the two designs is only 5%, but in
terms of their 75t percentiles, the concrete design is 11% superior, provided in Table 5-14.
Table 5-14: Probabilistic cost of low-volume scenario alternatives including material-specific price
projections
Asphalt Pavement
Concrete Pavement
Asphalt - Concrete
(millions of $'s)
(millions of $'s)
Asphalt
1.01
75 *
percentile
1.11
0.96
0.99
95 *
percentile
1.32
1.21
Mean Value
67
5%
11%
8%
Figure 5-8: Low-volume roadway cumulative distribution results including material-specific projections
100%
80%Concrete
*
*
I
I
I
I
I
I
I
I
60%
Asphalt
"E' 40%-
I
I
*
I
20%-
0%
0.75
0.95
1.15
1.35
Net Present Value (millions of $'s per mile)
5.4 Case Study Discussion
For the case study analysis, a concrete and asphalt pavement alternative were considered for
three different scenarios. For each scenario, four types of analyses were conducted: a
deterministic analysis (excluding and including material price projections) and a probabilistic
analysis (again, excluding and including material price projections). The following section
describes some of the key findings.
5.4.1 Influence of size of project
This thesis has statistically characterized the relationship between cost and bid volume. What
has not been addressed, however, is how state DOTs prefer to allocate costs for different project
sizes. Figure 5-9 presents the average percent contribution of initial cost to the total discounted
life-cycle cost for all three scenarios. As the size of the project decreases, so does the
contribution of initial cost to the total expected life-cycle cost for both the concrete and asphalt
alternatives. It is likely, therefore, the discount rate and material-specific price projection model
selected will become more important for lower volume roadways. This thesis thus turns its
attention to how sensitive the life-cycle cost results to discount rate and material-specific price
projection for the case study.
68
Figure 5-9: Percent contribution of initial cost to total life-cycle cost using a discount rate of 4%
100%1
-
0
Concrete
80% -
Asphalt
,al
60% -
:IU
U
0
40% 420% 0%
-
High-Volume
Medium-Volume
Low-Volume
5.4.2 Sensitivity analysis
As was shown in Figure 5-9, the concrete alternatives tend to have a larger portion of their lifecycle costs incurred initially relative to the asphalt designs. Since the asphalt designs tend to
incur a larger portion of life-cycle costs in the future, it is likely the results of this analysis are
sensitive to the discount rate selected. As discussed in Chapter 2, FHWA and state DOTs have
made it common practice to use a real discount rate of 4%. Historical real interest rates
published by OMB have historically declined, however, and there is undeniable uncertainty
when selecting one discount rate to use in an analysis. As such, a sensitivity analysis is
conducted for a discount rate of 2%, the most recent real interest rate published by OMB, and
5%. Figure 5-10 and Figure 5-11 present the contribution of initial cost for each design using the
two discount rates. The relative contribution of initial cost to total life-cycle cost discrepancy
between asphalt and concrete designs to tends to be reduced for a higher discount rate.
Figure 5-10: Percent contribution of initial cost to total life-cycle costs using a discount rate of 2%
100%
U
U
80%
I
+-
Concrete -I
Asphalt
60% 40% -
U
20% 0%
-
High Volume
Medium Volume
69
Low Volume
Figure 5-11: Percent contribution of initial cost to total life-cycle costs using a discount rate of 5%
100%
Q
U
04
80% -
U
Asphalt
iConcrete
-7--
0
60% T:
40% U
20% -0%
- -
H-
-
High Volume
Medium Volume
Low Volume
Table 5-16, Table 5-17, and Table 5-17 presents a summary table of the probabilistic life-cycle
cost for each design (accounting for material-specific price projections) using a discount rate of
2%, 4% (the discount rate selected in Section 5.3) and 5%. In general, the relative performance
of the asphalt alternatives improves, as expected, for a higher discount rate.
Table 5-15: Life-cycle cost of pavement designs using a discount rate of 2%
Asphalt Pavement
Concrete Pavement
(millions of $'s)
(millions of $'s)
High-Volume
2.54
Asphalt - Concrete
Asphalt
41%
Mean Value
4.29
75th percentile
4.79
2.60
46%
95th percentile
5.74
2.86
50%
Mean Value
3.52
3.05
13%
75th percentile
3.87
3.19
18%
95th percentile
4.61
4.20
9%
Medium-Volume
Low-Volume
Mean Value
1.40
1.32
5%
7 5 th
percentile
1.55
1.40
10%
9 5 th
percentile
1.91
1.81
5%
70
Table 5-16: Life-cycle cost of pavement designs using a discount rate of 4%
Asphalt Pavement
(millions of $'s)
Concrete Pavement
(millions of $'s)
High-Volume
2.43
2.55
Asphalt - Concrete
Asphalt
percentile
3.36
3.68
95 percentile
4.35
Mean Value
2.57
2.63
7%
12%
3.19
10%
Mean Value
7 5 th
2.60
28%
31%
40%
Medium-Volume
7 5 th
percentile
2.75
2.99
9 5 th
percentile
3.54
Low-Volume
Mean Value
1.01
0.96
5%
0.99
1.21
11%
7 5 th
percentile
1.11
th
percentile
1.32
9 5
8%
Table 5-17: Life-cycle cost of pavement designs using a discount rate of 5%
Asphalt Pavement
Concrete Pavement
(millions of $'s)
(millions of $'s)
High-Volume
2.40
Asphalt - Concrete
Asphalt
Mean Value
3.2
75th percentile
3.52
2.45
30%
95 percentile
4.19
2.55
39%
25%
Medium-Volume
Mean Value
2.57
2.44
5%
percentile
2.80
2.49
11%
95th percentile
3.26
2.93
10%
7 5 th
Low-Volume
Mean Value
0.90
0.86
4%
75th percentile
0.98
0.91
7%
95 percentile
1.15
1.08
6%
Additionally, a sensitivity analysis is conducted on the material-specific price selection
parameters. Figure 5-12and Figure 5-13 present the relative performance of the asphalt design
for different material-specific price growth rate projections for the medium and low volume
roadways. The volatility of the derived probabilistic forecast models in Equation 4-6 and
Equation 4-7, in addition to the concrete growth rate, are held constant while the asphalt growth
71
rate is varied. For both scenarios, adjusting the growth rate parameters from a value of 0.0075 to
-0.01, the asphalt design is approximately 10 to 15% cheaper for the medium-volume scenario
and 20% cheaper for the low-volume scenario. To give some perspective of the values used in
the scenario analysis, a growth rate of -0.01 assumes the price of a commodity will decrease by
40% over the next 50 years, while a growth rate of 0.0075 assumes prices will increase by 50%
Figure 5-12: Relative life-cycle cost savings of asphalt designs for medium-volume roadway for different
GBM growth rate parameters
20%
Asphalt calculated valueJ
Concrete value
75h percentile
a 10%
0%
- 0%
95h percentile
-0.01
-0.005
0
0.005
0.01
Asphalt GBM Growth Rate Selected
Figure 5-13: Relative life-cycle cost savings of asphalt designs for medium-volume roadwa y for different
GBM growth rate parameters
20%
Asphalt calculated value, )
Concrete value
i
10%
I
7 5h percentile
0%
'- 9 5th percentile
-10%
-20%
-0.01
Mean
-0.005
0
0.005
Asphalt GBM Growth Rate Selected
0. 01
A relevant question this thesis next will answer is how does changing the discount rate
impact, in addition to the inclusion of material-specific price projections, potentially alter the
likely pavement selection?
72
5.4.3
Parameters which impact pavement selection
Material-specific price projections
Conducting a probabilistic analysis that accounts for material-specific price changes over time,
has the implication of altering the final LCCA results as was shown in Chapter 5. Table 5-18
presents the likely pavement selection under four different types of analyses and/or selection
criteria from the previous analysis. The first set of results is the pavement selected in the
probabilistic analysis when future real-prices are held static. In the second set, a probabilistic
analysis is conducted accounting for material-specific price projections, and the pavement with
the lowest life-cycle cost is selected. The third set presents the pavement selection including
material-specific price projections, but an initially more expensive pavement is only selected if it
is lower in terms of life-cycle costs by at least 10%. The last set follows the same procedure, but
assumes at least a 20% life-cycle cost savings is needed.
As can clearly be seen, the inclusion of material-specific price projections can alter the likely
pavement decision. The ultimate pavement selection, however, changes as a function of decision
rule and risk-perspective. This can be illustrated by more closely evaluating the pavement
selection for the medium-volume scenario. For the third decision rule, the asphalt pavement is
superior from a risk-neutral perspective (i.e. mean value) but for more risk-averse perspectives
the concrete alternatives are better. Additionally, if the decision rule followed the fourth case,
where the threshold in life-cycle cost savings must be at least 20% instead of 10%, the asphalt
pavement is in fact superior for all risk-perspectives.
Table 5-18: Pavement selection for probabilistic analyses a) excluding material-specific price projections b)
including material-specific price projections and the lowest life-cycle cost pavement is selected c) including
material-specific price projections and selecting an initially more expensive pavement if it offers at least 10%
life-cycle cost savings and d) the same rule as (c) but for a threshold value of 20%. A discount rate of 4% is
used in all analyses.
High Volume
Medium Volume
Low Volume
a) Excluding material-specific price projections
Mean Value
Concrete
Asphalt
Asphalt
percentile
Concrete
Asphalt
Asphalt
95 percentile
Concrete
Asphalt
Asphalt
7 5 th
b) Including material-specific price projections where lowest
LCC pavement is selected
Mean Value
Concrete
Concrete
Concrete
7 5 th
percentile
Concrete
Concrete
Concrete
9 5 th
percentile
Concrete
Concrete
Concrete
73
High Volume
Medium Volume
Low Volume
c) Including material-specific price projections where initially
more expensive pavement is only selected if its LCC is at
least 10% less
Mean Value
Concrete
Asphalt
Asphalt
75th percentile
Concrete
Concrete
Concrete
95 percentile
Concrete
Concrete
Asphalt
d) Including material-specific price projections where initially
more expensive pavement is only selected if its LCC is at
least 20% less
Mean Value
Concrete
Asphalt
Asphalt
75' percentile
Concrete
Asphalt
Asphalt
9 5 th percentile
Concrete
Asphalt
Asphalt
Discount Rate
As was shown in Section 6.2.2 the comparative results of this analysis are sensitive to the real
discount rate selected. Table 5-19 presents the pavement selection using the three different types
of pavement selection criteria discussed for discount rates of 2% and 5%. For each decisionrule, the pavement selection can differ depending upon the discount rate which is selected.
Table 5-19: Pavement selection for probabilistic LCCA including material-specific price projections with a
discount rate of 2%. Pavements are selected by a) lowest life-cycle cost b) initially more expensive pavement
selected if its life-cycle cost savings are at least 10% and c) the same decision rule as (b) but the threshold
value is 20%
High Volume
Medium Volume
Low Volume
a) Lowest LCC pavement is selected
Mean Value
Concrete
Concrete
Concrete
7 5 th
percentile
Concrete
Concrete
Concrete
9 5 th
percentile
Concrete
Concrete
Concrete
b) Initially more expensive pavement is only selected if its
LCC is at least 10% less
Mean Value
Concrete
Concrete
Asphalt
75th percentile
Concrete
Concrete
Concrete
9 5 th percentile
Concrete
Asphalt
Asphalt
74
High Volume
Medium Volume
Low Volume
c) Initially more expensive pavement is only selected if its
LCC is at least 20% less
Mean Value
Asphalt
Asphalt
Asphalt
7 5 th
percentile
Concrete
Asphalt
Asphalt
9 5 th
percentile
Concrete
Asphalt
Asphalt
Table 5-20: Pavement selection for probabilistic LCCA including material-specific price projections with a
discount rate of 5%. Pavements are selected by a) lowest life-cycle cost b) initially more expensive pavement
selected if its life-cycle cost savings are at least 10% and c) the same decision rule as (b) but the threshold
value is 20%
High Volume
Medium Volume
Low Volume
a) Lowest LCC pavement is selected
Mean Value
Concrete
Concrete
Concrete
7 5 th
percentile
Concrete
Concrete
Concrete
9 5 th
percentile
Concrete
Concrete
Concrete
b) Initially more expensive pavement is only selected if its
LCC is at least 10% less
Mean Value
Concrete
Concrete
Asphalt
7 5 th
percentile
Concrete
Concrete
Asphalt
9 5 th
percentile
Concrete
Concrete
Asphalt
c) Initially more expensive pavement is only selected if its
LCC is at least 20% less
Concrete
Asphalt
Asphalt
percentile
Concrete
Asphalt
Asphalt
95th percentile
Concrete
Asphalt
Asphalt
Mean Value
7 5 th
5.5 Summary
This chapter has presented the life-cycle cost results for three different scenarios using a
deterministic and probabilistic approach (the latter based upon derived probabilistic values in
Chapter 4 and in the Appendix). For all three scenarios, the JPCP (concrete) pavement designs
cost initially more than the HMA (asphalt) designs. Conducting a deterministic analysis, the lifecycle cost of the JPCP alternative is lower for the high-volume scenario but lower in the medium
and low volume scenario. When accounting for uncertainty, however, the life-cycle cost of the
JPCP design is less or nearly equivalent in all three scenarios for all considered risk-perspectives.
75
This suggests accounting for uncertainty could potentially alter the pavement selection made by
a decision-maker, and in particular, the inclusion of material-specific price projections seems to
be the parameter which drives this change.
Implementing the statistically characterized inputs in a case study, the analysis showed the larger
the pavement project, the greater the tendency to spend heavily initially. Additionally, a larger
portion of life-cycle costs for concrete pavements is comprised of initial costs relative to asphalt,
and therefore, one would suspect using a higher discount rate would be more favorable to asphalt
pavement designs.
Lastly, the case study has shown the pavement selection from an LCCA is
both dependent upon one's decision-rule and risk-profile.
The following chapter will summarize the findings in chapters 4 and 5 and presents areas of
future work based upon the results.
76
6
CONCLUSIONS
This thesis has quantified uncertainty in the LCCA of pavements to allow for robust comparative
assessment. The quantified uncertainty has been applied to three scenarios to assess how the
inclusion of probabilistic inputs could potentially impact the final pavement selection of a
decision-maker. This section now revisits the posed questions in the Gap Analysis (Section 2.4)
to evaluate which questions were answered and what gaps still exist in this thesis.
6.1 Key Findings
Quantify cost uncertainty by characterizing possible relationships
Due to the limited amount of detailed cost information made publically available by state DOTs,
this thesis has only quantified the relationship between cost and quantity. All construction
processes evaluated showed a statistically significant relationship between cost and quantity
through regression analysis, which for almost all processes captured 50% to 70% of the
variability in the data. Therefore, although this analysis is only conducted at an aggregate level,
it is able to refine the variability that currently exists in cost data significantly.
Quantify uncertainty for input parameters with no empirical data
This thesis has implemented the pedigree matrix approach to account for uncertainty in
pavement LCCAs for input parameters with no data available to statistically characterize. By
qualitatively assessing input parameters with indicator scores, probabilistic values were derived
for uncertain parameters with no empirical data. Probabilistic values derived by the pedigree
matrix approach can be found in the Appendix.
Predicting future material prices based upon historical data
The goal of this thesis was to assess how asphalt and concrete have historically behaved and how
can an LCCA account this. Collecting historical real-price data made publically available by
BLS, the average growth rate and volatility of asphalt and concrete have been both not-static and
different, suggesting future price-projections should be made to account for this. The likely
explanation for this is a differential price-link has existed between asphalt and concrete with
different energy commodities. Asphalt has a much stronger price-link with oil than concrete, as
expected, given that oil is a petroleum based product, while concrete also shows statistically
significant long-run price equilibrium with another non-renewable energy commodity, coal,
which has historically been much more stable than natural gas or oil. As such, one would expect
the real-price of concrete to have behaved much less volatile historically.
Having understood asphalt and concrete have behaved in different manners, backcasting was
conducted using construction related time-series to assess if projecting the future is plausible.
The results showed predicting future prices, even by only considering a few simple models,
77
outperforms a forecast which does not project future prices up to 40-years into the future. A
simple forecasting model which is unlikely to lead to a large enough error in selecting the
optimal investment decision given the growth rate and volatility of paving materials has been
relatively constant is a Geometric Brownian Motion (GBM) process, and is used in this thesis.
Probabilistic price-projections show the expected real-price of asphalt to increase by 50% and
concrete to decrease by 10% over the next 50-years. Future work should consider other timeseries models, however, to improve upon the work presented in this thesis.
Does conducting a probabilistic analysis change the pavement selection, and if so, what is
driving that change?
This thesis implemented probabilistic input parameters in a case study to assess how the
inclusion of probabilistic input parameters impacts the pavement selection process. The results of
the analysis showed the inclusion of probabilistic input parameters could potentially change the
likely pavement selection, and the parameter driving that change is the projection of future
material prices. The results of the case study were sensitive to the discount rate, and in general,
the higher the discount rate, the more likely the asphalt pavement would be selected. The
analysis also showed the likely pavement selection is both a function of the decision rule for
when a pavement is superior and the risk-perspective of the decision-maker. In general, the
more risk-averse the decision, the higher the likelihood the concrete alternative would have been
selected.
6.2 Areas of future work
Although this thesis has answered some of the questions existing in the pavement LCCA
literature, others still exist. This section now presents some of those gaps in the current study
which future work could build upon.
Improved forecasting models
This thesis considered three relatively simple models to project future material-prices. Potential
issues in the analysis which future studies should address include:
"
Per Section 4.2.1, literature has shown the results of a real-price time-series analysis is
dependent upon the data used to deflate the series (Peterson and Tomek 2000). For this
work, the CPI was used to deflate all time-series. A sensitivity analysis should be
conducted, however, to test if the cointegration and material-specific price projection
work would have had the same results if a different inflation adjusting time-series was
used.
*
Considering more time-series data sets would further validate conclusions from the
backcasting analysis. In particular, certain findings, such as for the USGS Sand & Gravel
78
time-series 35-years of data outperformed using as much historical data as readily
available.
A more fundamental problem future studies should explore, however, is what the "best"
forecasting model is, given that this thesis only explores three simple time-series models in order
to prove one can outperform a forecast which assumes the real-price of commodities will remain
constant over time.
One potential finding in this thesis which future researchers could leverage is to use the long-run
price equilibrium derived between asphalt and oil and make long-term forecasts of oil as a proxy.
Given that significantly more data is available for oil and previous researchers have
comprehensively researched the topic, such a model may be appropriate. Figure 6-1 presents
historical real-price data for oil. Fitting the data to a quadratic ordinary least square (OLS)
regression, two characteristics, as previously discussed by Pindyck (1999), become apparent.
First, the real price of oil over the past 130 years has tended to revert back to a continually
shifting mean value over time. This mean-reversion is a reversion back to the marginal cost of
oil production, which has continually shifted over time in a quadratic fashion. Intuitively this
makes sense; initial oil production became cheaper as technology improved, but as supply,
demand, and cost to extract have increased over time, so has the marginal cost. Second, the time
it takes for the price of oil to revert back to its marginal cost can take up to a decade. Due to its
monopoly type nature, the price of oil has experienced short-run prices that are extremely
volatile and do not match expected competitive prices. Eventually those prices do revert back to
competitive levels, and given that pavement rehabilitations occur decades into the future, this
time to reversion should not be an issue given the scope of the analysis. It is perhaps likely,
therefore, a better model than the GBM process implemented in this thesis would be to project
the future price of oil as a proxy.
Figure 6-1: Historical real-price of oil (BP 2012)
2.5
2
1.5 -
0
0.o5
0~
1870
1890
1910
1950
1930
Year
79
1970
1990
2010
Incorporation of user cost
The scope of the above analysis only focuses on the cost to finance a roadway. The above model
should be expanded upon to include the user cost associated with a pavement decision.
Inclusion offlexibility in the analysis
One of the major limitations of the preceding analysis is the methodology has been applied to a
scenario that assumes future rehabilitation activities are fixed irrespective of future market
conditions. It is likely, for example, that a future rehabilitation activity would either change or
be delayed if material prices were significantly higher than expected. The LCCA model should
account for the flexibility of a decision-maker to change future actions depending upon future
events (known as real options), a current drawback from the above analysis.
Expand methodology for other types of infrastructure
The above methodology, although specifically applied to pavement projects, could be expanded
to consider other types of infrastructure. Forecasting of other relevant construction time-series
(i.e. timber, steel), characterization of unit-cost uncertainty, and others could be characterized to
allow for the implementation of this methodology for other infrastructure systems.
Does one make smarter decisions?
Lastly, the described methodology has shown that a probabilistic analysis could potentially
change the pavement selection of a decision-maker. The purpose, of course, for incorporating
probabilistic values in an LCCA is to make "smarter" investment decisions. A future study
should be conducted validating the above methodology by comparing the cost expectancy of
historical pavement projects to the actual costs to assess if the model leads one to the correct
selection.
80
7
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87
APPENDIX
Table A-1: Pedigree matrix uncertainty indicator scores for parameters with no empirical data
Reliability
Completeness
CTrrelton
Geograph
orlahi c
Te Further
Technological
CalTrans Design Inputs
Design Life
1
1
3
1
1
Time of Rehab
Lane Width
1
1
1
1
1
1
3
1
1
1
1
1
1
1
1
Shoulder Width
2
2
2
1
1
1
1
1
1
Length
Traffic
AADT
AADTT
1
1
1
1
1
1
Jointed Plain Concrete
Thickness
1
1
1
1
1
Unit Weight
1
1
1
1
1
Reinforcement
Dowel Bar Spacing
Longitudinal Joint
spacing
Tie Bar Spacing
1
1
1
1
1
3
5
5
5
4
3
5
5
5
4
Asphalt Concrete
Thickness Surface
1
1
1
1
1
Unit Weight Surface
1
1
1
1
1
3
5
5
5
4
1
1
1
1
1
3
5
5
5
4
1
1
1
1
1
Unit Weight Base
1
1
1
1
1
Primary/Tack Unit
Weight
3
5
5
5
4
Thickness
1
1
1
Primary/Tack Unit
Weight
Thickness Intermediate
Unit Weight
Intermediate
Primary/Tack Unit
Weight
Thickness Base
Lean Concrete Base
1
1
88
.p
Reliability
Completeness
1
1
Temporal
Correlation
rai
Further
Technological
1
1
Correlation
Correlation
Aggregate Sub-base
Thickness
1
Cement Treated Aggregate Base
Thickness
1
1
1
1
1
Unit Weight
1
1
1
1
1
1
1
Lime Stabilized
1
Area
1
1
Permeable Asphalt Treated Base
Thickness
1
1
1
1
1
Unit Weight
1
1
1
1
1
Aggregate Base
Thickness
1
1
1
1
1
Unit Weight
1
1
1
1
1
5
5
JPCP Maintenance
Diamond Grinding
Rehab
Patching
5
5
5
Additional Concrete
Thickness
1
1
1
1
1
Unit Weight
1
1
1
1
1
1
1
Additional Asphalt
1
Thickness
1
1
Unit Weight
Primary/Tack Unit
Weight
1
1
1
1
1
3
5
5
5
4
Table A-2: Variances calculated by pedigree matrix approach
Additional Uncertainty
Reliabil
ity
Completeness
Geographic
Temporal
Correlation
Correlation
C aatns
ein
t
Basic
Total
Further
Technological
Correlation
Basic
Total
CalTrans Design Inputs
Design Life
0
0
0.0002
0
0
0.006
0.0064
Time of Rehab
0
0
0.0002
0
0
0.006
0.0064
Length
0
0
0
0
0
0.006
0.006
0.006
0.0069
Lane Width
Shoulder
Width
0
0
0
0
0
0.006
0.0006
0.0001
0.0002
0
0
0.006
89
Reliability
Completeness
Tempraleogrphic
C ora n
Correlation
rla
Correlation
Further
Technological
Correlation
Basic
Total
0
0
0.006
0.006
0.006
0.006
0
0.006
0.006
0
0.006
0.006
Traffic
AADT
0
AADTT
0
0
0
Thickness
0
0
0
0
0
0
Jointed Plain Concrete
0
0
Unit Weight
0
0
0
0
0
0
0
0
0.006
0.006
0.002
0.008
0.04
0.002
0.04
0.006
0.098
0.002
0.008
0.04
0.002
0.04
0.006
0.098
Dowel Bar
Spacing
Longitudinal
Joint spacing
Tie Bar
Spacing
0
Reinforcement
Asphalt Concrete
Thickness
Surface
Unit Weight
Surface
Primary/Tack
Unit Weight
Thickness
Intermediate
Unit Weight
Intermediate
Primary/Tack
Unit Weight
Thickness
Base
Unit Weight
Base
Primary/Tack
Unit Weight
0
0
0
0
0
0.006
0.006
0
0
0
0
0
0.006
0.006
0.002
0.008
0.04
0.002
0.04
0.006
0.098
0
0
0
0
0
0.006
0.006
0
0
0
0
0
0.006
0.006
0.002
0.008
0.04
0.002
0.04
0.006
0.098
0
0
0
0
0
0.006
0.006
0
0
0
0
0
0.006
0.006
0.002
0.008
0.04
0.002
0.04
0.006
0.098
0
0.006
0.006
0
0.006
0.006
Lean Concrete Base
Thickness
0
0
0
0
Aggregate Sub-base
Thickness
0
0
Thickness
0
0
0
0
0
0.006
0.006
Unit Weight
0
0
0
0
0
0.006
0.006
0
0.006
0.006
0
0
Cement Treated Aggregate Base
Area
0
0
0
Lime Stabilized
0
90
Reliability
Completeness
GFurther
r
Technological
eograi
CTerretion
rmelatnAp
reatn BCorrelation
Basic
Total
Permeable Asphalt Treated Base
Thickness
0
0
0
0
0
0.006
0.006
Unit Weight
0
0
0
0
0
0.006
0.006
Aggregate Base
Thickness
0
0
0
0
0
0.006
0.006
Unit Weight
0
0
0
0
0
0.006
0.006
JPCP Maintenance
Diamond
Grinding
Rehab
Patching
0
0
0
0
0
0.006
0.006
0.002
0.008
0.04
0.002
0.04
0.006
0.098
Additional Concrete
Thickness
0
0
0
0
0
0.006
0.006
Unit Weight
0
0
0
0
0
0.006
0.006
Additional Asphalt
Thickness
0
0
0
0
0
0.006
0.006
Unit Weight
Primary/Tack
Unit Weight
0
0
0
0
0
0.006
0.006
0.002
0.008
0.0 4
0.002
0.04
0.006
0.098
91