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## Material Information- Title:
- Risk of Pavement Warranties to Contractors
- Creator:
- GUO, KELU
- Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Cost estimates ( jstor )
Estimation methods ( jstor ) Forecasting models ( jstor ) Highways ( jstor ) Modeling ( jstor ) Parametric models ( jstor ) Pavements ( jstor ) Standard deviation ( jstor ) Time series models ( jstor ) Unit costs ( jstor ) City of Tallahassee ( local )
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- University of Florida
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- University of Florida
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- Copyright Kelu Guo. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 12/31/2010
- Resource Identifier:
- 659560839 ( OCLC )
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PAGE 1 RISK OF PAVEMENT WARRANTIES TO CONTRACTORS By KELU GUO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 PAGE 2 Copyright 2006 by Kelu Guo PAGE 3 To my family PAGE 4 iv ACKNOWLEDGMENTS I would like to take this oppor tunity to thank everyone w ho, directly or indirectly, helped me complete this disse rtation. First, I would like to thank my advisor, Dr. R. Edward Minchin, for all the support, encour agement, and guidance he provided during my course of study. Many thanks also go to the members in my committee. Dr. Ralph Ellis and Dr. Charles Glagola, with their ri ch experience in civil engineering, have provided me valuable guidance and suggestions in developing this dissertation. Dr. Xueli Liu has provided me a lot of support in statistics. My ideas for this dissertation would not have been realized without training from Dr . Andy Naranjo and Dr. Wayne Archer in business and finance. I would also like to thank all the staff of the Florida Department of Transportation who provided me valuable information for th is research. Especially, I would thank Mr. Steve Guy for providing me the pavement perf ormance survey data which were essential to this study. This work could not have been done without his help. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................xi ABSTRACT ....................................................................................................................xiv CHAPTER 1 INTRODUCTION........................................................................................................1 Background...................................................................................................................1 Research Objective.......................................................................................................4 Research Methodology.................................................................................................5 Report Organization......................................................................................................7 2 LITERATURE REVIEW.............................................................................................8 Guidelines for Implementing Pavement Warranties.....................................................8 Impacts of Warranties on Involved Parties...................................................................9 Evaluating Effectiveness of Pavement Warranties.....................................................11 Warranty Analysis in Other Industries.......................................................................15 Construction Risks Analysis.......................................................................................17 3 OVERVIEW OF HIGHWAY CO NSTRUCTION BUSINESS.................................20 Industry Overview......................................................................................................20 Construction Bidding..................................................................................................23 Warranty Specifications..............................................................................................24 4 THE CONCEPT OF WARRANTY RISK.................................................................28 Definition of Warranty Risk.......................................................................................28 Factors for Warranty Risk..........................................................................................30 Implication of the Concept to Contractors..................................................................32 Summary.....................................................................................................................33 PAGE 6 vi 5 ANALYSIS OF ASPHALT PAVEMENT PERFORMANCE DATA......................34 Types of Asphalt Pavement Distress..........................................................................34 Florida Flexible Pavement Condition Survey.............................................................37 Pavement Performance Data Collection.....................................................................40 Performance Data Analysis........................................................................................44 Summary.....................................................................................................................56 6 MODELING PAVEMENT DISRESSES...................................................................58 Two Categories of Pavement Distresses.....................................................................58 Basics of Lifetime Distribution Models.....................................................................61 Failure Data Collection fo r Category I Distresses......................................................63 Modeling Failure Time of Category I Distresses.......................................................65 Modeling Failure Time of Category II Distresses......................................................71 Modeling Crack Growth.............................................................................................81 Summary.....................................................................................................................89 7 MODELING CONSTRUCTI ON COST ESCALATION..........................................90 Introduction.................................................................................................................90 Rationale for Price Forecast Errors.............................................................................92 Basic Concepts of ARIMA Models............................................................................95 Date Collection...........................................................................................................97 The Box-Jenkins Approach......................................................................................100 Model Identification.................................................................................................101 Model Estimation......................................................................................................107 Model Verification....................................................................................................108 Forecasting................................................................................................................117 Summary...................................................................................................................123 8 ASSESSMENT OF WARRANTY RI SK USING SIMULATION.........................124 Assumptions.............................................................................................................124 Simulation Methodology..........................................................................................125 A Case Study: Florida Turnpike...............................................................................134 Value at Risk (VaR) Measure of Warranty Risk......................................................139 Risk Attribution........................................................................................................142 The Effect of Multiple Projects................................................................................144 Summary...................................................................................................................148 9 CONCLUSIONS AND RECOMMENDATIONS...................................................149 Conclusions...............................................................................................................149 Limitations................................................................................................................151 Recommendations for Future Research....................................................................153 PAGE 7 vii APPENDIX A LIST OF ROAD SECTIONS IN THE SAMPLE.....................................................155 B COMPUTER PRINTOUTS FOR P AVEMENT FAILURE MODELING..............161 C COMPUTER PRINTOUTS FOR TIME SERIES ANALYSIS...............................183 LIST OF REFERENCES.................................................................................................187 BIOGRAPHICAL SKETCH...........................................................................................191 PAGE 8 viii LIST OF TABLES Table page 2-1 Comparison of pavement performance, WisDOT....................................................12 2-2 Comparison of pavement costs, WisDOT................................................................12 2-3 Warranty cost, Colorado DOT.................................................................................14 3-1 Size of highway, street and bridge construction section..........................................21 3-2 Legal forms of construction firms............................................................................22 5-1 Cracking codes and rating deduction.......................................................................39 5-2 Example of data format fo r ride rating, by calendar year........................................41 5-3 Investigation of short-life sections...........................................................................43 5-4 Construction completion year of the sample sections..............................................43 5-5 Observed pavement lives for the sampled sections..................................................43 5-6 Example data format for ride rating, by pavement age............................................45 5-7 Basic statistics of rut depth, by pavement age.........................................................46 5-8 Basic statistics of ride number (RN), by pavement age...........................................49 5-9 Basic statistics of crack rating, by pavement age.....................................................53 6-1 Performance requirements for warra nted interstate asphalt pavement....................59 6-2 Failure data for Category I distresses.......................................................................65 6-3 Category I distress failure model screening.............................................................66 6-4 Least squares estimates for pavement failure models, Weibull...............................67 6-5 Maximum likelihood estimates for pa vement failure models, Weibull...................67 6-6 Cumulative failure rates for ride, least square estimates..........................................69 PAGE 9 ix 6-7 Cumulative failure rates for rutting, least square estimates.....................................69 6-8 Failure data for Category II distresses......................................................................72 6-9 Model screening for Category II distresses..............................................................73 6-10 Least squares estimates of failure models for Category II distresses.......................74 6-11 Maximum likelihood estimates of fail ure models for Category II distresses..........74 6-12 Cumulative probabilities for cracking failure..........................................................77 6-13 Cumulative probabilities for raveling failure...........................................................78 6-14 Cumulative probabilities for bleeding failure..........................................................78 6-15 Cumulative probabilities for delamination failure...................................................79 6-16 Empirical CDF of pe rcent pavement cracked..........................................................86 6-17 Lognormal distribution of percen t pavement affected by cracking.........................87 6-18 Chi square goodness-of-fit test for year 1................................................................88 6-19 Chi square goodness-of-fit test for year 2................................................................88 6-20 Chi square goodness-of-fit test for year 3................................................................88 6-21 Chi square goodness-of-fit test for year 4................................................................88 6-22 Chi square goodness-of-fit test for year 5................................................................89 7-1 AIC for first difference series of log-PPI...............................................................107 7-2 Estimation of parameters for models AR(1) and AR(4)........................................108 7-3 Ljung-Box Portmanteau st atistic for AR(1) model................................................111 7-4 Ljung-Box Portmanteau st atistic for AR(4) model................................................113 7-5 Standard error of log-PPI forecasts........................................................................121 8-1 Model parameters for pavement failure/defect time simulation............................127 8-2 A sample probability table.....................................................................................130 8-3 Summary of the winning bid..................................................................................135 8-4 Project specific input data for simulation...............................................................136 PAGE 10 x 8-5 Assumed factors of repair ed area to distressed area...............................................136 8-6 Assumed distribution of size of distressed area.....................................................137 8-7 Expected values for risk factors.............................................................................142 8-8 Stand-alone risk fo r each risk factor.......................................................................143 8-9 Stand-alone risk fo r each distress type...................................................................144 8-10 Comparison of average repair costs for multiple projects......................................145 A-1 Road sections in the sample...................................................................................155 C-1 Augmented Dickey-Fuller (ADF) unit root test on ln(P).......................................183 PAGE 11 xi LIST OF FIGURES Figure page 3-1 Firm size by number of employees..........................................................................20 3-2 Size of highway construction firms, by value of business done..............................22 5-1 Distribution of lengths of sample sections...............................................................44 5-2 Boxplots of rut depths, by pavement age.................................................................46 5-3 Histograms of rut depth, by pavement age...............................................................47 5-4 Boxplots of ride rating, by pavement age................................................................48 5-5 Boxplots of ride number (RN), by pavement age....................................................49 5-6 Histograms of ride number (RN), by pavement age................................................50 5-7 Percent sections with reported raveling, by pavement age......................................51 5-8 Percent sections with reported patching, by pavement age......................................52 5-9 Boxplots of crack rating, by pavement age..............................................................53 5-10 Histograms of crack rating, by pavement age..........................................................54 5-11 Percent sections with reported bleeding/delamination, by pavement age................55 6-1 Probability plot for rutting failure, least square, Weibull........................................68 6-2 Probability plot for ride failure, least square, Weibull.............................................68 6-3 Cumulative failure plot fo r rutting failure, Weibull.................................................70 6-4 Cumulative failure plots for ride failure, Weibull....................................................70 6-5 Probability plot for cracki ng failure model, loglogistic...........................................75 6-6 Probability plot for raveli ng failure model, lognormal............................................76 6-7 Probability plot for bleedi ng failure model, lognormal............................................76 PAGE 12 xii 6-8 Probability plot for delami nation failure model, lognormal.....................................77 6-9 Cumulative defect probab ility plot for cracking......................................................79 6-10 Cumulative defect probab ility plot for raveling.......................................................80 6-11 Cumulative defect probab ility plot for bleeding......................................................80 6-12 Cumulative defect probabili ty plot for delamination...............................................81 6-13 Distributions of crack rati ng after cracking initialization........................................83 6-14 Estimating percent paveme nt affected by cracking..................................................84 6-15 Distribution in size of crack ed area after crack initiation........................................85 7-1 Time plot of highway and street construction PPI series.........................................99 7-2 Time plot of log-PPI...............................................................................................102 7-3 Autocorrelation function (ACF) for log-PPI..........................................................103 7-4 Partial autocorrelation func tion (PACF) for log-PPI............................................103 7-5 Time plot of firs t difference of log-PPI..................................................................105 7-6 Autocorrelation function (ACF) for log-inflation..................................................106 7-7 Partial autocorrelation functi on (PACF) for log-inflation......................................106 7-8 Residual plot of l og-inflation, ARIMA(1,1,0).......................................................109 7-9 ACF of residuals of log-inflation, ARIMA(1,1,0).................................................109 7-10 PACF of residuals of log-inflation, ARIMA(1).....................................................110 7-11 Residual plot of l og-inflation, ARIMA(4,1,0).......................................................111 7-12 ACF of residuals of loginflation ARIMA(4,1,0) model.......................................112 7-13 PACF of residuals of log-inflation ARIMA(4,1,0) model.....................................112 7-14 ACF of out-of-sample 1-step-forecast errors, ARIMA(1,1,0)...............................114 7-15 PACF of out-of-sample 1-st ep-forecast errors, ARIMA(1,1,0).............................114 7-16 ACF of out-of-sample 1-step-forecast errors, ARIMA(4,1,0)...............................116 7-17 PACF of out-of-sample 1-st ep-forecast errors, ARIMA(4,1,0).............................116 PAGE 13 xiii 7-18 Out-of-sample forecasts of log-inflation................................................................118 7-19 Out-of-sample forecasts of logPPI with 95% confidence interval........................119 7-20 Out-of-sample forecasts of PPI..............................................................................122 8-1 Remedial cost simulation flowchart.......................................................................126 8-2 Histogram of repair costs for the warranted pavement..........................................138 8-3 Cumulative distribution functi on for PV of repair costs........................................138 8-4 Probability distributi on of warranty gain/loss........................................................140 8-5 VaR measure of warranty risk................................................................................141 8-6 Histogram of average repair costs for two warranted projects...............................146 8-7 Histogram of average repair co sts for four warranted projects..............................146 8-8 Histogram of average repair costs for ten warranted projects................................147 PAGE 14 xiv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RISK OF PAVEMENT WARRANTIES TO CONTRACTORS By Kelu Guo December 2006 Chair: Robert Edward Minchin, Jr. Major: Civil Engineering The federal and state Departments of Tr ansportation are currently implementing warranties in highway construction projects with objectives to improve construction quality and encourage contractor innovation. The contractors reac t by increasing their bids to compensate for the additional risks that the warranties impose on their businesses. Risks have long been recognized and studied in the construction industry. However, a systematic study on risk of construction warranties basically does not exist. Many research efforts have been ma de to develop guidelines for implementing warranties and to evaluate their effectiveness. But all the studies are conducted from the ownerâ€™s point of view. No guideline has been developed to guide c ontractors through the warranty process. Survey results indicate that most contractors are not familiar with warranties. The objective of this research is to define a concept of warranty risk and develop an approach to measure the contractorâ€™s business risk exposure to pavement warranties. This PAGE 15 xv research is conducted from the contractorâ€™s pe rspective and begins w ith a review of the highway construction business. The risk of warr anties to the contractor is defined as the probability that actual warranty period repair co sts will exceed the estimated repair costs. Three factors for warranty risk are identifie d to be the uncertainty in repair work quantities, unexpected price escalation, a nd timing of repair. An investigation of historical pavement condition survey data is conducted to analy ze the uncertainty of pavement performance after construction. Life distribution models are fitted using the performance data to model the time for each t ype of distress to occur. Uncertainty in future price escalation is analyzed using th e historical highway construction cost index. The Box-Jenkins procedure is followed to bu ild a time series model for assessing future price uncertainty. The Monte Ca rlo simulation approach is a dopted to integrate all the models and produce a probability distribution for warranty ris k. Value at Risk (VaR) is used as a measure of warranty risk. A real case study is conducted to demonstrate and validate the simulation approach. PAGE 16 1 CHAPTER 1 INTRODUCTION Background Warranty Implementation in Highway Construction The use of warranties in highway construc tion has been accepted as an effective standard procedure in Europe but is still being evaluated as an innovative contracting practice in the United Stat es (Hancher 1994). Prior to 1991, the Federal Highway Administration (FHWA) had a l ongstanding policy that restrict ed the use of warranties on Federal-aid projects to electrical and mech anical equipment with the rationale of preventing Federal-aid funds from being used for maintenance costs (FHWA 2002). Warranty was first approved by the FHWA und er Special Experimental Project No. 14 (SEP-14) as an innovative contracting pract ice. Eight states participated in the evaluation of pavement warranties. Seventeen states evaluated warranty specifications (FHWA 2002). In 1991, the Intermodal Surface Transporta tion Efficiency Act (ISTEA) permitted states to use local procedures including warranty for Federalaid projects located off the National Highway System (NHS) (FHWA 2002). On August 25, 1995, the FHWA published an Interim Final Rule for warrantie s for projects on the NHS, which prohibits warranties for items not within the control of contractors. The warranty Final Rule was published on April 19, 1996. Warranty contracting is currently im plemented by state Departments of Transportation (DOTs) with objectives to reallocate performance risk, encourage PAGE 17 2 contractor innovation, increase product qualit y, and ultimately reduce the life cycle costs of highway projects (Anderson and Russell 2001). By 1999, at leas t 21 states had let about 240 highway construction projects with wa rranty provisions (Russell et al. 1999). It was recently estimated that more than half of the states have used warranties on highway construction projects (Bay raktar et al. 2004). Problems in Current Research Various studies have been conducted on the use of warranties in highway construction projects. Most of them focus on evaluating the effectiven ess of warranties or developing implementation guidelines. These studies provide valuable information for the DOTs to implement warranty contracting methods on highway construction projects. But they also have significant limitations. Warranty evaluation Many efforts have been made to evaluate the effectiveness of pavement warranties and their impact on the involve d parties, including state DOT s, contractors, and surety companies. The two major research approaches are 1. To compare the quality and co sts of warranted projects with those of nonwarranted projects. 2. To conduct questionnaire surveys or pe rsonal interviews with state DOTs, contractors, and surety companies. Surveys have provided clear re sults with regard to the c oncerns and reactions of the involved parties. Some typical survey ques tions for the contractors include the risk premium added to their bids, innovation impl emented, actual quality of the warranted projects, and ability to obtain warranty bonds , etc. The survey results can be objective and reliable, but also qualitative and intuit ive. Researchers usually provided a list of fragmentary survey results without furt her integration or explanation. PAGE 18 3 The comparisons of warranty project s with non-warranty projects provide diversified and sometimes contrary results (Bay raktar et al. 2004). Mo st states reported a bid price increase as a result of warranty clauses, but some st ates, such as Michigan, did not perceive significant bid price increases on warranted projects. Some states, like Wisconsin, reported significant improvement in quality, but others (Colorado) did not find any measurable difference; some states (Montana and Indiana) even reported lower performance on warranted projects. Besides the contrary results , the methodology of comparison is also questionable. A critical question is whether the warranted pr ojects and the non-wa rranted projects are actually comparable. Many factors may cause th e results of warranted projects to differ significantly from that of non-warranted targ ets. But they are usually ignored in the comparisons. Some of the f actors include the following: The warranted projects are usually selected from those with good site conditions to reduce the contractorâ€™s risk. Warranty specifications may set higher quality standards than non-warranty specifications. Due to the increased bonding requirements and liabilities on wa rranted projects, some contractors (usually small contractor s) are disadvantaged and reluctant to bid on warranted projects. As a result, large contractors are more likely to win the warranty contracts. Some bids on warranted projects were re jected because the bid prices were too high. Bid prices usually vary significantly am ong similar projects, whether or not under warranty. The observed difference in bi d price between warranted and nonwarranted projects may involve project specific variation. All the comparisons were conducted in a â€œt rialâ€ period when neith er the contractors nor the DOTs were experienced on warrantie s. Also, the motives of either party may cause biases in the results of comparison. PAGE 19 4 Due to the complexity inherent in the aforementioned research programs, it is difficult to reach a definite and reliable conc lusion on warranty evaluation via analysis of currently observed outcomes. Some other evaluation appr oaches might be needed. Guidelines for Implementing W arranties The Transportation Research Board (TRB) of the National Academies, the FHWA, and state DOTs have sponsored many research pr ojects to provide suggestions or develop guidelines and specifications for implementi ng warranty contracting. However, all these research projects are conducted from the pe rspective of the owners. No research was identified in the literature to provide guidel ines to the contractor s in bidding warranted projects. Current practices in highway wa rranties show that most contractors are inexperienced in warranty contra cting. One major concern of the contractors is the risk of warranties. However, few cont ractors understand the risk. Another important party i nvolved in highway construc tion projects is surety companies. It is widely known that surety companies are reluctan t to issue long-term warranty bonds due to the high underlying risks. Hastak et al. (2003) reported that surety companies currently do not have an appropria te procedure to underw rite warranty bonds because they do not understand the risks a ssociated with highway warranties. Research Objective Given the nature of current research on highway warranties, a systematic approach is needed to analyze highway constructi on warranties and assess their impacts on the involved parties, including state DOTs, contractors, and surety companies. The federal and state Departments of Tr ansportation aim for improved quality and, ultimately, reduced life cycle costs for warra nted projects. However, one major concern of the contractors and the surety companies is the risks associated with warranties. As a PAGE 20 5 result, the surety companies ask a high price for issuing warranty bonds, deny bonding applications from small contractors, and ar e reluctant to issue long-term warranty bonds. The contractors then submit a higher bid to tr ansfer the bonding costs to the DOTs and to compensate for the risks imposed on their businesses by warranty clauses. The reactions of the contractors and the surety companies to warranty clauses are driven by the risks associated with warranties. It is essent ial for them to understand the risk before accepting it. If the risks of wa rranties can be understood by all the parties involved, then the DOTs will be able to addr ess the warranty requirements appropriately to create a win-win situation. Unfortunately, the risks are perceived intuitively only. They have never been fully explored or even clearly defined. A comprehensive analysis of the risks associated with highway construction warranties is needed. This study tries to bridge the gap in research on highway construction warranties. In particular, the study will be conducted from the contractorâ€™s perspective, and analyze the risk of warran ties to the contractors. Since most of the current warranted highway projects are asphalt paving, the research is limited to asphalt pavement warranties. The obj ectives of this study are: to define a concept of warranty risk to the contractors, and to develop an approach to quantify th e risk of pavement warranties on the contractorsâ€™ business. Research Methodology This research was divided into six tasks. The first task was to conduct an extensive review of literature to identify the stateof-the-art research on pavement warranties a nd warranty risk analys is. The categories of literature review include PAGE 21 6 Guidelines for implementing pavement warranties Evaluation of pavement warranty effectiveness Risk analysis and modeling in the construction industry Warranty risk and cost anal ysis in other industries The second task was to develop a concept of warranty risk to the contractors and identify major elements of the risk. A revi ew of the constructi on business was conducted before the concept was developed. This re view included the indus try structure, the construction business process, and constr uction specifications including warranty specifications. Three factors of wa rranty risk were identified in this stage: uncertainty in quantities of repair items, timing of repair , and unexpected future unit repair cost escalation. The third task was to investigate the uncerta inty of pavement performance, which is directly related to repair works. The data us ed for this analysis are the asphalt pavement condition annual survey data for interstate projects in the state of Fl orida. Developing trends of various types of distresses ar e analyzed, including rutting, ride, cracking, raveling, and bleeding, etc. Further analysis of the results obtained in Task Three indicated that the results are biased estimates of pavement performance because failed pavements were excluded in later year analysis. Probability models were developed in Task Four using the performance results obtained in Task Three. These models, also called reliability or survival models, describe the probability di stribution of time when a certain type of distress occurs. The fifth task is to model the uncertainty of future unit cost escalation. Monthly data of the highway and street construction co st index, one of the PPI series, were used for this analysis. An ARIMA time series model was fitted in this stage. PAGE 22 7 The sixth and last task was to incorporate the models de veloped in Task Four and Five to obtain a profile of warranty risk. This was accomplished via Monte Carlo simulation. A case study with a real warranted project was performed to illustrate the simulation approach. Report Organization Chapter 2 summarizes prior studies by ot her researchers on highway construction warranties, construction risk management , and warranty risk analysis in the manufacturing industry. Chapter 3 briefly re views the highway construction business. Chapter 4 develops the concept of warranty risk with regard to cont ractors and identifies the major elements of the risk. Chapter 5 discusses the statistical aspects of asphalt pavement performance for interstate projects in the state of Florida. Chapter 6 explains in detail the modeling process and results of th e pavement distresses. Chapter 7 details the modeling of uncertainty in future price esca lation. The models developed in Chapters 6 and 7 are incorporated to reach a profile for warranty risk. This is accomplished via the Monte Carlo simulation and described in Chap ter 8. Chapter 9 closes the report with conclusions of the study and recommenda tions for future research. PAGE 23 8 CHAPTER 2 LITERATURE REVIEW This chapter documents previous studie s that may provide background information for this research. These studies can be divided into five groups: guidelines for implementing pavement warranties, impacts of pavement warranties on involved parties, evaluating warranty effectivene ss, construction risk analysis , and warranty risk analysis in the manufacturing industry. Guidelines for Implementi ng Pavement Warranties Anderson and Russell (2001) and Thompson et al. (2002) developed a process model to guide state DOTs in implementi ng warranty contracting method on highway construction projects. This proc ess model covers all the projec t delivery phases including conceptual planning, program planning, bidding, contract aw ard, construction, maintenance, and evaluation. Detailed guideline s were developed for each step, which the DOTs can follow in warranty implementation. Key elements of the guidelines include: Determine motivation for implementing warranties Review and understand industry best practice for warranty contracting Establish cooperation and communication betw een DOTs, contractors, sureties, etc. Prepare warranty specifications Select pilot projects Prepare bid documents Construction and project administration Evaluate effectiveness of warranties Review and refine the implementation process Anderson and Russell (2001) developed m odel warranty specifications for asphalt pavements. The key items addressed in the warranty specifications include PAGE 24 9 Length of warranty period Bonding requirements Maintenance responsibility Conflict resolution Contractor responsibilities Department responsibilities Performance indicators and threshold values Requirements for remedial actions Basis of payment The Transportation Research Board (TRB 2005) is currently sponsoring another research program to develop guidelines for a project-level applicati on of warranties. The guidelines will be able to assist state DOT s in determining whether warranties are the best option for a particular project. Warranty ev aluation criteria to be considered include initial construction costs, inspec tion and testing costs, life-cyc le costs, initial construction quality, and pavement performance. Impacts of Warranties on Involved Parties Hancher (1994) analyzed the impacts of warranties on DOTs, contractors, and surety companies. He concluded that the eff ect of warranties on bid prices is dependent on the contractorâ€™s knowledge of the conditions of the projects. If the contractor is unsure of project conditions, he or sh e may raise the bid price to co ver perceived risk. The risk involved with warranty contracting may exclude small contractors with weak financial conditions to obtain long-term warranty bonding. This may reduce the level of competition. Worischeck (2003) discussed local percepti ons of highway warranties in Utah. He believed that warranties could in crease quality and contractor awareness of their product and lower the risk of premature failure. But warranties are only as good as the contractor or the surety company. B onding capacities may limit the num ber of warranty projects PAGE 25 10 that the Utah DOT may have due to the lack of large contractors. The bid prices may be higher in the short term, but will fall when the contractors gain experience. Perceiving resistance from the contractors to warranty risk shift, Worischeck suggests that Utah DOT develop an asphalt pavement warranty as a long-term goal. Stephens et al. (1998) survey ed contractors and surety companies on various issues regarding pavement warranties. Findings of th e contractor survey include the following: The highway construction community in Montana has little experience with highway warranty contracts. Contractors will significantly increase their bids on warranted projects, in response to the shift of performance risk fr om the state to the contractors. Increase in initial costs may occur without substantial improvements in the quality of the projects. Small or medium contractors may find it di fficult to survive in a warranty market, due to their financial and bonding situation. The most favorable type of job for wa rranty is total reconstruction. For other projects, too many variables are beyond the control of the contractor to reasonably evaluate the contract orâ€™s performance. Major concerns of the surety companies include the following: The surety company has difficulty in estimating the financial condition of contractors several y ears in the future. Construction companies have the second highe st rate of bankruptcy of all types of business. Warranty bonding will impose great risk on surety companies. Retainage may provide more incentive for the contractor to do a good job than bonding. Hastak et al. (2003) surveyed all state DOT s, districts of Ohio DOT, contractors, and surety companies to identify the imp act of warranties on project cost, quality, bonding, time, etc. Initial bid price increase due to warranties was estimated to be between 0-15%, but there is no significant ch ange in maintenance cost and project life PAGE 26 11 cycle cost. Quantity improvement is not as sign ificant as bid price increase. Only slight quantity improvement is observed for warra nted projects. One common objective of pavement warranties is to encourage cont ractor innovation. Howeve r, it is found that innovative technologies and methods are not favored by contractors because of the associated risks. Evaluating Effectiveness of Pavement Warranties Anderson and Russell (2001) lis ted the important items that should be included in the evaluation of warranties, including The long-term performance of the project Personnel needs for design, testing, and inspection The use of DOT and outside expertise Risk distribution factors Total amount of claims and litigations Total project cost including cons truction and project administration The costs should be documente d to evaluate the life cycl e cost of the warranted project for comparison with those of traditi onal contracts. However evaluation of pilot projects may yield biased results if the pilo t projects are intentiona lly chosen with a high probability of success. Many state DOTs have participated in the evaluation of pavement warranties. The results vary significantly between states. Some of the results from selected states are summarized below. Wisconsin DOT (WisDOT 2001) reported its five-year experience with asphalt pavement warranties. Performance and cost da ta for warranted pavements were compared against the statewide average for projects of the same type and similar size. Based on the data collected, the warranted pavements perfor med better than typical pavements, as seen in Table 2-1. The costs of warranted projec ts were found to be lower than those of PAGE 27 12 standard projects, as seen in Table 2-2. It was also found that warra nted projects required less supervision and testing th an standard contract project s, and they reduced state construction delivery costs. Table 2-1. Comparison of pave ment performance, WisDOT International roughness index (IRI) Pa vement distress index (PDI) Pavement age Warranted State average Warranted State average New 0.81 1.11 0 0 1 year 0.87 1.17 1 5 2 year 0.89 1.29 2 11 3 year 0.89 1.33 6 16 4 year 0.94 1.37 12 21 5 year 0.94 1.45 9 26 Table 2-2. Comparison of pavement costs, WisDOT Period 1995-1999 2000 Standard contract Bid price (state average) Quality management State maintenance Total $25.05/ton $0.60/ton $2.07/ton $27.72/ton $28.58/ton $0.60/ton $2.07/ton $31.25/ton Warranty contract Bid price $24.34/ton $29.45/ton It should be noted that th e warranted projects used in this comparison were specially selected for adequate subgrade support. In addition, a few bids on warranted projects were rejected by Wisconsin DOT becau se the bids were significantly higher than the Engineerâ€™s estimates. Gallivan et al. (2003) evaluated the ef fectiveness of Indiana DOTâ€™s 5-year performance warranties. Performance of wa rranted asphalt pavements was compared with that of non-warranty interstate pavement s 4 to 6 years old. The data showed that warranted pavements had a lower and more consistent Intern ational Roughness Index (IRI). Less rutting was observed for warrant ed pavements; rut depths for warranted pavement were also less variab le. It was estimated that th e expected life of warranted pavements is 24 years years longer than non -warranted pavements. Initial bid prices PAGE 28 13 for warranted projects were about 5-10% higher than non-warranted projects. But Gallivan et al. (2003) predicted there would be a 27% cost saving in maintenance over 25 years. Ohio Department of Transportation ( ODOT 2001) observed a bid price increase on warranted projects. Bid prices for asphalt pa vements with 3-7 year warranties were 8.5% higher than similar non-warranty pavement. Bi d prices for PCC pavements with 7-year warranties were 11% higher than non-warra nted pavements. For pavement marking warranties, bid prices increased dram atically, about 90% on average. Johnson (2004) documented the observ ations on Minnesota DOTâ€™s asphalt warranty pilot projects. In all three warranted projects, the low bidderâ€™s unit prices for asphalt concrete were comparable to the averages of non-warranted projects. But the ranges of unit prices from all bidders were relatively large due to the fact that most contractors had no prior experi ence with pavement warranties. Feedback from the DOT project engineers indicated that there wa s no measurable difference in contractor behavior and material qu ality between warranted a nd non-warranted projects. Aschenbrener and Debios (2001) evaluated the cost-benefit of the Colorado 3-year asphalt pavement warranty projects. Each wa rranted project was compared with one or two control projects which used the traditional contract a pproach. The control projects were comparable to the warranty projects in terms of year of construction, overlay thickness, rehabilitation st rategy, traffic, and original pavement condition. The initial costs for warranty projects were compared in various ways. Subjective evaluation by members of the Cost Benef it Evaluation Committee concluded that the warranty cost was negligible. The contractor survey indicated that three contractors did PAGE 29 14 not consider any warranty cost in their bids; while another three cont ractors added a little additional cost for potential maintenance, bonding, and unknown risk. Warranty cost, though not explicitly define d, refers to the cost charged by the contractor to cover the potential maintenance wo rk cost, potential lane rental fees because of warranty work, the cost of a warranty bond, and the premium for the contractor to take the warranty risk. Four objective approaches were applied to calcula te warranty cost. The results are listed in Table 2-3. The estimated warranty cost, based on the average of the four approaches, was $-0.85 per ton, or -1.6% of bid price. Th is is an interesting result because it implies that the contractors prefer warranted projects and will lower their bid prices to take on the liabilities and risk inherent in warranty work. Table 2-3. Warranty cost, Colorado DOT Average of group 1 Average of group 2 Approach of Analysis $/ton % $/ton % Lump sum bid of warranty $1.68 5.1% Based on Engineerâ€™s estimate $1.23 3.9% Based on annual region avg. co st $-8.86 -20.4% $-1.29 -3.3% Based on control projects $0.81 2.6% $-1.13 -1.2% Based on avg. cost for all bidders $1.15 4.5% $0.43 2.2% Note: Each group consists of three warranty projects. Three-year maintenance costs were recorded for control projects. The average cost of maintenance was $7,753 per project. The main tenance cost is insi gnificant since the bid prices for the control projects range from, $3,472,988 to $4,634,123. It should be noted that before the 3-y ear warranty experiments, two warranty projects, one with a 5-year warranty a nd another with a 10-year warranty, were unsuccessfully bid in Colorado because the low bids were significantly higher than the Engineerâ€™s estimates. PAGE 30 15 Wienrank (2004) reported costs of paveme nt warranties in Illinois. Five-year warranties were implemented in Illinois fo r both bituminous and concrete pavements. The warranty was included as a separate bid item on each contract. The warranty pay item costs ranged from 0.0 to 0.43% of total pr oject cost for concrete pavement, 0.06% to 0.80% for bituminous pavements, and 1.14% to 2.38% for bituminous overlays. The data seems to indicate that the warranty cost is very low. But Wienrank (2004) claimed that the contractor might increase the cost of ot her pay items to hedge against the potential future cost for corrective work. Warranty Analysis in Other Industries Brennan (1994) developed detailed gu idelines for planning, analysis, and implantation of warranties in the consum er, commercial, and government business sectors. Methodologies of warra nty risk analysis and warrant y costing are highlighted as follows. Both supplier and customer assume risk in the warranty process. The supplierâ€™s major concerns of warranty risk are (a) to incl ude sufficient cost in th e sale price to cover the expected repair or replacement costs, and (b) to be competitive. The customerâ€™s major concern is the cost effectiveness of the wa rranty, or the value of the warranty to the customer. A Reliability Improvement Warranty (RIW) is an incentive warranty that has been used extensively by the government since th e 1970s, with the obj ectives of improving reliability and reducing support costs. Under a RIW, the contractor is paid a fixed price up front to perform repair services for an extended period of time. The potential risks of RIW to the government include High price for RIW coverage PAGE 31 16 Reduced self-sufficiency (tied in with one contractor for future repairs) Administrative complexity Potential transition of maintenan ce from contractor to government. The potential risks of RIW to the contractor include Possibility of large loss if the achieved reliability is lower than expected Fixed-price commitment with limited reliability data Pricing warranty too low (possi bility of loss on RIW option) Pricing warranty too high (po ssibility of losing contract) Reliability is an important consider ation in warranty requirements and implementation. The expected frequency of fa ilure is a critical parameter in warranty pricing. In addition, once the product is produced, the number of actual failures is essential in determining the achieved performance for performance guaranty determinations. The product being warranted can be eith er repairable or non-repairable. A repairable item can be restored to sa tisfactory operation by repair actions. A nonrepairable item will be removed permanently from the system when it fails. The underlying life distribution for a non-repairable product can be estimated using reliability models such as Weibull or lognormal distributio ns. The analysis of repairable systems is more complex than for non-repairable syst ems. The analysis can be performed as follows: If the time between failures does not exhibit a decreasing or increasing trend, then the system can be assumed to exhibit a re newal process. The product can be studied as if it were non-repairable. If there is a trend in failure time, a differ ent approach is used to model the repair rates, or Rate of Occurrence of Failures (ROCOF). If there is a trend in failure time and a detailed system analysis is desired, the system may be analyzed using a â€œb ottom-upâ€ approach which goes from component failure mode to system failure rate. PAGE 32 17 There are two major components for warrant y cost: warranty implementation cost and risk money. Warranty implementation cost is basically the cost needed for the contractor to perform the warranty work, in cluding repair, trans portation, administration, and field service. Risk money is a compen sation for the contractorâ€™s risk exposure associated with the warranty. Various approaches are available for form ulating a warranty cost-risk model. Some typical methods include Sensitivity analysis. Measure the amount of change in analysis results given a small change in an input parameter. Bounding technique. Estimate the limits asso ciated with each key cost driver and use these extremes to give the mi nimum and maximum warranty costs. Beta distribution approach. Estimate three different costs for each cost category: most likely cost, lowest cost, and highest cost. Monte Carlo simulation. Construction Risks Analysis The Construction Industry Institute (CII 1989) described the basic methodology of construction risk management. The risk mana gement approach includes three consecutive stages: risk identification, risk measurement, and risk control. Risk identification is the first step in risk management. The success of risk identification depends on the av ailability of historical info rmation, formalized checklists, and the experience of project personnel. The risks can be cataloged by source, such as technical uncertainties, contract ual risks, and financial risks. The CII also cataloged risks in terms of known, known-unknown, and unknow n-unknown situations or conditions. Risks from known conditions are the most co mmon risks in a project that have to be identified. Generally they involve a c ontinuous range of outcomes, have a relative PAGE 33 18 high frequency of occurrence, and low seve rity. Examples of known conditions include contract provisions, project schedule, quantities of work, site conditions, material and construction quality, labor productivity, etc. A detailed checklist should be used for risk identification to minimize the potential for overlooking some risk items. Risk from known-unknown conditions are ne ither explicit nor normally expected, but are foreseeable and possible. They genera lly tend to be discrete events, with a low frequency of occurrence and a high severity of impact when occurring. Examples of such events include extreme bad weather, extreme adverse labor activity, sudden labor shortages, or commodity s hortages in the project area. Known-unknown conditions are best identified through review of hist orical data on similar projects. Risks from unknown-unknown conditions are known as unforeseen risks. They cannot be identified in advance and th eir potential can only be acknowledged. Unforeseen risks normally have a low proba bility of occurrence, but have potential catastrophic effects. Risk measurement is the process to assess the potential loss associated with risks. However, risk measurements are usually difficult due to the following problems: There is usually a broad range of pot ential loss for each individual risk The potential losses for some risks are hard to estimate. Many risks are involved in the project. The CII listed four categories of met hods that are available for measuring construction risks, including Traditional methods that use allowa nces based on past experience Discrete event analysis including decision trees, influence diagrams, and utility theory PAGE 34 19 Analytic methods that use mathematics of probability to assess and combine the effects of individual risks into a an overall measure of risk Monte Carlo simulation Risk control is the last stage of risk ma nagement. There are two categories of risk control: advanced planning actions and in -process risk containment actions. The advanced planning actions are designed to pl ace risk exposure within controllable limits. Typical advance actions include risk avoidance, risk sharing, risk reduction, risk transfer to subcontractors, insurance, and risk accep tance with or without contingency. Risk containment actions are designed to reduce ac tual loss in the process of operation. A contingency account may be established and ma intained over the life of the project. PAGE 35 20 CHAPTER 3 OVERVIEW OF HIGHWAY CONSTRUCTION BUSINESS Industry Overview The construction industry is one of the la rge industries in the United States. In 2004, the construction industry accounted for 4.6% of the gross domestic product (GPD) and created a total added value of $541.4 billion (BEA 2005). In 2002, there were 710,307 construction firms nationally, with a total employment of 7.19 million (U.S. Census Bureau 2005). The construction industry is dominated by small firms. About 60% of the firms have less than five employees and 90% of the firms have less than 20 employees. Only 1.2% of the firms have 100 or more em ployees (U.S. Census Bureau 2005). The distribution of firm sizes by employment is illustrated in Figure 3-1. 0% 10% 20% 30% 40% 50% 60% 70% 1-45-910-1920-4950-99100-249259-499500-9991000+ Firm Size ( # em p lo y ee ) Percent of Total # firm s Construction Industry Highway, Street & Bridge Construction Figure 3-1. Firm size by number of employees PAGE 36 21 Highway, street, and bridge construction is a small se ction of the construction industry (see Table 3-1), with a total numb er of 11,348 firms and a total employment of 410,822 nationally in 2002 (U.S. Census Bureau 2004). Table 3-1. Size of highway, street and bridge construction section Construction Industry Highway, Street & Bridge construction (b) as % of (a) (a) (b) (c) Total employment 7,193,069410822 5.71% No. of firms 710,30711348 1.60% Net value of construction work $874,853,043,000$62,094,794,000 7.10% Source: US Census Bureau (2004) The highway, street, and bridge constructi on section is also dominated by small firms. But firm size in this section is relatively larger compared with the whole construction industry, as illustrated in Fi gure 3-1. Only 27.3% of total highway and bridge construction firms have less than 5 employees, compared with 59.4% of the whole construction industry. With 1.6% of the to tal number of firms in the construction industry, the highway, street a nd bridge construction secti on accounts for 5.7% of total employment and 7.1% of construction wo rk value, as shown in Table 3-1. In 2002, about 30% of highway construction firms did construction work less than $500,000 in value and 44.4% firms had construction business less than $1 million. Only 16% firms did construction work of $10 million or more in value. The distribution of business value is illustrated in Fi gure 3-2 (U.S. Census Bureau 2004). Of all highway, street, and bridge constr uction firms, 81.8% take the form of a corporation, higher than the 67.1% ratio for the whole construction industry (see Table 32). C corporation and S corpor ation are the two major cor porate forms for construction firms. Other legal forms include individual pr oprietorship and partners , etc. (U.S. Census Bureau 2005). PAGE 37 22 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% <$25K$25K$50K $50K$100k $100k250K $250K500k $500k$1M $1M$2.5M $2.5M$5M $5M$10M $10M+ Value business done% of total fir m Figure 3-2. Size of highway constructi on firms, by value of business done Table 3-2. Legal forms of construction firms Construction Industry Highway, Street & Bridge Construction Corporate 67.06% 81.69% Individual proprietorship 26.03% 10.97% Partners 5.88% 5.95% Other or unknown 1.03% 1.39% Total 100.00% 100.00% Source: U.S. Census Bureau (2005) The federal, state, and local government s collectively constitute an overwhelming majority in this section. In 2002, about 72.5% of the total value of roadway and bridge construction projects were owned by federal, state, and local governments. The other 27.5% were privately owned proj ects. So the roadway and bri dge construction business is highly controlled by the governments. PAGE 38 23 Construction Bidding The Competitive Low Bid System The two most common types of contract acquisition methods are competitive bidding and negotiation. In the private constructi on sector, the owner ha s great latitude in selecting the contractors, ranging from open bidding and in vited bidding to negotiation. In the public sector, however, construction co ntracts are generally awarded through open competitive bidding. All qualified contractor s can bid on any proposed project. For highway and street construction proj ects, the contractor â€™s bid is usually submitted on either a lump sum or unit price basis, as specified by the owner. The public owners, including federal and state DOTs, ar e generally required by law to award the contract to the lowest responsible bidder. Howe ver, the owner reserves the right to reject all the bids. Cost-based Construction Pricing The contractorâ€™s bid estimate consists of two basic elements: (1) direct construction costs including direct labor costs, material costs, equipment costs, and field supervision costs; and (2) the markup to cover gene ral overhead expenses and profit. The contractorâ€™s bid estimates are base d on the quantities for those construction items specified in the plans and specificati ons, which are furnished by the owner. The direct construction costs can be estimated using one or a combination of the following approaches (Hendrickson 2000): Subcontractor quotations. When a general contractor intends to use a specialty subcontractor, he or she may solicit price quotations for those work items to be subcontracted. Quantity takeoffs. The project is decomposed into several items. The quantities of the items are measured from the plans. The to tal cost of the project is calculated as the sum of products of the qua ntities and the associated unit prices of the items. PAGE 39 24 Construction procedures. If the actual construction procedures of a proposed project are considered, item s such as labor, material and equipment needed to perform various tasks are estimated and used for the cost estimates. The markup portion of the bid is the source of income to the contractor. A large markup means a higher profit. However, since only the bidder with the lowest price will win the contract, the markup is highly impact ed by bid competitions. In the construction industry, most contractors specialize in a subm arket of the industry and concentrate their work in particular geographic locations. So the level of demand in a submarket at a particular time can significantly influence the competition and ultimately the markups (Hendrickson 2000). The markups tend to be higher at times of low competition and lower at times of intense competition. The Construction Financial Management Association (CFMA 2003) reported that the na tional average gross margin for heavy and highway construction was 10.9% (or mar kup 12.2%) and the net margin was 3.2%. Warranty Specifications The traditional type of specifications for highway construction is referred to as method specifications, or prescriptive speci fications. A method specification specifies exactly the material, equipment, and construc tion method that the cont ractor is required to use in construction operat ions. Method specifications have significant limitations because they discourage contractor i nnovation and the owner assumes very high responsibility on the quality and perf ormance of the final product. To overcome the disadvantages of method sp ecifications and shif t responsibility for quality control to the contractors, DOTs adopt ed Quality Assurance (QA) specifications and performance specifications (i ncluding warranty specifications). Rather than specifying a method to constr uct a project, a warr anty specification defines the performance of the project. The c ontractor is bound by the warranty to repair PAGE 40 25 the product for a given time period after completion whenever certain performance criteria are not met. Accordingly, the contra ctor may have more freedom in selecting construction materials and methods. There are two basic types of warranties: the materials and workmanship warranty, and the performance warranty. Currently most highway and bridge warranties are for materials and workmanship. Under a materials and workmanship warranty, the contractor is responsible for correcting defects caused by poor materials or workmanship, which are within the contractorâ€™s contro l. The contractor assumes no re sponsibility for defects that are design-related or out of the control of the contractor. Under a performance warranty, however, the contractor is re sponsible for correcting all de fects even though the defects are attributable to factors out of his or her control. Some common elements of wa rranty specifications include Warranty term Performance criteria Remedial actions if perfor mance criteria are not met Warranty bond Conflict resolution Warranty term (or warranty period) is th e period of time following the completion of construction when the contractor is held accountable for repair and maintenance. Warranty terms are usually specified in calendar time. Some states, including New Mexico, Colorado, Ohio, Minnesota, and Florid a, use two dimensional terms, calendar time and traffic loads, whichever comes firs t (Hastak et al. 2003). The length of warranty terms varies by state as well as by project type. Typically, warranty terms range from 5 to 10 years for Portland cement concrete pavements and from 3 to 8 years for asphalt concrete pavements (FHWA 2000). New Mexico and Vi rginia also experienced 20-year pavement warranties for design-build projects. PAGE 41 26 Performance criteria are defined to eval uate the performance of the warranted product during the warranty term. The perf ormance criteria incl ude a selection of performance indicators and the associated threshold values for these indicators. Performance indicators, also ca lled performance parameters, are the types of distresses to be measured and evaluated. Remedial actions are needed if certain threshold values are exceeded. In the event that any of the performan ce criteria are not met during the warranty period, it must be determined whether or not the contractor is res ponsible for repairing the defects. If the contractor is held responsible for the defect, he/she shall perform the corrections at no cost to the ow ner. Approaches to remedial actions are usually stated in the warranty specifications. Bu t the contractor may have th e right to choose alternative approaches. A warranty bond is issued by a surety company which guarantees that the contractor will meet his/her warranty obligations. In case of contractor default on warranties, the surety company will be held liable for the required remedial works. Warranty bonds are required by most stat e DOTs that are implementing warranties. Florida is the only known state that has no warranty bonding requirements. The face values of warranty bonds are usually specifi ed as a fixed percen tage (ranging 10-100%) of the bid price or a percentage (up to 100%) of worst scenario repair costs. Disputes between the contractor and the owner arising out of warranties will be resolved through a conflict reso lution team which consists of representatives of the contractor and the owner. One major dispute between the contractor and the owner is the true cause and responsibility of product failures. Since the contractor is exempt from PAGE 42 27 repairing failures unrelated to his/her scope of work, the in ability to identify the true cause of the failures and the inclusion of method specifica tions within the scope of warranty may void the contractor â€™s obligation on warranties. PAGE 43 28 CHAPTER 4 THE CONCEPT OF WARRANTY RISK The federal and state DOTs are implementi ng pavement warranties with objectives to encourage contractor innovation and impr ove product quality. The contractors, on the other hand, react by increasing their bids to compensate for the additional risk imposed by warranty clauses. Risks have long been recognized and wellstudied in the construction industry. Many research efforts have been made to develop methodologies and guidelines in identifying, allocating, measuring, pricing, and controlling risks in the construction process. On the subject of highway (incl uding pavement) warranties, however, though the risks are widely recognized, they have not yet been well-analyzed, or even clearly defined. Beginning with this chapter, the risk of warranties will be analyzed from the contractorsâ€™ perspective by identifying th e risk of warranties to contractors and developing an approach to measure this risk. Definition of Warranty Risk What is Risk? Risk can be defined in different ways (CII 1989). It is generally defined as the probability that an adverse or unfavorable outcome may occur. Another widely accepted approach is to define risk in terms of un certainty. Since uncertainty is the set of all potential outcomes, both favorable and unfa vorable, it can be seen as a two-sided definition of risk, essentially including both risk (unfavorable outcomes) and opportunity (favorable outcomes). PAGE 44 29 Defining Warranty Risk Under a standard contract, the contractor is generally exempt from any liability on post-construction performance of the proj ect once it has been completed and accepted. Under a warranty contract, however, the cont ractor guarantees the performance of the product for a specified period of time and is re sponsible for the repair and replacement of any deficiencies. Pavement warranties are essentially a transf er of pavement failure risk as well as maintenance responsibility from the owner to the contractor. The contractor feels pavement warranties are risky because he/she has to, before the pavement is actually constructed, estimate and accept a fixed price fo r a highly uncertain obligation of future repair work. Performance warranties are popular for ma ny consumer products, which are usually produced in large quantities under identical and controlled conditions. Each highway project including pavement, however, is uniqu e with respect to factors such as site condition, design, materials, climate, and tr affic, all of which all may affect the performance of the highway af ter opening to traffic (Stephens et al. 2002). Various models have been developed to link highway performance to these factors. But these models by far provide the best estimates of th e average outcomes for all similar projects. No information is provided regarding how far the actual performa nce of a particular project can deviate from the estimated average. Thus it is difficult to exactly predict the actual performance of a highway after it is built. To develop a clear and definite concept fo r warranty risk, the following principles are considered and followed: The concept should reflect the contractorâ€™s major concerns about the warranties. PAGE 45 30 The concept should reflect the marginal e ffects of warranty clauses on construction risks. In other words, the warranty risk shoul d be or relate to th e additional risk that the warranty clauses add to the contractorâ€™s business. The concept should be developed at the pr oject level and should be able to be extended to a portfolio of proj ects or the corporate level. The risk to be defined should be measurable and quantifiable. The concept should be able to relate easil y to the contractorâ€™s decision regarding warranties, including bi dding and innovation. Considering the principles above, the risk of warranties to contra ctors is defined as the probability that the actual warranty cost will exceed the estimated warranty cost, or alternatively, the probability that the contract or will suffer a financial loss on warranties. It is important to distinguish between warrant y risk and warranty li ability. It is true that pavement warranties are risky to contra ctors because of the existence of warranty liability. But these concepts are not the same. As discussed in Chapter 3, contractor bid pr icing is cost-based. If the contractor is confident that he/she can predic t exactly his/her future warranty liability (the actual cost to repair), he/she can simply treat the liabil ity as a cost, add the amount to his/her bid and pass it on to the owner. In this case no contra ctor will feel the pressure of warranty risk. However, an accurate estimate of warranty liab ility (cost) is impossible, so the contractor is required to accept the liability in advance at a predetermine d price. This is the essence of warranty risk. A high level of warranty liab ility is not necessarily an indication of high warranty risk. Warranty risk is a result of the uncertainty in warranty liability. Factors for Warranty Risk Warranty specifications specify the types of distresses (def ects) that the contractor is responsible to repair. For example, unde r the FDOT (2005) asphalt pavement warranty PAGE 46 31 specifications, the types of distre sses that the contractor is responsible to repair include rutting, ride, cracking, raveling, delaminati on, bleeding, pot holes, and slippage areas. Assume the warranty term is W years and N types of pavement distresses are responsibility of the contractor. The current unit repair costs are C = Nc c c ...2 1, where ic is the unit repair cost for distress type i . The cost escalations are E = We e e ...2 1, where je is the cumulative cost escalation for jth warranty year. The quantities for repair items are Q = W N N N W Wq q q q q q q q q, 2 , 1 , , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1... ... ... ... ... ... ..., where j iq, is the quantity of type i distress repair at jth warranty year. Then the total re pair cost can be calculated as RC C Q Eâ€™ (4-1) where RC is the total repair cost, and Eâ€™ is the transpose of E . The present value of the repair costs can be calculated as N i W j j D j j i i RR e q c C11 , /) 1 ( (4-2) where /RC is the present value of repair cost, D is the construc tion duration, and R is the discount rate. The current unit repair costs are known at the time of bidding. The discount rate is a constant selected by the contractor. So in formula (4-2), ic and R can be assumed constant. But j iq, and je are unknown until they actually occur. An error in estimating either j iq, or je will result in an error in the estimate of warranty cost. If we assume the PAGE 47 32 quantities of repair items are uncorrelated to future unit cost escal ations. The error in estimate of repair cost can be expressed as N i W j j C j j i i RR e q c C11 , /0) 1 ( N i W j j C j j i iR e q c11 ,0) 1 ( (4-3) where /RC is the error in predicting total repair cost, j iq, is the error in predicting quantity for type idistress repair in thjwarranty year, and je is the error in predicting unit cost escalation for thjwarranty year repair. Thus three factors are identified for warranty risk: Uncertainty in quantities of future repair items Uncertainty in future unit repair cost escalations Timing of repair Timing of repair is included as a factor for warranty risk because it may affect the discounted total repair costs as well as the future unit price. Identification of factors for warranty risk is an important step in the an alysis of pavement wa rranty risk. To measure the risk of pavement warranty, an assessment of the uncertain natu re of its two risk elements is essential. This will be further discussed in later chapters. Implication of the Concept to Contractors The concept of warranty risk defines the contractorâ€™s business risk exposure to warranties. Understanding the risk is valuable in the analysis of the contractorâ€™s reaction to pavement warranties since it is an important factor in the contra ctorâ€™s decision-making process. One significant contribution of the concept is the partition of wa rranty cost, or the contractorâ€™s bid on warranty, into two parts: expected present value of future repair cost and risk premium. Expressively PAGE 48 33 RP C E Cr W ) (/ (4-4) where WC is the cost of warranty, ) (/rC E is the expected present value of future repair cost, and RP is the warranty risk premium. If warranty bonding is required and the bonding cost is incurred, formula (4-3) will be expanded to B RP C E Cr W ) (/ (4-5) where Bis the cost of th e warranty bond. The expected present value of future repair costs can be expressed as N i W j j C j j i i rR e E q E c C E11 , /0) 1 ( ) ( ) ( ) ( (4-6) where ) (, j iq E is the expected quantity for type idistress repair at the thjwarranty year, ) (je E is the expected unit cost escalation for the thjwarranty year repair. As a risk averse party, the contractor w ill increase his/her bid price to compensate for his/her increased business risk exposure. The risk premium is a compensation for the warranty risk that the contra ctor bears on a warranted proj ect. The premium is jointly determined by the level of warranty risk, the contractorâ€™s risk attitude, competition, and other factors. Summary In this chapter the concept of warranty risk is defined and the three factors of the risk â€“ uncertainty in quantities of future re pair items, uncertainty in future unit price escalations, and timing of repair â€“ are iden tified. In the next few chapters the two elements will be modeled and the warranty risk will be measured further. PAGE 49 34 CHAPTER 5 ANALYSIS OF ASPHALT PAVEMENT PERFORMANCE DATA One major element of warranty risk is the uncertainty in quantities of the future remedial work items. The actual quantities of remedial works will not be known until the repairs are actually performed or the warrant ies expire. However, the contractors are required to estimate the quantities and accept a fixed amount of cost for potential future remedial works at the time of bidding, duri ng which the pavement is not yet built. An accurate estimate of the quanti ties for future remedial works is difficult. But historical pavement performance data can be used to assess the uncertainty of future outcomes. The pavement condition surveys conducted by state Departments of Transportation provide a good source of pavement performan ce and distress data. In this chapter, the researcher analyzes the performance of asphalt pavements in the state of Florida using the pavement condition survey data . In particular, the objectiv e here is to analyze the variation as well as the mean le vels of various types of pave ment distresses at different pavement ages. This chapter will start with a description of the typical distresses for asphalt pavement, and then give a brief in troduction of the Florida Flexible Pavement Condition Survey program. The sampling proce ss and data analysis results will be discussed in detail. Types of Asphalt Pavement Distress Miller and Bellinger (2003) grouped aspha lt pavement distresses into five categories: Cracking PAGE 50 35 Patching and potholes Surface deformation Surface defects Miscellaneous distresses Cracking Cracking is the most common type of aspha lt pavement distress. It is further divided into several subtypes. Fatigue or alligator cracking is a series of interconnecting cracks caused by the fatigue failure of asphalt surface or stabilized ba se due to repeated tr affic loading. Cracks develop into many sided, sharp-angled pieces. Block cracking is caused mainly by the shrinkage of the asphalt and daily temperature cycling. The cracks divide the pavement surface into approximately rectangular pieces. Reflection cracking occurs in asphalt overlays over jointed concrete slabs. Cracks are caused by the movement of the slab beneath due to temperature and moisture changes. Longitudinal and transverse cracking. Cracks extend either parallel or transverse to the centerline of the pavement. Long itudinal cracks are generally related to construction defects while transv erse cracks are normally related to asphalt hardening. Patching and Potholes Patching is the replacement of a portion of the pavement surface after original construction. Potholes are bowl-shaped depressions in the ro ad surface, usually less than 3 feet in diameter. PAGE 51 36 Surface Deformation Two typical forms of surface deform ation are rutting and shoving. Rutting is the longitudinal depression in the wheel path. It can be caused by consolidation of one or more layers of the pavement. Shoving is a longitudinal displacement of th e pavement surface. It is usually located on hills or curves, or at inters ections, and generally caused by braking and accelerating vehicles. Surface Defects Three typical types of surface defects are raveling, bleeding, and polished aggregate. Raveling is the surface disintegration caused by th e loss of fine or coarse aggregate materials. Bleeding is usually found in the wheel path and is characterized by excess asphalt on the surface. The pavement may lose surf ace texture or form a shiny, glass-like, reflective, and tacky surface. Polished aggregate is the exposition of the aggreg ate caused by the wearing out of the asphalt surface binder. Miscellaneous Distresses Lane-to-shoulder drop-off is the difference in elevation between the travel lane and the outside shoulder. It is caused by the difference in settlement between the outside travel lane and the shoulder. Water bleeding and pumping is the seeping and ejecti on of water from beneath the pavement through cracks. PAGE 52 37 Roughness / Ride Quality Pavement roughness is an expression of i rregularities in the pavement surface that adversely affect the ride quality of a vehicle. Various statistical indexes have been de veloped to measure the roughness or ride quality of the road. These indexes include Present Serviceability Rating (PSR), Present Serviceability Index (PSI), International R oughness Index (IRI), Ride Number (RN), etc. PSR is a subjective rating given by panels of drivers and passengers who ride over sections of highways in passenger cars. PSI, on the other hand, estimates the pavement serviceability from objective physical measurements. IRI, now in common use, is a measurement of the cumulative vertical movement of the wheel divided by the distance traveled. It is a mathematical processing of the longitudinal profile generated by the pr ofiler, reported in units of m/km. Ride Number (RN) is also a mathematical processing of the longitudinal pavement profile generated by the profiler. The Ride Nu mber gives a pavement rating with a scale of 0 to 5. Florida Flexible Pavement Condition Survey The Florida Department of Transportati on (FDOT) conducts an annual pavement condition survey on state maintained roadway systems to evaluate surface distress and determine ride quality of the pavement. The survey program started in 1973. Currently both asphalt pavements and concrete pavement s are surveyed. In the 2006 survey year, 18,251.53 miles of asphalt pavements and 364. 39 miles of concrete pavements are surveyed, representing 44.7% of total asphalt pa vement miles and 36.7% of total concrete pavement miles (FDOT 2006). PAGE 53 38 Pavements under survey are divided into se ctions. Surface distre ss and ride quality items for each section evaluated under the Flexible Pavement Condition Survey Program include (FDOT 2003): Ride quality Cracking Rut depth Patching Raveling Ride Quality Both Ride Number (RN) and Internati onal Roughness Index (IR I) are calculated from the profiler data and reported as the av erage of the left and right wheel paths. Before 1999, the Ride Rating is calculated using IRI as IRI Rating Ride 1569 . 0 7576 . 99 Currently, the Ride Rating is ca lculated on a 0 to 100 scale as 20 RN Rating Ride Rutting Rut depths are measured by the profiler at highway speed. Manual rut depths are required only if the section cannot be survey ed by the profiler. Average rut depth is reported for each section. One point is deducted for each 1/8 inch of average rut depth. The Rut Rating, which is on a scale from 0 to 10, is obta ined by subtracting from 10 the deduct points associated with the rut depth. Cracking Cracks are grouped into three classes: Class IB. Hairline cracks that are no more than 1/8 inch wide in either the longitudinal or transverse direction. PAGE 54 39 Class II. Cracks 1/8 inch to 1/4 inch wide in either th e longitudinal or transverse direction. Also include alligator crack s that are less than 1/4 inch wide. Class III. Cracks greater than 1/4 inch wi de and cracks that are opened to the base or underlying material s. Also includes progressive class II cracks that result in severe spalling with chunks of pavement breaking out. Raveling and patching are also considered as Class III cracking. Cracks confined to wheel path (CW) and out side of wheel path (CO) are estimated separately. Square feet of the three classe s of cracks are added and recorded as the predominate type presented. Table 5-1 lists the codes for percentage. Cracking rating deduct values are also included in Table 51. Crack Rating is obt ained by subtracting from 10 the deduct values for both CW and CO. Table 5-1. Cracking codes and rating deduction Predominate cracking class Class IB Class II Class III Percent of pavement area affected by cracking Code CW deduct CO deduct CodeCW deduct CO deduct Code CW deduct CO deduct 00 â€“ 05 A 0.0 0.0 E 0.5 0.0 I 1.0 0.0 06 â€“ 25 B 1.0 0.5 F 2.0 1.0 J 2.5 1.0 26 50 C 2.0 1.0 G 3.0 1.5 K 4.5 2.0 51 + D 3.5 1.5 H 5.0 2.0 L 7.0 3.0 Source: FDOT (2003) Patching Patching is classified based on size of area as Light patching: less than 50 squa re feet per 100 feet of lane Moderate patching: 50 â€“ 100 square feet per 100 feet of lane Severe patching: more than 100 s quare feet per 100 feet of lane Patching is totaled with class III cracking. Raveling Patching is classified based on severity as Light raveling: The aggregate and/or bi nder has begun to wear away but has not progressed significantly. PAGE 55 40 Moderate raveling: the aggregate and/or binder has worn away and the surface texture becomes rough and pitted. Severe raveling: the aggregate and/or binde r has worn away and the surface texture is very rough and pitted. Raveling is totaled with class III cracking. Pavement Performance Data Collection The Pavement Condition Survey Database The asphalt pavement condition survey da ta are stored in Florida DOTâ€™s data library in the form of permanent flat file a nd area combined file. The permanent flat files are named as D5580954.FLEXxx.DATA , where xx is the year of survey. The area combined files are named as D5580954.FLEXxx.AREACOMB . The information stored in the two types of files is the same, although the data are coded differently. The coding of the data is not consistent over time due to the changes in survey methods, surveyed items, and recorded items. E ach file, either flat file or area combined file, consists of fixed-length records for su rvey results of each s ection in a given year. Data recorded for each survey section includes: Survey date. Roadway data, including roadway ID, roadwa y category, pavement type, number of lanes, speed. Location of the survey section, including direction, start and end mile mileposts. Performance and distress coding, including severity of raveling, cracking codes, patching severity, rutting de pth, IRI, RN, etc. Performance rating scores, including crack rating, ride rati ng, and rut rating. Remarks and other information. PAGE 56 41 Data Collection Procedure The source chosen for collecting aspha lt pavement performance data is the permanent flat files that FDOT keeps for fl exible pavement survey data. Performance data for interstates asphalt pavements are intented to be collected for analysis. The following procedures are followed to collect the data: 1. Retrieve the permanent flat files from F DOTâ€™s data library. This was done by staff personnel of FDOT upon request. The file s for survey data of 1991 to 2006 were obtained. Each file contains the survey data of all survey sections for a given year. 2. Convert the text format flat files into Microsoft Excel worksheets. Each worksheet contains the survey data of all survey sections for a given year. 3. Filter the survey data in the worksheet by the column of â€œSystem.â€ A subset of the data is selected for interstates (System = 4). 4. In the data subset, identify survey secti ons that are recorded as â€œnew pavementâ€ (Type = 7) in each year from 1992 to 1999. This is a complete list of interstate survey sections that were paved between 1991 and 1998. A section ID was assigned to each of the identified sec tions, called tentative candidates. 5. New worksheets were created for the tenta tive candidates based on type of data. Each new worksheet lists only one type of performance data (such as crack rating) for each tentative candidate from 1992 to 2006, as seen in Table 5-2. Those tentative candidates that merged with a la rger section or changed size significantly are eliminated from the list. Table 5-2. Example of data format for ride rating, by calendar year Year ID 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 A01 76 83 83 78 79 84 78 81 82 80 82 81 80 81 A02 82 85 85 79 81 87 77 82 83 82 81 82 82 81 A03 82 85 80 82 88 87 84 89 89 89 88 87 79 78 76 A04 81 82 80 80 85 82 81 89 88 88 88 87 79 76 76 A05 80 86 80 79 90 87 84 88 87 82 88 87 81 74 77 A06 79 82 77 80 88 84 81 90 88 90 89 88 82 81 81 B01 80 79 77 82 83 80 84 87 85 82 80 67 85 B02 80 76 77 83 78 79 86 85 85 83 83 71 78 B03 88 79 78 82 83 80 84 87 85 85 85 79 80 C01 79 80 83 78 83 87 86 87 86 84 74 76 72 C02 84 85 83 79 83 90 90 90 88 85 77 78 74 PAGE 57 42 6. Identify the service life of the pavement for each tentative candidate. The pavement service life of a tentative ca ndidate ended when it was late r recorded as either â€œnew pavementâ€ or â€œunder construction.â€ 7. Finalize the list of candidate sections. T hose tentative candidates with a pavement life of four years or more are selected as final candidates for further analysis. The criterion of four years minimum life fo r candidates is set because the current warranty term in Florida is three years. A total of 232 survey sections are included in the final candidate list. The Discarded Tentative Candidates The primary objective in this research is to investigate the variation of pavement performance or distresses after construction. T hus it is critical to avoid biases caused in the process of sampling. A total of 30 tentativ e candidate sections ar e eliminated from the final list because their actual service lives ar e less than three years. It is important to examine why these sections had a short record ed pavement life. In other words, for the sake of proper scientific research, it is important that no data were intentionally eliminated from the sample due to performance. A preliminary investigation of the eliminat ed tentative candidates is summarized in Table 5-3. Of the 30 survey sections with a recorded pavement life less than four years, 28 sections had no recorded pavement distre ss in their service lives. Light patching and light raveling were reported for two sections in the second year, but the recorded distresses are not severe enough to cause the fa ilure of pavement in two years. Based on the above investigation, it is reasonable to assume that no early failure sections are intentionally eliminated from the sample and the sampling process will not cause observable bias in the final results. The Sample A total of 232 survey sections, with a to tal length of 1249.3 m iles, are included in the sample, accounting for 46.5% of the total nu mber of sections and 46.4% of the total PAGE 58 43 length of all the survey sectio ns for interstate asphalt pave ments in 2006. A complete list of the sections in the sample is shown in Appendix A. The sample covers roadway sections from all seven districts of the F DOT, and from 38 of the 41 counties that have interstate asphalt survey sections in 2006. The pavements for these sample sections are constructed from 1991 to 1998, as seen in Tabl e 5-4. Lengths of the sections range from 0.075 to 25.462 miles, with an average of 5.385 miles. Figure 5-1 illustrates the distribution of section lengt hs within the sample. Table 5-3. Investigation of short-life sections No. of sections Reason for ending pavement life Performance 10 Merged with other section under construction Good 10 New construction, adding lane Good 4 Recorded as new pavement in two consecutive years Good 2 No survey data after two years Good 4 Under construction after two years, reason for construction not identified Two sections have recorded light raveling and light patching Table 5-4. Construction completi on year of the sample sections Year of completion 1991 1992 1993 1994 1995 1996 1997 1998 Total No. of sections 28 15 23 25 40 40 33 28 232 The observed pavement lives, or the tim e between two consequent construction efforts, for the sampled sections range from 4 to 15+ years. As seen in Table 5-5, a total of 60 sections, or 25.6% of the total sections in the sample, experienced another construction effort within seven years. Table 5-5. Observed pavement lives for the sampled sections Pavement life 4 5 6 7 8+ Total No. of sections 1 16 10 23 182 232 PAGE 59 44 Length of section (miles)No. of sections 26 24 22 20 18 16 14 12 10 8 6 4 2 0 50 40 30 20 10 0 Figure 5-1. Distribution of lengths of sample sections Performance Data Analysis This section will summarize the statistical aspects of the asphalt pavement performance and distress data for interstates in the state of Florida. The types of distresses analyzed include rutting, ride qua lity, raveling, patching, cracking, bleeding, and delamination. In particular, variation as we ll as the mean level of each distress type at different pavement age is of interest. Data Reorganization The data prepared in the process of data collection were orga nized by section ID and calendar year, as seen in Table 5-2. Befo re analysis, the performance/distress data are reorganized by section ID and paveme nt age, as seen in Table 5-6. PAGE 60 45 Table 5-6. Example data format for ride rating, by pavement age Age ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A01 76 83 83 78 79 84 78 81 82 80 82 A02 82 85 85 79 81 87 77 82 83 82 81 A03 82 85 80 82 88 87 84 89 89 89 88 87 79 78 76 A04 81 82 80 80 85 82 81 89 88 88 88 87 79 76 76 A05 80 86 80 79 90 87 84 88 87 82 88 87 81 74 77 A06 79 82 77 80 88 84 81 90 88 90 89 88 82 81 81 B01 80 79 77 82 83 80 84 87 85 82 80 67 B02 80 76 77 83 78 79 86 85 85 83 83 71 B03 88 79 78 82 83 80 84 87 85 85 85 79 C01 79 80 83 78 83 87 86 87 86 84 74 76 72 C02 84 85 83 79 83 90 90 90 88 85 77 78 74 Rutting Rut depths have been included in the surv ey since 1993. So for the sections paved in 1991, with section IDâ€™s from A01 to A30, there is no first year rut depth data. Rut depths are measured using a ro ad profiler at the highway speeds. An average rut depth is calculated for each survey section. The averag e rut depth is further converted to a rut rating score. As described in Table 5-7 and illustrate d in Figure 5-2, the mean level of the sample rut depths increase gra dually from 0.06 inch in the firs t year to 0.174 inch in the tenth year. The standard de viation and the interquartile range (IQR, the difference between the first and third quartile) of rut dept hs are relatively stable over time. However, the range becomes wider as pavement age in creases until age 7; after then the range narrows down because some sections with large rut depths were out of service. The rut depth data are positively skewed at all ages, with a longer higher tail. This is evidenced by all the positive skewness values in Table 5-7 and the large number of outliers at the higher end. Figure 5-3 illustrate s the distribution of rut depths at various ages. PAGE 61 46 The blue dashed line in Figure 5-2 and the red dotted lines in Figure 5-3 represent the rut depth threshold value of 0.25 inch fo r Florida interstate pavements under current 3-year asphalt pavement warranties. If th e rut depth data are evaluated using the threshold value, two sections in the first year , nine sections in the second year, and eleven sections in the third year will be identified as having a rutting problem. Table 5-7. Basic statistics of rut depth, by pavement age Age N Mean StDev Min Q1 Q2 Q3 MaxIQR Skew Kurtosis 1 204 0.061 0.060 0.00 0.01 0.05 0.10 0.30 0.09 1.16 1.48 2 232 0.090 0.069 0.00 0.04 0.08 0.12 0.33 0.08 1.09 1.31 3 232 0.101 0.080 0.00 0.05 0.09 0.14 0.42 0.09 1.43 2.94 4 230 0.117 0.082 0.00 0.07 0.10 0.15 0.50 0.08 1.52 3.72 5 231 0.132 0.091 0.00 0.07 0.12 0.17 0.47 0.10 1.19 2.09 6 215 0.138 0.091 0.00 0.08 0.12 0.17 0.48 0.09 1.39 2.81 7 205 0.152 0.083 0.00 0.10 0.14 0.19 0.57 0.09 1.47 4.62 8 182 0.161 0.075 0.00 0.12 0.15 0.20 0.49 0.08 1.03 2.12 9 149 0.165 0.071 0.00 0.11 0.16 0.20 0.35 0.09 0.43 -0.09 10 118 0.174 0.072 0.02 0.12 0.16 0.22 0.36 0.10 0.54 -0.09 Pavement AgeRut Depth (inch) 10 9 8 7 6 5 4 3 2 1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.25 Figure 5-2. Boxplots of rut depths, by pave ment age. The red dotted line is the mean connection line. PAGE 62 47 Rut Depth (in)Percent 40 30 20 10 0 40 30 20 10 0 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 40 30 20 10 0 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 10 Figure 5-3. Histograms of rut depth, by pavement age PAGE 63 48 Ride Quality Both ride number (RN) and Internationa l Roughness Index (IRI) are reported in the pavement condition survey. Before 1999, Ride Rating was calculated using IRI. Ride Numbers were not available until 1999 when th e Ride Rating began to be calculated as RN multiplied by 20. The boxplots of ride rating and RN are illust rated in Figures 5-4 and 5-5. These two figures are very similar due to the linear re lationship between RN and Ride Rating. For the same reason, only RN will be discussed in this subsection. As described in Table 5-8 and illustrated in Figure 5-5, the mean of RN decreases gradually as pavement age increases. The spr eads of the data, including range, standard deviation, and IQR, however, become wider as pavement age increases. The RN data are negatively skewed at all paveme nt ages, with the longer tail at the lower side, as seen in Table 5-8 and Figure 5-5. Figure 5-6 illustrate s the distribution of RN at various ages. Pavement AgeRide Rating 10 9 8 7 6 5 4 3 2 1 95 90 85 80 75 70 65 60 55 Figure 5-4. Boxplots of ri de rating, by pavement age PAGE 64 49 Pavement AgeRN 10 9 8 7 6 5 4 3 2 1 5.0 4.5 4.0 3.5 3.0 2.5 3.7 Figure 5-5. Boxplots of ride number (RN), by pavement age. The red dotted line is the mean connection line. Table 5-8. Basic statistics of ride number (RN), by pavement age Age N Mean StDev Min Q1 Q2 Q3 Max IQR Skew Kurtosis 1 28 4.44 0.14 3.93 4.36 4.50 4.54 4.58 0.18 -1.96 4.73 2 61 4.47 0.10 4.09 4.43 4.48 4.55 4.62 0.12 -1.24 2.16 3 101 4.45 0.12 3.95 4.42 4.49 4.53 4.61 0.12 -1.77 3.88 4 141 4.42 0.15 3.54 4.38 4.45 4.51 4.61 0.14 -2.30 8.97 5 165 4.37 0.15 3.51 4.30 4.42 4.48 4.57 0.18 -2.17 7.91 6 172 4.31 0.19 3.07 4.19 4.36 4.45 4.55 0.26 -2.15 10.46 7 177 4.24 0.20 3.55 4.13 4.27 4.40 4.55 0.27 -1.18 2.00 8 182 4.15 0.27 2.82 4.05 4.18 4.35 4.57 0.30 -1.42 3.24 9 149 4.08 0.27 3.19 3.94 4.13 4.27 4.54 0.33 -0.79 0.53 10 118 4.04 0.26 3.41 3.89 4.11 4.23 4.48 0.34 -0.55 -0.35 The blue dashed line in Figure 5-5 and the red dotted lines in Figure 5-6 represent the RN threshold value of 3.7 for Florida interstate pavements under current 3-year asphalt pavement warranties. If the RN data are evaluated using the th reshold value, it is obvious that no observed RN is lower than the threshold value in the first three years. PAGE 65 50 Thus the risk of ride failure to the cont ractors is very low under current warranty specifications. R N Pe r cent 50 40 30 20 10 0 50 40 30 20 10 0 4 . 8 4 . 5 4 . 2 3 . 9 3 . 6 3 . 3 3 . 0 50 40 30 20 10 0 4 . 8 4 . 5 4 . 2 3 . 9 3 . 6 3 . 3 3 . 0 4 . 8 4 . 5 4 . 2 3 . 9 3 . 6 3 . 3 3 . 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 10 Figure 5-6. Histograms of ride number (RN), by pavement age PAGE 66 51 Raveling Raveling is evaluated at thr ee severity levels: light, moderate, and severe. Only the predominate level is coded in the survey. The percent of pavement area affected by raveling is recorded at four levels: 15%, 6-25%, 26-50%, and more than 50%. The raveling area of all severity levels for the rated section is accumulated in the total percentage of class III cracking. The percentage in number of road sections with reported raveling is illustrated in Figure 5-7. It can be seen th at a raveling problem is not co mmon at early pavement age. Only 1.3% of survey sections have reporte d raveling distress at the pavement age of three. However, 40.9% of sections have re ported raveling at age ni ne. Of all the road sections with raveling distress, in most cases the affected areas are within 5% of total pavement area. 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 12345678910 Pavement AgePercent sections with raveling 1-5% roadway 6-25% roadway 26-50% roadway Total Figure 5-7. Percent sections with reported raveling, by pavement age PAGE 67 52 Patching Patching is recorded at three severity leve ls: light, moderate, and severe. However, no moderate or severe patching is identified on any survey sections. The only recorded level is light patching, with less than 50 s quare feet patching per 100 feet of lane. The percentage in number of sections with re ported patching is illustrated in Figure 5-8. About 2.6% of sections have reported pa tching at the pavement age of three. 0 2 4 6 8 10 12 14 012345678910 Pavement AgePercent sections (%) Figure 5-8. Percent sections with reported patching, by pavement age Cracking Cracking is a common type of distress for asphalt pavements. Cracking confined to wheel paths (CW) and cracking outside of wheel paths (CO) are estimated and coded separately. Cracks are recorded in three classes: class IB, cl ass II, and class III. Raveling and patching are included in class III cracking. PAGE 68 53 Crack rating is a single parameter used to measure the overall cracking condition of a survey section. Crack rati ng is obtained by subtracting fr om 10.0 the deduct values for both CW and CO cracking. Distributions of crack rating at various pavement ages are described in Table 5-9 and illustra ted in Figures 5-9 and 5-10. Table 5-9. Basic statistics of crack rating, by pavement age Age N Mean StDev Min Q1 Q2 Q3 Max IQR Skew Kurtosis 1 232 9.99 0.099.010.010.010.010.00.0 -10.7 113.47 2 232 9.97 0.208.010.010.010.010.00.0 -7.39 58.58 3 232 9.86 0.566.510.010.010.010.00.0 -4.83 24.11 4 232 9.74 0.714.510.010.010.010.00.0 -4.15 20.39 5 231 9.47 0.974.59.010.010.010.01.0 -3.06 11.4 6 215 8.95 1.353.58.59.510.010.01.5 -1.95 4.5 7 205 8.53 1.573.58.09.010.010.02.0 -1.33 1.5 8 182 8.04 1.841.07.08.59.510.02.5 -1.43 2.55 9 149 7.48 1.841.06.57.59.010.02.5 -0.74 1.04 10 118 7.11 1.911.06.57.08.310.01.8 -0.72 0.9 Pavement AgeCrac k Rating 10 9 8 7 6 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 0 Figure 5-9. Boxplots of crack rating, by pavement age. The red dotted line is the mean connection line. PAGE 69 54 Crack RatingPe r cent 100 80 60 40 20 0 100 80 60 40 20 0 9 . 0 7 . 5 6 . 0 4 . 5 3 . 0 1 . 5 100 80 60 40 20 0 9 . 0 7 . 5 6 . 0 4 . 5 3 . 0 1 . 5 9 . 0 7 . 5 6 . 0 4 . 5 3 . 0 1 . 5 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 10 Figure 5-10. Histograms of crack rating, by pavement age PAGE 70 55 Cracking grows at an increasing speed. This is evidenced by the downward curvature of the mean connection line of Cr ack Rating in Figure 5-9. The spreads of crack ratings are also an increasing function of pavement age, as seen in Figure 5-10, with wider distribution for older pavements. The crack rating for first year pavement ranges from 9 to 10, with 99.1% sections ha ving a rating of 10. The range widens from 1 to 10 at the pavement age of 8. The distribut ion of Crack Rating is negatively skewed at all pavement ages, with a l onger tail at the lower side. Bleeding and Delamination Bleeding and delamination are not official survey items in the pavement condition survey. Instead, they are noted in Remarks. Th e percentage in number of survey sections with raveling and delamination distresse s is illustrated in Figure 5-11. 0 2 4 6 8 10 12 14 012345678910 Pavement AgePercent sections (%) Delamination Bleeding Figure 5-11. Percent secti ons with reported bleeding/de lamination, by pavement age PAGE 71 56 The results indicate that delamination rare ly occurs at early pavement ages. No survey section was identified to have dela mination problem by the pavement age of five. Bleeding, however, is likely to occur at early pavement ages. About 2.2% of survey sections in first year and 6.9% of secti ons in the third year had a bleeding problem. Limitations of the Survey Data The pavement condition survey data descri bed above provide valuable information on asphalt pavement performance and distress. However, there are some limitations on the data. First, the results above are conditional on su rvival. Only those â€œsurvivingâ€ sections are included in the analysis for a given pavement age. If a section is reconstructed due to poor performance, it is excluded from later year analysis. As a result, the overall performance of all sections appears bette r because early failed pavements are not counted. Thus the results discussed above are conditional rather than absolute. Second, cracking data did not provide accura te estimates of the percent pavement area affected by cracking. Instead of an estimated definite number, the percent pavement affected by cracking is coded by ranges of 0-5%, 6-25%, 26-50%, or more than 50%. Furthermore, raveling and patchi ng are included in cracking data. Third, some types of distress are not included in the survey. Bleeding and delamination are evaluated under the warranty specifications. But they are not evaluated in the pavement condition surveys. Summary Typical pavement distresses for interstate projects in Florida are analyzed in this chapter using FDOT Pavement Condition Surv ey data. The variation of performance among survey sections and the developing tre nds of performance over time are discussed. PAGE 72 57 The types of distresses analyzed in this ch apter include rutting, Ride Number (RN), cracking, raveling, patching, bleeding, and delamination. There are some limitations on the available data. First of all, the data are on a survival basis and the results are conditional rather than absolute. Another significant limitation is that the cracking data do not provide accurate estimates of the percent pavement area affected by cracking. In the next chapter, the distresses will be modeled and the conditional data will be transformed into â€œabsoluteâ€ results of pavement performance. PAGE 73 58 CHAPTER 6 MODELING PAVEMENT DISRESSES The last chapter discussed the statisti cal aspects of various types of asphalt pavement distresses. One major limitation of th e data is that the results are conditional on survival. That is to say, for each pavement age, only those pavements in service are included in analysis. This survival effect causes the performance da ta to appear better because the poor-performing sections were re moved from the sample when they were reconstructed. One objective of this research is to estim ate the distribution of the absolute, not conditional, performance of asphalt pavements at various ages. Fortunately, the reliability theory provides a powerful tool to handle the conditional data. In this chapter we will try to model pavement performance and distresses using reliability theory. The pavement is considered as a repairable system which will be repaired when defects are identified. The modeling process is divided into two steps: a) to model the time that a certain type of distress occurs; and b) to model the distributi on and growth of a dist ress type after its initiation. Two Categories of Pavement Distresses Definition of the Two Distress Categories Table 6-1 lists the performance parameters , or the types of distresses, and the associated threshold values for evaluati ng the performance of warranted asphalt pavements on interstates in Florida. Based on their characteristics, the distress types can PAGE 74 59 be classified into two broad categories, namely Category I dist resses and Category II distresses. Table 6-1. Performance requirements for warranted interstate asphalt pavement Type of distress Threshold value (0.1 lane mile) Remedial work Rutting Rut Depth>0.25 in Remove and replace the distressed LOT(s) to the full depth of all layers, and to the full lane width Ride RN<3.7 Remove and repla ce the friction course for the full length and the full lane width of the distressed LOT(s) Cracking Cumulative length of cracking >30 ft for cracking > 1/8 in Remove and replace the distressed LOT(s) to the full depth of all layers, and to the full lane width Individual length >= 10 ft Remove and replace the distressed area(s) to the distressed depth and th e full lane width, for the full distressed length plus 50' on each end Raveling and delamination Individual length < 10 ft Patch the distressed area(s) to the full distressed depth and a minimum surface area of 150% of each distressed area Pot holes and slippage areas Observation by Engineer Patch the distressed area(s) to the full distressed depth and a minimum surface area of 150% of each distressed area Bleeding Individual length >=10 ft and >=1 ft in width Patch the distressed area(s) to the full distressed depth and a minimum surface area of 150% of each distressed area Source: FDOT (2005) Category I distresses include rutting and ride quality. A parameter value of a Category I distress is a measure of the aver age condition (other th an the size of the defected area) of the distress type. For example, rut depth is reported as the average of all measured rut depths for a survey section. It is an indicator of the average rutting performance of the section. But no informa tion is provided on what proportion of the pavement has rutting problems. A Category I defect is observed only when its parameter value exceeds the threshold value. Otherw ise, it is considered defect-free. PAGE 75 60 Category II distresses include all the distress types in Ta ble 6-6 other than rutting and ride. In addition to the severity of the distress, the measure of a Category II distress type also includes the size of the distress, in terms of linear foot, s quare foot, or percent of pavement affected. A Categor y II defect is observable even if its size is very small. The measure of distress size is directly re lated to the quantity of remedial work, if required. A threshold value for the distress type, if it exists, defines the minimum quantity condition that triggers a remedial action. It is not necessarily an indicator of defect initiation. In Florida, the threshold values for Ca tegory I distresses (rut ting and riding) are defined in warranty specifications so that the probability of pavement failure due to rutting or riding is very low, as illustrated in Figures 5-2 and 5-5. But once the threshold value is exceeded, a large portion of, or proba bly the entire pavement may need to be repaired under warranty provision. So the contra ctor will suffer a large loss if a Category I distress is identified. The situation is different for Category II distresses. A Category II distress is observable even if its size is very sma ll. Accordingly, under warranty contract, the contractor may be required to fix the distresses when only a small portion of the pavement is affected. Modeling Methods for the Two Distress Categories A pavement can be considered as a repair able system which involves various types of defects (distresses). When a certain type of distress occurs, the pavement will be repaired and continue in service until the end of its life. Modeli ng of the repairable system includes two steps. The first is to m odel the time that each type of distress occurs. The second is to model the severity or the si ze of the distressed area after initiation. PAGE 76 61 A measure of a Category I distress does not include any inform ation on the size of distress affected areas. From data of Category I distress, we can tell if a pavement is defined as defective based on certain given criteria. But we donâ€™t know what portion of the pavement is defective. The sizes of Ca tegory I distresses can onl y be estimated from other data sources. Measures of Category II di stress include the size of th e distress affected areas, which is directly related to the quantity of repair works. After a Category II distress is identified, the values of its measure become a description of the growth process of the distress. For a Category I distress, we can only mode l its distress initiation time, or the time that the observed value exceeds the threshold value. For a Category II distress, both modeling steps are achieved if the distress data are available. However, given the limitation of the survey data, the distress growth process will be modeled only for cracking. Additional data are needed to model the growth processes of other distress types. Basics of Lifetime Distribution Models Life distribution models are used to desc ribe the initiation tim es for each type of distress. Basics of life distribution models are intr oduced in this section. Lifetime Distribution Models The probability models used to describe the distributions of product lifetime, or failure time, are known as lifetime distribut ion models. A life distribution model is defined over the range of time , 0 t. The cumulative distri bution function (CDF) ) ( t Fgives the probability that a product or unit will fail by time t (NIST 2006). PAGE 77 62 The probability that a product will survive beyond time t is given by the Reliability Function ) ( t R , also known as Survival Function ) ( t S . We have ) ( 1 ) ( ) ( t F t S t R (6-1) The failure rate, denoted as ) ( t h , is defined as the inst antaneous rate of product failure for the survivors at time t . It is sometimes called the conditional failure rate since only survivors to time t are considered. The failure rate can be calculated as ) ( ) ( ) ( 1 ) ( ) ( t R t f t F t f t h (6-2) A lifetime distribution model can be any probability distribu tion function defined over time range , 0. Some widely used distributi on models include Exponential, Weibull, Smallest Extreme Value, Lognormal, and Gamma, etc. Life Data The data used for lifetime distribution analysis are life data or time-to-failure data of the products or units in the sample. The data are said to be complete if the exact failure time for each sample unit is known. However, in many cases, life data are incomplete, or called censored. There are three types of possible censoring: right censoring, left censoring, and interval censoring. Right censoring is the most common type of censoring. The units that did not fail before the observation or test period ends are right censored. In this case, the exact failure time is unknown, but it is known to be on the right of, or after , the time point. In the case of our pavement distress analysis, a road sec tion is said to be right censored if no failure is observed before it was taken out from the sample. PAGE 78 63 Similarly, left censoring means that the failure time is only known to be before a certain time. In other words, the actual fa ilure time could be any time between zero and the censoring time. Interval censoring is the result of periodi cal observations in a test. The failure time is known to be within a time interval be tween two consecutive observations. For our pavement distress analysis, since the pa vement condition surveys were performed annually, the life data ar e interval censored. For example, if the failure of a road section was not observed in the 4th year survey but observed in the 5th year survey, we know that the road section failed at some time between the 4th year survey and the 5th year survey. Life distribution models are generally used to model failure times of non-repairable systems where failed units are removed from the population instead of being repaired. Life distribution models are also able to model defect initiation times for repairable systems. In such cases, the initia tion of defect is treated as fa ilure and the â€œfailedâ€ unit is removed from the sample, although the unit is ac tually repaired and still in service. To be consistent with the terminology used in reliability theory , we call the time an initial defect occurs: failure time. Accordingly, a pavement is said to fail when a certain defect is observed. But it should be noted that the concept of â€œfailure,â€ as used here, is completely different from the concept of â€œpavement failureâ€ used in practice which means the pavement is functionally in sufficient and needs to be replaced. Failure Data Collection for Category I Distresses Definition of Failure As discussed before, the concept of fa ilure used here is based on warranty specifications and for purposes of modeling only. A Category I distress failure occurs when, within the warranty period, the paramete r value of a distress exceeds its threshold PAGE 79 64 value specified in the warrant y specifications. Mo re specifically, under current warranty specifications, a rutting failure occurs when the average rut depth of a pavement is greater than 0.25 inches within the 3-year warranty period; a ride failure occurs when the average RN of a pavement is less than 3.70 within the 3-year warranty period. The concept of failure is defined within the warranty period. For purposes of life distribution modeling, the concept has to be extended to an infinite time limit. That is to say, a distress failure occurs whenev er the threshold value is exceeded. Data Collection Method The data source used to generate failure data are the Rut Depth and RN data, which were analyzed in the prior chapter. Rut Depth and RN data are each stored in a spreadsheet and organized by survey section and pavement age. The same data collection procedure is repeated for each distress type. The procedure starts from pavement age 1 and repeats for each pavement age. For rutting failure at each pavement age t , the following procedure is repeated: Find the survey sections in the sample with average Rut Depth > 0.25. These sections failed by rutting between age t-1 and t . Record the number of the failed sections for time interval ( t-1 , t ). Remove the failed sections from the sample. Count the survey sections that are out of service from th is age due to other reasons. These sections are right censored at time t-1 . Record the number of right-censored sections. Remove the right-censored sections from the sample. Move to the next pavement age and repeat the procedure, or stop if the sample size is zero. The data collection procedure for ride failure is the same as that for rutting failure, except the failure criterion changes to RN < 3.7. PAGE 80 65 The collected rutting and ride failur e data are listed in Table 6-2. Table 6-2. Failure data for Category I distresses Time interval Start End Rutting Ride 0 1 2 0 1 2 7 0 2 3 4 0 3 4 3 1 4 5 4 1 5 6 6 0 6 7 3 4 7 8 10 7 8 9 6 9 9 10 7 11 10 11 2 4 11 12 3 6 4 * 1 1 5 * 11 16 6 * 6 10 7 * 18 23 8 * 21 27 9 * 26 22 10 * 30 25 11 * 32 32 12 * 30 33 Note: * in the column of End means right censoring. Modeling Failure Time of Category I Distresses Failure time modeling is performed in MiniTA B using the failure data in Table 6-2. The modeling process is divided into two step s. First, select a best fit model for each failure mode from a group of candidate mode ls; and second, fit a nd test the selected models. Model Selection A preliminary model screening is performed to select the most appropriate model for the given data. Six distribution models are tr ied and the results are listed in Table 6-3. PAGE 81 66 Table 6-3. Category I dist ress failure model screening Ride Rutting Distribution Model AD statistic Correlation AD statistic Correlation Weibull 82.546 0.994 129.482 0.989 Extreme Value 82.634 0.992 129.590 0.946 Exponential 82.735 * 129.537 * Lognormal 82.548 0.982 129.492 0.980 Logistic 82.590 0.994 129.540 0.954 Loglogistic 82.543 0.993 129.485 0.987 The Anderson-Darling (AD) statistic is a m easure of the deviation of the plot points from the fitted line in a pr obability plot. A smaller Ande rson-Darling statistic (AD) indicates that the distribution model fits th e data better. The Anderson-Darling statistic suggests that loglogistic is th e best candidate model for ride failure; Weibull is the best model for rutting failure. The Pearson correlation measures the strength of the linear relationship between the two variables on a probability plot. A higher correlation valu e indicates the distribution model fits the data better. Th e Pearson correlation suggests We ibull is the best model for both ride failure and rutting failure. Suggested by both Anderson-Darling stat istic and the Pearson correlation, the Weibull is selected for modeling rutting failur e. However, the Ande rson-Darling statistic and the Pearson correlation have suggested di fferent models for ride failure. We follow the suggestion from the Pearson correlation and choose the Weibull for modeling ride failure. The Weibull is a flexible life distributi on model. Its probability density function (PDF) ) ( t fand cumulative distri bution function (CDF) ) ( t F are as follows: t t t t f exp ) ( (6-3) PAGE 82 67 t t F exp 1 ) ( (6-4) where is the scale parameter and is the shape parameter. Model Fitting Two parameter estimation methods are used to fit life distribution models. The least squares method is achieved by regressing fa ilure time on the rank value to form a regression line in a probability plot. The maximum likelihood method is achieved by maximizing the likelihood function. The leas t squares and maximum likelihood estimates of the parameters are listed in Tables 6-4 a nd 6-5. Although slightly different, the results from the two estimation methods are consistent. Table 6-4. Least squares estimates for pavement failure models, Weibull 95% C.I. Failure Mode Parameter Estimate Standard Error Lower Upper Shape ( ) 4.45257 0.60696 3.40862 5.81626 Ride Scale ( ) 14.3533 0.78312 12.8977 15.9733 Shape ( ) 1.48121 0.19789 1.13998 1.92457 Rutting Scale ( ) 22.9150 3.43498 17.0814 30.7408 Table 6-5. Maximum likelihood estimates for pavement failure models, Weibull 95% C.I. Failure Mode Parameter Estimate Standard Error Lower Upper Shape ( ) 4.73200 0.598747 3.69267 6.06385 Ride Scale ( ) 14.0424 0.668777 12.7909 15.4163 Shape ( ) 1.55531 0.196767 1.21375 1.99299 Rutting Scale ( ) 21.7492 2.93836 16.6896 28.3428 As seen in Table 6-3, the least squares es timates yield a correlation coefficient of 0.994 for ride and 0.989 for rutting, indicati ng the linear regression models fit the data well for both failure modes. The probability plots for least square method (Figures 6-1 and 6-2) also indicate that the Weibull model fits the data well for both ride failure and rutting failure. PAGE 83 68 Pavement AgePercent 20 10 5 2 1 70 60 50 40 30 20 10 5 3 2 1 Figure 6-1. Probability plot for rutting failure, least square, Weibull Pa v ement A g ePercent 20 15 10 9 8 7 6 5 4 3 90 80 60 40 20 10 5 2 1 0.1 Figure 6-2. Probability plot for ride failure , least square, Weibull PAGE 84 69 Estimate of cumulative failure rates The failure probabilities for rutting and ride are calculated using the fitted models. The least square estimates are listed in Tabl es 6-6 and 6-7 and illustrated in Figures 6-3 and 6-4. The cumulative failure plot for rutti ng is approximately linear for pavement ages less than 20, indicating the failure rate is rela tively stable at different pavement ages. The cumulative failure plot for ride shows an S cu rve. The failure rates are very low in the first five years but the rates incr ease rapidly after that. Table 6-6. Cumulative failure rate s for ride, least square estimates 95% C.I. Pavement Age Failure Probability Lower Up 1 0.0000 0.0001 0.0000 2 0.0001 0.0011 0.0000 3 0.0009 0.0043 0.0002 4 0.0034 0.0111 0.0010 5 0.0091 0.0230 0.0036 6 0.0204 0.0420 0.0098 7 0.0401 0.0699 0.0228 8 0.0714 0.1092 0.0464 9 0.1176 0.1635 0.0840 10 0.1813 0.2383 0.1367 11 0.2635 0.3399 0.2016 12 0.3627 0.4688 0.2745 Table 6-7. Cumulative failure rates for rutting, least square estimates 95% C.I. Pavement Age Failure Probability Lower Up 1 0.0096 0.0236 0.0039 2 0.0266 0.0505 0.0140 3 0.0480 0.0789 0.0290 4 0.0726 0.1084 0.0483 5 0.0996 0.1392 0.0708 6 0.1284 0.1713 0.0956 7 0.1586 0.2051 0.1218 8 0.1897 0.2406 0.1486 9 0.2216 0.2779 0.1753 10 0.2538 0.3167 0.2016 11 0.2862 0.3568 0.2271 12 0.3186 0.3977 0.2519 PAGE 85 70 Pavement AgePercent 60 50 40 30 20 10 100 80 60 40 20 0 Figure 6-3. Cumulative failure pl ot for rutting failure, Weibull Pavement AgePercent 40 35 30 25 20 15 10 5 100 80 60 40 20 0 Figure 6-4. Cumulative failure plots for ride failure, Weibull PAGE 86 71 For a three-year warranty, the failure proba bilities for ride and rutting are estimated to be 0.09% and 4.80%, respectively. If the warranty period extends to five years and the threshold values stay unchanged, the failure probability for ride and rutting will increase to 0.91% and 9.96%. The results indicate that the contractorâ€™s ri sk exposure to ride failure is very low, but risk exposure to rutting failure is relatively high. Modeling Failure Time of Category II Distresses Failure Data Collection The data sources used to generate failur e data are the worksheets for Category II distresses that were analyzed in the last chap ter. The types of distresses to be considered include cracking, raveling, delamination, a nd bleeding. All these distress types are performance parameters in the FDOT warra nty specifications and are included in the Flexible Pavement Condition Survey. The failure time of each distress type for a roadway section is defined as follows: Cracking: the first year a nonA cracking code appeared for either within (CW) or outside (CO) of wheelpath. A non-A cracki ng code means either type I cracking predominates, but more than 5% of the ro adway is affected by cracking, or type II/III cracking predominates. Raveling: the first year raveling was coded. Delamination: the first year delaminati on is recorded in the field designated â€œRemarksâ€. Bleeding: the first year bl eeding is recorded in the field designated â€œRemarksâ€. It should be noted that cracking discu ssed here is actually a combination of cracking, raveling, and patching. As noted before, the FDOT accumulates raveling and patching into type III cracking in its Flex ible Pavement Condition Survey program. The failure time of raveling will be modeled separately , but it is considered in cracking also. PAGE 87 72 The failure data collection procedure for Category II distresses is the same as for Category I distresses. The procedure is repeated for each distress type. The failure data are summarized in Table 6-8. Table 6-8. Failure data for Category II distresses Time Interval Start End Cracking Raveling Bleeding Delamination 0 1 1 5 1 2 5 1 6 2 3 14 2 5 3 4 27 8 3 4 5 51 15 1 5 6 42 21 1 1 6 7 19 9 2 7 8 19 14 2 8 9 5 11 2 9 10 6 5 10 11 3 3 11 12 2 2 1 12 13 1 1 13 14 1 14 15 2 4 * 1 1 1 5 * 12 15 13 16 6 * 4 10 6 10 7 * 5 12 19 23 8 * 4 26 31 32 9 * 7 13 30 28 10 * 1 16 31 33 11 * 3 26 42 39 12 * 2 11 18 17 13 * 6 8 14 * 1 2 4 4 15 * 1 5 10 10 Sum 232 232 232 232 Note: a * in the column of End means right censoring. Model Selection A preliminary model screening is pe rformed using MiniTAB software. Six distribution models are tried and the results are listed in Table 6-9. PAGE 88 73 Table 6-9. Model screening for Category II distresses Cracking Raveling Bleeding Delamination Model AD Corr. AD Corr. AD Corr. AD Corr. Weibull 0.885 0.990 27.9190.980 131.2270.976 129.349 0.984 Lognormal 1.244 0.989 27.4750.995 131.2270.981 129.347 0.994 Exponential 4.609 28.236 131.227 129.354 Loglogistic 1.036 0.997 27.5510.989 131.2270.977 129.348 0.985 Extreme Value 3.303 0.892 31.5170.898 131.2280.889 129.351 0.964 Logistic 1.208 0.948 28.4770.923 131.2280.891 129.350 0.966 The Anderson-Darling (AD) statistic and the Pearson correlation both suggest the lognormal model as the best candidate for raveling, bleeding, and delamination. For cracking, the AD statistic s uggests the Weibull while the Pearson correlation suggests loglogistic. We follow the suggestion of the Pearson correlation and choose the loglogistic model for cracking. The probability density function (PDF) ) ( t f and the cumulative distribution function (CDF) ) ( t F of lognormal distribution are as follows: 2 22 ln exp 2 1 ) ( t t t f (6-5) dx x x t Ft 2 22 ln exp 2 1 ) ( (6-6) where u is the location Parameter and is the scale parameter. The PDF and CDF of the loglogisti c distribution ar e as follows: 2ln exp 1 ln exp ) ( t t t f (6-7) t t F ln exp 1 1 ) ( (6-8) PAGE 89 74 where u is the location Parameter and is the scale parameter. Model Fitting The least squares and maximum likeli hood estimates of the parameters are summarized in Tables 6-10 and 6-11. The re sults from the two estimation methods are consistent for cracking, rave ling, and delamination. Howe ver, there is a significant discrepancy between the estimation resu lts for bleeding. The maximum likelihood method is generally recommended when heavy censoring and interval data are present (ReliaSoft 2006). But the least squares estimate for bleeding is more conservative than the maximum likelihood result since it pr ovides higher cumulative failure rates. Table 6-10. Least squares estimates of failure models for Category II distresses 95% C.I. Defect type Model ParameterEstimate Standard Error Lower Upper Shape( ) 1.69323 0.03272 1.62910 1.75736 Cracking Loglogistic Scale( ) 0.28953 0.01998 0.25291 0.33145 Loc ( ) 2.39731 0.05869 2.28228 2.51234 Raveling Lognormal Scale( ) 0.62925 0.04985 0.53875 0.73496 Loc ( ) 4.90663 0.45337 4.01805 5.79521 Bleeding Lognormal Scale( ) 2.46632 0.36087 1.85141 3.28546 Loc ( ) 3.17491 0.18012 2.82188 3.52794 Delamination Lognormal Scale( ) 0.54698 0.10859 0.37068 0.80715 Table 6-11. Maximum likelihood estimates of failure models for Category II distresses 95% C.I. Defect type Model ParameterEstimate Standard Error Lower Upper Loc ( ) 1.69968 0.03044 1.64002 1.75935 Cracking Loglogistic Scale( ) 0.26396 0.01642 0.23366 0.29819 Loc ( ) 2.38193 0.05765 2.26894 2.49493 Raveling Lognormal Scale( ) 0.63519 0.05089 0.54288 0.74319 Loc ( ) 7.57542 1.39716 4.83704 10.3138 Bleeding Lognormal Scale( ) 4.00969 0.92332 2.55332 6.29677 Loc ( ) 3.34360 0.26158 2.83092 3.85628 Delamination Lognormal Scale( ) 0.62194 0.14637 0.39213 0.98644 PAGE 90 75 As seen in Table 6-10, the least squares es timation yields correlat ion coefficients of 0.997, 0.995, 0.981, and 0.994 for cracking, raveling, bleeding, and delamination. The high correlation coefficients indicate that the models fit the data very well. The probability plots for the maximum likelihood es timates (see Figures 6-5 to 6-8) also suggest that the models fit the data we ll for all four types of distresses. Pavement Age P ercen t 10 5 2 1 99 95 90 80 70 60 50 40 30 20 10 5 1 0.1 Figure 6-5. Probability plot for cracking failu re model, loglogistic PAGE 91 76 Pavement AgePercent 10 5 2 1 80 50 20 5 1 0.01 Figure 6-6. Probability plot for raveling failu re model, lognormal Pavement AgePercent 10.0 1.0 0.1 30 20 10 5 1 Figure 6-7. Probability plot for bleeding failu re model, lognormal PAGE 92 77 Pavement AgePercent 10 8 5 4 2 20 5 1 0.01 Figure 6-8. Probability plot for delamination failure model, lognormal Estimate of Cumulative Defect Probability The cumulative failure probabilities for each distress type are calculated using the fitted failure models. The maximum likelihood results are listed in Tables 6-12 to 6-15 and illustrated in Figures 6-9 to 6-12. Table 6-12. Cumulative probabilities for cracking failure 95% C.I. Pavement Age Failure Probability Lower Up 1 0.00160 0.00071 0.00356 2 0.02160 0.01314 0.03530 3 0.09303 0.06754 0.12685 4 0.23375 0.19020 0.28378 5 0.41535 0.36137 0.47144 6 0.58633 0.52909 0.64134 7 0.71765 0.66242 0.76701 8 0.80825 0.75857 0.84974 9 0.86818 0.82543 0.90170 10 0.90755 0.87165 0.93417 11 0.93371 0.90387 0.95475 12 0.95142 0.92668 0.96811 PAGE 93 78 For a three-year warranty, th e probabilities for a roadway section to have each type of distress are estimated to be 9.30% for cracking (including raveling and patching), 2.17% for raveling, 5.31% for bleeding, and 0.015% for delamination. The results indicate that cracking is the most common distress type for asphalt pavements. Raveling and bleeding also have consider ably high rates of occurrence at early pavement ages. But the risk of delamination is very low. Table 6-13. Cumulative probabi lities for raveling failure 95% C.I. Pavement Age Failure Probability Lower Up 1 0.00009 0.00001 0.00061 2 0.00392 0.00128 0.01068 3 0.02167 0.01101 0.03999 4 0.05850 0.03739 0.08802 5 0.11196 0.08141 0.14995 6 0.17641 0.13811 0.22082 7 0.24622 0.20098 0.29640 8 0.31696 0.26484 0.37298 9 0.38560 0.32658 0.44741 10 0.45029 0.38466 0.51732 11 0.51002 0.43846 0.58127 12 0.56439 0.48786 0.63858 Table 6-14. Cumulative proba bilities for bleeding failure 95% C.I. Pavement Age Failure Probability Lower Up 1 0.02943 0.01417 0.05637 2 0.04304 0.02425 0.07216 3 0.05312 0.03200 0.08404 4 0.06135 0.03827 0.09409 5 0.06839 0.04352 0.10301 6 0.07459 0.04802 0.11114 7 0.08016 0.05196 0.11865 8 0.08524 0.05545 0.12566 9 0.08991 0.05858 0.13226 10 0.09425 0.06142 0.13850 11 0.09831 0.06402 0.14444 12 0.10212 0.06641 0.15010 PAGE 94 79 Table 6-15. Cumulative probabi lities for delamination failure 95% C.I. Pavement Age Failure Probability Lower Up 1 0.00000 0.00000 0.00013 2 0.00001 0.00000 0.00120 3 0.00015 0.00000 0.00375 4 0.00082 0.00005 0.00794 5 0.00265 0.00037 0.01383 6 0.00630 0.00150 0.02156 7 0.01231 0.00421 0.03145 8 0.02105 0.00911 0.04417 9 0.03265 0.01626 0.06075 10 0.04708 0.02509 0.08238 11 0.06418 0.03483 0.10992 12 0.08369 0.04490 0.14345 Pavement AgePercent 20 18 16 14 12 10 8 6 4 2 0 100 90 80 70 60 50 40 30 20 10 0 Figure 6-9. Cumulative defect probability plot for cracking PAGE 95 80 Pavement AgePercent 20 18 16 14 12 10 8 6 4 2 0 100 90 80 70 60 50 40 30 20 10 0 Figure 6-10. Cumulative defect probability plot for raveling Pavement AgePercent 20 18 16 14 12 10 8 6 4 2 0 100 80 60 40 20 0 Figure 6-11. Cumulative defect probability plot for bleeding PAGE 96 81 Pavement AgePercent 20 18 16 14 12 10 8 6 4 2 0 100 90 80 70 60 50 40 30 20 10 0 Figure 6-12. Cumulative defect pr obability plot for delamination Modeling Crack Growth In the prior sections the failure times of various distress types have been modeled. In this section the severity of distresses after initiation will be modeled. The percent of pavement affected by distresses was estimated and coded for distresses of raveling, patching, and cracki ng in the FDOT Flexible Pavement Survey program. But no such information was provided for bleeding and delamination. So modeling growth of bleeding or delamination is not possible at this time without other sources of data. Since the areas affected by raveling and patching ar e already included in the estimates of cracking areas, we need only to model the growth of cracking. But we should keep in mind that the so-called cracking area here is actually the sum of cracking, raveling, and patching areas. PAGE 97 82 Cracking Defect Rating The cracking data are reorganized based on crack initiation time . Beginning at the point of crack initiation, the cracking data for each year are stored in the same column in the worksheet. So the new worksheets for cr acking data are organi zed by section ID and year after crack initiation. The distributions of cracking rating scores for each year after cracking initialization are illustrated in Figure 6-13. Observations gathered are as follows: In the first year after crack initiation, mo st roadway sections ha ve a rating score of 8.5 or higher. However, there are also some road sections that have a sharp drop in cracking rating (all the sections have a crack ing rating score of 10 in the prior year), indicating a sudden and rapid pave ment condition deterioration. The range of the rating scores becomes wi der over time, reflecting the difference in cracking growth rates among s ections. Cracking grows slowly in some sections, but very quickly in others. The distributions of cracking rating scor es are negatively skewed, with a longer lower tail. Estimating Cracking Affected Area A cracking rating score is a comprehens ive measure of both the severity of cracking and the size of cracking affected areas . To estimate the cracking affected areas, we have to look at the cracking codes (see Tabl e 5-1) used to calcula te the cracking rating scores. Two cracking codes, confined to wheel path (CW) and outside of wheel path (CO), are given for each road section. However, th e codes do not provide an accurate estimate of the percent pavement affected, since the represented percentage ranges are very large for each code. The estimation of cracking a ffected areas is achieved by fitting the cracking codes to a growth curve. An example is illustrated in Figure 6-14. PAGE 98 Crack R atin g Percent sections 40 30 20 10 0 9 8 7 6 5 4 3 2 1 40 30 20 10 0 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Figure 6-13. Distributions of crack rating after cracking initialization 83 PAGE 99 84 0 10 20 30 40 50 60 01234567 Years after cracking initiationPercent area affecte d A I I FJ J K K Figure 6-14. Estimating percent pavement affected by cracking Modeling Cracking Affected Areas A histogram illustration of percent areas a ffected by cracking is shown in Figure 615. The cracking affected areas show simila r distributional characteristics with cracking rating (Figure 6-13) except with th e reverse order in the x-axis. Empirical cumulative distribution function Without being fitted to a particular parame tric distribution model, the distribution of the percent area affected by cracking can be estimated empirically. The empirical cumulative distribution function (CDF) gi ves the values of CDF such that ) ( x F represents the proportion of observations in a sample less than or equal to x . The results are listed in Table 6-16. PAGE 100 Cracking affected area (%)Percent in No. of sections 75 60 45 30 15 0 50 40 30 20 10 0 75 60 45 30 15 0 50 40 30 20 10 0 50 40 30 20 10 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Figure 6-15. Distribution in size of cracked area after crack initiation 85 PAGE 101 86 Table 6-16. Empirical CDF of percent pavement cracked Percent area cracked Year 1 Year 2 Year 3 Year 4 Year 5 0 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.05128 0.00000 0.00000 0.00000 0.00000 2 0.22051 0.11236 0.05848 0.03521 0.00847 3 0.49231 0.21910 0.11696 0.08451 0.05085 4 0.66667 0.37079 0.16959 0.13380 0.11017 5 0.76923 0.53933 0.31579 0.16901 0.14407 6 0.81026 0.63483 0.38012 0.21831 0.16949 7 0.90256 0.70225 0.50877 0.28169 0.20339 8 0.94872 0.75843 0.59064 0.35211 0.23729 9 0.96410 0.78090 0.64327 0.44366 0.30508 10 0.97949 0.82584 0.66667 0.48592 0.36441 11 0.98462 0.87640 0.70760 0.56338 0.39831 12 0.99487 0.90449 0.74269 0.64085 0.46610 13 1.00000 0.92135 0.78363 0.67606 0.48305 14 0.92697 0.81287 0.71831 0.54237 15 0.93820 0.83626 0.71831 0.58475 16 0.95506 0.84795 0.74648 0.64407 17 0.96067 0.85965 0.77465 0.66949 18 0.96629 0.85965 0.80282 0.69492 19 0.97191 0.87719 0.81690 0.72034 20 0.98315 0.88304 0.82394 0.75424 21 0.98315 0.90643 0.83803 0.78814 22 0.98876 0.93567 0.83803 0.80508 23 0.99438 0.94152 0.83803 0.83051 24 1.00000 0.94152 0.83803 0.83898 25 0.95322 0.84507 0.84746 26 0.97076 0.84507 0.84746 27 0.97661 0.85211 0.84746 28 0.98246 0.86620 0.85593 29 0.98246 0.87324 0.85593 30 0.98830 0.90141 0.85593 31 0.98830 0.90845 0.85593 32 0.99415 0.91549 0.85593 33 0.99415 0.95070 0.85593 34 0.99415 0.96479 0.86441 35 0.99415 0.96479 0.87288 36 0.99415 0.96479 0.87288 37 1.00000 0.97887 0.88983 38 0.97887 0.88983 39 0.97887 0.91525 PAGE 102 87 Table 6-16. Continued Percent area cracked Year 1 Year 2 Year 3 Year 4 Year 5 40 0.98592 0.91525 41 0.98592 0.93220 42 0.98592 0.94915 43 0.98592 0.97458 44 0.98592 0.97458 45 0.98592 0.97458 46 0.99296 0.97458 47 0.99296 0.97458 48 0.99296 1.00000 49 0.99296 50 1.00000 Parametric analysis A preliminary screening indicates that lognormal is the best fitting model for the given data. The estimated parameters are liste d in Table 6-17. A comparison of the fitted curves with the observed data is shown in Figure 6-15. Table 6-17. Lognormal dist ribution of percent paveme nt affected by cracking 95% C.I. Years after cracking initiation Parameter Estimate Standard Error Lower Upper Loc ( ) 0.99928 0.05112 0.89910 1.09947 1 Scale ( ) 0.71382 0.03608 0.64650 0.78815 Loc ( ) 1.51647 0.05578 1.40714 1.62580 2 Scale ( ) 0.74422 0.03985 0.67008 0.82656 Loc ( ) 1.91934 0.06000 1.80174 2.03693 3 Scale ( ) 0.78457 0.04285 0.70493 0.87322 Loc ( ) 2.25146 0.06777 2.11863 2.38428 4 Scale ( ) 0.80755 0.04832 0.71820 0.90802 Loc ( ) 2.48503 0.07261 2.34272 2.62734 5 Scale ( ) 0.78873 0.05176 0.69353 0.89699 Results of chi-square goodne ss-of-fit tests are listed in Tables 6-18 to 6-22. The associated p-values for year 1 through year 3 are all greater than 5%, indicating the fitted lognormal models are adequate for the data of th e first three years. Fo r year 4 and year 5, however, the associated p-values are less than 1%, indicating an inade quate fit. Thus the PAGE 103 88 lognormal models can be used to model the di stribution of percent pavement affected by cracking for year 1 to 3 after crack initiati on. For years 4 and 5, only the empirical CDF can be used. Table 6-18. Chi square g oodness-of-fit test for year 1 x F(x) ProbabilityExpectedObservations2 P-value 2.5 0.45372 0.45372 88.47682 7.4566 0.0587 5.0 0.80366 0.34994 68.23868 (DF=3) 7.5 0.92260 0.11894 23.19335 10.0 0.96606 0.04346 8.4756 12.5 0.98376 0.01770 3.4513 15.0 0.99166 0.00791 1.5421 Table 6-19. Chi square g oodness-of-fit test for year 2 x F(x) ProbabilityExpectedObservations2 P-value 4 0.43057 0.43057 76.64266 3.7909 0.2850 8 0.77531 0.34474 61.36469 (DF=3) 12 0.90342 0.12811 22.80326 16 0.95428 0.05086 9.0539 20 0.97658 0.02230 3.9695 24 0.98721 0.01064 1.8943 Table 6-20. Chi square g oodness-of-fit test for year 3 x F(x) ProbabilityExpectedObservations2 P-value 5 0.34642 0.34642 59.23854 10.3291 0.0664 10 0.68739 0.34097 58.30660 (DF=5) 15 0.84262 0.15522 26.54329 20 0.91496 0.07234 12.3718 25 0.95118 0.03622 6.19312 30 0.97054 0.01936 3.3116 35 0.98148 0.01094 1.8711 40 0.98795 0.00647 1.1071 Table 6-21. Chi square g oodness-of-fit test for year 4 x F(x) ProbabilityExpectedObservations2 P-value 8 0.41566 0.41566 59.02450 15.3548 0.0040 16 0.74064 0.32498 46.14756 (DF=4) 24 0.87439 0.13375 18.99313 32 0.93367 0.05927 8.41611 40 0.96246 0.02880 4.08910 48 0.97756 0.01510 2.1441 56 0.98598 0.00842 1.1951 PAGE 104 89 Table 6-22. Chi square g oodness-of-fit test for year 5 x F(x) ProbabilityExpectedObservations2 P-value 6 0.18971 0.18971 22.38520 27.9257 0.0000 12 0.49994 0.31023 36.60735 (DF=5) 18 0.69635 0.19641 23.17627 24 0.81021 0.11386 13.43617 30 0.87730 0.06709 7.9172 36 0.91815 0.04085 4.8212 42 0.94388 0.02573 3.0369 48 0.96058 0.01671 1.9716 Summary The failure time of each distress type is modeled in this chapter using life distribution models. In particul ar, Weibull models are fitted for ride and rutting failure; the loglogistic model is fitted for cracking fa ilure; lognormal models are fitted for failures of raveling, bleeding, and delamination. The growth and distribution of percent pavement affected by cracking are also modeled in this chapter. Both empirical CD F and lognormal models are used to estimate the distribution of percent pavement affect ed by cracking. The ch i square goodness-of-fit test indicates that the lognormal model fits the data well for th e first three years but is not adequate for the data of the 4th and 5th years. PAGE 105 90 CHAPTER 7 MODELING CONSTRUCTI ON COST ESCALATION The two elements of warranty risk for cont ractors are the uncertainty in quantities of repair works and the uncertainty in future unit repair costs. The uncertainty in repair work quantities has been analyzed and modele d in previous chapters. This chapter will deal with the uncertainty in future cost escalation. The uncertainty in future cost escalati on will be assessed us ing the historical construction cost index. The Box-Jenkins appr oach is adopted to model the index series and predict the probability distribution of future unit costs. Introduction Existing Forecast Methods A review of literature showed that a variet y of future costs forecasting models have been developed or are in use in the constr uction industry. These forecasting methods can be broadly classified into th ree categories (Chatfield 2001): Judgmental forecasts based on subjective judgment, intuition, and commercial knowledge. Univariate time series methods where fo recasts depend only on current and past values of a historical co nstruction cost index. Relational methods where for ecasts depend, at least part ly, on the value of other macro economic variables, called predictors, indicators, or explanatory variables. Akintoye et al. (1998) identified that unemployment level, construction output, industrial production, and the ratio of price to cost i ndices in manufacturing are consistently the leading indicators of constr uction contract prices; while gross national PAGE 106 91 domestic product is a coinci dental indicator. Herbsman (1986) found that the total volume of contract bids in a particular year, as well as ma terial, labor, and equipment costs, affect construction bid price. Regression analysis, due to its simplicity in both concept and application, has been widely used to model the relationship between future construction costs and other explanatory va riables (Goh and Teo 2000). Neural network technique was also used to model construction costs (Williams 1994; Wilmot and Mei 2005). Relational models allow the impact of vari ous alternative inputs to be evaluated. However, this approach requires information on several explanatory variables in addition to the variable to be predicted. Future valu e of explanatory variables may be needed to make long-term forecasts. By contrast, univari ate time series models predict future cost based solely on the past values of the cost in dex, which is much easier to collect. Typical time series forecasting techniques include the smoothing method and the Box-Jenkins approach (Goh and Teo 2000). Despite the existence of various forecasti ng models, their application in practice by contractors is limited. Herbsman (1986) found that contractors and suppliers mainly forecast future construction costs based on the intuition of experienced professionals. The Proposed Approach Most forecast literature is concerned w ith methods for calculating point forecasts, which provide single number predictions. Pr ediction intervals, also called confidence intervals, are sometimes calculated but not given sufficient attention. A common use of prediction intervals is to validate the fitted model. Given the particular objectives of this st udy on warranty risk analysis, the interest herein is to assess the uncertainty of future construction costs. More specifically, the aim PAGE 107 92 is to develop probabilistic forecasts for futu re unit costs. An appropriate historical construction cost index will be selected and used for this analysis. However, a model that gives accurate poi nt forecasts may not provide accurate probabilistic forecasts. Time series met hods are concerned with the estimation of difference equations containi ng stochastic components, and they are concerned with uncovering the dynamic aspect of a series. Time series models are able to uncover the dynamic aspects of a series and to model both its systematic variation and unexpl ained variation. The systematic variation facilitates the computation of point for ecasts while the description of unexplained variation helps to model the uncertainty. The Box-Jenkins approach is adopted to model the index series and predict probabilistic distributions of future unit costs. Rationale for Price Forecast Errors Where Does the Forecast Error Come From? Forecasts of future unit costs generally start from current unit costs. Let 0P be the current unit cost and tP be the future unit cost at time t . The corresponding one-period inflation tI for time period t can be calculated as: 1 1 t t t tP P P I (7-1) The equation can be rewritten as: ) 1 (1 t t tI P P (7-2) Expressing tP in term of0P, renders ) 1 )...( 1 )( 1 (2 1 0 t tI I I P P (7-3) or PAGE 108 93 t j j tI P P1 0) 1 ( (7-4) If we denote the predicte d one-period inflation as tI , the predicted future cost tP can be expressed as t j j tI P P1 0) 1 ( (7-5) An error in the forecast exists because tI can not be accurately predicted, ort tI I . The forecast error can be expressed as t j j t j j t t tI I P P P E1 1 0) 1 ( ) 1 ( (7-6) The Concept of Log-inflation If we take the natural logarithm at both sides, equation (7-4) becomes t j j tI P P1 0) 1 ln( ln ln (7-7) or t j j tp p p1 0 (7-8) where t tP pln 0 0lnP p ) 1 ln(j jI p The jp defined above is similar to the c ontinuously compounded return or the log-return that is widely used in finance literature. It is actually another form of PAGE 109 94 expression for inflation. It is called log-inflation. To expl ain this definition, see the Taylor series: 1 01 ) 1 ( ) 1 ln( n n nx n x for 1x (7-9) For the case of log-inflation, n j n n j jI n I p 1 11 ) 1 ( ) 1 ln( ... 5 4 3 25 4 3 2 j j j j jI I I I I (7-10) Since generally 1 jI , ignoring the higher order terms renders j jI p (7-11) As an example, if %, 4 jI then % 922 . 3 %) 4 1 ln( jp . These two numbers are very close. So jp can be considered as an alternative form of inflation. However, the difference between jp and jI gets larger when jI deviates significantly from zero. For example, when%, 200 jI % 86 . 109 %) 4 1 ln( jp ; when%, 100 jI jp . The concept of log-inflation has obvious adva ntages over the simple inflation. First, the multiperiod log-inflation is simply the sum of the one-period log-inflations of all the periods, as illustrated in e quation (7-8). This additive form is much easier to handle in practice than the multiplicative form in equati on (7-5). Second, the theoretical range for jI is ) , 1 [ , while the range for jp is ) , ( . Thus jp is more likely to follow a symmetric distribution and its statis tical properties are more tractable. PAGE 110 95 Forecast Error under the Concept of Log-inflation If the log-inflation is estimated as ti for period t , then the predicted log-cost at the end of period t can be expressed as: t j j ti p p1 0 (7-12) The error in predicting log-cost tp is t j j j t t tp p p p e1) ( (7-13) For the unit cost series, we have t t te P P ln ln (7-14) or ) exp( t t te P P (7-15) or ) exp( 1 t t t te P P P (7-16) Thus the error in predicting tP can be obtained through the error in predicting tp . Basic Concepts of ARIMA Models Stationarity Stationarity is the founda tion of time series analysis. A stochastic process {yt} is said to be stationary if its probability distri bution at a fixed time is invariant for all times. However, this strict condition is hard to verify empirically. A weaker version of stationary, called weak stationa ry, secondary stationary, or co variance stationary, is often PAGE 111 96 assumed. A series {yt} is weakly stationary if (a) the mean of yt is constant and (b) the covariance between yt and yt-s only depends on s, where sis an arbitrary integer. Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) The autocorrelation of a time series is its correlation with itself at different points of time. The autoco rrelation between yt and yt-s is called the lag-s autocorrelation. It is denoted as s and defined as: )( ) ( ) (s t t s t t sy Var y Var y y Cov (7-17) The autocorrelation (ACF) between yt and yt-s , when s>1, is the sum of (a) the direct correlation between yt and yt-s , and (b) the indirect correlation effect through yt-1 to yt-s+1. So partial autocorrelation (PACF) is us ed to eliminate the indirect effects of intervening values and reflect the direct correlation between yt and yt-s . White Noise A white noise series {t } is a sequence of independent and identically distributed random variables with finite mean and variance. The characteristics of white noise series include (a) zero mean, (b) constant variance 2, and (c) all the ACFs are zero. Autoregressive Models The autoregressive model of order p for a time series {yt} is denoted as AR(p) and expressed as: t p i i t i ty y 1 0 (7-18) Where, 0 p are parameters of the model, 0 is a constant; t is a white noise. Stationarity requires 1i for all p i ... 1 . PAGE 112 97 Moving Average Models The moving average model of order q for a time series {yt} is denoted as MA(q) and expressed as: t q i i t i ty 1 (7-19) where, 0 p are parameters of the model, } {t is a white noise series. Moving average models are always stationary. Autoregressive Moving Average (ARMA) Models An autoregressive moving average model ARMA(p,q) is a combination of models AR(p) and MA(q). t q i i t i p i i t i ty y 1 1 0 (7-20) Like a pure AR model, the stationa rity of ARMA models requires 1i for p i ... 1 . Autoregressive Integrated Moving Average (ARIMA) Models An autoregressive integrated moving aver age (ARIMA) model, referred to as the ARIMA(p,d,q) model, is a generalization of an ARMA model. An ARIMA(p,d,q) series can be transformed into a stationary ARMA(p,q ) series after d differences. If d is zero, then the model is equivale nt to an ARMA model. Date Collection Availability of Highway Maintenance Cost Index To assess the uncertainty of future un it costs for remedial work on warranted highway projects, an appropria te highway maintenance cost index would be ideal for the research. However, little inform ation can be found on such indices. PAGE 113 98 The Federal Highway Administration us ed to publish an annual Highway Maintenance and Operations Cost Index. Th is index was establis hed in 1947 but it was discontinued in the early 1990s due to the sm all number of users (Kyte and Gillespie, 1998). Virginia is the only state identified to currently have an official highway Maintenance Cost Index (MCI). The Virginia Department of Transportation uses this index to make adjustments on its allocation of street payments to localities. Kyte and Gillespie (1998) surveyed 15 states, but no state other than Virginia has a formal maintenance cost index. Highway Construction Cost Index as a Substitute Since an appropriate highway maintenance co st index is basically unavailable, the highway construction cost index may be tried as a substitute due to their high correlation. Another reason for using a construction co st index is the c ontractors, who are experienced with construction estimating, may use a construction index too, if they need to do this analysis. The FHWA publishes a Composite Bid Pr ice Index on National Highway System projects. This index was established in 1972 a nd is updated quarterly. Annual data since 1972 and quarterly data for recent years ar e published on the FHWAâ€™s website. Some states, such as California, Colorado, Ore gon, and Washington, also develop highway construction cost indexes fo r their respect ive states. The Bureau of Labor Statistics (BLS) of the U.S. Department of Labor publishes a Producer Price Index (PPI) for highway a nd street construction. The index was established in June 1986 and is reported m onthly. Both monthly and annual data are available. PAGE 114 99 Comparatively, the sample size for BLS monthly data is much bigger than the FHWA and other state indices, and may provide more reliable statistical results. So the BLS monthly PPI is chosen for this study. The BLS Highway and Street Construction PPI Data Set The monthly data of the highway and street construction PPI seri es range from June 1986 to December 2005, 235 months in total. The data set is divided into two parts. The data from June 1986 to Jun 2006, called the fit set, are used to build and fit the model. The data from July 2000 to December 2005, cal led the test set, are kept back for comparing with the out-of-sample prediction from the fitted model. Figure 7-1 is a time plot of the index. PPI Year Month 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 Figure 7-1. Time plot of highway and street constr uction PPI series PAGE 115 100 The Box-Jenkins Approach Box and Jenkins (1976) developed a th ree-stage method for selecting an appropriate ARIMA model for uni variate time series analyses . The three stages suggested by Box and Jenkins are: Model identification Model estimation Model checking Model Identification In time series analysis, it is common to fo rmulate a broad set of candidate models and then select one that best fits the data. One way to identify the candidate models is to examine the time plot, the ACF, and the PA CF of the raw data and of the suitably differenced series. A non-stationa ry series may be transformed into a stationary series by differencing or detrending. Formal procedures may be applied to test for the presence of seasonality, or of non-sta tionarity (unit root). In modeling a time series, increasing the orders of the model will necessarily reduce the sum of squares of the es timated residuals and increase the R2. However, the inclusion of extraneous coefficients may re duce the degrees of freedom and the forecast performance of the fitted model. Various model selection criteria have been developed that trade off a reduction in sum squares of residuals for a more parsimonious model. The most widely used criterion is the Akaike Information Criterion (AIC). Model Estimation In this stage, each of the tentatively select ed models is fitted to the data. Statistical software, such as SAS, MiniTab, and SPSS, will be helpful for model estimation and diagnostic checking. In this study, th e data are analyzed using MiniTab. PAGE 116 101 Model Verification This stage is to ensure that the fitted mode l is consistent with the properties of the given data. Diagnostic checks on the residuals , which are the one-step-ahead forecasting errors, will be conducted. The ACF and the Lj ung-Box portmanteau test are usually used to test for adequacy. If the model is adequate , the residual should form a random series. If the checks fail, the original m odel should be further modified. Out-of-sample predictions pr ovide another way to valid ate the model. One common approach is to divide the data into two parts, fit the model to the fit set, and compare the prediction from the model to the test set. Model Identification Data Transformation The time plot of the index in Figure 7-1 s hows a significant trend of increase with an upward curve. Now we take the natural l ogarithm of the PPI, ca ll it log-PPI and denote it as {tp}. We have ) ln(t tPPI p (7-21) A time plot of the log-PPI series {tp} is shown in Figure 7-2. The log-PPI time plot removes part of the upward curvature in the PPI plot, so the log-pr ice series is more likely to follow a linear model than the PPI series. Unit-Root Test The upward trend in the log-PPI series s uggests it is unit-root non-stationary since there is no fixed level for the series. The AC F in Figure 7-3 and PACF in Figure 7-4 also indicate that there is a single near unit signifi cant PACF at lag-1. It is called a unit-root non-stationary because unit roots exist in the ARIMA process. Recall that stationarity PAGE 117 102 requires the autoregression parameters to be less than 1 in absolute value. If this requirement is violated, the random proce ss will be non-stationary and become an ARIMA process. The well-known Augmented Dickey-Fuller (ADF) test is used to verify the existence of unit roots in ARIMA models. A p-th order autoregressi ve model AP(p) can be written as: t p i i t i t ty y c y 1 1 1 (7-22) where tcis a constant or zero and 1 j j jy y y is the first difference of ty. ln(PPI) Year M onth 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Figure 7-2. Time plot of log-PPI PAGE 118 103 LagAutocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-3. Autocorrelation function (ACF) for log-PPI LagPartial Autocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-4. Partial autocorrelati on function (PACF) for log-PPI PAGE 119 104 The hypothesis is H0: 1 versus Ha: 1 . The t-ratio of 1 can be calculated as: ) ( 1 std ADF where is the least squares estimate of . Equation (7-22) can be rewritten as: t p i i t i t c ty y c y 1 1 1 (7-23) where 1 c. Then the equivalent hypothesis is H0: 0 c versus Ha: 0 c . The ADF unit-root test is applied to the fit set of the log-PPI series using Excel add-in developed by Annen (2005). We choose1 p. Other values of p are also used, but they do not alternate the c onclusion of the test. With1 p, the resulting ADF test statistic is -0.0195 with p-value 0.9547, which indica tes that the unit-root hypothesis cannot be rejected at any reasonable significance level. Thus the log-PPI series is a unit-root nonstationary series. Difference is needed to transform it into a stationary series. Model Identification Taking the first difference of the series {tp}, we obtain a new series {tp }, where 1 t t tp p p (7-24) is the log-inflation in time period t. A time plot of the {tp } series is illustrated in Figure 7-5. There is no obvious trend in the mean le vel of the series. So it is reasonable to assume this series to be weakly stationary. There are two general approaches for dete rmining the order of ARMA models. One is to use the ACF and PACF, and the other is to use some information criteria. PAGE 120 105 Year Month 2000 1998 1996 1994 1992 1990 1988 1986 Jun Jun Jun Jun Jun Jun Jun Jun 0.03 0.02 0.01 0.00 -0.01 -0.02 Figure 7-5. Time plot of first difference of log-PPI The ACF and PACF of the {tp } series are shown in Figur es 7-6 and 7-7. The two dash dotted lines of the plot denote an a pproximate 95% confidence interval. It can be seen that the lag-1 PACF is significant at the 95% confident level, lag-9 and lag-13 PACFâ€™s are only marginally significant. Thus the PACF suggests we consider models AR(1), AR(9), and AR(13) for the {tp } series. Several information criteria are availa ble for determining the order of an autoregressive model. The well-known Akaike Information Criterion (AIC) is defined as: ) ( 2 ) ln( 2 parameters of number N likelihood N AIC (7-25) where N is the sample size (Tsay 2005). Th e model with the lowest AIC value is preferred by the criterion. PAGE 121 106 LagAutocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-6. Autocorrelation f unction (ACF) for log-inflation LagPartial Autocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-7. Partial autocorrelation function (PACF) for log-inflation PAGE 122 107 Table 7.1 gives the AIC for autoregressive models for {tp } series with order p = 1.13. The minimum AIC value -9.4069 occurs at p = 4, suggesting that the AR(4) model is preferred by the criterion. Table 7-1. AIC for first di fference series of log-PPI p 1 2 3 4 5 6 7 AIC -9.4028 -9.4007 -9.3961 -9.4069 -9.3925 -9.3883 -9.3839 P 8 9 10 11 12 13 AIC -9.3665 -9.3764 -9.3648 -9.3451 -9.3330 -9.3506 Considering the results from both approaches, AR(1) and AR(4) models are selected as tentative candidates for the {tp } series. AR(4) is suggested by the AIC. AR(1) is suggested by the PACF with an AIC value which is very close to the minimum. It should be noted that the autoregressive (AR) model mentioned above is for the differenced series {tp }. For the log-price series {tp }, the two corresponding candidate models are called ARIMA(1,1,0) and ARIMA(4,1,0). Model Estimation Parameters for the two candidate models are estimated using MiniTab. Printouts from MiniTab are included in Appendix C. The ARIMA(1,1,0) model is fitted as: t t tp p 13925 . 0 001129 . 0 005623 . 0 (7-26) The ARIMA(4,1,0) model is fitted as t t t t t tp p p p p 4 3 2 10994 . 0 1369 . 0 1859 . 0 4517 . 0 001307 . 0 005589 . 0 (7-27) The t-statistics for the parameters are show n in Table 7-2. All parameters of the ARIMA(1,1,0) model are statistically signifi cant at the 1% confidence level. For model PAGE 123 108 ARIMA(4,1,0), only 0 , 1 , and 2 are significant at the 5% confidence level; other parameters, 3 and 3 are not significant from zero. Table 7-2. Estimation of parameters for models AR(1) and AR(4) Model ParameterCoefficientSE CoefT P ARIMA(1,1,0) 0.0011290.0004342.60 0.010 0.39250.07195.46 0.000 ARIMA(4,1,0) 0.0013070.0004313.03 0.003 0.45170.07835.77 0.000 -0.18590.0855-2.18 0.031 0.13690.08631.59 0.115 -0.09940.0803-1.24 0.218 Model Verification In Sample Diagnostic Check Diagnostic checks on the residuals are pe rformed to check for possible model inadequacies. If a fitted model is adequate, the residuals should form a random series, or called a white noise. The ACF and the Ljung-Box st atistics of the residuals can be used to check the closeness of the residuals series to a white noise. The residual plot for the ARIMA(1,1,0) m odel is shown in Figure 7-8. It can be seen that the residuals distri bute randomly around the zero level. The pattern in the distribution of the residuals does not change over time. The normal plot shows that the residuals follow approximately a normal distribution, but the tails deviate from the fitted line. The ACF and PACF of the residuals of the ARIMA(1,1,0) model are shown in Figures 7-9 and 7-10. No ACF at any lag is significant at the 5% confidence level. The PACF, however, is marginally si gnificant at lags 15 and 36. PAGE 124 109 ResidualPercent 0.02 0.01 0.00 -0.01 -0.02 99.9 99 90 50 10 1 0.1 Fitted ValueResidual 0.3 0.2 0.1 0.0 0.02 0.01 0.00 -0.01 -0.02 ResidualFrequency 0.018 0.012 0.006 0.000 -0.006 -0.012 -0.018 30 20 10 0 Observation OrderResidual 160 140 120 100 80 60 40 20 1 0.02 0.01 0.00 -0.01 -0.02 Figure 7-8. Residual plot of log-inflation, ARIMA(1,1,0) LagAutocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-9. ACF of residuals of log-inflation, ARIMA(1,1,0) PAGE 125 110 LagPartial Autocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-10. PACF of residuals of log-inflation, ARIMA(1) The Ljung-Box Portmanteau test is used to test jointly that several autocorrelations are zero. The hypothesis is H0: 0 ...2 1 m versus Ha: 0 i for some } ,..., 1 { m i . The statistic is: m l ll N N N m Q1 2 ) 2 ( ) ( (7-28) where l is the autocorrelation of lag l , N is the sample size. The decision rule is to reject H0 if 2) ( m Q . If the p-value is provided, th e decision rule is to reject H0 if p-value is less than , the significance level. The Ljung-Box statistics for selected lags ar e listed in Table 7-3. The statistic is significant at 5% confident level when mtakes the values of 5, 12, 24, or 48. It is nonsignificant when m = 36. Simulation studies suggest the choice of ) ln( N m provides PAGE 126 111 better power performance (Tsa y 2005). In the case, N=168, we choose m=5, then the pvalue is 0.171, indicating insignificant serial correlations in the residual series. Table 7-3. Ljung-Box Portmant eau statistic for AR(1) model Lag 5 12243648 Chi-Square 5.0 14.931.249.256.8 DF 3 10223446 P-Value 0.171 0.1370.0930.0440.133 The residual plot for the ARIMA(4,1,0) m odel is shown in Figure 7-11. It can be seen that the residuals distributed randomly around the zero level without significant pattern. The normal plot shows that the residuals follow approximately a normal distribution, but the tails deviat e from the fitted line. The ACF and PACF of the residuals of the ARIMA(4,1,0) model are shown in Figures 7-12 and 7-13. No ACF at any lag is significant at a 5% confidence level. The PACF, however, is marginally si gnificant at lags 17 and 36. ResidualPercent 0.02 0.01 0.00 -0.01 -0.02 99.9 99 90 50 10 1 0.1 Fitted ValueResidual 0.3 0.2 0.1 0.0 0.02 0.01 0.00 -0.01 -0.02 ResidualFrequency 0.018 0.012 0.006 0.000 -0.006 -0.012 -0.018 40 30 20 10 0 Observation OrderResidual 160 140 120 100 80 60 40 20 1 0.02 0.01 0.00 -0.01 -0.02 Figure 7-11. Residual plot of log-inflation, ARIMA(4,1,0) PAGE 127 112 LagAutocorrelation 40 35 30 25 20 15 10 5 1 1. 0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-12. ACF of residuals of log-inflation ARIMA(4,1,0) model LagPartial Autocorrelation 40 35 30 25 20 15 10 5 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-13. PACF of residuals of log-inflation ARIMA(4,1,0) model PAGE 128 113 The Ljung-Box statistics for selected valu es of m are listed in Table 7-4. The statistic is significant at the 5% confidence level when the value of m is 6, 12, or 24. It is non-significant when m is 36 or 48. If we c hoose m=6 for better power performance, the p-value is 0.052, indicati ng no significant serial correlation in the residuals. Table 7-4. Ljung-Box Portmant eau statistic for AR(4) model Lag 6 12243648 Chi-Square 3.79 11.828.751.262.5 DF 1 7193143 P-Value 0.052 0.1070.0700.0130.028 In summary, the residuals from both the ARIMA(1,1,0) and the ARIMA(4,1,0) models distribute randomly around the zero leve l without significant pattern. The LjungBox tests indicate no significant correlation within the residuals. Both the ARIMA(1,1,0) and ARIMA(4,1,0) models are ad equate for the fit set data. Out-of-sample Model Verification The out-of-sample model verification is approached by means of forecast monitoring, where the out-of-sample one-step-fo recast errors are analyzed one at a time as each new observation is available. The fo rmula used to calculate the out-of-sample one-step-forecast errors is the same as for th e in-sample residuals, but the forecast is on an out-of-sample basis. The out-of-sample one-step-forecasts are made using the test set data ranging from July 2000 to December 2005. The ACF and PACF of the out-of-sample one-step-forecast errors for model ARIMA(1,1,0) are plotted in Figures 7-14 and 7-15. It can be seen that both ACF and PACF are non-significant at a ny lag at the 5% conf idence level. This indicates that the out-of-sample one-step-fo recast errors form a random series, and the ARIMA(1,1,0) model is adequate on the test set data. PAGE 129 114 LagAutocorrelation 24 22 20 18 16 14 12 10 8 6 4 2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-14. ACF of out-of-sample 1step-forecast errors, ARIMA(1,1,0) LagPartial Autocorrelation 24 22 20 18 16 14 12 10 8 6 4 2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-15. PACF of out-of-sample 1-step-forecast errors, ARIMA(1,1,0) PAGE 130 115 The ACF and PACF of the out-of-sample one-step-forecast errors for model ARIMA(4,1,0) are plotted in Fi gures 7-16 and 7-17. The ACF is significant at lag 6; the PACF is significant at lags 6 and 12 at the 5% confidence level. This indicates that the out-of-sample one-step-forecast errors do not form a random series, thus the ARIMA(4,1,0) model is inadequate on the test set data. Final Selection of Model From the above model checking it is seen that both candidate models passed the diagnostic test. The ARIMA(1,1,0) model is ad equate for the test set of data, but the ARIMA(4,1,0) is inadequate. So the ARIMA(1,1, 0) model is selected as the final model. The fitted model is t t tp p 1 1 0 (7-29) or t t t tp p p 1 1 0 1 (7-30) where, 001129 . 00 3925 . 01 005623 . 0 PAGE 131 116 LagAutocorrelation 16 14 12 10 8 6 4 2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-16. ACF of out-of-sample 1step-forecast errors, ARIMA(4,1,0) LagPartial Autocorrelation 16 14 12 10 8 6 4 2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Figure 7-17. PACF of out-of-sample 1-step-forecast errors, ARIMA(4,1,0) PAGE 132 117 Forecasting Point Forecast Using ARIMA Model Future values of the time series are forecasted recursively from the start time point which is called the forecast origin. The total number of forecast periods is called the forecast horizon. Suppose we are at forecast origin h and interested in forecasting l periods. All of the past and current realizations of {tp }, {tp }, and {t } are available at time h . Let ) ( l ph be the forecast of l hp , ) ( l phbe the forecast of l hp, and hF be the collection of information available at time h . Then the one-step-ahead forecast is h h l h hp F p E p 1 0) | ( ) 1 ( (7-31) ) 1 ( ) 1 ( h h hp p p (7-32) The two-step-ahead forecast is ) 1 ( ) | ( ) 2 ( 1 0 2h h h hp F p E p (7-33) ) 2 ( ) 1 ( ) 1 ( h h h hp p p p (7-34) The l -step-ahead forecast can be obtained as ) 1 ( ) ( 1 0 l p l ph h (7-35) l i h h hi p p l p1) ( ) ( (7-36) The fit set data from June 1986 to June 2000, 169 observations in total have been used to fit the model. Forecasts must be made for the period from July 2000 to December 2005, 66 months in total. Setting June 1986 as time period 1, the forecast origin is h = 169 (June 2000) and the forecast horizon is l = 66 (months). Figure 7-18 illustrates the out-of-sample forecasts of l hp using the fitted ARIMA(1,1,0) model. The two dashed lines denote the two standard-erro r limits of the forecasts. PAGE 133 118 log-in f lation Year Month 2004 2001 1998 1995 1992 1989 1986 Jun Jun Jun Jun Jun Jun Jun 0.03 0.02 0.01 0.00 -0.01 -0.02 0 Observations Forecasts Figure 7-18. Out-of-sample forecasts of log-inflation It is worth noting that the forecasts of l hp approach the steady level of 0.001858. Actually this is one of the important propert ies of stationary autoregressive models. In this case, the expectation of tp can be calculated as 001858 . 0 3925 . 0 1 001129 . 0 1 ) (1 0 tp E The non-zero expectation in the {tp } series indicates that there is a stochastic trend in the {tp} series. The log-PPI, ortp, is expected to increase by 0.001858 per period in the long run. This is reflec ted in the out-of-sample forecasts of tp, as seen in Figure 7-19. PAGE 134 119 l og-PPI Year Month 2004 2002 2000 1998 1996 1994 1992 1990 1988 1986 Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Observations Forecasts Figure 7-19. Out-of-sample forecasts of log-PPI with 95% confidence interval Estimating Forecast Error The ARIMA(1,1,0) model renders 1 1 0 1 h h hp p (7-37) The point forecast of 1hp is calculated using formula (7-31) as h hp p 1 0) 1 ( Then the associated error in one-step-ahead forecast of {tp } is 1 1) 1 ( ) 1 ( h h h hp p e (7-38) For two-step ahead forecast of {tp }, we have 2 1 1 0 2 h h hp p (7-39) The point forecast of 2hp is calculated using formula (7-33) as ) 1 ( ) 2 ( 1 0h hp p PAGE 135 120 Then the associated error in two-step ahead forecast of {tp } is 2 1 1 2)] 1 ( [ ) 2 ( ) 2 ( h h h h h hp p p p e 1 1 2 2 1) 1 ( h h h he (7-40) In general, the error in the l-step ahead forecast of {tp } is 1 1 1 3 3 1 2 2 1 1 1... ) ( h l l h l h l h l h hl e (7-41) Since {t } is a white noise with mean zero and variance 2, the mean of the forecast error is 0 )] ( [ l e Eh (7-42) and the variance is ] ... 1 [ )] ( [) 1 ( 2 1 2 1 2 1 2 l hl e Var (7-43) Since the expected value of the fo recast error is zero, the forecast ) ( l ph is an unbiased estimate of l hp, but the forecast is necessarily ina ccurate, especially when the forecast horizon l is long. By definition, l i i h h l hp p p1 (7-44) The point forecast of l hp, as expressed in Formula (7-36), is l i h h hi p p l p1) ( ) ( The associated forecast error can be calculated as l i h l i h i h h l h hi e i p p l p p l e1 1 *) ( ] ) ( [ ) ( ) ( (7-45) PAGE 136 121 Combining formulas (7-41) and (7-45) produces 1 0 1 1 1 *1 1 ) (l j j l h j hl e (7-46) It is reasonable to assume that ) (*l eh follows a normal distribution with zero mean. Its variance can be calculated as 2 1 0 1 1 1 2 *1 1 )] ( [ l j j hl e Var (7-47) In the fitted ARIMA(1,1,0) model, 3925 . 01 , 005623 . 0 . The estimated standard errors of l hp forecasts for select l â€™s are listed in Table 7-5. Table 7-5. Standard erro r of log-PPI forecasts Forecast horizon 1 12 24 36 48 60 Std error forecast 0.00562 0.03055 0.04429 0.05467 0.06338 0.07103 Since the mean of ) (*l eh is zero, the forecast of l hp is unbiased. However, the variance of the forecast error is an increasing function of l , meaning more uncertainty in long-term forecasts than in short-term for ecasts. As illustrated in Figure 7-19, the 95% confidence interval for log-PPI forecasts gets wider as the forecast horizon increases. Forecasting PPI Since the log-PPI series {tp } is a transform of the PPI series {tP }, the forecasts of PPI can be easily obtained using the forecasts of log-PPI. The l -step-ahead forecast of PPI can be calculated as )] ( exp[ ) ( l p l Ph h (7-48) However, cautions should be taken. Since it is assumed the forecast errors of l hp follow a normal distribution, the forecast errors of l hPshould follow a lognormal PAGE 137 122 distribution. Thus the ) ( l Ph calculated above is the maximum likelihood, and biased, estimate of l hP. Denoting the variance of l hp forecast error as 2 ) (l ph produces an unbiased estimate of l hP given by 2 ) ( exp ) ( 2 ) ( *l p h hhl p l P (7-49) The variance of l hp forecast error is given by 1 ) exp( ) ( 2 exp )] ( [2 ) ( 2 ) ( * l p l p h h l hh hl p l P P Var (7-50) The out-of-sample forecasts along with the observed values of PPI are illustrated in Figure 7-20. Both the biased and the unbiased po int forecasts are included. It can be seen that the biased forecasts are always lower than the unbiased forecasts. Observations Biased forecast Unbiased forecast 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Jun-86Jun-88Jun-90Jun-92Jun-94Jun-96Jun-98Jun-00Jun-02Jun-04PPI Figure 7-20. Out-of-sample forecasts of PPI PAGE 138 123 The observed values of the index within the forecast horizon deviate significantly from the point forecasts, either biased or unbiased. Of the 66 forecasts points, four observations are on or below the lower 2.5% line and one observati on is above the upper 2.5% line. There were two unusual change s in the economic conditions during the forecast horizon, which were not observed during the time period of the fit set data. One is the unusual economic depression after September 2001 which caused a sudden drop in the index. Another is the oil price shocks si nce 2004 which caused a fast increase in the index. Summary This chapter presents an empirical study of the use of the Box-Jenkins approach to modeling and forecasting future cost es calation. The objective was to provide probabilistic forecasts of future unit costs. Th e data used for this study is the highway and street construction PPI developed by the Bur eau of Labor Statistics. The ARIMA (1,1,0) model is found to fit the data well. PAGE 139 124 CHAPTER 8 ASSESSMENT OF WARRANTY RISK USING SIMULATION In the prior two chapters we have seen the development of probability-based models for quantifying the uncertainty in pa vement performance and the uncertainty in future unit costs. In this ch apter we will try to incorporate all these models and produce a probability distribution of the costs of future remedial works for warranted asphalt paving projects. Due to the complexity of the system, deriving formulae to assess the uncertainty of future repair costs is very difficult. Thus a Monte Carlo simulation approach is adopted for this analysis. Models developed in Chapters 6 and 7 will be used to generate outcomes of future remedial activities and un it prices. The simulation approach will be illustrated in a case study of a real warranted project. Assumptions Risk associated with pavement warra nties can be estimated in many ways, including intuitive judgment and analysis of historical data. However, an accurate assessment of warranty risk is almost impossible at this time due to the lack of available information. Many parameters necessary fo r the assessment are currently unknown and have to be estimated empirically or even in tuitively. Assumptions are also needed to simplify the modeling process since the wa rranty requirements and future pavement performance are usually very complicated. The simulation approach developed in th is chapter is based on the following assumptions: PAGE 140 125 The estimation method is based on historical data. In particular, the method uses the pavement distress models developed in Chapter 6 and the price uncertainty model developed in Chapter 7. Two basic types of warrantie s are the materials and workmanship warranty and the performance warranty. Under a performa nce warranty, the contractor usually assumes full responsibility on remedial work, even if the defects are due to factors out of the control of the contractor. Under a materials and workmanship warranty, the contractorâ€™s responsibility is limited to factors with in his control. However, no information on the causes of distress is available given the current data used in this analysis. Even in practice, the determin ation of distress cause s is difficult and sometimes inconclusive. So, in this assess ment, we assume that all the identified distresses are the potential responsibility of the contractor. Each project is independent of any other. There is no correlati on between the actual outcomes of warranted repair work between two projects. The contractor has no pavement performance data on the projects built by the company. So the estimate of exposure to warranty risk is based on the Florida DOT survey data which are from pr ojects done by many contractors. Various types of distresses may occur w ithin the warranty period. Each distress type occurs independently. The condition of warranted pavement will be evaluated annually at the end of each year within the warranty period. Remedial work, if needed, will be performed immediately after the evaluation or at the end of the warranty period. Evaluation of warranted pavement is conducted LOT by LOT which is usually defined as 0.1 mile per lane. The models developed in Chapter 6, however, are based on pavement survey sections which are much larger than the size of LOT. Since no information on within-project performance variation is available, the pavement evaluation is assumed to be made at the project level. Simulation Methodology Unlike many formula-based approaches which calculate the â€œexpectedâ€ results using formulae, the Monte Carlo simulation a pproach is achieved by generating a large number of â€œactualâ€ future outcomes and an alyzing the statistical aspects of these outcomes. An outline of the simulation approach is illustrated in Figure 8-1. PAGE 141 126 Figure 8-1. Remedial co st simulation flowchart Determine Failure Time for Each Distress Type In Chapter 6 probability distribution models for various types of distresses were developed. The cumulative distribution function, ) ( t F, is the probability that a pavement will fail by time t. ) ( t F is a strictly monotonic incr easing continuous function of t . If its inverse function, ) (1F , is well-defined for 1 0 , then Data input Failure occurs? Generate quantities of remedial work items Generate future unit costs for remedial work items Calculate total cost of remedial work Continue simulation? Statistical analysis of simulation data No remedial work needed, zero remedial cost N o Yes Yes N o PAGE 142 127 ) (1 F t (8-1) is the time that a pavement has a probability of to fail, and is uniformly distributed on ] 1 , 0 [, called a ) 1 , 0 ( U random variable. Thus the inversion of the CDF can be used to sample the failure time of a distress type. The Weibull distribution is fitted to mode l rutting and ride failure time in Chapter 6. The inversion of the CDF fo r Weibull can be expressed as / 1) 1 ln( t (8-2) where is the scale parameter and is the shape parameter. Loglogistic models are fitted in Chapte r 6 to model cracking failure time. The inversion of the loglogistic CDF can be expressed as 1 1 ln exp t (8-3) where is the location parameter and is the scale parameter. Lognormal is used to model initiation time of raveling, bleeding, and delamination. The exact formula for inverse lognormal CD F is very complicated. Fortunately, many software products provide func tions to calculate the inverse value of lognormal models. The values of the parameters for the models discussed above can be found in Chapter 6. The maximum likelihood estimates of the parameters are restated in Table 8-1. Table 8-1. Model parameters for pa vement failure/defect time simulation Failure/defect Model Rutting Weibull 1.55531 21.7492 Ride Weibull 4.73200 14.0424 Cracking Loglogistic 1.69968 0.26396 Raveling Lognormal 2.38193 0.63519 Bleeding Lognormal 7.57542 4.00969 Delamination Lognormal 3.34360 0.62194 PAGE 143 128 The warranty term will be denoted as W . In Florida, the current warranty term for asphalt pavements is three years, or . 3 W For any distress type, its failure time t can be simulated using the method discussed a bove. If a failure time is greater than the warranty term, or W t, then this type of distress does not occur within the warranty term. However, if W t, then the specific type of distress is formed within the warranty term and the contractor is responsib le for repairing the defects. The failure time simulated using the me thod discussed above can be any positive real number. However, pavement condition surveys are usually c onducted annually. If it is assumed that the failure or defect can be identified only at the time of survey and the contractor will be asked to perform remedial actions immediately after the survey, then the repair time can be assumed to be the survey time. To simplify the analysis, it is further assumed that the completion of th e construction and annual surveys are on the same date in each year; then the repair time will be an integer which can be calculated as eger an not is t if t eger an is t if t t int 1 ) int( int ' (8-4) where ' tis the time that the failure is identified, t is the time the failure â€œactuallyâ€ formed. Generate Size of Distressed Areas If a certain type of distress failure oc curs within the warranty term, then it is necessary to generate the size of the distress area that has to be repaired. It is ideal to develop a probability model for the size distribution of the defect area for each failure/defect type. However, this is not easy due to the availability of field data. In Chapter 6, we have devel oped size distribution models for cracking affected areas (actually it is the sum of cracking, raveling, and patching). But no su ch model exists for PAGE 144 129 other distress types, including rutting, ri de, bleeding, and delamination. Subjective judgments are needed to estimate these factors. Data input Three types of input data are allowed fo r simulating the size of distressed areas, including single values probability tables parametric models A single value is just a single estimate of the size of the dist ressed area. It is essentially assumed that the size of a distress ed area is a constant. For example, the area affected by bleeding, if it exists, is estim ated to be 1% of the entire pavement. A probability table provides an empirical estimate of the probability distribution of the size of the defective area. Table 6-16 is an example of a probability table for percent pavement cracked. A parametric model, if well-developed, can be used to simulate the size of defected areas. Lognormal models were fitted in Chap ter 6 to model the di stribution of percent pavement affected by cracking. The table-look-up method The sampling methods applied depend on the types of input data. For a single value estimate, the sampling results will always be equal to the single input value. For example, if a bleeding affected area is assumed to be 1% of the entire pavement, the sampling results will be 1% anyway. For a probability table input , a â€œtable-look-upâ€ method is applied for sampling. The algorithm is explained as follows. PAGE 145 130 Assume that a random variable x takes the discrete values1X, 2X, , nX , with probability ) Pr(i iX x q for n i ... 1 . The probability table is illustrated in Table 8-2. To simplify the description, a dd to the table a dummy outcome0X with zero probability, or0 ) Pr(0 X x . The cumulative distribution function (CDF) can be constructed by i j j i j i i iq X x X x X F1 1) Pr( ) Pr( ) ( (8-5) Table 8-2. A sample probability table Outcome ( x ) Probability ) Pr(iX x CDF ) Pr(iX x 0X 0 0 1X 1q 1q 2X 2q 2 1q q 3X 3q 3 2 1q q q nX nq 1.00 Select a ) 1 , 0 ( U random variable, . The table-look-up algorithm works by choosing a value for and finding the probability interval that lands. More specifically, if 1 0 0i j j i j jq q then we set 1iX x . Sampling method for size of cracking affected area As discussed in Chapter 6, after cracking initially forms, the size of cracked areas tends to grow over time. Cracking initiation time t can be simulated using Formula (83). If W t, then the cracking defect appears within the warranty period. The time t is translated into an integer number ' t using Formula (8-4). PAGE 146 131 If W t ', then cracking is formed in the last year of the warranty period. The size of the cracked area at the end of the warrant y period follows the fi rst-year distribution model as listed in Tables 6-16 and 6-17. If W t ', then cracking is formed at least one year before the warranty expires. The size of the cracked area will grow further over time. At the end of the warranty period, the size of the cracking affected area would have been growing for ' t W years and following a ) 1 ' ( t W -year distribution model. Table 6-16 gives empirical probability di stributions of the sizes of cracking affected areas from Year 1 to Year 5 afte r cracking initiation. Lognormal models were found to fit the distribution data well for the fi rst three years (see Table 6-17). Therefore, for the 1st, 2nd, or 3rd year distribution, bot h the inversion of the lognormal model in Table 6-17 and the table-look-up of the empirical CDF in Table 6-16 can be used to sample the size of cracking affected areas. For the 4th and 5th year distribution, however, only the table-look-up method can be used. It should be noted that the simulation so ftware can handle warranty terms for five years or less because the probability tables for the size of cracking affected areas were developed only for the first five years. Generate Future Unit costs An ARIMA(1,1,0) model was fitted for the monthly log-PPI series in Chapter 7. The fitted model (Formula 7-29) is restated as follows m m mp p 1 1 0 (8-6) where mis the time period measured in months, 001129 . 00 , 3925 . 01 , 005623 . 0 . PAGE 147 132 Suppose that the forecast origin h is the bid submission date, the construction is completed 0T months later. If remedial actions are performed at the end of the ' t year, then the forecast horizon l (in months) is calculated as ' 120t T l (8-7) The future values of the log-PPI series can be generated recursively from the forecast horizon h using formula (8-6). The simulation approach is significantly different from the forecast method described in Chapte r 7. First, the forecast method yields an expected, or average, value of future outco mes, while the simulation approach produces â€œactualâ€ values of future outcomes. Second, the forecast method sets all future shocks m to be zero, while the simulation approach monitors the â€œactualâ€ future shocks. The simulation approach is explained as follows. For each time period mfrom 1 h to l h , a random value is generated for variable m , which follows a normal distribution wi th a mean of zero and a standard deviation of . The results are listed as follows: 1 1 0 1 h h hp p 2 1 1 0 2 h h hp p l h l h l hp p 1 1 0 The â€œactualâ€ log-PPI at period l h is l h h m m h l hp p p (8-8) and the â€œactualâ€ future unit cost at period l h will be PAGE 148 133 l h h m m h l hp P P exp (8-9) where l hP is the future unit cost and hP is the current unit cost. Calculate Cost of Remedial Work When the â€œactualâ€ sizes of future defect areas and the â€œactualâ€ future unit costs are simulated, the calculation of repair cost is simple and straightforward. But caution should be taken because the required repair area is usually larger than the defect area. As an example, for bleeding defect, the FDOT (2005) warranty specifi cations require the contractor to â€œ[r]emove and repl ace the distressed area(s) to the full distressed depth, and to a minimum surface area of 150% of each distressed area.â€ The future cost of repair can be calculated using the following formula l h rP A x C (8-10) where, rC is the future cost of repair is a constant greater than 1 x is the percentage of pavement defected A is the total area of warranted pavement l hP is the future unit cost of repair The present value of the future repair cost is 12 / /) 1 ( ) (l r r rR C C PV C (8-11) Where, /rC is the present value of rC R is the discount rate l is forecast horizon in months. PAGE 149 134 The method described above can be used to simulate repair cost for any type of distress. If two or more types of distresses occur within the warranty period, the cost of each type of repair is calculated individually. The total cost of remedial work is the sum of all types of repairs. A Case Study: Florida Turnpike The realization of the simulation approach is illustrated in this section on a real warranted asphalt pavement resu rfacing project. This project is located on the Florida Turnpike. It is not an inters tate project, but since the de signated construction criteria for the Turnpike are identical to the interstate criteria, it is appropriate to analyze its pavement performance using interstate performance data. It should be noted that the purpose of this simulation is to demonstrate the methodology developed in this research. The simulation results of this analysis are undoubtedly inaccurate and probably biased be cause they involve the researcherâ€™s subjective judgments which may deviate signi ficantly from the reality. The simulated results cannot, therefore, be appl ied to any project, directly or indirectly, as an assessment of warranty risk. Overview of the Construction Project The project is located on Floridaâ€™s Tu rnpike from milepost 275.677 to milepost 281.834, with a total roadway length of 6.157 miles. The improvements under this contract consisted of milling, resurfacing, base work, drainage improvements, upgrading guardrail, highway signing, pavement marking, etc. The resurfaced pavement was subject to a three year warranty. It consisted of 2 existing southbound lanes for 6.157 miles in cluding shoulders. The southbound lanes were milled 2â€ average depth (the milling ranged from 1â€ to 4â€, depending on typical PAGE 150 135 section and location), an asphalt rubber memb rane interlayer was pl aced and 3â€-6.25â€ of traffic level â€œCâ€ with polymer modified asphalt was placed depending on the station location and proposed cross-slope. Six different typical sections were given with different milling depths and pavement layer thickness for each. A 0.75â€ FC-5 friction course with rubber was placed atop the structural course. The project was let on October 12, 2 004. Construction began on December 13, 2004 and was completed on September 20, 2005. The programmed budget was $14,599,025 and the winning bid was $12,666,667. A breakdown of the winning bid is listed in Table 8-3. If the costs of mobilization, maintenance of traffic, site clearing, and contingency are proportioned to the other ca tegories, the adjusted total cost for the warranted pavements (including shoulders) is $4,246,168.25, accounting for 33.52% of the total bid. Table 8-3. Summary of the winning bid No. Category Amount 1 Mobilization and maintenance of traffic 1,741,132.30 2 Clearing Construction site 78,035.85 3 Earthwork 1,545,529.88 4 Base 88,301.40 5 Asphalt surface courses (warranted) 3,586,056.20 6 Structures 362,054.23 7 Incidental construction 5,042,678.76 8 Traffic control 72,878.04 9 initial contingency amount 150,000.00 Sum 12,666,666.66 Data Input All the project specific data needed for the simulation are designed as inputs to the simulation program. These data include constr uction schedule, projec t size, unit costs of repair, etc. A detailed list of the input data is included in Table 84. The discount rate is the contractorâ€™s debt rate. PAGE 151 136 Table 8-4. Project specific input data for simulation Category Variable Value Remarks Bid letting October 2004 Construction completion September 2005 Schedule Warranty duration 3 years Pavement area 85,424 SY Project size Shoulder & median area 51,178 SY Ride $5.13/SY Rutting $36.22/SY Cracking (incl. raveling) $31.08/SY Pavement&shoulder Bleeding $31.08/SY Pavement&shoulder Current unit cost for repair Delamination $5.13/SY Pavement&shoulder Financial Discount rate 5% Debt rate Additional Assumptions Models for various types of pavement distre sses have been developed in Chapter 6. But there are still many parameters unknown. Before accurate estimates of these parameters are available, subjective estimates of these unknown parameters are needed to run the simulation. When a distressed area is to be repaired, th e repaired area is usually larger than the distressed area. The assumed factors of repair ed area to distressed area are listed in Table 8-5. Table 8-5. Assumed factors of re paired area to distressed area Type of distress Repaired area / distressed area Rutting, ride 110% Cracking, raveling, delamination, bleeding, pot holes, etc. 150% Distributions of the size of the affected area for cracking (including raveling and patching) are well monitored in Chapter 6. For other distress types, however, no such data are currently available. For the purpose of simulation, additional assumptions are made on the sizes of distressed ar eas, as listed in Table 8-6. PAGE 152 137 Table 8-6. Assumed distribution of size of distressed area Type of distress % pavement distressed Probability Rutting 10% 25% 40% 50% 35% 15% Ride 20% 35% 50% 50% 35% 15% Cracking (incl. raveling & potholes) (see models) Bleeding 1% Delamination 2% The warranted pavements under this cont ract include both travel lanes and shoulders. For shoulders, no requirements on rutti ng and ride distresses are specified in the warranty specifications. But the contractor st ill has to repair dist resses of other types including cracking, raveling, delamination, bl eeding, pot holes, etc. The distress models were developed using travel lane pavement performance data. No data were collected on shoulder distresses. In this simulation, we assume the shoulder and the travel lane pavements have similar performance so the pavement models can be applied to the shoulders too. The Simulation Results The simulation is performed in Micros oft Excel worksheets using an Excel simulation add-in called Simtools, which wa s developed by Professo r Roger Myerson at the Northwestern University. The output of the simulation is the present value (PV) of repair costs for each simulated outcome. A total of 50,000 simulation outcomes are collected for the given project. The simulated PVs of pavement repair costs range from $0 to $2,167,184, with an average of $59,129 and a standard deviation of $185,797. The distribution of the discounted repair costs is illustrated in Figures 8-2 and 8-3. PAGE 153 138 PV of repair costs (in $1,000)Probability (%) 2100 1800 1500 1200 900 600 300 0 90 80 70 60 50 40 30 20 10 0 Figure 8-2. Histogram of repair costs for the warranted pavement PV of repair costs (in $1,000)Cumulative Probability 2100 1800 1500 1200 900 600 300 0 100 80 60 40 20 0 Figure 8-3. Cumulative di stribution function for PV of repair costs PAGE 154 139 The simulated range of repair costs is reas onable if it is compared with two extreme scenarios. In the best case, the pavement is perfect without any repair within the warranty period. In the worst case, the pavement may fail completely and the contractor has to replace the entire pavement at his own e xpense (current cost is $4,246,168). Some findings from the simulated results are summarized as follows: The expected PV of future repair costs is about 1.39% of the (adjusted) bid price for milling and resurfacing work, and 0.47% of the total bid. The range of the outcomes is very large. The lower bound of repair costs is zero; while the theoretical upper bound is approxi mately the total m illing and resurfacing construction costs. The distribution of the discounted repair costs is highly skewed to the right. In most cases, the discounted repair cost s are very low. About 81.8% simulated outcomes yield a discounted repair cost of less than 1% of the bid price for the warranted work. However, there are also some unlikely scenarios that the repair cost is very high. Value at Risk (VaR) Measure of Warranty Risk Let us assume the contractor estimated th e future repair cost of the warranted pavement as its expected value, or the averag e of all of its possible future outcomes. The contractor will suffer a loss on the warranty if the actual discounted repair cost is higher than the expected value. On the other hand, the contractor will have a gain on the warranty if the actual repair cost is lower. The gain or loss on warranty can be calculated by subtracting the actual discounted repair cost from the expected cost. A positive number indicates a gain while a negative number means a loss. For the simulated results of the Florida Turnpike project, if we subtract each of the simulated repair costs from $59,129, the averag e repair cost, we can get the distribution of the contractorâ€™s gain/loss on the warranty, as illustrated in Figure 8-4. The figure indicates that it is very likely (84.68% probability) that the contractor will have a gain on PAGE 155 140 the warranty. But the maximum gain is only $59,129, when no repair work is needed. On the other hand, the probability of loss is relative low (15.32% ), however the loss tends to be large. Gain/loss on warranty (in $1,000)Probability (%) 0 -300 -600 -900 -1200 -1500 -1800 -2100 90 80 70 60 50 40 30 20 10 0 Loss region Gain region Figure 8-4. Probability distri bution of warranty gain/loss Standard deviation is a tr aditional and popular single m easure of uncertainty or risk. For the simulated warranty project discussed above, the simulated standard deviation of discounted warrant y period repair cost is $185,797. However, the use of standard deviation to measure risk ha s significant limits. A general assumption underlying this application is that the outco mes conform to a normal distribution. But obviously the distribution of the simulated result s, as illustrated in Figure 8-4, is neither normal nor symmetric. Value at Risk (VaR) is another popular, and probably the best, single riskmeasurement technique available. VaR is wi dely used by banks, securities firms, PAGE 156 141 commodity merchants, energy merchants, a nd other trading organi zations to measure their portfoliosâ€™ exposure to market risk. Value at Risk is defined as the value that is expected to be lost under severe adverse conditions. It is based on the probability di stribution for the value of an asset or a portfolio. In the contex t of our simulated pavement warrant y project, if a severe loss on warranty is defined as a loss that has a % probability to occur, then we call it a )% 1 ( VaR. It is equivalent to say that there is a % chance the actual loss will be greater than the )% 1 ( VaR. If the present value of warranty period repa ir cost is estimated to be $59,129, the 99% VaR and 95% VaR are calculated as $930,251 and $293,862, respectively (see Figure 8-5). That is to say, th ere is a 1% probability the lo ss on warranty is greater than $930,251, and there is a 5% probability the loss on warranty is greater than $293,862. Gain/loss on warranty (in $1,000)Probability (%) 0 -300 -600 -900 -1200 -1500 -1800 -2100 90 80 70 60 50 40 30 20 10 0 99% VaR 95% VaR Figure 8-5. VaR measure of warranty risk PAGE 157 142 Risk Attribution The simulation approach for estimating th e distribution of warranty risk and its VaR is described in the previ ous sections. However, this al one is not sufficient for risk measurement. It is usually necessary to decompose the overall risk and analyze the contribution of each risk component to overa ll risk exposure. The simulation approach used for measuring overall warranty risk can be further extended to analyze the risk contribution of each risk component. Stand-alone Risk for Each Risk Factor Three factors of warranty risk are iden tified in Chapter 4, including uncertain quantities of repair work, uncertain price esca lation for repair work, and timing of repair. The stand-alone risk for any risk factor is estimated by setting the variations on all other risk factors equal to zero. In other words, the stand-alone risk for a risk factor is calculated by replacing the variables of other risk factors with their expected values. The expected values for variables of all ri sk factors are summarized Table 8-7. The simulated resulting stand-alone risks for each ri sk factor are summarized in Table 8-8. Table 8-7. Expected values for risk factors Risk factor Variable Year 1 Year 2 Year 3 Total Rutting 0.1924 0.3402 0.4278 0.9605 Ride 0.0003 0.0042 0.0237 0.0282 Bleeding 0.0294 0.0136 0.0101 0.0531 Delamination 0.0000 0.0000 0.0003 0.0003 Cracking 0.0067 0.0883 0.3556 0.4506 Quantities (percent pavement defective) Raveling included in cracking Rutting 2.2451 Ride 2.8298 Bleeding 1.6357 Delamination 2.9333 Cracking 2.7742 Timing (year of repair) Raveling included in cracking Escalation Pt/P0 1.0588 1.0826 1.1071 PAGE 158 143 Table 8-8. Stand-alone risk for each risk factor Risk factor Escalation Quantity Timing All factors No. of replications 25000 50000 50000 50000 Mean $60,343 $60,552 $60,242 $59,129 Standard deviation $3,288 $185,185 $701 $185,797 Skewness 0.158 4.323 0.497 4.483 Minimum $47,574 $0 $59,360 $0 Maximum $75,327 $2,166,570 $62,589 $2,167,184 99% VaR $8,098 $876,343 $1,617 $930,251 95% VaR $5,573 $290,784 $916 $293,862 90% VaR $4,294 $173,672 $912 $129,054 The simulated results in Table 8-8 show th at the quantity of repair work constitutes the single largest factor of warranty risk. In fact, it constitutes almost the entire warranty risk. Comparatively, under the current th ree-year pavement warranty, the risk contributions from price escalation and tim ing of repair are minor and ignorable. Stand-alone Risk for Each Distress Type The stand-alone risk for a di stress type is the warranty risk if we ignore all other distress types. It is estimated by setting th e quantities of repair work for all other distresses equal to zero. The simulated results of distress-type-specific, stand-alone risks are summarized in Table 8-9. Some observations are as follows: The expected repair cost of the project ($59,129) is the sum of the expected repair costs of each individual di stress type. Rutting and cr acking (including raveling) contribute approxiamately 94.5% to the tota l repair cost. Bleeding contributes to 5.3% of the total repair cost. The contribu tions of ride and delamination are trivial and ignorable. Compared with the average level, the sta ndard deviation of repair costs for each individual distress type is very high, meaning the repair cost is highly uncertain for each individual distress type. Rutting and cracking (includi ng raveling), which contri bute significantly to the total expected cost, have also the high est risk contribution to the total risk (represented by both standard deviation a nd VaR). On the other hand, ride and delamination have the lowest risk contribution to the total risk. PAGE 159 144 Table 8-9. Stand-alone risk for each distress type Ride Rutting Bleeding Delamin. Cracking Total No. replications 50,000 50,00050,00050,00050,000 50,000 Mean $87 $30,129$3,142$4$25,768 $59,129 Std. deviation $3,501 $155,524$13,215$262$100,440 $185,797 Skewness 44.792 6.0293.98774.5965.905 4.483 Minimum $0 $0$0$0$0 $0 Maximum $236,648 $1,486,144$73,373$20,245$2,167,184 $2,167,184 99% VaR $0 $816,041$58,763$0$452,180 $930,251 95% VaR $0 $0$50,104$0$188,982 $293,862 90% VaR $0 $0$0$0$0 $129,054 In summary, rutting and cracking (includi ng raveling) account for most of the total warranty risk as well as the total expected re pair cost, thus constituting the major sources of warranty risk. The contractorâ€™s risk expos ure due to ride and delamination failure is trivial and ignorable. The Effect of Multiple Projects Up to now the analysis on warranty risk has been limited to one warranted project only. Since many states ar e trying to move pavement warranties from innovative contracting to standard practice, more and more contractors will be engaged in the construction of two or more warranted projects. This section will focus on the risk of warranty for multiple projects. Suppose a contractor won n bids on warranted pavement projects. The present value of warranty period repa ir costs for each project is iw , . ,..., 1 n i Each iw is a random variable following a distribution simila r to that illustrated in Figure 8-2. The average repair cost can be calculated as n i i r rC n C1 ) (1 (8-12) where rC is the average repair cost, ) (i rC is the repair cost for project i . PAGE 160 145 The expected value of wcan be easily calculated as n i i r rC E n C E1 ) () ( 1 ) ( (8-13) However, the distribution of rC is complicated due to the complexity in the distribution for ) ( i rC and the potential correl ation between projects. The simulation approach developed in the pr ior sections can be extended to analyze warranty risk for multiple projects. â€œActua lâ€ repair costs can be simulated for each project using the simulation prog ram, and the average repair cost can be calculated using Formula (8-12). With a large number of replications, a sample for rC can be obtained and its distribution can be easily analyzed. As an example, let us analyze the impact of multiple warranted projects on the winning contractor of the Tur npike warranty project. Suppose the contractor wins several bids on warranted paving projects. For simplic ity, assume all the warranted projects are identical in all aspects including size, desi gn, physical condition, schedule, and bid (although this will never happen). We fu rther assume no correlation between the outcomes of the projects. Statistics of the simulated results are summarized in Table 8-10. The distributions for total repair costs of tw o, four, and ten project s are illustrated in Figures 8-6 to 8-8. Table 8-10. Comparison of average re pair costs for multiple projects No. of projects 1 2 4 10 No. of replications 20,000 20,000 20,000 20,000 Mean $60,182 $59,354 $59,410 $59,772 Standard deviation $185,664 $131,755 $92,658 $58,934 Skewness 4.379 3.204 2.177 1.432 99% VaR $934,710 $592,578 $333,666 $195,425 95% VaR $292,226 $316,297 $210,004 $114,773 PAGE 161 146 PV of repair costs (in $1,000)Probability (%) 2100 1800 1500 1200 900 600 300 0 90 80 70 60 50 40 30 20 10 0 Figure 8-6. Histogram of average repa ir costs for two warranted projects PV of repair costs (in $1,000)Probability (%) 2100 1800 1500 1200 900 600 300 0 90 80 70 60 50 40 30 20 10 0 Figure 8-7. Histogram of average repa ir costs for four warranted projects PAGE 162 147 PV of repair costs (in $1,000)Probability (%) 2100 1800 1500 1200 900 600 300 0 90 80 70 60 50 40 30 20 10 0 Figure 8-8. Histogram of average repa ir costs for ten warranted projects As seen in Table 8-10, in this simplified example, the expected value of average repair costs is the same as that for a single project. However, the standard deviation and VaR decreases as the number of projects incr eases. The standard deviation and VaR for nprojects are approximately n / 1 of the corresponding values for a single project. The above observations can be explai ned by the Central Limit Theorem. The theorem states that the average of n observations from any population with mean and standard deviation will approach normal distribution as n increases. The mean of the sum is and the standard deviation of the sum is given by n /. This effect of multiple projects is called di versification. As more projects are put in a portfolio, the total risk of the portfolio is lo wer than the sum of risks for each individual project in the portfolio. Thus diversification reduces the total risk in a portfolio. PAGE 163 148 Some implications of the diversifica tion effect are summarized as follows: A large contractor engaged in many warrant ed projects can gather more benefits from diversification and thus have significant advantages over small contractors in doing warranty business. A small contractor , on the other hand, has higher risk exposure on warranties. The benefit of diversificati on is limited when a small number of warranted projects are in the portfolio. Currently, pavement warranties are regarded as an innovative contracting practice by most states. Only a few warranted projects are let each year in a state. As a result, even the larg e contractors canâ€™t en joy the benefit of significant diversification. The construction industry is dominated by sm all contractors. Th e total number of construction projects finished by each cont ractor is limited. Comparatively, each state DOT holds a much larger portfolio of projects than the contractors. So essentially, the pavement warranties tran sfer project failure risk from the well diversified parties to the less diversified parties. As noted above, each contractor holds a very limited number of construction projects. This is significantly different from the manufacturing industry where a manufacturer can produce thousands or milli ons of units of products each year. The benefits of diversificatio n can be realized in the manufacturing industry when warranties are implemented. But the diversif ication effect will be always limited in the construction industry. Summary This chapter details the simulation appro ach to quantifying warranty risk using the models developed in prior chapters. The simulation methodology is illustrated on an actual pavement warranty project on Florida Turnpike. The simulati on is further extended to analyze the impact of multiple projects on warr anty risk. Value at risk (VaR) is used to measure the warranty risk. PAGE 164 149 CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS Conclusions The federal and state Departments of Transportation (DOTs) are implementing warranties in highway construc tion projects, with an objecti ve to encourage contractor innovation, and to improve construction qua lity and highway performance. Various research projects have been conducted to de velop guidelines for im plementing warranties and evaluating the effectiveness of warrantie s. All these studies are from the ownerâ€™s perspective and provide guidelines to the DOTs. Prior research indicated that few contractors are familiar with warranties. One major concern of the contractors re garding pavement wa rranties is risk. Unfortunately the risk of pavement warran ties is not well-understood. Risks have long been recognized and well-studied in the c onstruction industry. But a comprehensive study on the risk of pavement warranties basically does not exist. This research is conducted from the contra ctorâ€™s perspective, with objectives to define clearly the risk of pavement warran ties for the contractors, and to develop an approach to model and measure this risk. The risk of warranties, from the contractorâ€™s viewpoint, is defined as the pr obability that the actual warranted repair cost exceeds the estimated repair cost, or the probability that the contractor will suffer a financial loss on warranties. Two major elements of warranty risk are identified: the uncertainty in quantities of repair items and the uncertainty in future escalations of unit repair costs. PAGE 165 150 A study on the uncertainty in pavement performance afte r construction was conducted using Florida DOTâ€™s annual asphalt pavement condition survey data on interstates. The data show that as the paveme nt ages increase, the average performance of the sample deteriorates and the dispersion of the performance usually increases, implying higher risk for long-term warranties. However, the statistical results of the performance data was found to be distorted and conditiona l because failed pavements were excluded from later year statistics. To reflect th e â€œtrueâ€ performance of the pavements, life distribution models were fitted to model th e occurring time for each type of pavement defect. Statistical distribution models were al so fitted to model the growth and dispersion sizes of cracking areas. The uncertainty in unit repair cost escal ation was assessed using historical cost data. The data used were the monthly PPI se ries for highway and street construction. Time series models are able to model both the systematic variation and the unexplained variation in the series. The systematic vari ation facilitates the computation of point forecasts while the description of unexplaine d variation helps to model the uncertainty. The Box-Jenkins approach was followed to f it the time series models. An ARIMA(1,1,0) model was found to fit the data well and was able to compute the probabilistic distribution of future escalations. The Monte Carlo simulation was adopted to quantify the risk of warranties using the pavement performance models and time series model developed in this study. The basic steps to generate a simulated outcome are as follows: Generate initiation time for each distress type using the life distribution models If a type of defect occurs within the wa rranty period, generate the severity of the defect (Percent pavement affected) PAGE 166 151 If repair is needed, generate the escalati on of unit repair cost using the fitted time series model Compute total repair costs. If no re pair needed, the cost is zero. Value at Risk (VaR) was adopted as a meas ure of warranty risk. A real project case study is performed to demonstr ate the approach developed in this study. The simulated results indicate that under the current threeyear asphalt pavement warranties in Florida, in most cases the warranty period repair cost s are trivial compared to the project costs; however, there are considerable chances th at the contractor may suffer huge loss on warranties. The concept of warranty risk is directly related to the risk premium that the contractor adds to his/her bi d on the warranted project. The approach developed in this study may act as a guide to the contractors in assessing their business risk exposure to pavement warranties. It may also help the DOTs in addressing wa rranty requirements and evaluating warranty effectiveness. Limitations Total validation of the results of this research may have been somewhat compromised by the following limitations. First, the pavement performance data were not prepared for the particular purpose of this research. The main purposes of the Florida DOTâ€™s pavement condition survey include rating pavement conditions, estim ating rehabilitation funding needs, and prioritizing rehabilitation projects. The survey provides valuable information for this study, but some data such as the percen t pavement affected by cracking are not satisfactorily accurate. Some necessary data like the sizes of bleeding and delamination areas are not available. PAGE 167 152 Second, under warranty specifications, the pavements should be evaluated for each LOT which is defined as 0.1 lane-mile of trav el way. However, the source data used for pavement performance analysis were organi zed by survey sections which ranged from less than 1 mile to more than 25 miles. No information was provided on the variation of pavement performance between LOTs within a construction project or a survey section. Third, the data sources used to asse ss pavement performance are from nonwarranted projects. Due to the limited numb ers of pavement warranty projects and the availability of performance data on warrant ed pavements, data from non-warranted pavements may act as a substitute for curr ent research. However, the performance of warranted pavements may deviat e from that of non-warranted. Fourth, the pavement distress models are developed using pavement performance data from projects done by many contractors. However, a contractorâ€™s performance may deviate significantly from the average level of all contractors. The variation of pavement performance from projects done by one contract or may also be lower than that from a pool of projects within the state. Fifth, it is assumed in the case study that the contractor is fu lly responsible for repair of all defects occurring within the warranty period. Since the contractorâ€™s warranty liability is limited to the sc ope under his/her control, this may overestimate the risk as well as the expected repair cost. Sixth, a key assumption in the analysis is that the performance of each individual project is independent. This assumption simp lified the analysis. But it should be noted that correlation of performance may exist when multiple projects are done by a single contractor. PAGE 168 153 Seventh, the fitted time-to-defect models are project-type specific and tied to the threshold values for the distresses to be warra nted. The data were from Florida interstate asphalt pavement projects. The results may not fit other categories of highway or interstate sections in other states. The models for rutting and ride failures are tied to the current threshold values of these distresses as specified in the warranty specifications. The models must be refitted if the thre shold values change. However, the methodology developed in this study is able to model proj ect defects for other t ypes of pavements or non-pavement projects. In addition, the distress models are built us ing data from interstate projects in the state of Florida. The approach developed in th is research can be used to model pavement performance of non-interstate projects or intersta te projects in other states. But the fitted models are appropriate for inters tate projects in Florida only. Finally, the modeling of pavement distresses is not complete. Due to the limitation of available information, some parameters neces sary for warranty risk assessment are still unknown. Subjective estimates of these parameters were used in the simulation. Recommendations for Future Research This study proposed a comprehensive appro ach to modeling and measuring the risk of warranties for contractors. Due to limitati ons in available data, the set of pavement distress models is not complete. Further studies are needed to model some unknown parameters, including but not limited to Variation of pavement performan ce within a construction project. Growth and dispersion of percent pavement affected by each distress type other than cracking. PAGE 169 154 The pavement distress models were de veloped using performance data from Florida interstate asphalt pavement re surfacing projects. However, the methodology developed in this study is able to handle th e performance modeling of other pavement or non-pavement projects. An expansion of th is approach to other warranted highway construction projects is recommended. Hopefully this work presents just a begi nning of an on-going effort to analyze the impact of warranties from the contractorâ€™s perspective. As it stands now, the contractorsâ€™ reactions to warranties, including decision on bidding and innovation, are highly affected by their risk exposure to warra nties. The success and effectiv eness of warranties, from the ownerâ€™s perspective, depend at least partia lly on the contractorsâ€™ reactions to it. Thus this study may build a foundation for a series of future studies on pavement warranties. Following are some potential research topics th at may relate to the concept of warranty risk developed in this study: Relationship between the contractorâ€™s warra nty risk exposure and the risk premium he/she charges, Incentives for contractor innovation, Warranty bonding issues, including the su rety companiesâ€™ risk exposure to warranty bonding and underwriting proce ss, and the DOTsâ€™ policy on warranty bonding, Optimizing length of warranty period consid ering the contractorâ€™s risk exposure, Determination of threshold values for performance parameters. PAGE 170 155 APPENDIX A LIST OF ROAD SECTIONS IN THE SAMPLE Table A-1. Road sections in the sample ID Completion year Roadway ID US Rd. RoadwayBMP EMP Length (miles) A01 1991 101900002752 1.200 4.112 2.912 A02 1991 101900002753 1.200 4.112 2.912 A03 1991 131750002752 1.307 3.566 2.259 A04 1991 131750002753 1.406 3.553 2.147 A05 1991 131750002752 3.884 5.327 1.443 A06 1991 131750002753 3.898 5.327 1.429 A07 1991 70220000952 14.335 21.453 7.118 A08 1991 70220000953 14.378 14.682 0.304 A09 1991 7528000043 0.000 7.698 7.698 A10 1991 7528000042 1.828 2.236 0.408 A11 1991 7528000042 2.236 7.647 5.411 A12 1991 7528000042 7.647 8.954 1.307 A13 1991 7528000043 7.698 8.747 1.049 A14 1991 7528000043 8.747 14.825 6.078 A15 1991 7528000042 8.954 10.752 1.798 A16 1991 7528000042 10.752 14.715 3.963 A17 1991 78080000952 15.300 19.950 4.650 A18 1991 78080000953 15.300 19.950 4.650 A19 1991 78080000952 31.500 34.855 3.355 A20 1991 78080000953 31.700 34.855 3.155 A21 1991 860950005953 9.971 11.171 1.200 A24 1991 870040001952 3.916 4.910 0.994 A25 1991 870040001953 3.916 4.910 0.994 A26 1991 87075000753 0.000 1.018 1.018 A27 1991 87075000752 1.018 5.442 4.424 A28 1991 87075000753 1.018 5.442 4.424 A29 1991 93220000952 0.000 4.509 4.509 A30 1991 93220000953 0.000 4.509 4.509 B01 1992 151700002752 4.247 5.495 1.248 B02 1992 151700002753 4.247 5.495 1.248 B03 1992 151700002752 5.495 7.310 1.815 B04 1992 151700002753 5.495 7.310 1.815 B05 1992 17075000753 34.445 37.160 2.715 B08 1992 7716000042 7.400 8.161 0.761 PAGE 171 156 Table A-1. Continued ID Completion year Roadway ID US Rd. RoadwayBMP EMP Length (miles) B09 1992 7716000043 7.400 8.900 1.500 B10 1992 7716000042 8.161 8.900 0.739 B11 1992 78080000952 0.000 15.300 15.300 B12 1992 78080000953 0.000 15.300 15.300 B13 1992 78080000952 19.950 26.180 6.230 B14 1992 78080000953 19.950 26.180 6.230 B15 1992 7911000042 14.668 25.274 10.606 B16 1992 7911000043 14.668 25.274 10.606 B20 1992 87075000752 0.000 0.333 0.333 C01 1993 3175000752 0.000 24.274 24.274 C02 1993 3175000752 24.274 30.089 5.815 C03 1993 15190000752 14.631 16.649 2.018 C06 1993 32100000752 0.000 8.874 8.874 C07 1993 32100000753 0.000 8.874 8.874 C08 1993 50001000102 20.437 31.538 11.101 C09 1993 50001000103 20.437 31.538 11.101 C10 1993 53002000102 13.609 19.504 5.895 C11 1993 53002000103 13.609 19.504 5.895 C12 1993 55320000102 15.630 19.755 4.125 C13 1993 55320000103 15.630 19.755 4.125 C14 1993 60002000102 18.100 24.061 5.961 C15 1993 60002000103 18.100 24.061 5.961 C16A 1993 70225000952 13.797 16.468 2.671 C16B 1993 70225000952 16.468 22.509 6.041 C17A 1993 70225000953 13.797 16.468 2.671 C17B 1993 70225000953 16.468 22.509 6.041 C18 1993 72290000952 4.314 10.468 6.154 C19 1993 72290000953 4.314 10.468 6.154 C20 1993 74160000952 0.000 11.989 11.989 C21 1993 74160000953 0.000 11.989 11.989 C22 1993 93220000952 15.565 17.000 1.435 C23 1993 93220000953 15.565 17.000 1.435 D01 1994 101900002752 0.000 1.200 1.200 D02 1994 101900002753 0.000 1.200 1.200 D04 1994 130750002752 3.750 8.288 4.538 D05 1994 130750002753 3.750 8.288 4.538 D06 1994 130750002752 12.896 15.723 2.827 D07 1994 130750002753 12.896 15.723 2.827 D08 1994 151900002753 14.631 16.649 2.018 D09 1994 1632000042 31.947 32.022 0.075 D10 1994 1632000043 31.947 32.022 0.075 PAGE 172 157 Table A-1. Continued ID Completion year Roadway ID US Rd. RoadwayBMP EMP Length (miles) D15 1994 50001000103 1.160 11.896 10.736 D16 1994 50001000102 1.278 11.896 10.618 D17 1994 53002000102 10.351 13.609 3.258 D18 1994 53002000103 10.351 13.609 3.258 D19 1994 58002000102 2.586 5.491 2.905 D20 1994 58002000103 2.586 5.491 2.905 D21A 1994 70220000952 0.000 14.335 14.335 D22 1994 73001000952 0.000 18.729 18.729 D23 1994 73001000953 0.000 18.729 18.729 D26 1994 87004000952 2.664 3.916 1.252 D27 1994 87004000953 2.664 3.916 1.252 D30 1994 9213000042 6.821 7.885 1.064 D31 1994 9213000043 7.058 7.885 0.827 D32 1994 93220000952 4.275 8.400 4.125 D33 1994 93220000953 4.275 8.400 4.125 D34 1994 93220000952 8.400 9.252 0.852 E01 1995 8150000752 0.000 3.846 3.846 E02 1995 8150000753 0.000 3.846 3.846 E05 1995 131750002753 0.000 1.406 1.406 E06 1995 131750002752 0.470 1.406 0.936 E07 1995 14140000752 0.000 8.173 8.173 E08 1995 14140000753 0.000 8.173 8.173 E09 1995 17075000752 29.039 34.340 5.301 E10 1995 17075000753 29.510 34.340 4.830 E13 1995 36210000752 0.000 4.094 4.094 E14 1995 36210000753 0.000 4.094 4.094 E15 1995 36210000752 5.968 13.140 7.172 E16 1995 36210000753 5.968 13.140 7.172 E17 1995 36210000752 18.664 22.500 3.836 E18 1995 36210000753 18.664 22.500 3.836 E19 1995 52002000102 16.627 21.224 4.597 E20 1995 52002000103 16.627 21.224 4.597 E21 1995 53002000102 0.000 8.680 8.680 E22 1995 53002000103 0.000 8.680 8.680 E23 1995 53002000102 8.680 10.351 1.671 E24 1995 53002000103 8.680 10.351 1.671 E25 1995 55320000102 4.573 8.576 4.003 E26 1995 55320000103 4.573 8.576 4.003 E27 1995 55320000102 8.576 15.630 7.054 E28 1995 55320000103 8.576 15.630 7.054 E29 1995 61001000102 12.908 17.380 4.472 PAGE 173 158 Table A-1. Continued ID Completion year Roadway ID US rd. No. RoadwayBMP EMP Length (miles) E30 1995 61001000103 12.908 17.380 4.472 E31 1995 61001000102 17.380 23.969 6.589 E32 1995 61001000103 17.380 23.969 6.589 E33 1995 70220000952 21.453 25.360 3.907 E34 1995 70220000952 25.360 31.252 5.892 E35 1995 70225000952 5.332 13.797 8.465 E36 1995 70225000953 5.332 13.797 8.465 E37 1995 72270000102 16.388 17.050 0.662 E38 1995 72270000103 16.388 17.162 0.774 E39 1995 7528000042 2.130 3.347 1.217 E40 1995 7528000043 2.130 3.347 1.217 E41 1995 86070000952 8.750 10.956 2.206 E42 1995 86070000953 8.750 10.956 2.206 E43 1995 87270000952 13.441 13.846 0.405 E44 1995 87270000953 13.441 13.846 0.405 F01 1996 101900002752 4.112 6.110 1.998 F02 1996 101900002753 4.112 6.110 1.998 F03 1996 14140000752 8.173 20.386 12.213 F04 1996 14140000753 8.173 20.386 12.213 F05 1996 151900002752 13.451 14.631 1.180 F06 1996 151900002753 13.451 14.631 1.180 F07 1996 18130000752 21.730 28.996 7.266 F08 1996 18130000753 21.740 28.996 7.256 F09A 1996 27090000102 0.000 9.439 9.439 F09B 1996 27090000102 9.439 25.462 16.023 F10 1996 27090000103 0.000 25.462 25.462 F11 1996 29180000752 19.369 26.718 7.349 F12 1996 29180000753 19.369 26.718 7.349 F13 1996 36210000752 4.094 5.968 1.874 F14 1996 36210000753 4.094 5.968 1.874 F15 1996 36210000752 13.140 18.664 5.524 F17 1996 57002000102 8.201 17.041 8.840 F18 1996 57002000103 8.201 17.041 8.840 F21 1996 70225000952 22.509 26.506 3.997 F22 1996 70225000953 22.509 26.506 3.997 F23 1996 70225000952 26.902 31.190 4.288 F24 1996 70225000953 26.902 31.190 4.288 F25 1996 720010002952 19.960 21.340 1.380 F26 1996 720010002953 19.960 21.490 1.530 F29 1996 74170000102 0.000 0.710 0.710 F30 1996 74170000103 0.000 0.710 0.710 PAGE 174 159 Table A-1. Continued ID Completion year Roadway ID US Rd. RoadwayBMP EMP Length (miles) F31A 1996 79002000952 0.000 6.771 6.771 F31B 1996 79002000952 6.771 27.402 20.631 F32A 1996 79002000953 0.000 6.771 6.771 F32B 1996 79002000953 6.771 27.402 20.631 F35 1996 79002000952 35.397 35.982 0.585 F36 1996 79002000953 35.397 35.982 0.585 F37 1996 89095000952 8.500 11.600 3.100 F38 1996 89095000953 8.450 11.500 3.050 F39 1996 89095000952 11.600 24.967 13.367 F40 1996 89095000953 11.500 24.967 13.467 F43 1996 93220000952 34.750 36.232 1.482 F44 1996 93220000953 34.750 36.232 1.482 F45 1996 94001000952 0.000 15.379 15.379 F46 1996 94001000953 0.000 15.379 15.379 G01 1997 13075000752 8.288 10.307 2.019 G02 1997 13075000753 8.288 10.307 2.019 G03 1997 13075000752 11.049 12.896 1.847 G04 1997 13075000753 11.049 12.896 1.847 G07 1997 151909002752 0.000 1.139 1.139 G08 1997 151909002753 0.000 1.139 1.139 G09 1997 1632000042 12.636 31.947 19.311 G10 1997 1632000043 12.636 31.947 19.311 G11 1997 26260000752 34.740 35.190 0.450 G12 1997 26260000753 34.740 35.190 0.450 G13 1997 29170000102 10.105 20.690 10.585 G14 1997 29170000103 10.105 20.690 10.585 G19 1997 29180000752 9.369 19.450 10.081 G20 1997 29180000753 9.369 19.450 10.081 G21 1997 36210000752 22.500 23.506 1.006 G22 1997 36210000753 22.500 38.282 15.782 G23 1997 36210000752 23.506 38.282 14.776 G24 1997 37120000102 15.099 25.523 10.424 G25 1997 37120000103 15.099 25.523 10.424 G26 1997 70220000952 31.252 41.503 10.251 G27 1997 70220000953 31.252 41.503 10.251 G28 1997 70225000952 3.195 4.505 1.310 G29 1997 70225000953 3.195 4.505 1.310 G30 1997 720010002952 4.970 5.698 0.728 G31 1997 720010002953 4.970 5.698 0.728 G32 1997 720010002952 8.786 9.628 0.842 G33 1997 720010002953 8.786 9.678 0.892 PAGE 175 160 Table A-1. Continued ID Completion year Roadway ID US Rd. RoadwayBMP EMP Length (miles) G34 1997 7528000042 18.775 19.197 0.422 G35 1997 7528000043 18.955 19.460 0.505 G36 1997 79002000952 27.402 31.528 4.126 G37 1997 79002000953 27.402 31.528 4.126 G38 1997 79002000952 35.982 45.804 9.822 G39 1997 79002000953 35.982 45.804 9.822 H01 1998 18130000752 15.329 21.730 6.401 H02 1998 18130000753 15.329 21.740 6.411 H03 1998 32100000752 19.175 28.746 9.571 H04 1998 32100000753 19.175 28.746 9.571 H05 1998 35090000102 0.000 11.333 11.333 H06 1998 35090000103 0.000 11.333 11.333 H07 1998 37120000102 5.861 15.099 9.238 H08 1998 37120000103 5.861 15.099 9.238 H09 1998 37130000752 0.000 3.277 3.277 H10 1998 37130000753 0.000 3.277 3.277 H12 1998 70225000953 0.000 3.342 3.342 H13 1998 70225000952 4.505 5.332 0.827 H14 1998 70225000953 4.665 5.332 0.667 H15 1998 72270000102 0.000 2.427 2.427 H16 1998 72270000103 0.000 2.623 2.623 H17 1998 72270000102 3.220 5.102 1.882 H18 1998 72270000103 3.547 13.233 9.686 H19 1998 72270000102 6.083 15.864 9.781 H20 1998 7911000042 0.503 14.668 14.165 H21 1998 7911000043 0.503 14.668 14.165 H22 1998 9213000043 3.226 3.836 0.610 H23 1998 9213000042 3.226 3.887 0.661 H24 1998 9213000043 4.557 5.300 0.743 H25 1998 9213000042 4.618 5.175 0.557 H26 1998 9213000042 5.175 6.250 1.075 H27 1998 9213000043 5.300 6.301 1.001 H28 1998 29180000752 26.718 30.447 3.729 H29 1998 29180000753 26.718 30.447 3.729 PAGE 176 161 APPENDIX B COMPUTER PRINTOUTS FOR PAVEMENT FAILURE MODELING Distribution Analysis, Rutting, Least Squares Variable Start: Start End: End Frequency: Rutting Censoring Information Count Right censored value 175 Interval censored value 57 Estimation Method: Least Squares (failure time(X) on rank(Y)) Distribution: Weibull Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 1.48121 0.197887 1.13998 1.92457 Scale 22.9150 3.43498 17.0814 30.7408 Log-Likelihood = -254.266 Goodness-of-Fit Anderson-Darling (adjusted) = 129.482 Correlation Coefficient = 0.989 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 20.7188 3.39406 15.0288 28.5631 Standard Deviation 14.2320 3.91784 8.29741 24.4111 Median 17.8919 2.23785 14.0020 22.8624 First Quartile(Q1) 9.88138 0.893639 8.27633 11.7977 Third Quartile(Q3) 28.5686 4.98077 20.2996 40.2060 Interquartile Range(IQR) 18.6873 4.51621 11.6368 30.0095 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.1 0.216217 0.110538 0.0793828 0.588918 1 1.02646 0.316757 0.560617 1.87938 PAGE 177 162 2 1.64459 0.409575 1.00942 2.67945 3 2.16990 0.466399 1.42391 3.30672 4 2.64423 0.505798 1.81752 3.84698 5 3.08495 0.534900 2.19613 4.33350 6 3.50141 0.557358 2.56299 4.78341 7 3.89933 0.575358 2.92007 5.20700 8 4.28259 0.590349 3.26866 5.61104 9 4.65393 0.603367 3.60964 6.00033 10 5.01539 0.615191 3.94362 6.37843 20 8.32400 0.761298 6.95798 9.95822 30 11.4248 1.07428 9.50185 13.7368 40 14.5602 1.57003 11.7864 17.9867 50 17.8919 2.23785 14.0020 22.8624 60 21.6017 3.10371 16.3000 28.6277 70 25.9744 4.24913 18.8494 35.7926 80 31.5973 5.87702 21.9447 45.4958 90 40.2402 8.64462 26.4120 61.3084 91 41.4743 9.06192 27.0268 63.6446 92 42.8332 9.52721 27.6982 66.2384 93 44.3492 10.0532 28.4403 69.1573 94 46.0689 10.6582 29.2738 72.4998 95 48.0638 11.3710 30.2302 76.4181 96 50.4526 12.2390 31.3614 81.1655 97 53.4542 13.3512 32.7626 87.2139 98 57.5525 14.9060 34.6422 95.6144 99 64.2530 17.5307 37.6402 109.682 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.990378 0.976369 0.996098 2 0.973367 0.949466 0.986046 3 0.951976 0.921111 0.970955 4 0.927407 0.891575 0.951716 5 0.900431 0.860823 0.929229 6 0.871622 0.828679 0.904422 7 0.841439 0.794925 0.878213 8 0.810260 0.759407 0.851422 9 0.778409 0.722122 0.824689 10 0.746162 0.683256 0.798433 11 0.713759 0.643159 0.772878 12 0.681408 0.602274 0.748112 13 0.649288 0.561080 0.724149 14 0.617556 0.520042 0.700964 15 0.586345 0.479589 0.678523 Distribution Analysis, Rutting, Maximum Likelihood Variable Start: Start End: End Frequency: Rutting Censoring Information Count Right censored value 175 Interval censored value 57 PAGE 178 163 Estimation Method: Maximum Likelihood Distribution: Weibull Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 1.55531 0.196767 1.21375 1.99299 Scale 21.7492 2.93836 16.6896 28.3428 Log-Likelihood = -254.186 Goodness-of-Fit Anderson-Darling (adjusted) = 129.482 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 19.5541 2.84683 14.6999 26.0113 Standard Deviation 12.8418 3.19993 7.87994 20.9281 Median 17.1831 1.95071 13.7553 21.4652 First Quartile(Q1) 9.76220 0.832127 8.26022 11.5373 Third Quartile(Q3) 26.8319 4.20254 19.7393 36.4729 Interquartile Range(IQR) 17.0697 3.78295 11.0557 26.3553 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.1 0.256276 0.119129 0.103046 0.637361 1 1.12961 0.319066 0.649385 1.96497 2 1.76967 0.405161 1.12983 2.77187 3 2.30430 0.457025 1.56212 3.39909 4 2.78168 0.492690 1.96581 3.93615 5 3.22156 0.518936 2.34938 4.41754 6 3.63447 0.539178 2.71746 4.86091 7 4.02681 0.555432 3.07292 5.27680 8 4.40288 0.569013 3.41768 5.67210 9 4.76573 0.580847 3.75306 6.05165 10 5.11760 0.591619 4.08003 6.41903 20 8.29108 0.720267 6.99302 9.83008 30 11.2092 0.983289 9.43855 13.3120 40 14.1213 1.39630 11.6335 17.1412 50 17.1831 1.95071 13.7553 21.4652 60 20.5605 2.66585 15.9466 26.5093 70 24.5063 3.60564 18.3670 32.6976 80 29.5343 4.93058 21.2924 40.9666 90 37.1821 7.16034 25.4926 54.2318 91 38.2672 7.49448 26.0689 56.1735 92 39.4605 7.86648 26.6977 58.3244 93 40.7894 8.28628 27.3923 60.7390 PAGE 179 164 94 42.2944 8.76838 28.1717 63.4968 95 44.0368 9.33515 29.0652 66.7203 96 46.1187 10.0239 30.1210 70.6131 97 48.7282 10.9040 31.4271 75.5536 98 52.2798 12.1304 33.1766 82.3827 99 58.0609 14.1912 35.9612 93.7418 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.991720 0.979637 0.996645 2 0.975859 0.954073 0.987378 3 0.955126 0.926068 0.972929 4 0.930698 0.896209 0.954019 5 0.903374 0.864643 0.931460 6 0.873772 0.831315 0.906146 7 0.842405 0.796093 0.879001 8 0.809710 0.758874 0.850887 9 0.776069 0.719692 0.822515 10 0.741815 0.678761 0.794384 11 0.707246 0.636468 0.766792 12 0.672623 0.593308 0.739888 13 0.638173 0.549817 0.713732 14 0.604100 0.506529 0.688334 15 0.570577 0.463937 0.663686 Distribution Analysis, Ride, Least Squares * NOTE * 17 cases were used * NOTE * 4 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Ride Censoring Information Count Right censored value 189 Interval censored value 43 Estimation Method: Least Squares (failure time(X) on rank(Y)) Distribution: Weibull Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 4.45257 0.606958 3.40862 5.81626 Scale 14.3533 0.783123 12.8977 15.9733 Log-Likelihood = -168.651 Goodness-of-Fit PAGE 180 165 Anderson-Darling (adjusted) = 82.546 Correlation Coefficient = 0.994 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 13.0904 0.637544 11.8987 14.4016 Standard Deviation 3.33244 0.539415 2.42650 4.57661 Median 13.2192 0.613776 12.0693 14.4786 First Quartile(Q1) 10.8500 0.377864 10.1341 11.6165 Third Quartile(Q3) 15.4459 0.967153 13.6620 17.4627 Interquartile Range(IQR) 4.59585 0.778636 3.29726 6.40586 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.1 3.04252 0.525658 2.16856 4.26871 1 5.10816 0.533950 4.16188 6.26959 2 5.97539 0.505836 5.06185 7.05380 3 6.55258 0.480702 5.67502 7.56584 4 6.99801 0.459007 6.15380 7.95804 5 7.36625 0.440246 6.55201 8.28169 6 7.68318 0.423978 6.89556 8.56076 7 7.96329 0.409873 7.19914 8.80854 8 8.21556 0.397686 7.47194 9.03318 9 8.44599 0.387230 7.72013 9.24009 10 8.65879 0.378358 7.94809 9.43304 20 10.2483 0.356536 9.57280 10.9715 30 11.3867 0.412050 10.6071 12.2236 40 12.3434 0.503556 11.3949 13.3709 50 13.2192 0.613776 12.0693 14.4786 60 14.0743 0.739152 12.6976 15.6002 70 14.9644 0.883876 13.3285 16.8010 80 15.9724 1.06172 14.0213 18.1950 90 17.3102 1.31645 14.9131 20.0926 91 17.4851 1.35111 15.0277 20.3442 92 17.6736 1.38884 15.1508 20.6165 93 17.8793 1.43038 15.2845 20.9145 94 18.1070 1.47682 15.4320 21.2457 95 18.3641 1.52984 15.5977 21.6212 96 18.6629 1.59216 15.7892 22.0595 97 19.0251 1.66876 16.0201 22.5938 98 19.4985 1.77049 16.3196 23.2965 99 20.2261 1.93034 16.7754 24.3864 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.99999 0.999882 1.00000 2 0.99985 0.998860 0.99998 3 0.99906 0.995695 0.99980 PAGE 181 166 4 0.99662 0.988935 0.99897 5 0.99090 0.976966 0.99642 6 0.97963 0.958026 0.99018 7 0.95995 0.930132 0.97720 8 0.92860 0.890827 0.95365 9 0.88237 0.836524 0.91600 10 0.81868 0.761667 0.86328 11 0.73652 0.660079 0.79840 12 0.63730 0.531214 0.72553 13 0.52549 0.385806 0.64748 14 0.40863 0.244479 0.56632 15 0.29618 0.129760 0.48431 Distribution Analysis, Ride, Maximum Likelihood * NOTE * 17 cases were used * NOTE * 4 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Ride Censoring Information Count Right censored value 189 Interval censored value 43 Estimation Method: Maximum Likelihood Distribution: Weibull Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 4.73200 0.598747 3.69267 6.06385 Scale 14.0424 0.668777 12.7909 15.4163 Log-Likelihood = -168.530 Goodness-of-Fit Anderson-Darling (adjusted) = 82.552 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 12.8522 0.546335 11.8248 13.9688 Standard Deviation 3.09554 0.453713 2.32261 4.12570 Median 12.9958 0.532210 11.9935 14.0819 First Quartile(Q1) 10.7918 0.348400 10.1301 11.4967 Third Quartile(Q3) 15.0459 0.818153 13.5249 16.7380 Interquartile Range(IQR) 4.25411 0.654601 3.14652 5.75158 PAGE 182 167 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.1 3.26216 0.499348 2.41663 4.40353 1 5.31187 0.497567 4.42094 6.38234 2 6.15641 0.470424 5.30012 7.15105 3 6.71443 0.447312 5.89254 7.65096 4 7.14307 0.427778 6.35198 8.03269 5 7.49621 0.411092 6.73227 8.34683 6 7.79930 0.396737 7.05922 8.61698 7 8.06657 0.384348 7.34736 8.85617 8 8.30680 0.373664 7.60579 9.07242 9 8.52585 0.364490 7.84057 9.27102 10 8.72783 0.356675 8.05602 9.45566 20 10.2277 0.333731 9.59410 10.9032 30 11.2934 0.373760 10.5841 12.0502 40 12.1840 0.444585 11.3431 13.0873 50 12.9958 0.532210 11.9935 14.0819 60 13.7853 0.633191 12.5985 15.0839 70 14.6042 0.750498 13.2049 16.1518 80 15.5280 0.895030 13.8693 17.3852 90 16.7489 1.10203 14.7225 19.0543 91 16.9080 1.13017 14.8319 19.2748 92 17.0795 1.16079 14.9495 19.5131 93 17.2665 1.19449 15.0771 19.7738 94 17.4733 1.23215 15.2178 20.0632 95 17.7068 1.27513 15.3759 20.3909 96 17.9776 1.32561 15.5585 20.7729 97 18.3058 1.38761 15.7785 21.2379 98 18.7340 1.46987 16.0637 21.8482 99 19.3911 1.59891 16.4974 22.7924 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 1.00000 0.999939 1.00000 2 0.99990 0.999281 0.99999 3 0.99933 0.996940 0.99985 4 0.99738 0.991433 0.99920 5 0.99248 0.980943 0.99704 6 0.98227 0.963343 0.99147 7 0.96358 0.936168 0.97935 8 0.93260 0.896426 0.95644 9 0.88529 0.840057 0.91835 10 0.81825 0.761172 0.86291 11 0.72986 0.653313 0.79220 12 0.62168 0.516101 0.71065 13 0.49947 0.362337 0.62206 14 0.37314 0.216692 0.52969 15 0.25503 0.104765 0.43711 PAGE 183 168 Distribution Analysis, Cracking, Least Squares * NOTE * 22 cases were used * NOTE * 5 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Cracking Censoring Information Count Right censored value 40 Interval censored value 192 Estimation Method: Least Squares (failure time(X) on rank(Y)) Distribution: Loglogistic Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 1.69323 0.0327192 1.62910 1.75736 Scale 0.289530 0.0199759 0.252910 0.331453 Log-Likelihood = -469.416 Goodness-of-Fit Anderson-Darling (adjusted) = 1.036 Correlation Coefficient = 0.997 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 6.26601 0.244588 5.80450 6.76420 Standard Deviation 4.02699 0.519504 3.12731 5.18550 Median 5.43703 0.177895 5.09930 5.79712 First Quartile(Q1) 3.95567 0.153287 3.66636 4.26781 Third Quartile(Q3) 7.47313 0.299174 6.90917 8.08311 Interquartile Range(IQR) 3.51745 0.279612 3.00998 4.11048 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 0.0511270 0.0164877 0.0271738 0.0961944 0.0001 0.0995820 0.0275608 0.0578893 0.171302 0.001 0.193960 0.0448347 0.123297 0.305120 0.01 0.377793 0.0701686 0.262517 0.543688 0.1 0.736032 0.103538 0.558673 0.969697 1 1.43736 0.138502 1.18999 1.73614 2 1.76197 0.146737 1.49662 2.07437 3 1.98734 0.150467 1.71327 2.30525 4 2.16645 0.152496 1.88726 2.48694 PAGE 184 169 5 2.31805 0.153670 2.03561 2.63969 6 2.45121 0.154351 2.16661 2.77319 7 2.57103 0.154723 2.28498 2.89289 8 2.68076 0.154894 2.39373 3.00220 9 2.78254 0.154930 2.49487 3.10339 10 2.87792 0.154874 2.58984 3.19806 20 3.63954 0.153280 3.35118 3.95271 30 4.25423 0.154451 3.96203 4.56798 40 4.83479 0.161623 4.52817 5.16217 50 5.43703 0.177895 5.09930 5.79712 60 6.11428 0.207776 5.72031 6.53538 70 6.94867 0.259621 6.45801 7.47661 80 8.12224 0.354147 7.45696 8.84688 90 10.2717 0.571722 9.21013 11.4557 91 10.6238 0.611484 9.49047 11.8925 92 11.0272 0.658232 9.80970 12.3958 93 11.4978 0.714289 10.1797 12.9866 94 12.0599 0.783253 10.6184 13.6970 95 12.7526 0.871053 11.1547 14.5794 96 13.6450 0.988384 11.8391 15.7265 97 14.8748 1.15718 12.7712 17.3249 98 16.7774 1.43281 14.1916 19.8343 99 20.5664 2.02633 16.9548 24.9473 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.997123 0.993533 0.998723 2 0.969355 0.949994 0.981367 3 0.886325 0.846142 0.917042 4 0.742714 0.690093 0.789131 5 0.571853 0.516457 0.625504 6 0.415736 0.361675 0.471904 7 0.294689 0.244655 0.350211 8 0.208512 0.164593 0.260496 9 0.149221 0.112090 0.195937 10 0.108648 0.077891 0.149580 11 0.080630 0.055361 0.116017 12 0.060977 0.040233 0.091397 13 0.046940 0.029855 0.073066 14 0.036729 0.022580 0.059207 15 0.029169 0.017374 0.048574 Distribution Analysis, Cracking, Maximum Likelihood * NOTE * 22 cases were used * NOTE * 5 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Cracking Censoring Information Count Right censored value 40 PAGE 185 170 Interval censored value 192 Estimation Method: Maximum Likelihood Distribution: Loglogistic Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 1.69968 0.0304412 1.64002 1.75935 Scale 0.263958 0.0164217 0.233657 0.298189 Log-Likelihood = -468.309 Goodness-of-Fit Anderson-Darling (adjusted) = 0.984 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 6.15363 0.213330 5.74940 6.58628 Standard Deviation 3.46245 0.370797 2.80690 4.27110 Median 5.47222 0.166581 5.15528 5.80865 First Quartile(Q1) 4.09471 0.141266 3.82699 4.38116 Third Quartile(Q3) 7.31314 0.265102 6.81157 7.85163 Interquartile Range(IQR) 3.21842 0.232874 2.79289 3.70880 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 0.0777063 0.0205705 0.0462514 0.130553 0.0001 0.142697 0.0324207 0.0914158 0.222745 0.001 0.262044 0.0497345 0.180642 0.380126 0.01 0.481218 0.0734287 0.356827 0.648972 0.1 0.883901 0.102325 0.704472 1.10903 1 1.62704 0.129775 1.39157 1.90236 2 1.95895 0.135604 1.71041 2.24360 3 2.18614 0.138078 1.93160 2.47424 4 2.36508 0.139344 2.10715 2.65458 5 2.51551 0.140028 2.25550 2.80549 6 2.64691 0.140392 2.38557 2.93689 7 2.76463 0.140568 2.50240 3.05433 8 2.87199 0.140628 2.60918 3.16128 9 2.97125 0.140618 2.70804 3.26004 10 3.06396 0.140567 2.80048 3.35223 20 3.79529 0.140312 3.53001 4.08050 30 4.37555 0.143296 4.10352 4.66562 40 4.91680 0.151398 4.62884 5.22267 50 5.47222 0.166581 5.15528 5.80865 60 6.09038 0.192002 5.72545 6.47856 70 6.84375 0.233895 6.40034 7.31788 PAGE 186 171 80 7.89009 0.307923 7.30908 8.51729 90 9.77336 0.474027 8.88707 10.7480 91 10.0783 0.503978 9.13741 11.1161 92 10.4266 0.539062 9.42184 11.5385 93 10.8315 0.580963 9.75068 12.0322 94 11.3133 0.632273 10.1395 12.6229 95 11.9042 0.697249 10.6132 13.3523 96 12.6614 0.783526 11.2152 14.2941 97 13.6977 0.906654 12.0311 15.5951 98 15.2864 1.10555 13.2661 17.6143 99 18.4047 1.52657 15.6432 21.6536 99.9 33.8784 4.03428 26.8264 42.7844 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.998405 0.996440 0.999286 2 0.978400 0.964697 0.986857 3 0.906967 0.873154 0.932463 4 0.766253 0.716225 0.809803 5 0.584651 0.528562 0.638629 6 0.413667 0.358663 0.470913 7 0.282353 0.232986 0.337576 8 0.191746 0.150264 0.241428 9 0.131825 0.098301 0.174567 10 0.092450 0.065833 0.128351 11 0.066288 0.045247 0.096129 12 0.048578 0.031895 0.073324 13 0.036332 0.023018 0.056900 14 0.027685 0.016969 0.044859 15 0.021454 0.012751 0.035879 Distribution Analysis, Raveling, Least Squares * NOTE * 25 cases were used * NOTE * 2 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Raveling Censoring Information Count Right censored value 137 Interval censored value 95 Estimation Method: Least Squares (failure time(X) on rank(Y)) Distribution: Lognormal Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper PAGE 187 172 Location 2.39731 0.0586878 2.28228 2.51234 Scale 0.629249 0.0498535 0.538746 0.734955 Log-Likelihood = -341.521 Goodness-of-Fit Anderson-Darling (adjusted) = 27.475 Correlation Coefficient = 0.995 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 13.4004 1.07949 11.4432 15.6924 Standard Deviation 9.34003 1.56338 6.72775 12.9666 Median 10.9936 0.645189 9.79904 12.3337 First Quartile(Q1) 7.19142 0.349784 6.53752 7.91072 Third Quartile(Q3) 16.8059 1.38424 14.3006 19.7503 Interquartile Range(IQR) 9.61452 1.21623 7.50327 12.3198 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 0.417126 0.0965734 0.264970 0.656656 0.0001 0.552238 0.115881 0.366025 0.833184 0.001 0.750986 0.139858 0.521328 1.08182 0.01 1.05879 0.169533 0.773594 1.44912 0.1 1.57269 0.205396 1.21751 2.03147 1 2.54329 0.244939 2.10581 3.07166 2 3.01920 0.256197 2.55659 3.56550 3 3.36633 0.262395 2.88940 3.92198 4 3.65353 0.266672 3.16652 4.21543 5 3.90512 0.270021 3.41018 4.47189 6 4.13286 0.272888 3.63118 4.70386 7 4.34346 0.275513 3.83568 4.91845 8 4.54111 0.278042 4.02758 5.12010 9 4.72866 0.280573 4.20952 5.31182 10 4.90814 0.283173 4.38336 5.49575 20 6.47353 0.320300 5.87523 7.13275 30 7.90371 0.388018 7.17865 8.70200 40 9.37355 0.493692 8.45420 10.3929 50 10.9936 0.645189 9.79904 12.3337 60 12.8936 0.858520 11.3161 14.6910 70 15.2914 1.16910 13.1634 17.7633 80 18.6697 1.66582 15.6743 22.2375 90 24.6241 2.66620 19.9157 30.4456 91 25.5587 2.83500 20.5647 31.7655 92 26.6143 3.02902 21.2931 33.2653 93 27.8254 3.25581 22.1230 34.9977 94 29.2433 3.52677 23.0871 37.0409 95 30.9487 3.86004 24.2370 39.5192 96 33.0800 4.28711 25.6596 42.6461 97 35.9022 4.86951 27.5214 46.8351 98 40.0300 5.75291 30.2034 53.0538 PAGE 188 173 99 47.5205 7.44033 34.9630 64.5884 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.999930 0.999512 0.99999 2 0.996618 0.990759 0.99890 3 0.980486 0.963934 0.99008 4 0.945940 0.918568 0.96546 5 0.894730 0.858768 0.92355 6 0.832060 0.789215 0.86885 7 0.763425 0.714171 0.80761 8 0.693275 0.637508 0.74489 9 0.624748 0.562578 0.68383 10 0.559829 0.491910 0.62603 11 0.499629 0.427081 0.57219 12 0.444643 0.368840 0.52253 13 0.394961 0.317322 0.47702 14 0.350421 0.272272 0.43549 15 0.310714 0.233208 0.39769 Distribution Analysis, Raveling, Maximum Likelihood * NOTE * 25 cases were used * NOTE * 2 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Raveling Censoring Information Count Right censored value 137 Interval censored value 95 Estimation Method: Maximum Likelihood Distribution: Lognormal Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 2.38193 0.0576517 2.26894 2.49493 Scale 0.635188 0.0508904 0.542882 0.743189 Log-Likelihood = -341.438 Goodness-of-Fit Anderson-Darling (adjusted) = 27.500 Characteristics of Distribution PAGE 189 174 Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 13.2456 1.05448 11.3320 15.4823 Standard Deviation 9.33793 1.56375 6.72522 12.9657 Median 10.8258 0.624127 9.66914 12.1209 First Quartile(Q1) 7.05338 0.346332 6.40622 7.76591 Third Quartile(Q3) 16.6159 1.34916 14.1713 19.4823 Interquartile Range(IQR) 9.56256 1.19886 7.47925 12.2261 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 0.398271 0.0952665 0.249213 0.636483 0.0001 0.528674 0.114742 0.345498 0.808966 0.001 0.721031 0.139085 0.494037 1.05232 0.01 1.01985 0.169476 0.736357 1.41250 0.1 1.52053 0.206677 1.16492 1.98469 1 2.47012 0.248585 2.02794 3.00871 2 2.93709 0.260733 2.46805 3.49526 3 3.27814 0.267412 2.79378 3.84648 4 3.56057 0.271959 3.06552 4.13556 5 3.80815 0.275439 3.30482 4.38814 6 4.03240 0.278331 3.52217 4.61654 7 4.23986 0.280895 3.72356 4.82775 8 4.43466 0.283291 3.91277 5.02615 9 4.61958 0.285624 4.09235 5.21472 10 4.79661 0.287968 4.26414 5.39556 20 6.34297 0.320246 5.74535 7.00274 30 7.75891 0.380917 7.04712 8.54259 40 9.21665 0.479315 8.32350 10.2056 50 10.8258 0.624127 9.66914 12.1209 60 12.7160 0.831447 11.1864 14.4546 70 15.1050 1.13656 13.0339 17.5053 80 18.4769 1.62841 15.5457 21.9608 90 24.4336 2.62592 19.7928 30.1625 91 25.3700 2.79481 20.4432 31.4840 92 26.4279 2.98909 21.1733 32.9865 93 27.6421 3.21640 22.0052 34.7228 94 29.0642 3.48820 22.9721 36.7720 95 30.7757 3.82286 24.1254 39.2592 96 32.9157 4.25219 25.5529 42.3999 97 35.7515 4.83843 27.4218 46.6113 98 39.9030 5.72912 30.1157 52.8710 99 47.4465 7.43435 34.9004 64.5027 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.999912 0.999387 0.99999 2 0.996078 0.989324 0.99872 3 0.978328 0.960011 0.98899 4 0.941498 0.911985 0.96261 5 0.888040 0.850055 0.91859 PAGE 190 175 6 0.823590 0.779178 0.86189 7 0.753785 0.703599 0.79902 8 0.683043 0.627020 0.73516 9 0.614396 0.552593 0.67342 10 0.549708 0.482678 0.61534 11 0.489977 0.418733 0.56154 12 0.435608 0.361418 0.51214 13 0.386625 0.310813 0.46704 14 0.342813 0.266625 0.42601 15 0.303830 0.228352 0.38876 Distribution Analysis, Bleeding, Least Squares * NOTE * 18 cases were used * NOTE * 9 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Bleeding Censoring Information Count Right censored value 211 Interval censored value 21 Estimation Method: Least Squares (failure time(X) on rank(Y)) Distribution: Lognormal Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 4.90663 0.453368 4.01805 5.79521 Scale 2.46632 0.360870 1.85141 3.28546 Log-Likelihood = -115.459 Goodness-of-Fit Anderson-Darling (adjusted) = 131.227 Correlation Coefficient = 0.981 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 2829.91 3650.86 225.755 35473.7 Standard Deviation 59173.2 128506 838.653 4175111 Median 135.183 61.2877 55.5923 328.723 First Quartile(Q1) 25.6136 7.35539 14.5892 44.9684 Third Quartile(Q3) 713.468 477.070 192.401 2645.70 Interquartile Range(IQR) 687.855 471.490 179.490 2636.04 Table of Percentiles PAGE 191 176 Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 0.0003646 0.0005555 0.0000184 0.0072199 0.0001 0.0010952 0.0014949 0.0000754 0.0158981 0.001 0.0036539 0.0043563 0.0003531 0.0378090 0.01 0.0140426 0.0140527 0.0019753 0.0998301 0.1 0.0662134 0.0518763 0.0142578 0.307495 1 0.435662 0.231139 0.154009 1.23240 2 0.853361 0.381324 0.355446 2.04876 3 1.30733 0.519337 0.600133 2.84791 4 1.80196 0.653311 0.885398 3.66736 5 2.33940 0.787289 1.20961 4.52442 6 2.92139 0.924152 1.57152 5.43074 7 3.54968 1.06638 1.97004 6.39591 8 4.22607 1.21630 2.40411 7.42883 9 4.95253 1.37624 2.87270 8.53817 10 5.73113 1.54854 3.37480 9.73269 20 16.9611 4.49665 10.0876 28.5181 30 37.0879 11.7049 19.9799 68.8447 40 72.3702 27.5975 34.2740 152.811 50 135.183 61.2877 55.5923 328.723 60 252.514 134.185 89.1165 715.503 70 492.734 304.980 146.473 1657.56 80 1077.43 780.888 260.297 4459.77 90 3188.63 2790.45 573.716 17721.9 91 3689.93 3304.62 637.821 21347.0 92 4324.22 3968.98 715.538 26132.6 93 5148.21 4851.61 811.884 32645.1 94 6255.40 6066.82 934.789 41859.7 95 7811.61 7821.37 1097.68 55591.2 96 10141.4 10529.1 1325.45 77595.3 97 13978.4 15149.7 1670.83 116946 98 21414.7 24510.0 2272.35 201813 99 41946.5 52040.0 3686.98 477221 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.976674 0.950458 0.990086 2 0.956220 0.923464 0.976597 3 0.938707 0.902240 0.963561 4 0.923262 0.884080 0.951452 5 0.909370 0.867893 0.940299 6 0.896699 0.853121 0.930029 7 0.885020 0.839436 0.920550 8 0.874168 0.826627 0.911767 9 0.864020 0.814549 0.903600 10 0.854480 0.803100 0.895974 11 0.845470 0.792201 0.888827 12 0.836930 0.781790 0.882106 13 0.828807 0.771821 0.875764 14 0.821061 0.762251 0.869762 15 0.813654 0.753049 0.864065 PAGE 192 177 Distribution Analysis, Bleeding, Maximum Likelihood * NOTE * 18 cases were used * NOTE * 9 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Bleeding Censoring Information Count Right censored value 211 Interval censored value 21 Estimation Method: Maximum Likelihood Distribution: Lognormal Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 7.57542 1.39716 4.83704 10.3138 Scale 4.00969 0.923315 2.55332 6.29677 Log-Likelihood = -111.139 Goodness-of-Fit Anderson-Darling (adjusted) = 131.229 Characteristics of Distribution 95.0% Normal CI Estimate Standard Error Lower Upper Mean(MTTF) 6042017 30476307 307.375 1.18767E+11 Standard Deviation 1.87240E+10 1.63603E+11 683.804 5.12703E+17 Median 1949.69 2724.02 126.096 30146.0 First Quartile(Q1) 130.442 108.671 25.4846 667.659 Third Quartile(Q3) 29141.6 58179.1 582.304 1458400 Interquartile Range(IQR) 29011.2 58076.9 573.534 1467475 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 0.0000017 0.0000060 0.0000000 0.0016693 0.0001 0.0000103 0.0000319 0.0000000 0.0044865 0.001 0.0000730 0.0001938 0.0000004 0.0132898 0.01 0.0006512 0.0014070 0.0000094 0.0449630 0.1 0.0081035 0.0129468 0.0003538 0.185621 1 0.173337 0.163451 0.0273044 1.10040 2 0.517119 0.378756 0.123067 2.17290 3 1.03460 0.636392 0.309879 3.45425 4 1.74317 0.943634 0.603336 5.03643 5 2.66463 1.31872 1.01013 7.02903 PAGE 193 178 6 3.82385 1.78954 1.52807 9.56881 7 5.24856 2.39191 2.14845 12.8220 8 6.96931 3.16752 2.85972 16.9847 9 9.01966 4.16226 3.65082 22.2838 10 11.4363 5.42546 4.51305 28.9802 20 66.7384 47.2975 16.6390 267.684 30 238.111 226.756 36.8275 1539.53 40 705.975 831.906 70.1054 7109.31 50 1949.69 2724.02 126.096 30146.0 60 5384.44 8723.74 224.928 128895 70 15964.3 29721.4 415.365 613577 80 56957.9 122288 847.291 3828912 90 332387 846075 2264.38 48790782 91 421444 1095442 2583.72 68743809 92 545431 1449620 2981.65 99775161 93 724252 1971496 3489.99 150298933 94 994098 2777553 4160.40 237532703 95 1426571 4103023 5082.95 400378584 96 2180664 6482407 6430.47 739494170 97 3674147 11358448 8584.36 1572552180 98 7350872 23886645 12599.7 4288612882 99 21930041 76731079 23055.2 2.08598E+10 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 0.970573 0.943632 0.985830 2 0.956956 0.927842 0.975747 3 0.946876 0.915957 0.968001 4 0.938650 0.905907 0.961729 5 0.931610 0.896987 0.956479 6 0.925408 0.888865 0.951976 7 0.919837 0.881355 0.948042 8 0.914762 0.874341 0.944554 9 0.910088 0.867741 0.941422 10 0.905749 0.861497 0.938582 11 0.901692 0.855563 0.935984 12 0.897878 0.849905 0.933591 13 0.894275 0.844493 0.931372 14 0.890859 0.839302 0.929303 15 0.887607 0.834314 0.927366 Distribution Analysis, Delamination, Least Squares * NOTE * 18 cases were used * NOTE * 9 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Delamination Censoring Information Count Right censored value 221 Interval censored value 11 PAGE 194 179 Estimation Method: Least Squares (failure time(X) on rank(Y)) Distribution: Lognormal Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 3.17491 0.180121 2.82188 3.52794 Scale 0.546982 0.108586 0.370676 0.807145 Log-Likelihood = -61.465 Goodness-of-Fit Anderson-Darling (adjusted) = 129.347 Correlation Coefficient = 0.994 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 27.7852 6.53640 17.5214 44.0612 Standard Deviation 16.4089 7.52480 6.67938 40.3110 Median 23.9246 4.30933 16.8084 34.0537 First Quartile(Q1) 16.5432 1.95196 13.1276 20.8475 Third Quartile(Q3) 34.5996 8.59389 21.2642 56.2981 Interquartile Range(IQR) 18.0564 6.83120 8.60202 37.9020 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 1.39231 0.569083 0.624915 3.10206 0.0001 1.77690 0.641963 0.875261 3.60734 0.001 2.32121 0.718745 1.26516 4.25875 0.01 3.12887 0.790450 1.90699 5.13366 0.1 4.41323 0.832762 3.04886 6.38815 1 6.70221 0.789371 5.32066 8.44249 2 7.77990 0.756162 6.43046 9.41252 3 8.55185 0.742974 7.21288 10.1394 4 9.18263 0.745120 7.83244 10.7656 5 9.72989 0.759625 8.34936 11.3387 6 10.2213 0.784231 8.79423 11.8800 7 10.6726 0.817094 9.18545 12.4004 8 11.0935 0.856693 9.53530 12.9063 9 11.4907 0.901784 9.85247 13.4014 10 11.8689 0.951370 10.1433 13.8880 20 15.0979 1.58284 12.2936 18.5419 30 17.9586 2.34945 13.8968 23.2077 40 20.8287 3.24210 15.3521 28.2590 50 23.9246 4.30933 16.8084 34.0537 60 27.4807 5.64340 18.3750 41.0988 70 31.8726 7.42409 20.1905 50.3139 PAGE 195 180 80 37.9117 10.0734 22.5218 63.8179 90 48.2258 15.0347 26.1765 88.8478 91 49.8131 15.8402 26.7096 92.9007 92 51.5967 16.7573 27.3006 97.5149 93 53.6317 17.8188 27.9650 102.855 94 55.9994 19.0731 28.7256 109.169 95 58.8277 20.5976 29.6175 116.847 96 62.3337 22.5247 30.6998 126.564 97 66.9314 25.1110 32.0834 139.630 98 73.5726 28.9567 34.0173 159.123 99 85.4028 36.0959 37.2999 195.540 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 1.00000 0.999980 1.00000 2 1.00000 0.999557 1.00000 3 0.99993 0.997896 1.00000 4 0.99946 0.994286 0.99997 5 0.99790 0.988273 0.99973 6 0.99428 0.979528 0.99871 7 0.98768 0.967688 0.99593 8 0.97740 0.952178 0.99034 9 0.96306 0.932096 0.98139 10 0.94462 0.906292 0.96935 11 0.92228 0.873809 0.95510 12 0.89643 0.834431 0.93956 13 0.86760 0.788849 0.92333 14 0.83637 0.738409 0.90672 15 0.80331 0.684743 0.88991 Distribution Analysis, Delamination, Maximum Likelihood * NOTE * 18 cases were used * NOTE * 9 cases contained missing values or was a case with zero frequency. Variable Start: Start End: End Frequency: Delamination Censoring Information Count Right censored value 221 Interval censored value 11 Estimation Method: Maximum Likelihood Distribution: Lognormal Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Location 3.34360 0.261576 2.83092 3.85628 PAGE 196 181 Scale 0.621942 0.146369 0.392127 0.986444 Log-Likelihood = -61.150 Goodness-of-Fit Anderson-Darling (adjusted) = 129.346 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 34.3640 11.9735 17.3587 68.0283 Standard Deviation 23.6158 14.8036 6.91236 80.6825 Median 28.3210 7.40809 16.9611 47.2893 First Quartile(Q1) 18.6177 3.20828 13.2813 26.0980 Third Quartile(Q3) 43.0816 15.3313 21.4476 86.5379 Interquartile Range(IQR) 24.4640 12.2971 9.13403 65.5227 Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 0.00001 1.11619 0.584183 0.400167 3.11339 0.0001 1.47293 0.676400 0.598812 3.62303 0.001 1.99589 0.777132 0.930482 4.28120 0.01 2.80273 0.875000 1.51998 5.16800 0.1 4.14399 0.935179 2.66273 6.44927 1 6.66421 0.873509 5.15439 8.61628 2 7.89549 0.835240 6.41702 9.71461 3 8.79216 0.836021 7.29721 10.5934 4 9.53319 0.866916 7.97688 11.3931 5 10.1818 0.920671 8.52817 12.1561 6 10.7685 0.991359 8.99069 12.8979 7 11.3107 1.07442 9.38928 13.6253 8 11.8193 1.16652 9.74048 14.3417 9 12.3016 1.26531 10.0557 15.0493 10 12.7630 1.36917 10.3429 15.7495 20 16.7796 2.55101 12.4558 22.6043 30 20.4393 3.91170 14.0463 29.7420 40 24.1924 5.49492 15.5003 37.7586 50 28.3210 7.40809 16.9611 47.2893 60 33.1543 9.83536 18.5364 59.2996 70 39.2421 13.1329 20.3651 75.6168 80 47.8008 18.1442 22.7163 100.585 90 62.8439 27.8014 26.4059 149.563 91 65.2010 29.3975 26.9444 157.776 92 67.8620 31.2234 27.5415 167.211 93 70.9134 33.3475 28.2128 178.242 94 74.4838 35.8719 28.9813 191.428 95 78.7759 38.9597 29.8827 207.667 96 84.1355 42.8924 30.9767 228.520 97 91.2267 48.2183 32.3754 257.057 98 101.587 56.2299 34.3309 300.601 99 120.356 71.3612 37.6511 384.733 PAGE 197 182 Table of Survival Probabilities 95.0% Normal CI Time Probability Lower Upper 1 1.00000 0.999867 1.00000 2 0.99999 0.998801 1.00000 3 0.99985 0.996252 1.00000 4 0.99918 0.992063 0.99995 5 0.99735 0.986171 0.99963 6 0.99370 0.978443 0.99850 7 0.98769 0.968547 0.99579 8 0.97895 0.955829 0.99089 9 0.96735 0.939254 0.98374 10 0.95292 0.917622 0.97491 11 0.93582 0.890084 0.96517 12 0.91631 0.856547 0.95510 13 0.89471 0.817625 0.94493 14 0.87135 0.774363 0.93478 15 0.84658 0.727984 0.92465 PAGE 198 183 APPENDIX C COMPUTER PRINTOUTS FOR TIME SERIES ANALYSIS Table C-1. Augmented Dickey-Fuller (ADF) unit root test on ln(P) Null Hypothesis: logP ha s a unit root Exogenous: Constant Lag Length: 1 (Fixed) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -0.019506 0.954686 Test critical values: 1% level -3.469737 5% level -2.878723 10% level -2.576009 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(logP) Method: Least Squares Date: 7/18/2006 Time: 1:32:40 AM Included observations: 167 af ter adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob logP(-1) -0.000104 0.005334 -0.019506 0.984461 D(logP(-1)) 0.361362 0.070871 5.098894 0.000001 C 0.001384 0.000804 1.720891 0.087156 R-squared 0.138810 Mean dependent var 0.002032 Adjusted R-squared -0.435317 S.D. dependent var 0.005848 S.E. of regression 0.005460 Akaike info criterion -7.576855 Sum squared resid 0.004889 Schwarz criterion -7.539514 Log likelihood 634.667432 F-statistic 13.217043 Durbin-Watson stat 1.867131 Prob(F-statistic) 0.000371 Note: This table was created using Excel add-in developed by Kurt Annen. PAGE 199 184 MiniTab Printout for ln(P) ARIMA(1,1,0) Model Estimation Estimates at each iteration Iteration SSE Parameters 0 1.36858 0.100 0.092 1 0.01379 0.126 0.009 2 0.00616 0.276 0.003 3 0.00534 0.358 0.002 4 0.00531 0.385 0.001 5 0.00531 0.391 0.001 6 0.00531 0.392 0.001 7 0.00531 0.393 0.001 Relative change in each estimate less than 0.0010 Final Estimates of Parameters Type Coef SE Coef T P AR 1 0.3925 0.0719 5.46 0.000 Constant 0.0011291 0.0004339 2.60 0.010 Differencing: 1 regular difference Number of observations: Original series 169, after differencing 168 Residuals: SS = 0.00524920 (backforecasts excluded) MS = 0.00003162 DF = 166 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 14.9 31.2 49.2 56.8 DF 10 22 34 46 P-Value 0.137 0.093 0.044 0.133 Correlation matrix of the estimated parameters 1 2 0.013 Forecasts from period 169 95 Percent Limits Period Forecast Lower Upper Actual 170 0.325478 0.314454 0.336502 171 0.329079 0.310180 0.347978 172 0.331622 0.306169 0.357075 173 0.333749 0.302737 0.364760 174 0.335713 0.299867 0.371559 175 0.337613 0.297466 0.377760 176 0.339488 0.295440 0.383535 177 0.341353 0.293718 0.388988 178 0.343214 0.292241 0.394187 PAGE 200 185 179 0.345074 0.290967 0.399180 180 0.346933 0.289864 0.404001 181 0.348791 0.288908 0.408675 182 0.350650 0.288077 0.413223 183 0.352509 0.287358 0.417660 184 0.354367 0.286736 0.421999 185 0.356226 0.286203 0.426249 186 0.358085 0.285748 0.430421 187 0.359943 0.285365 0.434521 188 0.361802 0.285048 0.438556 189 0.363661 0.284791 0.442531 190 0.365519 0.284589 0.446450 191 0.367378 0.284438 0.450318 192 0.369237 0.284334 0.454139 193 0.371095 0.284275 0.457915 194 0.372954 0.284258 0.461650 195 0.374813 0.284279 0.465346 196 0.376671 0.284337 0.469005 197 0.378530 0.284429 0.472630 198 0.380388 0.284554 0.476223 199 0.382247 0.284710 0.479784 200 0.384106 0.284895 0.483317 201 0.385964 0.285107 0.486821 202 0.387823 0.285347 0.490299 203 0.389682 0.285611 0.493752 204 0.391540 0.285899 0.497181 205 0.393399 0.286211 0.500587 206 0.395258 0.286544 0.503971 207 0.397116 0.286899 0.507334 208 0.398975 0.287273 0.510676 209 0.400834 0.287668 0.514000 210 0.402692 0.288081 0.517304 211 0.404551 0.288511 0.520590 212 0.406409 0.288960 0.523859 213 0.408268 0.289425 0.527112 214 0.410127 0.289906 0.530348 215 0.411985 0.290403 0.533568 216 0.413844 0.290915 0.536774 217 0.415703 0.291441 0.539964 218 0.417561 0.291982 0.543141 219 0.419420 0.292536 0.546304 220 0.421279 0.293104 0.549454 221 0.423137 0.293684 0.552591 222 0.424996 0.294277 0.555715 223 0.426855 0.294882 0.558827 224 0.428713 0.295499 0.561928 225 0.430572 0.296127 0.565017 226 0.432431 0.296767 0.568094 227 0.434289 0.297417 0.571161 228 0.436148 0.298078 0.574218 229 0.438006 0.298749 0.577264 230 0.439865 0.299430 0.580300 231 0.441724 0.300122 0.583326 232 0.443582 0.300822 0.586343 233 0.445441 0.301532 0.589350 234 0.447300 0.302251 0.592348 235 0.449158 0.302979 0.595338 PAGE 201 186 MiniTab Printout for ln(P) ARIMA(4,1,0) Model Estimation Estimates at each iteration Iteration SSE Parameters 0 0.623534 0.100 0.100 0.100 0.100 0.061 1 0.007737 0.214 0.003 0.138 0.049 0.005 2 0.005288 0.364 -0.132 0.149 -0.042 0.002 3 0.005144 0.438 -0.184 0.139 -0.105 0.001 4 0.005142 0.449 -0.186 0.138 -0.101 0.001 5 0.005142 0.451 -0.186 0.137 -0.100 0.001 6 0.005142 0.452 -0.186 0.137 -0.099 0.001 7 0.005142 0.452 -0.186 0.137 -0.099 0.001 Relative change in each estimate less than 0.0010 Final Estimates of Parameters Type Coef SE Coef T P AR 1 0.4517 0.0783 5.77 0.000 AR 2 -0.1859 0.0855 -2.18 0.031 AR 3 0.1369 0.0863 1.59 0.115 AR 4 -0.0994 0.0803 -1.24 0.218 Constant 0.0013065 0.0004314 3.03 0.003 Differencing: 1 regular difference Number of observations: Original series 169, after differencing 168 Residuals: SS = 0.00509149 (backforecasts excluded) MS = 0.00003124 DF = 163 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 11.8 28.7 51.2 62.5 DF 7 19 31 43 P-Value 0.107 0.070 0.013 0.028 Correlation matrix of the estimated parameters 1 2 3 4 2 -0.402 3 0.133 -0.417 4 -0.061 0.139 -0.420 5 0.009 0.010 -0.004 0.022 PAGE 202 187 LIST OF REFERENCES Akintoye, A, Bowen, P., and Hardcastle, C. (1998). â€œMacro-economic leading indicators of construction contract prices.â€ Construction Management and Economics , 16, 159-175. Anderson, S.D., and Russell, J.S. (2001). â€œGuidelines for warranty, multi-parameter, and best value contracting.â€ NCHRP Report 451, Transportation Research Board, National Research Council, Washington, DC. Annen, K. (2005). â€œUnit root test (adf test) add-in.â€ PAGE 203 188 Federal Highway Administration (FHWA). (2002) . â€œSpecial experimental projects No. 14 innovative contracting (SEP-14).â€ Briefing , FHWA, U.S. Department of Transportation. Washington, DC. Florida Department of Transportation (FDOT). (2003). Flexible Pavement Condition Survey Handbook , Florida Department of Transportation, Tallahassee, FL. Florida Department of Transportation (F DOT). (2005). â€œSection 338: value added asphalt pavements.â€ Standard Specifications for Highway and Bridge Construction , Florida Department of Transportation, Tallahassee, FL. Florida Department of Transportation (FDOT). (2006). History of Florida Pavement Condition Survey , Florida Department of Transportation, Tallahassee, FL. Gallivan, V.L., Huber, G.R., and Flora, W.F. (2003). Benefit of Warranties to Indiana . Submitted for consideration of presen tation and publication at the 2004 Annual Meeting of the Transportation Research Board. Goh, B.H., and Teo, H.P. (2000). â€œForecasti ng construction industry demand, price and productivity in Singapore: the Box-Jenkins approach.â€ Construction Management and Economics , 18, 607-618. Hancher, D.E. (1994). â€œUse of wa rranties in road construction.â€ NCHRP Synthesis 195 , Transportation Research Board, Nationa l Research Council, Washington, D.C. Hastak, M., Minkarah, I.A., Cui, Q., and Bayraktar, M.E. (2003). The Evaluation of Warranty Provisions on ODO T Construction Projects , University of Cincinnati and Purdue University. Hendrickson C. (2003). Project Management for Construction . PAGE 204 189 National Institute of Standard s and Technology (NIST). (2006). NIST/SEMATECH eHandbook of Statistical Methods. PAGE 205 190 Worischeck, M.C. (2003). â€œInvestigation of warranty specificati on implementation for hot mix asphalt in the state of Utah.â€ Report No. UT-03.09 , Utah Department of Transportation, Salt Lake City, UT. Wisconsin Department of Tr ansportation (WisDOT). (2001). Asphalt Pavement Warranties: Five-year Progress Report . Wisconsin Department of Transportation, Madison, WI. PAGE 206 191 BIOGRAPHICAL SKETCH Kelu Guo was born in Shandong, China. He earned a Bachelor of Engineering in transportation from Tongji University in July 1993. He worked as a civil engineer in China for seven years before he went to Flor ida. He received a Master of Engineering in civil engineering from the University of Fl orida in August 2003. He continued his study at the University of Florida and graduated in December 2006 with the degree of Doctor of Philosophy in civil engineering. He is l ooking forward to furthe r contribution to the construction industry in China in the future. |