by William Colby Dunn Undergraduate in Materials Science and Engineering Massachusetts Institute of Technology

Transcription

1 A More Comprehensive Life Cycle Cost Analysis of Pavement Materials Alternatives by William Colby Dunn Undergraduate in Materials Science and Engineering Massachusetts Institute of Technology Submitted to the Department of Materials Science and Engineering in Partial Fulfillment of the Requirements for the Degrees of Bachelors of Science in Materials Science and Engineering at the Massachusetts Institute of Technology June Massachusetts Institute of Technology. All rights reserved. Signature of author Department of Materials Science and Engineering Materials Systems Laboratory 5/3/2013 Certified by Joel P. Clark Materials Systems Laboratory Faculty Director Thesis Advisor Accepted by Jeffrey C. Grossman Undergraduate Committee Chairman Correspondence: W. Colby Dunn Phone:

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3 ACKNOWLEDGMENTS This research was implemented in tandem with the Materials Systems Laboratory at MIT. The author is grateful to Omar Swei for his extensive technical input, and to Randolph Kirchain and Joel Clark for their technical and literary support and mentorship. Additionally, the author would like to thank MIT for seamless software implementation and Applied Research Associates, Inc., which helped developed pavement designs used in the analysis. 3

4 TABLE OF CONTENTS I. LIST OF FIGURES...5 II. III. IV. LIST OF TABLES...6 ABSTRACT....7 INTRODUCTION. 8 V. LITERATURE REVIEW a. Literature Review b. Gap Analysis.. 15 i. Part 1: Out-of-sample Forecasting. 15 ii. Part 2: Unit-cost Variability Due to Location and Cost. 16 iii. Part 3: Case Study Methodology VI. VII. VIII. IX. METHODOLOGY.. 17 a. Part 1: Out-of-sample Forecasting. 17 i. Section A: Stationary or Non-Stationary Classification ii. Section B: Forecasting Based on Data Classification iii. Section C: Accuracy of the Forecast vs. Baseline Case b. Part 2: Unit-cost Variability Due to Location and Cost.21 c. Part 3: Case Study Methodology i. Section A: Unit Cost Relationship ii. Section B: Uncertainty Characterization without Historical Data iii. Section B: Perform and Interpret Simulations RESULTS & ANALYSIS 24 a. Part 1: Out-of-sample Forecasting. 24 i. Section A: 40-year Results ii. Section B: 60-year Results b. Part 2: Unit-cost Variability Due to Location and Cost.28 c. Part 3: Case Study Methodology...36 CONCLUSIONS AND FUTURE WORK.42 WORKS CITED X. APPENDICES

5 LIST OF FIGURES 1. Figure 1: Two Pavement Designs with Different Net Present Values (NPV) and Risks Figure 2: The Life-Cycle of a Highway Construction Project. page Figure 3: The Initial and Life-Cycle Costs Associated with Highway Construction Projects Figure 4: The Historical Price Behavior of Dimensioned Stone Figure 5: MAPE vs. Year into the Future (40-year Analysis) Figure 6: MAPE vs. Year into the Future (60-year Analysis) Figure 7: Florida linear regression analysis of unit-cost of HMA winning pavement bids with respect to bid volume Figure 8: Aggregate (FL & CO) linear regression analysis of unit-cost of JPCP winning pavement bids with respect to bid volume Figure 9: Colorado linear regression analysis of unit-cost of HMA winning pavement bids with respect to bid volume Figure 10: Colorado linear regression analysis of unit-cost of JPCP winning pavement bids with respect to bid volume Figure 11: ARA Pavement Design Specifications Figure 12: Initial, discounted rehabilitation, and life-cycle costs of JPCP and HMA pavement designs in Florida and Colorado Figure 13: Florida probabilistic life-cycle cost of urban interstate road alternatives (gray line represents HMA design, orange represents JPCP design). The dashed lines represent the mean value from the analysis Figure 14: Colorado probabilistic life-cycle cost of urban interstate road alternatives (gray line represents HMA design, orange represents JPCP design). The dashed lines represent the mean value from the analysis Figure 15: 40-year MAPE Analysis with Real dollars, No-change Model Figure 16: 60-year MAPE Analysis with Real dollars, No-change Model

6 LIST OF TABLES 1. Table 1: Florida quantification of unit-cost uncertainty for significant input parameters. Values in parenthesis represent the standard error of the regression coefficients Table 2: Colorado quantification of unit-cost uncertainty for significant input parameters. Values in parenthesis represent the standard error of the regression coefficients Table 3: MEPDG based JPCP and HMA pavement designs for the urban interstate and local road case studies Table 4: Maintenance schedule for JPCP and HMA pavement designs at MEPDG specified 90% reliability for the urban interstate and local road case studies Table 5: Florida Pavement Specifications

7 ABSTRACT Life Cycle Cost Analysis (LCCA) is a commonly used tool in analyzing the economic viability of highway construction investments. The initial and life-cycle materials costs associated with highway construction involve a high level of uncertainty and therefore warrant extensive and dynamic cost analysis. These uncertainties derive from extensive materials usage costs. Despite the advantages of implementing a probabilistic approach to cost analysis, many state departments of transportation (DOTs) continue to employ a deterministic model, thereby misjudging, and often altogether neglecting the underlying uncertainty and risks. The goals of this paper are twofold: first, to validate forecasting as a viable method to predict future materials prices, and second, to explore economies of scale as a potential driver of uncertainty. The paper will then apply these results to a case study methodology, looking at a comparative LCCA of two materials alternative, asphalt vs. concrete pavement designs for two states: Florida and Colorado. Endeavoring in this light, the author has characterized uncertainty in a way that will be comprehensible by practitioners. This research has successfully validated out-of-sample forecasting as a superior method of forecasting materials prices, characterized uncertainty related to project quantity, and delivered results using a relatable case study approach. 7

8 INTRODUCTION Analyzing highway construction projects deterministically is a cursory methodology in that it oversimplifies input parameters by assigning them a single, static value. These simplified values effectively mask any uncertainty related to the investment decision on hand, as shown in Figure 1 (Gransberg 2009). While this approach does provide the user with a simple selection process, it bases its cost assessment off a limited number of possible outcomes. Realistically, there are a vast multitude of nuanced scenarios that could play out; the risk associated with such long-term investments therefore is much more complicated than a single output. A probabilistic analysis will account for a range of outcomes, thereby better incorporating the underlying risks associated with a project (Swei 2013). With a better understanding of the risks associated with different highway implementations practitioners will be better able to integrate risk-threshold levels into their highway construction decisions. Net Present Value Design A Design B Figure 1: Two pavement designs with different net present values (NPV) and different associated risks (error bars). A deterministic approach would effectively mask the underlying uncertainty here. 8

9 The concept of implementing a probabilistic LCCA is by no means novel. In fact, the methodology has come strongly recommended by the Federal Highway Administration (FHWA) since 1998 (Smith and Walls 1998; Temple and William 2004). The implementers of such models, however, have been hesitant to switch from the traditional deterministic model for several reasons. The first of which is that the implementation of a probabilistic model is far more complex and multi-faceted than is its deterministic counterpart. Secondly, practitioners have been provided little guidance regarding how to characterize uncertainty regarding input parameters (Tighe 2001). Life Cycle Cost Analysis (LCCA) is a commonly used analytical investment assessment tool that accounts for total project costs beginning with the initial building and extending to the final removal of the venture, as shown in Figure 2 (Lee 2002; Guo 2010). All costs incurred in-between (e.g. routine maintenance and rehabilitation expenses) are included (Smith 1998). It is therefore no surprise that projects warrant extensive materials usage, and the importance of materials in the LCCA, therefore, cannot be stressed enough. While highway investments extend far into the future (some reaching 50-years), recent data has shown that practitioners continue to place a disproportionate weighting on initial costs rather than their life-cycle counterparts (Rangaraju 2008). One explanation for this phenomenon is that the level of uncertainty is far greater when forecasting costs that extend far into the future. It is therefore reasonable to abstract that should practitioners be provided with a framework to better understand potential future cost behavior, they are likely to weigh life-cycle costs when selecting a paving material (Frangopol 2001). Beyond uncertainties in future costs, investments in highways also warrant extensive 9

10 characterization of initial parameters, such as unit-cost of paving material inputs, which likely vary as a function of quantity, location, and more as shown in Figure 3. With better characterization of the impact of these two variables on cost, practitioners will be more confident in their understanding of the variability associated with a specific pavement materials selection. Figure 2: The three main constituents of life-cycle costs in highway construction projects. The final removal costs and user costs are excluded. Figure 3: The initial and life-cycle costs associated with highway construction investments. (Swei 2012) 10

11 The overarching theme of this paper is to provide LCCA practitioners with an improved structure for understanding uncertainty related to highway costs, particularly uncertainty related to future materials costs. In this vein, the paper will take a two-pronged approach in first classifying uncertainty related to future materials prices, and testing whether forecasting prices decades into the future is a plausible endeavor. The research will then shift to characterizing the unit-cost of construction inputs associated with highway construction, exploring how much of the variability can be explained through a likely relationship which exists between cost and quantity and testing this for multiple locations. Finally, the author has taken a case study approach in order to test the implications of the characterized inputs on a probabilistic LCCA for pavement design alternatives that are considered functionally equivalent. Specifically, an LCCA between two materials alternatives, asphalt and concrete pavement designs in Florida and Colorado will be compared. In so doing, drivers of uncertainty will be elucidated in a way relatable to LCCA practitioners, thereby leading to an improved methodology for understanding the uncertainty embedded in such long-term investments. 11

12 LITERATURE REVIEW Economic feasibility is a fundamental concern when assessing any long-term investment proposition, and highway construction projects are certainly no exception. With historical LCCA model implementation ranging from thirty to fifty-five years, highways represent just how important financial viability is to infrastructure projects (Frangopol 2001). Culminating in what was a revolutionary ordinance in 1995, the Federal Highway Administration (FHWA) forced all state DOTs to incorporate LCCA models into highway projects that had estimated costs of over $25mm (National Highway System Designation Act of 1995). The FHWA further emphasized the importance of sound highway investment by outlining the shortcomings of a deterministic LCCA (Walls and Smith 1998). Stressing the relative merits of a probabilistic dynamic model, the FHWA has established itself as a prominent source of funding for such research. After the 1995 legislation, teams of researchers set to work developing a more robust LCCA model to accurately assess the economic performance of highway construction projects. Specifically, Zimmerman and Peshkin focused their efforts on specifying optimal maintenance schedules for pavements (Zimmerman and Peshkin 2003). Embacher and Snyder evaluated the relative economic merits of asphalt (HMA) vs. concrete (RMC) in low-volume pavement applications (Embacher and Snyder 2001). Oberlander used factor analysis to make predictions for the cost of construction projects (Oberlander 2003). The above research endeavors, while having contributed to the development and implementation of LCCA in the field, are fundamentally flawed, as they have taken a deterministic approach. In so doing, said research has failed to address the 12

13 uncertainty and risks previously discussed (Swei et al. 2013a). With that said, one parameter that has traditionally been ignored is the forecasting of future material prices. Before forecasting prices, however, it is first necessary to verify that statistical methods exist, which are a superior method of predicting such uncertainties with a relatively high level of precision. In order to forecast future cost behavior, one risk that must be considered is how construction prices will evolve over time. To that extent, research has been geared towards using statistics in assessing the merits of forecasting future materials prices (i.e. whether or not historical data is a good predictor of future data). One such methodology, known as backcasting, takes out-of-sample data and forecasts said data where the future price point is known. Multiple studies have implemented these methods. Ashuri and Lu (2010) created a Construction Cost Index (CCI) that uses an Autoregressive Integrated Moving Average (ARIMA) model to forecast future costs. The two used a 12- month out-of-sample forecast in tandem with Mean Square Error (MSE), Mean Absolute Error, and Mean Absolute Percent Error (MAPE) metrics. Hwang built upon such research, using Mean Absolute Error (MAE) to access the precision of forecasting materials prices for a 12-month and 24-month period using the CCI (Hwang 2009; Hwang 2011). Xu and Moon (2013) used Mean Square Error (MSE) to determine the validity of their 54-month vector autoregression (VAR) forecasts that established a relationship between CCI and the Consumer Price Index (CPI). MIT s Concrete Sustainability Hub (CSH) have followed a similar methodology as Xu and Moon, conducting cointegration between multiple time-series to construct a forecasting model 13

14 for asphalt and concretes (Swei 2013b). Moreover, backcasting techniques have been employed, resulting in Mean Average Percent Error (MAPE) values that prove forecasting future pavement commodity costs (asphalt and concrete constituent materials) as a valid endeavor to reduce uncertainty (Swei et al. 2013b). In addition to long-term costs, highway construction projects also warrant extensive risk characterization in shortterm inputs; accordingly, it is crucial to understand the developmental timeline of a highway. When assessing the economic viability of highway construction projects, it is important to account for the substantial cash outflows that span over the highway lifespan. The outflows can be separated into three distinct phases: the initial costs, use costs, and the end-of-life costs. The initial costs include materials extraction and production, transportation, and any costs associated with construction (i.e. equipment and labor). After the highway is constructed, maintenance costs are incurred throughout the lifespan of the highway. And, finally, the end-of-life costs (i.e. pavement removal and recycling) are incurred when the highway is no longer in working condition (Swei et al. 2013a). It is therefore clear that highway construction is not only a long-term investment venture but also a multi-faceted project susceptible to uncertainties at phase. Accordingly, it is crucial to develop as comprehensive an LCCA model as possible to ensure practitioners are not only aware of the expected costs associated with a project but also cognizant of expected uncertainties associated with such costs. This research focuses on the cost to the agency, thereby excluding the user costs, in an attempt to focus the scope of the analysis. 14

15 Due to the increased recognized importance of accounting for uncertainty, researchers have recently begun to endeavor in extensive research that is geared towards accessing the uncertainties rooted in the classical deterministic methodology (Swei 2013b). Specifically, Touran (2003) incorporated uncertainty in bid vs. actual construction costs by accessing the data from a probabilistic standpoint. Tighe (2001) further contributed by accessing pertinent inputs and overall construction costs with an emphasis on probability distributions. Additionally, Osman (2005) employed a Weibull distribution to elucidate some uncertainty related to pavement performance and maintenance over its life span. In other words, much attention has been focused on the risks inherent in the input parameters, with the goal of arriving at a more robust output. This improved model would better indicate the uncertainty associated with particular highway constructions. In so doing, the model will provide practitioners with a more robust and holistic perspective, thereby enabling them to align their investment decisions with their risk thresholds (Swei et al. 2013c). GAP ANALYSIS Part 1: Out-of-sample Forecasting While a CCI is an invaluable and widely used dataset for forecasting, it is incomplete in that it is a weighted average index comprised of material, labor, and construction costs. This consolidation, as stated by Hwang et al., wrongfully assumes that all cost items will grow at the same rate. (Hwang et al. 2011) It is therefore prudent to delineate each constituent cost and forecast separately. Moreover, prior research has been limited in that it has assessed forecasting for a short time period (i.e. 5-years); the novel aspect of this research is that it will determine the relative merits of forecasting over extended periods 15

16 of time (i.e. 40 years). The research will also explore how much empirical data is needed in order to forecast with a reasonable level of precision. Part 2: Unit-cost Variability due to Location and Scale Location with respect to state will be accessed through analyzing two distinct states. In other words, this research will see if parameter characterization holds true from state to state, and if not, by how much they differ. Secondly, establishing a relationship between unit-cost and quantity for materials used in recent highway construction projects will assess economies of scale as a potential driver of variability. This data will prove not only worthwhile for academia but also relatable to practitioners in the field, thereby achieving the penultimate goal of the research. Part 3: Case Study Methodology After validation of forecasting long-term costs and characterization of initial costs, this paper will deliver the results using a case study methodology. Accordingly, state DOTs will be able to better understand the implications of variable characterization on a probabilistic LCCA. 16

17 METHODOLOGY Part 1: Out-of-sample Forecasting In order to validate backcasting as a viable means of forecasting, it is crucial to first establish the historical behavior of the underlying data. In general terms, if the statistical properties of a data set do not change over time, the data is deemed stationary, and otherwise is classified as non-stationary. In other words, a joint distribution (X t1, X tn ) is the same as the joint distribution (X, X t1 +r t n +r ) for all n and r, thereby depending only on the distance between t 1 and t n. (Nason et al. 2006). If a time series does not exhibit stationarity as outlined above, it is deemed non-stationary. After establishing said behavior, an appropriate forecast can be initiated using empirical data. The model used in this paper will not be set to the strict standards of stationarity as outlined above, but will rather adhere to a less rigid criterion, which will be discussed in the coming sections. After initiating the forecast, the model will then be evaluated on the basis of accuracy using MAPE against a baseline model that assumes future real prices that are in line with inflation (Gneiting 2010; Baumeister 2012). Finally, the accuracy of said forecasts will be compared on the basis of the time extent of the data set. In other words, the author will assess the accuracy of the model for different sizes of the dataset, which will lead to a better understanding of how much historical data is sufficient to produce superior results. Section A: Time-Series Classification Stationary or Non-Stationary There are various tests that serve to characterize the historical behavior of a data set, including the Augmented Dickey-Fuller (ADF) and the Phillip-Perron (PP) test. (Leybourne 1999) Although the former are more common, the implementation is quite time consuming, in particular for this research given that 40 different samples will be 17

18 analyzed for the time-series. As such, this research employs a threshold test that utilizes the univariate ARIMA (p, d, q) 1 model, seen in equation 1:!!! 1!!!! (1!)! =! +!! 1!!!!!!!!!! (1)!! : price in year t!! : error term in year t!! : autoregressive coefficient!! : moving average coefficient C: constant drift term L: lag operator If an ARIMA (1, 0, 0) model is used, it equates to the following, as shown in equation 2:!! =!!!!! +! +!! (2) Interestingly, the simplified, first-order auto-regressive process, ARIMA (1,0,0), as shown in equation 2 bears a resemblance to the Ornstein-Uhlenbeck mean-reverting process, shown in discretized form in equation 3:!! = 1!!!!! +!" +!! (3) K: speed of mean reversion C: mean reversion value!! : white noise error term The first step is to determine the value of the autoregressive coefficient,! in equation 2. It is important to note that! corresponds to 1-K, seen in equation 3, where K is the rate of 1 p: number of autoregressive terms d: number of non-seasonal differences q: number of moving average terms 18

19 mean reversion. The inverse of K symbolizes the time it takes for reversion to occur in a data set; therefore, the higher the value of!, the lower the value of K, and the longer it takes for reversion. Thus, this research uses a threshold test, in which the data is deemed to exhibit autocorrelation and is classified as non-stationary if! is greater than or equal to 0.95 (Selke et al. (1999). It is also worth noting that the error term will be disregarded in this analysis, as this research is only concerned with the precision of the forecast, and so a confidence interval is not pertinent to the scope of this research. Additionally, the 0.95 threshold, which implies a process takes 20 years to revert back to its mean, is in line with previous research that has found commodities can take up to a decade to show reversion (Pindyck 1999). Parameters are estimated using Stata and JMP 10, two statistical software packages (Stata 2012; JMP 2012). Excel will also be used extensively to prepare data for result analysis. Section B: Forecasting Based on Data Classification After characterizing the behavior of the time-series empirical data, the forecast will then be implemented. If a sample set is classified as stationary, the ARIMA (1,0,0) (i.e. Ornstein Uhlenbeck mean-reverting) model will be used, as was previously shown in equation 3. If, however, the data are classified as non-stationary, this research will use an ARIMA (1,1,0) model will be used. The ARIMA (1,1,0) is a first-order autoregressive model that includes one order of non-seasonal differencing and a constant term, as shown in equation 4. 19

20 !! =!!!!! (1!) +! +!! (4) Section C: Accuracy of the Forecast vs. Baseline Case After collating the results, the future prices will be compared to the baseline case using MAPE, as defined in equation 5 and, as previously discussed, is one of the more prevalent scale-independent measures of precision (Ahlburg 1992; Armstrong 2001; Hyndman 2006; Rayer 2008; Swanson 2011)!"#$ = 100%!!!!!!"#$%&!"#$%&'#$!"#$%& (5) n: number of years The baseline case used is the no change model that assumes prices grow with inflation. Thus, the forecasting model will be compared to the baseline case (i.e. inflation rate model) to determine the relative merits of projecting future materials prices. Finally, this research will vary the sample size of the data set that is used in each forecast, thereby analyzing how much data is sufficient for forecasts that are more accurate than the baseline case. Specifically, two simulations will be run, in which one trial uses at least 40-years of empirical data is used to estimate parameters, representing the amount of available asphalt data, while the other uses at least 60-years of data to forecast future prices and determine the precision as a function of the extent of the data set. A well performing model will have MAPE values of approximately 10%. (Fan 2010) 20

21 Part 2: Unit-cost Variability due to Location and Scale Basic economic theory implies that as the quantity associated with a project increases, the corresponding unit cost will decrease. It is therefore prudent to apply this to the highway construction data using publically-available bid data provided by state DOTs. Total costs provided by the DOTs typically encompass both labor and materials into a single number. It is therefore difficult to further separate the data based on cost specification; nevertheless, it will likely prove valuable to establish what is likely to be a relationship between unit price for materials-intensive construction activities and the quantity of those activities. Specifically, the log-transformation of the average unit-cost will be plotted as a function of the log-transformation of the bid volume associated with the project. Statistical significance will then be gauged by the reported p-value of the dependent variable through the use of a univariate regression analysis. In statistical analysis, the p- value represents the probability of observing an extreme statistic in a data set, with the assumption that the null hypothesis is warranted. In other words, for the scope of this research, any statistical anomalies (i.e. outside of the 5% threshold) will be deemed statistically insignificant. Construction projects that are deemed statistically significant in respect to unit-cost vs. quantity will then be used to project initial costs. It is important to note, however, that this model does not effectively incorporate seasonality, location, and other more qualitative drivers that may affect unit cost. These factors will instead be modeled using the standard error of the univariate regression in the Monte Carlo simulations. In the case 21

22 that the relationship is not statistically significant, a chi-square best-fit log-normal distribution will be fitted to the data. Part 3: Case Study Methodology Section A: Monte Carlo Simulations A Monte Carlo simulation will be used to build a probabilistic distribution that incorporates the characterized uncertainty while maintaining the structural integrity of the model. It is thus of paramount importance that the model incorporates inter-variable correlations and dependencies. For example, if concrete prices were being compared between two projects, we would assume that the growth rate is constant between the two alternatives. Along that vein, we assume the distance between concrete processing plants are the same, and thus transportation costs associated with using concrete are constant between projects. By accurately considering these relationships in the data, the model will be able to select reasonable values that are representative of what one would expect in the real world. There are many ways to interpret the results from the aforementioned Monte Carlo simulation. The more common method of visualization in most fields is a probability distribution function (PDF), but this research will use a cumulative distribution function (CDF), as is standard in cost analysis. This research has incorporated additional sources of uncertainty, such as pavement deterioration and maintenance, inputs that lack historical data, and future materials prices. Specifically, in terms of future maintenance schedules, the recently developed Mechanistic Empirical Pavement Design Guide (MEPDG) combines pavement design parameters (e.g. pavement type, thickness, etc.) with contextual conditions (e.g. climate, 22

23 traffic flows, etc.) into a predicted level of pavement performance (15). This design guide uses models validated from the Long-Term Pavement Performance (LTPP) program (21). This performance is measured through distress levels that incorporate top-down and bottom-up cracking. Furthermore, variables that lack historical data, such as layer thickness, have been quantified by using a pedigree matrix approach, a method used by the life cycle assessment (LCA) community. Section B: Case Study The aforementioned results will then be collated and applied to two urban interstate case studies located in Florida and Colorado. The case studies will be analyzing the relative economic merits of concrete vs. asphalt pavement designs using the recently developed Mechanistic-Empirical Pavement Design Guide (MEPDG) software, the new state of the art pavement design tool. 23

24 RESULTS AND ANALYSIS The results and subsequent analysis can be separated into this paper s three inexorably linked pieces: out-of-sample forecasts, unit-cost relationship, and the case study methodology. These three sections will discuss and analyze explanatory drivers of uncertainty in the highway selection process and validate the probabilistic model as a more comprehensive and prudent implementation of LCCA. Part 1: Out-of-sample Forecasting Data for dimensioned stone has been collected through publically available databases at the United States Geological Service (USGS), as shown in Figure 4. Dimensioned stone was selected as part of this analysis since it is a common input for construction activities and because there is sufficient data that extends as far back as 1900, making it possible to backcast (Kelly 2012). 400 Price ('98$/ton) Price (98$/t) Year Figure 4: Dimensioned Stone historical price behavior. 24

25 Out-of-sample forecasting has been conducted using at least 40-years of empirical data using the previously discussed ARIMA (1,0,0) for stationary time-series, ARIMA (1,1,0) for non-stationary time-series, as well as a real-dollars, no-change model. The MAPE of each model was used to gauge precision. Out-of sample forecasts have been constructed between 1945 and 1986, with the minimum amount of data used being 40 years (for the 1945 forecast) and the maximum being 80 years (for the 1985 forecast). In each forecast, the precision of the forecast is measured for up to 40 years, if possible. Of course, it is impossible to track an out-of-sample forecast made after 1970 for 40 years, and so the sample size of the MAPE reduces for further years out. The selection of 40-years as a bottom threshold is because there are only 40-years of empirical data for asphalt. Having said this, the calculated MAPE is conducted using the forecast errors made between 1966 and 1986, in order to understand if increasing the sample size improves the relative performance of the forecasting model. Section A: 40-year Results Figure 5 shows the calculated MAPE values of the data set over time with at least 40- years of empirical data. It is clear that the no-change performs substantially better for the first 30-years. After this midpoint, however, the ARIMA (1,0,0) begins to outperform its counterpart, representing the stationary characteristics of the data set. 25

26 30% 20% MAPE 10% DS No Change ( ) DS (1,0,0) ( ) 0% Years into the Future Figure 5: MAPE vs. Years into the Future (at least 40-years of empirical data) Section B: 60-year Results Figure 6 shows a similar analysis but for at least 60-years of empirical data. The two models are shown again, as outlined in the graphic. It is clear to see a similar trend with the 40-year data: the real-dollars no change outperform the ARIMA (1,0,0) model at the onset of forecasting. However, after 20-years (unlike 30-years previously), the ARIMA (1,0,0) model begins to outperform. Therefore, not only is it more accurate to use more empirical data, but it also takes less time for to observe reversion in the 60-year data set than it does in the 40-year data set. This is an interesting observation, as it shows that with more empirical data, practitioners will be better able to access future price trends of 26

27 materials costs, thereby reducing uncertainty and incorporating their risk threshold into pavement selection % DS No Change ( ) 30.00% DS (1,0,0) ( ) MAPE 15.00% 0.00% Years into the Future Figure 6: MAPE vs. Years into the Future (at least 60-years of empirical data) Not only do these results show that with more empirical data, more accurate forecasts can be generated, but these data also validate forecasting as a worthwhile endeavor to understand future price behavior that extends decades into the future (i.e. a timeline in line with a highway s life-cycle). Furthermore, it shows that by applying such models going forward, practitioners will be able to better quantify the uncertainty related to materials prices relevant to their projects. 27

28 Part 2: Unit-cost Variability Due to Location and Scale Tables 1 and 2 show the linear regression analysis of construction bid projects for Florida and Colorado, respectively. Materials used in the projects were broken down into Concrete, Basestone, Asphalt, and repair costs (i.e. patching, milling, and grinding), in accordance with outlines set forth in Table 5. The repair and basestone cost show very little statistical significance, while most other material constituent costs show a determination coefficient of at least 0.50, implying that 50% of the variation in the data can be explained by economies of scale. In the Florida data, there were not enough JPCP data points to create a statistically meaningful regression; therefore, a regression was run on all JPCP data (Colorado and Florida), as shown in Figure 8. This aggregation will likely increase uncertainty, but with such a low JPCP construction rate in Florida, the endeavor is uncertain as it is. It is also worth noting that one of the base stones in the Colorado data is the only material for which we found a p-value greater than the 0.05 threshold value. Figures 7-10 show the regression analysis of the unit-price of HMA and JPCP pavement designs as a function of bid quantity over a 36-month span in Florida and Colorado, respectively. Figure 7 shows the regression data of winning bids of HMA pavement designs in Florida. The corresponding equation shows that approximately 65% of the uncertainty related to HMA pavement development projects can be explained by economies of scale. 28

29 6.00 Natural Log of Unit Price ($/CY) LN (unit-cost) = * LN (quantity) R² = Natural Log of Quantity (Cubic Yards) Figure 7: Florida linear regression analysis of unit-cost of HMA winning pavement bids with respect to bid volume. Figure 8 then shows the same result for the aggregate JPCP designs in Florida and Colorado. This aggregation is a result of Florida not having statistically sufficient data for analysis, as was previously discussed. As shown in Figure 8, 38% of uncertainty in this aggregation of JPCP designs can be explained by economies of scale. 29

30 Natural Log of Unit Price ($/CY) LN (unit-cost) = * LN (quantity) R² = Natural Log of Quantity (Cubic Yards) Figure 8: Florida and Colorado linear regression analysis of unit-cost of JPCP winning pavement bids with respect to bid volume. Figures 9 and 10 show the same results for HMA and JPCP, respectively for winning bids in Colorado. As shown in figure 9, approximately 57% of the uncertainty in HMA data can be explained by economies of scale. Analogously, figure 10 shows that 44% of uncertainty in JPCP bids can be attributed to cost-quantity relationships. 30

31 8.00 Natural Log of Unit Price ($/CY) LN (unit-cost) = * LN (quantity) R² = Natural Log of Quantity (Cubic Yards) Figure 9: Colorado linear regression analysis of unit-cost of HMA winning pavement bids with respect to bid volume. Natural Log of Unit Price ($/CY) LN (unit-cost) = * LN(quantity) R² = Natural Log of Quantity (Cubic Yards) Figure 10: Colorado linear regression analysis of unit-cost of JPCP winning pavement bids with respect to bid volume. 31

32 In both states, regression coefficients are similar, while the determination coefficients vary dramatically between the HMA and the JPCP selections. This discrepancy represents the need for probabilistic LCCA models, as it has been proven that an uncertain cost is not the only factor to consider when assessing highway construction projects. Moreover, the figures show a clear relationship between cost and quantity, accounting for at least 44% of the variance in bid prices. Looking at the data from an inter-state perspective, it can be observed that economies of scale play a more prominent role in the Colorado data than it does in the Florida data. Tables 1 and 2 have collated the statistical results for all constituent materials related to winning bid prices for Florida and Colorado, respectively. As shown, bid data are delineated into concrete and base materials, asphalt designs, and maintenance expenses relevant to the two designs. Table 1 shows the data and pertinent statistical parameters in the winning bids in Florida. As was previously discussed, the data lacked a sufficient sample size to analyze JPCP bids, while the HMA bids mostly show high determination coefficients, owing to the role economies of scale has in pavement construction projects. 32

33 Table 1: Florida quantification of unit-cost uncertainty for significant input parameters. Values in parenthesis represent the standard error of the regression coefficients. Input Units P-Value R 2 Regression Equation Ln(P)=a*Ln(Q)+b Concrete and Bases JPCP Small Sample Size, Aggregate Regression Conducted Base Group 06 Square a = (0.053) < Yards b = 4.65 (0.38) Base Group 09 Square a = (0.042) < Yards b = 4.29 (0.33) Base Group 15 Square a = (0.038) < Yards b = 5.61 (0.29) Initial Asphalt Design Asphalt Inc. Bit. a = (0.019) Tons < (PG: 76-22) b = 5.88 (0.15) Asphalt Traffic B a = (0.010) Tons < (PG: 76-22) b = 4.92 (0.077) Asphalt Traffic C a = (0.010) Tons < (PG: 76-22) b = 5.58 (0.080) Asphalt Traffic D a = (0.014) Tons < (PG: 76-22) b = 5.02 (0.091) Asphalt Superpave a = (0.017) Tons < Traffic A b = 5.73 (0.12) Asphalt Superpave a = (0.020) Tons < Traffic B b = 5.69 (0.15) Asphalt Superpave a = (0.0092) Tons < Traffic C b = 5.33 (0.069) Asphalt Superpave a = (0.018) Tons < Traffic D b = 5.49 (0.13) Asphalt Traffic E Tons < a = (0.039) b = 6.22 (0.36) Tack Coat Asphalt Milling Concrete Grinding Concrete Removal Concrete Slab Replacement Maintenance Specific Input Parameters Cubic a = (0.017) < Yards b = 5.42 (0.11) Square a = (0.020) < Yards b = 4.03 (0.22) Square a = (0.030) < Yards b = 3.78 (0.19) Cubic a = (0.028) < Yards b = 6.58 (0.19) Best-fit Log- Normal Distribution 33

34 Table 2 shows similar data for winning bids in Colorado. Again, these data have been separated into concrete and bases, asphalt designs, and maintenance expenses. Statistical parameters are shown, and it is worth noting that the final base stone was the only material that did not fit within the p-value threshold. These final base stone data have thus been fit to a lognormal distribution, as opposed to the linear regression. It is also worth noting that the uncertainty in HMA designs attributable to economies of scale varies much more than does the Florida data, which will prove interesting when accessing the comparative LCCA from a probabilistic standpoint. Finally, most of the maintenance data show very little statistical significance. 34

35 Table 2: Colorado quantification of unit-cost uncertainty for significant input parameters. Values in parenthesis represent the standard error of the regression coefficients. Input Units P-Value R 2 Regression Equation Ln(P)=a*Ln(Q)+b Concrete and Bases Concrete Cubic a = (0.015) < Yards b = 6.14 (0.098) Base Stone Cubic a = (0.023) < Class 6 Yards b = 4.60 (0.15) Base Stone Cubic Specification Yards Initial Asphalt Design Asphalt a = (0.053) Tons < (PG: 58-28) b = 7.18 (0.42) Asphalt a = (0.067) Tons < (PG: 58-34) b = 9.16 (0.52) Asphalt a = (0.045) Tons < (PG: 64-22) b = 5.95 (0.36) Asphalt Traffic D a = (0.16) Tons < (PG: 64-28) b = (1.37) Asphalt Traffic D a = (0.044) Tons < (PG: 76-28) b = 7.22 (0.35) Tack Coat Gal < a = (0.033) b = 3.18 (0.27) Maintenance Specific Input Parameters Asphalt Patching Tons < a = (0.020) b = 5.86 (0.10) Concrete Grinding Square a = (0.045) Yards b = 3.82 (0.32) Concrete Patching Cubic a = (0.12) < Yards b = 8.07 (0.50) Best-fit Log- Normal Distribution Mean = 3.04 St. Dev. =

36 Part 3: Case Study Methodology A third-party associate, namely the Applied Research Associates (ARA), have developed Hot Mix Asphalt (HMA) and Joint Plain Concrete Pavement (JPCP) designs that although made with different materials, are functionally equivalent (Fig. 11). The goal of this section is to apply the characterized uncertainty values in realistic pavement decisions in order to understand how they could potentially impact the likely pavement selection. Hot Mixed Asphalt E"#;0AB6/<##F+<.59## >"#?60.## Jointed Plain Concrete Pavement (JPCP)!!"#$%&%## J6<.5*6/## 89:"#;445.46<.#123= 360.# # # # 36

37 Figure 11: Applied Research Associates (ARA) Pavement Design Specifications Tables 3 and Table 4 show these designs and their respective maintenance schedules with a corresponding reliability of 90%. The MEPDG software has been constructed for a major highways consisting of three lanes of traffic in each direction with an expected Annual Daily Truck Traffic (AADTT) of 8,000. Finally, the life-cycle costs have been discounted at a 4% annual rate (consistent with FHWA ordinances) over a maintenance schedule that extends 50-years into the future. Deterministic Analysis: A deterministic model has been implemented to predict the likely pavement selection under the current state DOT methodology. Historical bid data was collated for Florida and Colorado for the past 36-months using Oman bid tabs database. Figure 3 presents the initial, discounted rehabilitation, and discounted life-cycle costs for HMA and JPCP designs in both Florida and Colorado. The analysis shows that all costs associated with HMA design selection are lower than their JPCP counterparts. Additionally, the differences between the initial costs in Florida are statistically insignificant, while all other costs (Florida and Colorado) vary significantly. As was previously noted, however, the deterministic analysis lacks sufficient characterization of risk. It is therefore prudent to endeavor in the same way using a probabilistic model in order to observe the differences. 37

38 3.5 Concrete Asphalt Net Present Value (millions of $'s per mile) FL Initial Costs FL Life Cycle Costs FL Discounted Rehab. Costs CO Initial Costs CO Life Cycle Costs CO Discounted Rehab. Costs Figure 12: Initial, life cycle, and discounted rehabilitation costs of JPCP and HMA pavement designs in Florida and Colorado Probabilistic Analysis: Figures 13 and 14 show the analysis for the same data using a probabilistic model for Florida and Colorado, respectively. The analysis shows that the expected net present value (ENPV) of the HMA design is, in fact, more expensive than its JPCP alternative. Given that rehabilitation costs account for a larger proportion of HMA pavement investment, a probabilistic LCCA, which considers all reliability levels rather than 90% as does its deterministic counterpart, could potentially favor the HMA mean values. 38

39 Figure 13 shows the CDF plots for JPCP (gray) and HMA (black) life-cycle costs in Florida. The symmetry in the plot shows that one s pavement design selection should not change as one s risk threshold changes. Moreover, one can see the mean costs associated with each pavement design; specifically, the mean cost per mile for JPCP design is $1.83mm, while its HMA counterpart $2.52mm per mile. Perhaps the budget-conscious state DOTs would benefit from a risk-threshold analysis, wherein practitioners could analyze the cost of a project with a level of certainty. For instance, an LCCA practitioner could use these data to say with 95% certainty that the highest JPCP life-cycle cost per mile would be $2.44mm, while its asphalt counterpart would be $3.25mm. Cumula1ve Probability 100% 80% 60% 40% 20% 0% Asphalt Concrete Life Cycle Cost (millions of $ s per mile) Figure 13: Florida probabilistic life-cycle cost of urban interstate road alternatives (black line represents HMA design, gray represents JPCP design). The dashed lines represent the mean value from the analysis Figure 14 shows the same plot for the results from the Colorado data. The probabilistic results are much less symmetric than are the data in Florida, indicating that design selection could potentially be impacted by one s risk threshold. It can be seen that the 39

40 mean cost for a JPCP design is $2.21mm per mile, while the mean cost associated with a HMA design is $2.72mm per mile. From a risk-threshold perspective, one can say with 95% certainty that the JPCP design would be $2.87mm per mile, while the equivalent metric for HMA design is not included on the graph due to the asymmetry of the distribution. This discrepancy in the distribution, as compared to the Florida data, can be attributed to the difference in standard errors in the regression data. Specifically, the average standard error in the HMA design was approximately 43% higher in the Colorado data than it was in the Florida data. Cumula1ve Probability 100% 80% 60% 40% 20% 0% Asphalt Concrete Life Cycle Cost (millions of $ s per mile) Figure 14: Colorado probabilistic life-cycle cost of urban interstate road alternatives (black line represents HMA design, gray represents JPCP design). The dashed lines represent the mean value from the analysis These data have successfully classified various uncertainties relevant to highway pavement construction projects. Additionally, the results have been presented in a case study framework that will be easy to interpret for the average pavement LCCA practitioner. It was this case study that found that the JPCP design was a cost effective 40

41 alternative to HMA for all risk thresholds, and while data varied between Florida and Colorado, both states showed JPCP as a superior alternative (both in the deterministic and the probabilistic case). Of course there are more factors that play a role in pavement design selection, including government ordinances, environmental implications, surrounding environment, single project as opposed to highway network perspective, and more. These results will provide practitioners with a cost analysis that will prove useful in the larger selection process. 41

42 CONCLUSIONS AND FUTURE WORK This research has validated out-of-sample, long-term (i.e. multiple decades) forecasting as a viable methodology for predicting future materials prices. Moreover, economies of scale have been quantitatively classified as a driver of variability in bid prices for materials-intensive activities. By collating results and implementing the MEPDG framework, this paper has assessed the relative merits of a probabilistic LCCA model. In so doing, this research has established the shortcomings of the deterministic LCCA approach, and has established the importance of characterizing uncertainties and risks in any project involving materials selection and investment. One of the limitations of this research is that the forecasts neglected a confidence interval. Further research into the reliability of such forecasts would prove valuable to augment the validation of backcasting. Furthermore, while state variation was briefly mentioned, it would prove valuable to look at additional drivers of uncertainty, such as seasonality, timing of construction (i.e. night vs. day), intrastate location (i.e. county or north vs. south), and more. Additional characterization of such inputs would inevitably lead to a more comprehensive understanding of the probabilistic implementation of the LCCA model for highway construction projects. 42