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Determining Optimum Sample Sizes with Respect to Cost for Multiple Acceptance Quality Characteristics

Permanent Link: http://ufdc.ufl.edu/UFE0042101/00001

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Title: Determining Optimum Sample Sizes with Respect to Cost for Multiple Acceptance Quality Characteristics
Physical Description: 1 online resource (95 p.)
Language: english
Creator: Cho, Dooyong
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: acceptance, aqc, optimization, payment, quality, sample
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: There has been little and limited research done to determine the optimum sample size n for highway construction acceptance plans containing pay adjustment provisions. To optimize the sample size of Acceptance Quality Characteristics (AQCs) with respect to the cost to the agency, an understanding is required of costs of testing and of the cost consequences associated with making a wrong acceptance decision. This study developed an optimization model to determine the sample size. Actual state agency?s data were used not only to check the validity of model?s assumptions but also draw the conclusions. Among the key conclusions were: (1) The optimum sample size is generally small (n=3) for Percent Within Limits acceptance plans; (2) The degree of correlation among AQCs has a negligible effect on the optimum sample size; and (3) The highway agency tends to underpay contractors through use of its pay adjustment provisions. Some cautions are presented for state highway agencies planning to use small sample sizes. Agencies are encouraged to perform their own optimization calculations. By doing so, they will gain a better understanding of their acceptance plan systems and associated costs, and have greater confidence in applying economic decision analysis principles to minimize expected costs and optimize statistical acceptance risks.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Dooyong Cho.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Najafi, Fazil T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042101:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042101/00001

Material Information

Title: Determining Optimum Sample Sizes with Respect to Cost for Multiple Acceptance Quality Characteristics
Physical Description: 1 online resource (95 p.)
Language: english
Creator: Cho, Dooyong
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: acceptance, aqc, optimization, payment, quality, sample
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: There has been little and limited research done to determine the optimum sample size n for highway construction acceptance plans containing pay adjustment provisions. To optimize the sample size of Acceptance Quality Characteristics (AQCs) with respect to the cost to the agency, an understanding is required of costs of testing and of the cost consequences associated with making a wrong acceptance decision. This study developed an optimization model to determine the sample size. Actual state agency?s data were used not only to check the validity of model?s assumptions but also draw the conclusions. Among the key conclusions were: (1) The optimum sample size is generally small (n=3) for Percent Within Limits acceptance plans; (2) The degree of correlation among AQCs has a negligible effect on the optimum sample size; and (3) The highway agency tends to underpay contractors through use of its pay adjustment provisions. Some cautions are presented for state highway agencies planning to use small sample sizes. Agencies are encouraged to perform their own optimization calculations. By doing so, they will gain a better understanding of their acceptance plan systems and associated costs, and have greater confidence in applying economic decision analysis principles to minimize expected costs and optimize statistical acceptance risks.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Dooyong Cho.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Najafi, Fazil T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042101:00001


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1 D ETERMINING OPTIMUM SAMPLE SIZES WITH RESPECT TO COST FOR MULTIPLE ACCEPTANCE QUALITY CHA RACTERISTICS By DOOYONG CHO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF TH E REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

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2 2010 DOOYONG CHO

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3 I would like to dedicate this dissertation t o my supporting parents, Joonhee Cho and SungOck Shin and to my wife Hyeseung Chung and to my son, Leo Seungjae Cho.

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4 ACKNOWLEDGMENTS It is a great pleasure for me to thank and acknowledge the many individuals who assisted me and supported me during the course of my doctorial program. I begin b y expressing my gratitude to Dr. Fazil T. Najafi, my advisory committee chairman, for his continuing encouragement, patience, and support throughout my studies at the U niversity of Florida. He has been treating me as his son in my life in United States of America as well as in my school life. I will always be grateful for lessons learned under his tutelage. I am greatly indebted to Mr. Peter A. Kopac, P.E., Senior R esearch E ngineer for the F ederal Highway Administration, who helped me select this research topic and contribute toward fulfilling some of the FHWA research needs. I would like to thank him for his invaluable assistance, patience, advice, and cri tique throughout this research. I want to express my gratitude to Dr. Nasir G. Gharaibeh, from Texan A&M University for assisting me on a program that became an excellent starting point for this research. I would also like to thank the rest of my committee members, Dr. Mang Tia, Dr. Reynaldo Roque, and Dr. Ian Flood, for their support, guidance, and help in accomplishing my work. I would have not been able to r each this milestone if not for their advice, guidance, and support. I would like to thank Eric Chavez of the Colorado Department of Transportation for providing the data in a form suitable for performing the analyses in this study I would also like to ext end my thanks to all of my friends for their support in the progress and completion of my study. Finally, I express my deepest gratitude to my parents my wife, and my son for their love and support and for many sacrifices they have provided me with the opportunities that enabled me to pursue my higher education at the University of Florida.

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5 I will always be grateful for everything they have done and owe them a debt that can never be repaid.

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6 TABLE OF CONTEN TS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 8 LIST OF FIGURES .......................................................................................................... 9 ABSTRACT ................................................................................................................... 11 CHAPTER 1 INTRODUCTION .................................................................................................... 13 1.1 Background ....................................................................................................... 13 1.2 Problem Statement ........................................................................................... 14 1.3 Objectives ......................................................................................................... 15 1.4 Scope ................................................................................................................ 16 1.5 Research Approach .......................................................................................... 16 1.5.1 Task 1 ...................................................................................................... 16 1.5.2 Task 2 ...................................................................................................... 16 1.5.3 Task 3 ...................................................................................................... 16 1.5.4 Task 4 ...................................................................................................... 17 1.5.5 Task 5 ...................................................................................................... 17 1.5.6 Task 6 ...................................................................................................... 17 2 LITERATURE REVIEW .......................................................................................... 18 2.1 Sample Size ...................................................................................................... 18 2.1.1 Centr al Limit Theorem ............................................................................. 18 2.1.2 Statistical Equations ................................................................................ 20 2.1.3 Practical Consideration ............................................................................ 22 2.2 Optimization with respect to Cost ...................................................................... 22 2.3 Determining Sample Size for PWL Specification .............................................. 25 2.3.1 Percent Within Limit ................................................................................. 25 2.3.2 Determining Optimum Sample Size ......................................................... 27 3 OPTIMUM SAMPLE SIZE ...................................................................................... 28 3.1 Introduction ....................................................................................................... 28 3.2 Optimization Model ........................................................................................... 28 3.3 Discussion of Assumptions ............................................................................... 33 4 PRELIMINARY OPTIMIZATION MODEL AND ANALYSIS .................................... 36

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7 4.1 Introduction ....................................................................................................... 36 4.2 Preliminary Analysis .......................................................................................... 37 4.2.1 Single AQC .............................................................................................. 37 4.2.2 Independent Double AQCs ...................................................................... 40 4.2.3 D ependent Double AQCs ........................................................................ 42 5 DATA COLLECTION .............................................................................................. 45 5.1 Introduction ....................................................................................................... 45 5.2 Colorado State Acceptance Plans .................................................................... 45 6 DATA ANALYSIS .................................................................................................... 48 6.1 Acceptance Quality Characteristics Data Analysis ............................................ 48 6.1.1 Asphalt Content Data Analysis ................................................................ 48 6.1.2 InPlace Density Data Analysis ............................................................... 52 6.1.3 Multi Characteristics Analysis .................................................................. 53 6.2 Correlation Between Two AQCs ....................................................................... 54 6.3 P ay Adjustments ............................................................................................... 58 6.3.1 CDOT Data with Agencys Desired Pay Factor ....................................... 59 6.3.2 Contractors Expected Pay Factor ........................................................... 62 6.3.3 Pay Adjustment Acceptance Plan ........................................................... 63 6.4 C ost of Sampling and Testing ........................................................................... 68 6.6 Cost of Contractor Reaction .............................................................................. 72 7 CONCLUSIONS RECOMMENDATIONS ............................................................... 75 7.1 Summary and Conclusions ............................................................................... 75 7.2 Recomme ndations ............................................................................................ 77 APPENDIX A PWL QUALITY MEASURE ..................................................................................... 79 A .1 Estimating PWL ................................................................................................ 80 A .2 Calculation and Rounding Procedures ............................................................. 80 A .3 Quality Index and PWL ..................................................................................... 81 A.4 Example ........................................................................................................... 90 LIST OF REFERENCES ............................................................................................... 93 BIOGRAPHICAL SKETCH ............................................................................................ 95

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8 LIST OF TABLES Table page 3 1 Acceptable Risk Levels, based on the Criticality of the Inspected Product (AASHTO 1995). ................................................................................................ 31 3 2 Probability of Accepting Poor Quality Lot Considering Two AQCs ..................... 33 3 3 Inputs for Determining Optimum Sample Sizes for Two AQCs .......................... 34 4 1 Fictitious Data ..................................................................................................... 36 4 2 Category of Fictitious Data ................................................................................. 37 5 1 Formulas for calculating PF based on sample size ............................................ 46 5 2 W Factors for Various Elements ....................................................................... 47 6 1 Asphalt Content Proportion ................................................................................ 48 6 2 In Place Density Proportion ................................................................................ 52 6 3 Classified AQCs PWL Data with Pay Factor ....................................................... 60 6 4 Agencys Desired Pay F actor for A L ot H aving any C ombination of AQC PWLs .................................................................................................................. 61 6 5 Difference Between Agencys Desired Pay and Contractors Expected Pay for n=3 ................................................................................................................ 64 A 1 Quality Index Values for Estimating PWL I ......................................................... 82 A 1 Quality Index Values for Estimating PWL II ........................................................ 83 A 2 PWL Estimation Table for Sample Size n = 5 ..................................................... 84 A 3 Another PWL Estimation Table for Sample Size n = 5 ..................................... 875 A 4 Areas Under the Standard Normal Distribution ................................................... 87

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9 LIST OF FIGURES Figure page 2 1 Relationship between standard error of the mean and sample size ................... 20 2 2 Relationship among costs in sampling adapte d from Lapin ................................ 24 2 3 Percent Within Limits. LSL = Lower Specified Limit, USL = Upper Specified Limit, PD = Percent Defected, PWL = Percent Within Limits .............................. 25 2 4 Three Examples of Symmetric Beta Distributions ............................................... 27 3 1 Graphical depiction of the sample size optimization concept. ............................ 29 3 2 T ypical OC Curve for Accept/Reject Acceptance Plan ....................................... 31 4 1 AQC1 Histogram ................................................................................................ 37 4 2 AQC2 Histogram ................................................................................................ 37 4 3 Proportion of prior lots of rejectable quality ........................................................ 38 4 4 Venn Diagram for single AQC ............................................................................ 39 4 5 Single AQC model for Optimum Sample Size .................................................... 40 4 6 Venn Diagram for Independent Double AQCs .................................................... 41 4 7 Independent Double AQCs model for Optimum Sample Size ............................ 42 4 8 Venn Diagram for Dependent Double AQCs ...................................................... 43 6 1 Proportion Histogram at each sample size category for Asphalt Contents I ....... 50 6 2 Proportion Histogram at each sample size category for Asphalt Contents II ...... 51 6 3 Proportion Histogram for Asphalt Contents ........................................................ 51 6 4 Proportion Histogram for InPlace Density ......................................................... 53 6 5 Plot of Density and Asphal t Content Lot PWL Pairs ........................................... 56 6 6 Plot s of Randomly Assigned Density and Asphalt Content Lot PWL Pairs ......... 57 6 7 Pay Equation Curves f or Asphalt Content and Density in SpecRisk software .... 62 6 8 Pay Equation Curves for Asphalt Content and Density with a low threshold ...... 63

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10 6 9 Application of optimization model with negative agency acceptance costs ........ 65 6 10 Application of optimization model with negative agency acceptance costs (with Threshold) .................................................................................................. 66 6 11 Comparison of a composite pay equation and its expected pay curve ............... 67 6 12 National Average Cost of Sampling and Testing for unmolded HMA (Binder Content and Gradation together) (2008 Prices) .................................................. 69 6 13 National Average Cost of Sampling and Testing for HMA Density Cores (2008 Prices) ...................................................................................................... 69 6 14 Distribution of CDOT asphalt content PWL estimates for lots with n=3, 9, and 1720 .................................................................................................................. 74 A 1 Illustration of the Calculation of the Z statistic ................................................... 89 A 2 Illustration of Positive Quality Index Values ........................................................ 91 A 3 Illustration of a Negative Quality Index Value ..................................................... 92

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11 Abst ract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy D ETERMINING OPTIMUM SAMPLE SIZES WITH RESPECT TO COST FOR MULTIPLE ACCEPTANCE QUALITY CHA RACTERISTICS By D ooyong C ho A ugust 2010 Chair: Fazil T. Najafi Major: Civil Engineering T here ha s been little and limited research done to determine the optimum sample size n for highway construction acceptance plans containing pay adjus tment provisions To optimize the sample size of A ccepta nce Quality Characteristics (AQCs) with respect to the cost to the agency, an understanding is required of costs of testing and of the cost consequences associated with making a wrong acceptance decis ion This study developed an optimization model to determine the sample size. Actual state agency s data were used not only to check the validity of model s assumptions but also draw the conclusions Among the key conclusions were: (1) The optimum sample size is generally small (n=3) for Percent Within Limits acceptance plans; (2) The degree of correlation among AQCs has a negligible effect on the optimum sample size; and (3) Th e highway agency tends to underpay contractors through use of its pay adjustme nt provisions Some cautions are presented for state highway agenc ies planning to use small sample sizes. Agencies are encouraged to perform their own optimization calculations. By doing so, they will gain a better understanding of their acceptance plan systems and

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12 associated costs, and have greater confidence in applying economic decision analysis principles to minimize expected costs and optimize statistical acceptance risks.

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13 CHAPTER 1 INTRODUCTION 1.1 Background There has been little research done to establish the optimum sample sizes for todays highway construction acceptance plans containing pay adjustment provisions. One notable recent research project was conducted for the Federal Highway Administration by the Transtec Group, Inc. The purpose o f that research was to determine whether todays commonly used sample sizes (i.e., n = 4 7 for most acceptance quality characteristics [AQCs]**) are too small, too large, or about right. The larger the sample size, the lower the risk of making wrong decisi ons about rejecting, accepting, or assigning a pay factor to a lot. On the other hand, the more sampling and testing performed, the greater the cost and personnel required. If this sample size is too small, the cost consequences in the long run of making erroneous acceptance or pay adjustment decisions would be too high for state highway agencies. The approach that was taken in that research was one that involved optimization of sample size with respect to minimizing the cost of acceptance plans to the hi ghway agencies. Numerous assumptions were made, and the research conclusions/findings were general in nature. Also, it was anticipated specific questions will need to be answered through further research before appropriate implementation of findings can be done by highway agencies. Within the highway construction community, the term sample size refers to the number of samples obtained from a unit of materials or construction. It should not be confused with the physical siz e or amount of material or construction that is sampled. ** A quality characteristic is an attribute of a unit or product that is actually measured to determine conformance with a given requirement (e.g., asphalt content, density). When the quality charact eristic is measured for unit or product acceptance purposes, it is an acceptance quality characteristic.

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14 To optimize the sample size of AQC(s) with respect to the cost to the agency, an understanding is required of costs of testing and the cost consequences associated with making a wrong acceptance decision. An understanding of specific AQCs is also required, including the correlation among AQCs in an acceptance system composed of several individual AQC acceptance plans, as correlation among AQCs may affect optimum sample size s in an acceptance system. The research described in thi s thesis deals with determining the optimum sample sizes for multiple AQCs in a pay adjustment acceptance system* and will result in specific recommendations for implementation. 1.2 Problem Statement The following are some questions that will be addressed: What is the optimum sample size n for a selected twoAQC acceptance system such as to minimize the cost to a state highway agency? T odays commonly used sample sizes range between 4 and 7 for most AQCs This range of sample size is typically established based on practical considerations such as personnel and time constraints. While a sample size within this range may be practical, it is unclear if it is economically optimal. What effect does the degree of correlation among AQCs have on the optimum sample size? If two AQCs are correlated, are their optimum sample sizes greater than, the same as, or less than if they were independent AQCs? Quality assurance experts have encouraged state highway agencies to employ independent AQCs, but many agencies still us e some correlated AQCs. What effect do pay adjustments have on optimum sample size determination? A major difference between quality management systems in highway construction and those in other applications is that units of production submitted f or acceptance purposes are either rejected, accepted, or accepted with a pay adjustment (either positive or negative).

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15 Do pay increases and pay decreases necessarily balance out in the long run such that the highway agency neither gains nor loses money through use of its pay adjustment provisions, or is there some gain or loss to the agency? Although most state highway agencies have pay adjustment provisions that include pay increases (incentives) as well as decreases (disincentives), for some AQCs only pay decreases are possi ble in some states. How do the costs of sampling and testing compare against the expected cost of the consequences to the state highway agency as a result of a wrong acceptance decision? State highway agencies need an optimization method they can apply on their own acceptance systems. The use of any optimization method to minimize costs requires an understanding of economic decision theory and of costs and cost models. 1.3 Objectives The objectives of this study are to: 1. Collect real state highway agenc y data and use the data to determine the optimum sample sizes for two AQCs commonly employed in current highway construction acceptance plans for hot mix asphalt (HMA). 2. E valuate the degree of correlation between the two AQC s. 3. Determine the effect o f having correlated versus uncorrelated AQCs on the determination of optimum sample size. 4. Evaluate pay adjustment system. 5 Compare findings against current practices and make practical suggestions for implementation.

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16 1.4 Scope The study consists of developing procedures that can be used to determine optimum sample sizes for AQCs and evaluating correlation between AQCs and its effect on optimum sample size The results from this study are to be compared with state highway agency practice. The overal l goal of this study is to minimize, through the use of optimal sample sizes, the cost of state highway agencies acceptance plans. 1.5 Research Approach The research approach that was followed in order to fulfill the research objectives mentioned in Subhe ading 1.3 is described below: 1.5.1 Task 1 Conduct a literature review to (a) identify how A A SHTO and s tate highway agencies determine sample sizes for their acceptance plans (b) critically evaluate the findings from the Transtec Group research, and (c) es tablish the AQCs to be analyzed for purposes of this study and the degree to which the AQCs are correlated. 1.5.2 Task 2 (1) Drawing on the Transtec Group research, develop an advanced model that will allow determination of optimum sample sizes for HMA concrete. (2) Develop a procedure that will be used to determine optimum sample size n for a single AQC and multiple (at least two) AQCs 1.5.3 Task 3 Obtain the necessary state highway agency data needed to apply the advanced optimization model developed in Task 1 in keeping with the procedure developed in Task 2.

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17 1.5.4 Task 4 Analyze the state highway agency data to determine the degree of correlation that is present between the AQCs selected in Task 1. 1.5.5 Task 5 Determine the optimum sample sizes for the state highway agencys single and multiple AQC acceptance plans. 1.5.6 Task 6 Draw conclusions and develop recommendations for how to minimize the cost to state highway agency through the use of optimum sample sizes.

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18 CHAPTER 2 2 LITERATURE REVIEW 2.1 S a mple Size S ample size refers to the number of samples obtained from a unit of materials or construction. State highway agencies commonly use a sample size between 3 and 7 units per lot (Russell et al. 2001, Mahoney and Backus 2000). This range of sample si ze is typically established based on practical considerations such as personnel and time constraints. While a sample size within this range may be practical, it is unclear if it is economically optimal. When highway construction QA specifications were fir st being developed in the 60s and 70s, the issue of acceptance sample size was approached several ways. 2.1.1 Central Limit Theorem In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem also justifies the approximation of largesample statistics to the normal distribution in controlled experiments. Equati on 1 shown in FHWA report Cost Effectiveness of Sampling & Testing Programs, FHWA/RD 85/030 explained that if a population has a means x of sampl es of size n from that population approaches a normal distribution n / as the sample size n increases. The term n / is also called the standard error of the mean ( M ).

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19 n x Z 1 Where, Z = standardized statistic with a mean of zero and standard deviation of one x = sample mean n = number of samples Equation 1 can be rewritten as: nM/ 2 The assumption of a normal distribution for sample means from a normal parent population does not hold for small n; these obey a distribution called a Students t distribution. Small n here might be considered to be n less than 20 shown in Figure 21 The me thodology d oes make this distinction in actual practice, using the t statistic rather than z statistic.

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20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 Sample Size, n Standard Error of the Mean (Normalized) Figure 21. Relationship between standard error of the mean and sample size 2.1.2 Statistical Equations Many statistic text books present Equation 3. The sample size (per lot) required to meet allowable tolerance in the mean (typically stated in the construction computed as follows: 2 e Z Z n 3 Where n sample size Z standard normal distribution value for the required sellers risk

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21 Z st andard normal distribution value for the required buyers risk standard deviation of the population e tolerable error allowed by the specifications ASTM E122 09 Standard Practice for Calculating Sample Size to E stimate, With Specified Precision, the Average for a Characteristic of a Lot or Process is intended for use in determining the sample size required to estimate, with specified precision, a measure of quality of a lot or process. The practice applies when quality is expressed as either the lot average for a given property, or as the lot fraction not conforming to prescribed standards. The level of a characteristic may often be taken as an indication of the quality of a material. If so, an estimate of the av erage value of that characteristic or of the fraction of the observed values that do not conform to a specification for that characteristic becomes a measure of quality with respect to that characteristic. ASTM E122 present equation of the following form f or the required sample size, n: 23 E n Where: n sample size E = the maximum acceptable difference between the true average and the sample average. The multiplier 3 is a factor corresponding to a low probability that the difference between the sample estimate and the r esult of measuring (by the same methods) all the units in the lot or process is greater than E. The value 3 is recommended for general use. With the multiplier 3, and with a lot or process standard deviation equal to the advance estimate, it is practically certain that the sampling error will not exceed E.

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22 These equations have the following limitations to apply to Highway construction Q uality A ssurance: 1. It does not consider cost. 2. In its current form, it is meant only accept/reject acceptance plan. 3. It applies to estimates of the average, and is not meant for Percent Within Limit (PWL) estimates. 4. It is for a single acceptance plan (one AQC) and not for an acceptance system (more than one AQC). As the number of AQCs increase, the probability of rej ecting the lot increases. Thus, as the number of AQCs increases, the probability of rejecting good construction ( risk) increases, and the probability of accepting poor construction ( risk) decreases. 2.1.3 Practical Consideration Todays commonly used sample size is typically established based on practical considerations such as personnel and time constraints From a practical standpoint, a reasonable acceptance sample size is the number of tests the technician(s) can perform in one day (assuming the lot represents onedays production). This guidance typically results in n = 5 for many AQCs. While a sample size may be practical, it is unclear if it is economically optimal. 2.2 O ptimization with respect to Cost Lapin addressed that whether a sample size is appropriate depends upon (1) the disadvantage of erroneous estimates, and (2) the costs of obtaining the sample. Larger samples are more reliable, but they are also more costly. The costs of obtaining the sample must therefore be balanced against any potential damage from error, as reliability and economy are competing ends.

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23 The risks of error can be considered as having two components: the probability of making them and the consequences they cause. In a sense, implicit costs are incurred when an estimate is erroneous if it leads to consequences that may be damaging. (Lapin, 1975) In highway construction, it is also very difficult to place cost on the results of an erroneous estimate. Also, the seriousness of an error will vary. A more reliable sample wi ll yield even smaller chances of major error. Ideally, one should then select a sample size that achieves the most desirable balance between the chances of making errors their costs, and the costs of sampling. Figure 21 illustrates the concepts involved i n finding the optimal sample size, which minimizes the total cost of sampling. The costs of collecting the sample data increase with the value of n But larger samples are more reliable, so that the risks of loss from chance sampling error decline. The tot al cost of sampling the sum of collection costs and error costs will achieve a minimum value for some optimal n This is the sample size that should be used.

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24 n Total cost of sampling Cost of collecting sample Sampling error costs Optimal sample size Possible sample size Cost with 100% sampling Minimum cost with sampling Maximum savings with sampling Figure 22 Relationship among costs in sampling adapted from Lapin Because of the difficulties associated with finding the costs of sampling error, the above procedure is not usually used to determine the required sample size in traditional applications. Instead, the focus has been upon a single number that delineates insignificant errors from decidedly undesirable ones, i.e., e in equation 3. This is called the tolerableerror level. In determining this level, one acknowledges that all error is undesirable, but that potential error must be accepted as the price for us ing a sample n instead of the entire population N. There are no guarantees that the tolerable error will not be exceeded. However, large errors may be controlled by keeping the chances of their occurrence small. Furthermore, as indicated earlier, reducing the chance of error increases reliability. Thus, one speaks of reliability in terms of the probability that the estimate will differ from the parameters true value by no more than the tolerable error.

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25 2.3 D etermining Sample Size for PWL Specification As indicated earlier, previous approaches to determine required sample size are based on average as the measure of quality rather than the PWL measure or any other measure of quality. PWL is considered a preferred measure of quality because it considers both the central tendency and variability in a statistically sound way and Federal Highway Administration (FHWA) also promoted. 2.3.1 Percent Within Limit The PWL is the percentage of the lot falling above the lower specified limit (LSL), below the upper speci fied limit (USL), or between the specified limits, as seen in Figure 2 2. PWL may refer to either the population value or the sample estimate of the population value. The PWL quality measure uses the mean and standard deviation in a normally distributed cu rve to estimate the percentage of population in each lot that is within the specified limit (TRB, 2005). Figure 23 Percent Within Limits. LSL = Lower Specified Limit, USL = Upper Specified Limit, PD = Percent Defected, PWL = Percent Within Limits

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26 In practice, it has been found that statistical estimates of quality are reasonably accurate provided the sampled population is at least approximately normal (i.e., bell shaped and not bimodal or highly skewed). The PWL is calculated using the following equation: 100 1 2 5 0 1 100 N N Q B PD PWLiL i ...5 Where PD = percent defective B ( ( ) = shape parameters of the distribution iL Q = lower quality index for an AQC N = number of samples per lot Unlike the normal distribution, which is a single distribution that uses the z statistic parameter to calculate areas below the distribution, the beta distribution is a family of distributions with four parameters alpha ( ) and beta ( ). The PWL calculation uses the symmetrical beta distribution. For symmetric distributions, the alpha and beta are the same. Figure 23 shows three examples of a symmetric beta distribution. As and values increase the distributions become more peaked. The uniform distribution has alpha and beta both equal to one. This does not have a well defined mode because every point has the same probability. Distributions with alpha and beta less than one are bathtub shaped curves and generally not useful for statistical modeling (Ramanathan, 1993)

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27 Figure 24 Three Examples of Symmetric Beta Distributions 2.3.2 Determining Optimum Sample Size Recent research (Gharaibeh, 2009) addressed the optimum sample size n for pay adjustment acceptance plans and acceptance systems (two or more AQCs) rather than accept/reject acceptance plans and single acceptance plans (one AQC) in previous research. And PWL quality measure was used in this research to determine the optimum sample size for hot mix asphalt concrete (HMAC) pavement. PWL based (or PD based) plans are considered to be advantageous because they take produc t variability into account and thus promote uniform quality.

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28 CHAPTER 3 3 OPTIMUM SAMPLE SIZE 3.1 I ntroduction In this Chapter, recent research by Gharaibeh et al. was used to understand how to determine optimum sample sizes with respect to minimizing the cost of acceptance plans to the highway agencies. At first, his methodology was thoroughly verified. And then, through his optimization model and underlying assumptions, the findings and many questions raised were closely examined. This thesis summarizes that research. 3.2 O ptimization Model Generally a sample size is determined depending upon the disadvantage (cost) of erroneous estimates and the costs of obtaining the sample. The model used by Gharaibeh was come from this concept. To optimize sample size with respect to State A gencys cost, a quantitative relationship must be established between sample size and th is cost. Figure 31 illustrates the State Agencys total cost of accepting a construction or material lot consist ing of two components (Expected cost of erroneous acceptance decision and Cost of sampling & testing).

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29 Sample Size, n Expected Cost, $ Optimum Sample Size Cost of Sampling & Testing Expected Cost of Erroneous Acceptance Decision Total Cost of Lot Acceptance F igure 3 1 Graphical depiction of the sample size optimization concept. From this model, total cost of lot acceptance can be formulated follows: Total Cost of Lot Acceptance = CT + CD ......6 Where, Cost of sampling and testing (CT) Expected cost of erroneous acceptance decisions (CD) The cost of sampling and testing increases with the value of sample size. But larger samples are more reliable, so that the expected cost of erroneous acceptance decision decline. The total cost of lot acceptance achieves an optimum sample size n As the cost of sampling and testing increases (i.e. the slope of this cost increases), the curve of total cost of lot acceptance changes and then the optimum sample size would move to the left resulting in a smaller optimum n Similarly, if the expected cost of erroneous acceptance decision were greater, the curve of total cost of lot acceptance would move to the right resulting in a larger optimum n In his research, the cost of sampling and testing is relatively straightforward and can be computed as follows:

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30 CT = n m c .......................................................................................... ...............7 Where n is s ample size, m is number of replicates, and c is unit cost of testing. The relationship between n and cost of sampling and testing may not be exactly linear in some cases; in those cases where a linearity assumption is not deemed to be appropriate, a discre te function can be applied. It is also important to understand that while an agency can apply the general model to either a single AQC (a plan) or to multiple AQCs (a system of plans), true optimization can occur only when the agency applies it to the whol e system rather than just to a component of the system. Most, if not all, highway agencies have multi AQC systems. The risks associated with lot acceptance of multi AQCs are different from those of a single AQC. The more AQCs, the greater the contractor s Operating characteristic (OC) curves should be constructed for each quality characteristic to determine if the buyers and sellers risks associated with the sampling erroneously rejecting or assigning a payment decrease to a lot that indeed should be the risk of erroneously accepting or not assigning a payment decrease to a lot that indeed should be rejected or assigned a pay decrease. The sellers risk represents the contractors risk and the buyers risk represents the highway agencys risk. The grap hical representation of an OC curve for accept/reject acceptance plans is shown in Figure 3 2.

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31 Figure 32 T ypical OC Curve for Accept/Reject Acceptance Plan AASHTO R9 05 (AASHTO 2005) suggests that highway agencies should design acceptance plans that control these risks at suitable levels. AASHTO R 9 does not provide specific recommendations for acceptable risk levels, but it states The more critical the application, the lower should be the buyers risk. But only under rare circumstances should the buyers risk be lower than the sellers risk. AASHTO R990 (AASHTO 1995), however, indicates that these risks can be set based on the criticality of the measured property as it affects safety, performance, or durability as shown in Table 3 1 Table 31 Acce ptable Risk Levels, based on the Criticality of the Inspected Product (AASHTO 1995).

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32 A conventional OC curve represents the relationship between PWL (or PD) and probability of acceptance. The statistical procedure for developing OC curves is well docume nted in the statistics and probability literature (see for example Duncan 1986). Perhaps one of the most established software tools in the highway construction arena for developing OC curves is the OCPLOT simulation software (Weed 1996). For acceptance pla ns with a single AQC, a lot is considered of poor quality if that particular AQC is rejectable [i.e., PWL is at (or below) the rejectable quality level (RQL)]. Thus, the agencys expected cost due to erroneous acceptance decisions is computed as follows: CD = RQL PRQL B S ..................................................................................8 Where RQL = buyers risk (i.e., probability of erroneously accepting a lot that has a true PWL equal to or less than the RQL, computed at midpoint betw een zero and RQL). PRQL = proportion of prior lots of rejectable quality throughout the state. This prior distribution of quality can be obtained from the agencys construction quality databases or paper records of past construction projects. B = unit bid price S = lot size For an acceptance plan that considers multiple AQCs, a lot is considered of poor quality if at least one of the AQCs is rejectable (i.e., PWL is at or below RQL). Thus, the agencys expected cost due to erroneous acceptance decisions is computed as follows: CD = S B P P Pj l j k k i i i i 1 1 1.... where l is the number of combinations of P for all AQCs, where e = RQL and P = PRQL for at least one AQC. Combinations that do not have all AQCs at the RQL are influenced by the acceptable quality level (AQL). Suppose an acceptance plan includes

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33 two AQCs, a rejectable quality lot can occur under any one of the five possible scenarios shown in Table 32 Table 32 Probability of Accepting Poor Quality Lot Considering Two AQCs The acceptance plans that have been developed for S tate H ighway A gencies are u sed to determine accept/reject/pay for all lots that are delivered by the contractor to the state. To optimize n, we need information on how the quality of submitted lots varies within a st ate. Some states have construction quality databases that can be used to determine the proportions of lots submitted at various estimated quality levels. From these construction quality databases, indications are that few lots are estimated to be RQL, th us few lots are rejected; most lots are estimated to be close to or above the AQL 3.3 D iscussion of Assumptions Gharaibeh assumed some items to balance the complexity of the developed method with the level of accuracy needed to achieve the studys objectiv es. In this chapter, his assumptions were introduced and further discussion in details will be considered in Chapter 5.

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34 Gharaibeh used three different anticipated lot quality categories in his calculations shown in Table 33 With these lot quality assumpt ions, Gharaibeh found optimum n to be very small (3 or less) for all but a few cases under the regular and poor quality categories for all but a few cases under the regular and poor quality categories. These categories were based on likely conservative assumptions because of the absence of specific data (real state agencys data). Table 33 Inputs for Determining Optimum Sample Sizes for Two AQCs He assumed that erroneous acceptance decision is defined as accepting a lot that should have been reje cted. It is assumed that erroneous decisions associated with assigning the wrong pay adjustment cancel each other out (i.e., the cost to the state DOT of paying more for a lot than it is worth is offset by the gain to the state DOT of paying less for a lot than it is worth). He essentially treated pay adjustment acceptance plans as if they were accept/reject acceptance plans. The validity check that the assumption of zero cost of erroneous pay decisions is appropriate will be performed in Chapter 6. He assu med that AQCs are independent or weakly dependent. This assumption is not unrealistic since contractors tend to pay attention to individual AQCs, rather than combined measures of quality. In other words, if one AQC has poor quality, it does not

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35 necessarily indicate t hat other AQCs also have poor quality. In Chapter 6, the effect of correlated AQCs on optimum n will be investigated. He assumed a linear relationship between the cost of sampling and testing and sample size n. This cost is hard to model because there are many different possible cost scenarios. The assumption of linearity tends to average all the possible cost scenarios that exist within the state and/or within the acceptance plan system. He also assumed that the agencys expected cost due to erroneous acceptance decisions is based on bid price, and does not consider other costs such as user costs or future maintenance and rehabilitation costs. An important third cost element (CR) was missed from the Gharaibeh model. The CR can be called as the cost of contractor reaction. This cost can be separate from the two elements: c ost of sampling and testing (CT) and expected cost of erroneous acceptance decisions (CD). If an agency switches to a smaller n than the one it currently uses, this cost can aff ect lot cost and quality.

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36 CHAPTER 4 4 PRELIMINARY OPTIMIZA TION MODEL AND ANALY SIS 4.1 I ntroduction To optimize sample size with respect to the cost to the agency, an understanding is required of costs of testing and the cost consequences associated with making a wrong acceptance decision. An understanding of AQCs is also required, and particularly of the correlation among AQCs in an acceptance system composed of several individual AQC acceptance plans. Correlation among AQCs may affect optimum sample size s in an acceptance system. Due to absence of state agencys real database preliminarily fictitious data was created and used as examples of a modified Gharaibeh approach.. These data are presented in T able 4 1. T able 4 1 Fictitious Data AQC Lot Number AQC 1 (PWL) AQC2 (PWL) 1 70 75 2 100 98 3 90 88 4 100 100 5 95 75 6 80 60 7 100 97 8 91 100 9 100 100 10 80 45 11 48 93 12 99 95

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37 4.2 P reliminary Analysis 4.2.1 Single AQC The data are categorized to 3 parts of proportion for 2 AQCs respectively s ince we will use them as input value to compute proportion of prior lots of rejectable quality shown in T able 4 2 and F igure 4 1 and 2 as histograms. T able 4 2 Category of Fictitious Data AQC1 AQC2 Proportion Class Proportion Class 1/6 (17%) RQL (0~70 PW L) 1/6 (17%) RQL (0~70 PWL) 1/3 (33%) AQL (71 91 PWL) 1/4 (25%) AQL (71 91 PWL) 1/2 (50%) 95 PWL (above 91 PWL) 7/12 (68%) 95 PWL (above 91 PWL) 0 17 % 50 70 91 100 RQL 33 % 50 % AQL 95 PWL F igure 4 1 AQC1 Histogram 0 17 % 50 70 91 100 RQL 25 % 68 % AQL 95 PWL F igure 4 2 AQ C2 Histogram

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38 To optimize sample size with respect to State A gencys cost, a quantitative relationship must be established between sample size and these costs. State Agencys total cost of accepting a construction or material lot consists of two components introduced in Equation 6. For acceptance plans with a single AQC, State A gencys expected cost due to erroneous acceptance decisions is computed as follows: CD = Size Lot Price Unit xx poor) is (AQC r P ...10 Where, Pr = proportion of prior lots of rejectable quality xx = probability of accepting a lot having xx PWL Proportion of prior lots of rejectable quality can be found in Figure 43. B Area A Area A Area poor) is quality estimated (AQC1 r P Figure 43 Proportion of prior lots of rejectable quality S tate Agency has an acceptance plan ( AQC1, asphalt binder content) in use, and the database shows that 17 percent of the submitted asphalt content lots are in RQL class, 33 percent are in AQL class and 50 percent are 95 PWL class. The assumption is made here that there is only one event that can lead to State making an RQL Say 60 PWL PWL estimates Area A Area B (not shaded) 90 95 100

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39 acceptance decision error that has significant negative consequences. That event occurs when the state accepts an RQL lot. FIGURE 5 represents the proportion of lots (0.17) that are RQL. A 17% AQC is not RQL AQC is RQL Figure 44 Venn Diagram for single AQC Equation 10 can be computed for cost of making erroneous decisions as follow: CD = size Lot price Unit 0.17 50 .. 11 In order to obtain probability of making erroneous decision OC PLOT (AASHTO R 9, 1990) is used with input values ( AASHTO Quality Assurance Guide Specifications 1996) as follows: Acceptable Quality Level (AQL) = 90% Rejectable Quality Level (RQL) = 60% Acceptance Limit (M) = 70% S ampling and testing costs (CT) for asphalt binder content was approximately estimated from an anonymous contractor. Total cost of lot accepting was computed by Equation 6. These procedures have been programmed in an Excel spreadsheet for rapid computation. As can be seen from Figure 4 5 optimum sample size n for AQC1 is 19.

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40 Sample Size vs. Cost $0 $2,000 $4,000 $6,000 $8,000 $10,000 $12,000 3 8 13 18 23 28 Sample Size Cost Expected cost of wrong decision Cost of Sampling and Testing Expected total cost Figure 45 Single AQC model for Optimum Sample Size 4.2.2 Independent Double AQCs State A gencys expected cost due to erroneous acceptance decisions for multiple AQCs consist of two parts. If multiple AQCs are no t correlated ( i.e. independent), cost equation is computed as follows: CD = Size Lot Price Unit ) AQC each xx AQC each r (P ..12 Where, AQC each r P = proportion of prior lots of rejectable quality for each AQC AQC each xx = probability of accepting a lot having xx PWL for each AQC State Agency has an acceptance plan system (asphalt binder content and air void, AQC1 and AQC2 respectively ) in use, and the database shows that 17 percent of the submitted asphalt binder content lots are in RQL class, 33 p ercent are in AQL class and 50 percent are 95 PWL class for AQC1 and 17 percent of the submitted air void lots

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41 are in RQL class, 25 percent are in AQL class and 68 percent are 95 PWL class. All events that can occur are shown in Figure 46 as Venn Diagr am. 0 50% 100% 17% 17% 42% 100% A D E F G I C B H Figure 46 Venn Diagram for Independent Double AQCs All events that can lead to any consequential acceptance decision error by the state (i.e., acceptance of a lot that should be rejected) are as follows: Event A: Bo th AQCs come from RQL class Event B: AQC1 is from AQL class, AQC2 is from RQL class Event C: AQC1 is from 95 PWL class, AQC2 is from RQL class Event D: AQC1 is from RQL class, AQC2 is from AQL class Event E: AQC1 is from RQL class, AQC2 is from 95 PWL class Note that events F, G, H, and I are inconsequential to us. We have assumed that the only way the State agency can make an acceptance decision error is by accepting RQL class material. The State cannot make an

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42 acceptance decision error when event F, G, H, or I occur. The State can, however, make an acceptance decision error when both AQCs are RQL (event A), when only AQC1 is RQL (events D and E) or when only AQC2 is RQL (events B and C) For this double AQC case, cost of making erroneous decisions is computed as follows: CD 5050 5090 5095 9050 + 9550] size Lot price Unit 13 As can be seen from Figure 8 optimum sample size n for independent double AQCs is 15. This is the total n for both AQC1 and AQC2 and is considerably smaller than the optimum n of about 19 determined in the single AQC1 example. Sample Size vs. Cost $0 $2,000 $4,000 $6,000 $8,000 $10,000 $12,000 $14,000 $16,000 $18,000 $20,000 3 8 13 18 23 28 Sample Size Cost Expected cost of wrong decision Cost of Sampling and Testing Expected total cost Figure 47 Independent Double AQCs model for Optimum Sample Size 4.2.3 Dependent Double AQCs If multiple AQCs are correlated (i.e., dependent), cost equation is computed as follows:

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43 CD = Size Lot Price Unit ) AQC each xx AQC_given each r (P 4 Where, AQC_given each r P = proportion of prior lots of rejectable quality for AQC given the quality level proportions of other correlated AQC s AQC each xx = probability of accepting a lot having xx PWL for each AQC Conditional probabilities from fictitious dada are used to make Venn diagram. T he database shows that 0 percent of the air void lots are in RQL class 50 percent are in A QL class, and 50 percent are 95 PWL class for AQC2 when asphalt binder content lots (AQC1) are in RQL class and 0 percent of the submitted asphalt binder content lots are in RQL class, 100 percent are in AQL class and 0 percent are 95 PWL class when air v oid lots (AQC2) are in RQL class. 0 50% 100% 17% 17% 50% 100% D E F G I B H Figure 48 Venn Diagram for Dependent Double AQCs

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44 Events A and C do not occur in fictitious data. Events F, G, H, and I are inconsequential to us. Acceptance decision error can not be m ade when neither AQC is RQL. For this double AQC case, cost of making erroneous decisions is computed as follows: CD = [(1/6)(1/ 2 509 0 + (1/6)(1/ 2 509 5 + (1/6)( 1 9 05 0] size Lot price Unit 5 As can be seen from Figure 4 9 optimum sample size n for independent double AQCs is 21. This size is larger than 15 for independent. Sample Size vs. Cost $0 $5,000 $10,000 $15,000 $20,000 $25,000 $30,000 3 8 13 18 23 28 Sample Size Cost Expected cost of wrong decision Cost of Sampling and Testing Expected total cost Figure 49 Dependent Double AQCs model for O ptimum Sample Size

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45 CHAPTER 5 5 DATA COLLECTION 5.1 I ntroduction A data collection effort was required to obtain information necessary to apply the advanced optimization model that will allow determination of sample sizes for HMA concrete. For the collecti on of data, Colorado State Agency was selected because of following reasons. First its database was well organized lots of construction project data from as early as 2000. Second, it was the PWL base acceptance plan as Federal Highway Administration (FHWA) promoted to use PWL base. Third, it allowed additional research to better understand Quality Assurance issues. For example, a couple of AQCs were employed in current state highway construction to evaluate correlation between the multiple AQCs and determin e the effect of having correlated versus uncorrelated AQCs on the determination of optimum sample size. 5.2 C olorado State Acceptance Plans As a part of Colorado Department of Transportations (CDOT) Standard Specifications for Road and Bridge Construction, subsection C onformity to the Contract of Hot Mix Asphalt governs the Quality Control (QC)/Quality Assurance (QA) calculations. Colorado States hot mix asphalt gradation acceptance final report has asphalt content, gradation, inplace density, and join t density acceptance quality characteristics. This thesis was focused on the asphalt content and the inplace density characteristics with test sample sizes and pay factors (PF). A few years ago, a standard provision revises or modifies CDOTs Standard Sp ecifications for Road and Bridge Construction. That provision has gone through a formal review and approval process and has been issued by CDOTs Project their own

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46 quality of hot bituminous pavement. CDOT u se this standard special provision on projects wi th 5000 or more metric tons (5000 or more tons) of hot bituminous pavement when acceptance is based on gradation, asphalt content and inplace density In this standard provision, sample size (n) in each process determines the final pay factor PF is compu ted using the formulas designated in Table 5 1 (CDOTs own table). Appendix A describes PWL quality measure calculation. Table 5 1 Formulas for calculating PF based on sample size n (sample size) Formulas for calculating PF Maximum PF 3 0.31177 + 1. 57878 (QL/100) 0.84862 (QL/100) 2 1.025 4 0.27890 + 1.51471 (QL/100) 0.73553 (QL/100) 2 1.030 5 0.25529 + 1.48268 (QL/100) 0.67759 (QL/100) 2 1.030 6 0.19468 + 1.56729 (QL/100) 0.70239 (QL/100) 2 1.035 7 0.16709 + 1.58245 (Q L/100) 0.68705 (QL/100) 2 1.035 8 0.16394 + 1.55070 (QL/100) 0.65270 (QL/100) 2 1.040 9 0.11412 + 1.63532 (QL/100) 0.68786 (QL/100) 2 1.040 10 to 11 0.15344 +1.50104 (QL/100) 0.58896 (QL/100) 2 1.045 12 to 14 0.07278 + 1.6428 5 (QL/100) 0.65033 (QL/100) 2 1.045 15 to 18 0.07826 + 1.55649 (QL/100) 0.56616 (QL/100) 2 1.050 19 to 25 0.09907 + 1.43088 (QL/100) 0.45550 (QL/100) 2 1.050 26 to 37 0.07373 + 1.41851 (QL/100) 0.41777 (QL/100) 2 1.055 38 to 69 0.10586 + 1.26473 (QL/100) 0.29660 (QL/100) 2 1.055 70 to 200 0.21611 + 0.86111 (QL/100) 1.060 > 201 0.15221 + 0.92171 (QL/100) 1.060 If the PF is less than 0.75, the Engineer may: 1. Require complete removal and replacement with specificati on material at the Contractors expense; or

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47 2. Where the finished product is found to be capable of performing the intended purpose and the value of the finished product is not affected, permit the Contractor to leave the material in place. CDOT uses weigh t factors in its composite pay equation. Table 32 shows the weights CDOT uses for four AQCs. Because only two of four CDOTs AQCs were analyzed in this thesis, weights in composite pay equation had to be adjusted. Table 52 W Factors for Various Element s HOT BITUMINOUS PAVEMENT ELEMENT W FACTOR Gradation 15 Asphalt Content 25 In place Density 45 Joint Density 15 Since this thesis was focused on the asphalt contents and the inplace density, W factors were recalculated at 0.357 for asphalt content and 0.643 for density to be applied to data analysis.

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48 CHAPTER 6 6 DATA ANALYSIS 6.1 A cceptance Quality Characteristics Data Analysis 6.1.1 Asphalt Content Data Analysis CDOTs report for asphalt content has start date, grading, region number, project number, sample size, lot size, price per ton, quality level, pay factor, Incentive/Disincentive Payment (I/DP), standard deviation, etc. From this report, asphalt content data was resorted by sample sizes in 2000 year through 2007 year to check quality levels in actual practice. All data were classified by 25 classes (quality levels) and distribution of quality PWLs were calculated at each class shown in Table 6 1 Table 6 1 Asphalt Content Proportion Class (PWL) Frequency Lot Numbers Proportion (Percent) 0 4 1 0.16 4 8 0 0.00 8 12 0 0.00 12 16 0 0.00 16 20 0 0.00 20 24 2 0.32 24 28 0 0.00 28 32 1 0.16 32 36 2 0.32 36 40 1 0.16 40 44 3 0.48 44 48 4 0.65 48 52 7 1.13 52 56 7 1.13 56 60 7 1.13 60 64 8 1.29 64 68 13 2.10 68 72 21 3.39 72 76 2 0 3.23

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49 Class (PWL) Frequency Lot Numbers Proportion (Percent) 76 80 29 4.68 80 84 40 6.45 84 88 56 9.03 88 92 65 10.48 92 96 93 15.00 96 100 240 38.71 Total 620 100.00 Table 6 1 Continued Asphalt Content Proportion From analysis of CDOT database, at least 50 percent of lots were estimated to be better than AQL (about 85 PWL depending on pay equation). Based on the CDOT data and supported by discussions with FHWA official, there is sufficient reason to believe that most states have lot quality distributions that are much better than the distribution Gharaibeh assumed for his good category. The CDOT database contains lots with n ranging from 3 to well over 100. The comparison consisted of grouping lots by sample size and comparing PWL estimates shown in Figure 6 1 and 6 2. (Kopac, 2010)

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50 Figure 61 Proportion Histogram at each sample size category for Asphalt Contents I Histogram (n=3) 0 10 20 30 40 50 60 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=4) 0 10 20 30 40 50 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=5) 0 10 20 30 40 50 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=6) 0 5 10 15 20 25 30 35 40 45 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=7) 0 5 10 15 20 25 30 35 40 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=8) 0 5 10 15 20 25 30 35 40 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=9) 0 5 10 15 20 25 30 35 40 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=10) 0 5 10 15 20 25 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=11) 0 5 10 15 20 25 30 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=12) 0 10 20 30 40 50 60 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n=13) 0 5 10 15 20 25 30 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (14<=n<=16) 0 5 10 15 20 25 30 35 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion

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51 Figure 62 Proportion Histogram at each sample size category for Asphalt Contents II As sample si zes decrease, the spread of PWL estimates increases; thus, the smaller n distributions contain not only more low quality estimates but also more highquality estimates. This does not mean that the true quality is different, only that the quality estimates have different distributions. Figure 6 3 shows all data proportion versus quality level for asphalt contents. The 38.71 percent of lots range between 96 and 100 PWL. Histogram (All Data) 0 5 10 15 20 25 30 35 40 45 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Figure 63 Proportion Histogram for Asphalt Contents Histogram (17<=n<=20) 0 5 10 15 20 25 30 35 40 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (n>20) 0 5 10 15 20 25 30 35 40 45 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion

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52 6.1.2 In Place Density Data Analy sis From CDOT database, Inplace density of asphalt data was also categorized by 25 quality levels in actual practice. The distribution of quality PWLs were calculated at each class shown in Table 6 2 and Figure 6 4. Table 6 2 In Place Density Proportion C lass (PWL) Frequency (Lot Numbers) Proportion (Percent) 0 4 0 0.00 4 8 0 0.00 8 12 0 0.00 12 16 0 0.00 16 20 0 0.00 20 24 0 0.00 24 28 0 0.00 28 32 0 0.00 32 36 2 0.33 36 40 1 0.16 40 44 0 0.00 44 48 0 0.00 48 52 1 0.16 52 56 2 0.33 56 60 4 0.65 60 64 4 0.65 64 68 5 0.82 68 72 10 1.64 72 76 13 2.13 76 80 28 4.58 80 84 37 6.06 84 88 53 8.67 88 92 86 14.08 92 96 111 18.17 96 100 254 41.57 Total 611 100.00 The distribution of InPlace density data had a similar tendency as the o ne of asphalt contents. The 82.49 percent of lots were estimated to be better than AQL (85 PWL).

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53 The anticipated quality distribution of incoming lots is an important variable in determining optimum n since the highway agency is using its acceptance system to make acceptance decisions on all incoming lots, not just on a lot of a given quality level (not on a typical lot, for example). If incoming lots are anticipated to be generally high quality (i.e., good means, low standard deviations), less testing is needed. This is because the joint probability of a contractor delivering and the agency accepting a poor quality lot is smaller when most incoming lots are of high quality; the expected cost of erroneous acceptance decisions is smaller for highquality lo ts as shown in Figure 3 1 model. Histogram (All Data) 0 5 10 15 20 25 30 35 40 45 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Figure 64 Proportion Histogram for InPlace Density 6.1.3 Multi Characteristics Analysis It is important to understand that while an agency can apply the general model either a single AQC (a plan) or to multiple AQCs (a system of plans), true optimization can occur only when the agency applies it to the whole system rather than just to a

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54 component of the system. Most, if not all, highway agencies have multi AQC systems. CDOT has also multiAQCs such as asphalt contents in place density, gradation, joint density, etc. in the HMA. For this study, asphalt contents and inplace density AQCs were chosen to determine the required sample sizes to correspond to the desirable level of risk. These AQCs were also efficiency to ex plain correlation as a view of AQC lot quality measures (e.g., lot PWLs). Based on CDOT database, for these AQCs, only lots (528 lots) that had same mix design number were selected to concern muti AQC system risks. 6.2 C orrelation Between Two AQCs In previ ous research, optimization model assumed AQCs were independent or weakly dependent. The assumption seemed logical considering highway agencies are generally discouraged from using correlated AQCs (Burati, 2005) Even when agencies do use correlated AQCs, contractors tend to pay attention to individual AQC quality rather than to combined measures of quality; if one AQC is poor quality, it does not necessarily indicate other AQCs are also poor quality (Gharaibeh, 2010) In th is study the effect of correlat ed AQCs on optimum n was investigated. The study showed first, that for purposes of the optimization, the issue is not whether the AQC test results are correlated but whether the AQC lot quality measures (e.g., lot PWLs) are correlated. From the CDOT dat a asphalt content and inplace density AQCs were organized with same physical lots and then used to determine the degree of correlation that might be expected between AQC lot PWLs, and the effect of AQC lot PWL correlation on optimum n. The plot of densit y lot PWLs versus corresponding asphalt content lot PWLs were shown in Figure 65 Each data point represents a pair of PWLs that came from the

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55 same physical lot. In Place d ensity and asphalt content were sampled at different frequencies, but the average sample size (average n = 24 for density and average n = 13 for asphalt content) is large enough that the PWL estimates can be considered to be fair estimates of the population PWLs. Figure 6 5 is typical of the plots one obtains with the various combinati ons of CDOT AQC lot PWLs. Data points are clustered at the upper right, indicating very high lot quality; and they also tend toward either axis, indicating a lack of correlation. W here R2 is the correlation coefficient, y is the trendline of the correlati on. Because conventional correlation techniques should not be used to determine the degree of correlation between nonnormally distributed variables whose values contain testing error and are topheavy, a procedure to calculate a new measure of correlat ion, the relative correlation ratio, was developed as an aside in this study specifically for use in determining the degree of correl ation between AQC lot PWLs. This procedure compares the plot from actual PWL data (e.g., Figure 6 5) against a plot using the same PWL data but where the PWL pairs are assigned randomly, thus representing independence (i.e., Figure 66 ). Because data points are clustered at the upper right, one can count data points which are less or equal than 80 PWL. In figure 66, 21 da ta points were at less than 80 PWL, on the other hands, in Figure 66, 9 data points were counted for both 2 figures. The relative correlation ratio was developed as 2.333 (21/9) for both cases. As this ratio is closer to 1, one can say two AQC lots are m ore strongly correlated.

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56 y = 0.0915x + 83.492 R2 = 0.0222 0 20 40 60 80 100 0 20 40 60 80 100 Asphalt Content PWL Density PWL Figure 65 Plot of Density and Asphalt Content Lot PWL Pairs

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57 y = -0.0398x + 95.036 R2 = 0.0042 0 20 40 60 80 100 0 20 40 60 80 100 Asphalt Content PWL Random Density PWL y = -0.0354x + 94.656 R2 = 0.0033 0 20 40 60 80 100 0 20 40 60 80 100 Random Asphalt Content PWL Density PWL Figure 66 Plot s of Randomly Assigned Density and Asphalt Content Lot PWL Pairs

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58 The degree of correlation was found to have a negligible effect on the total optimum n For the degree of correlation in Figure 6 5 (where the plot exhibited mild correlation using the relative correlation ratio), the optimization procedure arrived at the same optimum n = 3 for each AQC (total n = 6) as that arrived under an assumption of independence. Even if the density and asphalt content lot PWLs were strongly correlated, the optimization procedure still arrives at an optimum n = 3 for each AQC. However, according to the optimization procedure (and logic), if the total optimum n for two perfectly correlated AQCs is 6, the lowest cost to the agency occurs when n = 6 for the AQC having the lower sampling and testing and n = 0 for the AQC having the higher sampling and testing cost. Thus, unless the agency needs to use both AQC s, as it might when each AQC controls different key distresses, the recommended course of action to achieve optimization is for the agency to drop the higher cost AQC from its acceptance system. The optimization procedure thus provides an economic argument against the use of highly correlated AQCs. 6.3 P ay A djustments The theory behind acceptance plan development that was initially presented to the highway community in the 60s dealt strictly with accept/reject acceptance plans ( Military Standard 414, 1957, Development of Guidelines for Practical and Realistic Construction Specifications 1967). In the intervening years, the theory had to be expanded as state highway agencies adopted first pay decreases (penalties) then also pay increases (bonuses) associat ed with different levels of estimated quality. Although the vast majority of acceptance plan systems now contain pay adjustment provisions, there are still gaps in our understanding of pay adjustment development and function.

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59 In the previous research, Gha raibeh assumed that the cost consequences of erroneous pay decisions were such that, in the long run, the positive cost consequences incurred when the agency underestimates quality and pays less than it should cancel out the negative cost consequences incurred when it overestimates quality and pays more than it should; in other words, he e ssentially treated pay adjustment acceptance plans as if they were accept/reject acceptance plans. 6 3 1 CDOT Data with Agency s Desired Pay Factor To calculate the cost o f erroneous pay decisions in the this study it was assumed that the agency s desired pay factor for a lot having any combination of AQC PWLs is, by definition, whatever the agency wants it to be for that particular AQC PWL combination. Obviously, what the agency wants it to be is spelled out in the agency pay equations. The CDOT pay equations ( which are a function of sample size, see Table 51) were thus used to determine the agency s desired multi AQC pay. The SpecRisk software program was used to deter mine the contractors multi AQC expected pay. The difference between the contractor s expected pay and the agency s desired pay is the cost of erroneous pay decisions. Table 63 show s classified AQCs PWL CDOT data with pay factors. The agency s desired multi AQC pay factor is calculated in Table 64.

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60 Table 63 Classified AQCs PWL D ata with Pay F actor Class Representative PWL AC. Proportion Den. Proportion Pay Factor (n=12~14), 1.045 Pay Factor (n=15~18), 1.050 Pay Factor (n=19~25), 1.050 Pay Factor (n=3), 1.025 Class 1 30 PWL 0 45 PWL 2.08% 0 45 PWL 0.19 % 0.50711 0.49425 0.48734 0.70903 Class 2 58 PWL 45 65 PWL 5.68% 45 65 PWL 1.70% 0.80686 0.79057 0.77575 0.94199 Class 3 75 PWL 65 80 PWL 12.88% 65 80 PWL 8.52% 0.93911 0.92716 0.91601 1.01851 Class 4 87 PWL 80 90 PWL 20.83% 80 90 PWL 22.92% 1.00982 1.00388 0.99917 1.04299 Class 5 97 PWL 90 100 PWL 58.52% 90 100 PWL 66.67% 1.05445 1.05536 1.05844 1.04472

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61 Table 64 Agency s Desired Pay F actor for A L ot H aving any C ombination of AQC PWLs Possible Events Desired Pay D ecision reject or pay PF, n=3 Maximum PF (1.025) PF, n=15~18 Maximum PF (1.050) Combined PF (AQC1:n=12~14, AQC2:n=19~25) Maximum PF (1.048215) AQC1 in class 1 and AQC2 in class 1 reject or pay 0.70903 0.70903 0.49425 0.49425 0.49440 0.49440 AQC1 in class 1 and AQC2 in class 2 reject or pay 0.85882 0.85882 0.68478 0.68478 0.67984 0.67984 AQC1 in class 1 and AQC2 in class 3 reject or pay 0.90802 0.90802 0.77261 0.77261 0.7700 3 0.77003 AQC1 in class 1 and AQC2 in class 4 reject or pay 0.92376 0.92376 0.82194 0.82194 0.82350 0.82350 AQC1 in class 1 and AQC2 in class 5 reject or pay 0.92488 0.92488 0.85504 0.85504 0.86162 0.86162 AQC1 in class 2 and AQC2 in class 1 reject or pay 0.79219 0.79219 0.60004 0.60004 0.60141 0.60141 AQC1 in class 2 and AQC2 in class 2 pay 0.94199 0.94199 0.79057 0.79057 0.78686 0.78686 AQC1 in class 2 and AQC2 in class 3 pay 0.99119 0.99119 0.87840 0.87840 0.87704 0.87 704 AQC1 in class 2 and AQC2 in class 4 pay 1.00693 1.00693 0.92773 0.92773 0.93051 0.93051 AQC1 in class 2 and AQC2 in class 5 pay 1.00804 1.00804 0.96083 0.96083 0.96863 0.96863 AQC1 in class 3 and AQC2 in class 1 reject or pay 0.81951 0 .81951 0.64880 0.64880 0.64862 0.64862 AQC1 in class 3 and AQC2 in class 2 pay 0.96930 0.96930 0.83933 0.83933 0.83407 0.83407 AQC1 in class 3 and AQC2 in class 3 pay 1.01851 1.01851 0.92716 0.92716 0.92426 0.92426 AQC1 in class 3 and A QC2 in class 4 pay 1.03425 1.02500 0.97649 0.97649 0.97773 0.97773 AQC1 in class 3 and AQC2 in class 5 pay 1.03536 1.02500 1.00959 1.00959 1.01584 1.01584 AQC1 in class 4 and AQC2 in class 1 reject or pay 0.82825 0.82825 0.67619 0.67619 0 .67387 0.67387 AQC1 in class 4 and AQC2 in class 2 pay 0.97804 0.97804 0.86672 0.86672 0.85931 0.85931 AQC1 in class 4 and AQC2 in class 3 pay 1.02725 1.02500 0.95455 0.95455 0.94950 0.94950 AQC1 in class 4 and AQC2 in class 4 pay 1.04299 1.02500 1.00388 1.00388 1.00297 1.00297 AQC1 in class 4 and AQC2 in class 5 pay 1.04410 1.02500 1.03698 1.03698 1.04109 1.04109 AQC1 in class 5 and AQC2 in class 1 reject or pay 0.82887 0.82887 0.69457 0.69457 0.68980 0.68980 AQC1 in c lass 5 and AQC2 in class 2 pay 0.97866 0.97866 0.88510 0.88510 0.87525 0.87525 AQC1 in class 5 and AQC2 in class 3 pay 1.02786 1.02500 0.97293 0.97293 0.96543 0.96543 AQC1 in class 5 and AQC2 in class 4 pay 1.04361 1.02500 1.02226 1.02226 1.01890 1.01890 AQC1 in class 5 and AQC2 in class 5 pay 1.04472 1.02500 1.05536 1.05000 1.05702 1.04822

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6 2 6 3 2 Contractor s Expected Pay Factor The SpecRisk software program was used to determine the contractors multi AQC expected pay Figure 6 7 show s pay equation curves for asphalt content and in place density as graphical input values. Figure 67 Pay Equation Curves for Asphalt Content and Density in SpecRisk software

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63 Figure 68 Pay Equation Curves for Asphalt Content and Density with a low threshold To compare the estimated PWL pay curve and the expected pay curve, pay equation curve with low thresholds were considered sh own in Figure 68. These pay curves both have a minimum acceptable estimated quality level of 45 PWL. 6 3 3 Pay Adjustment Acceptance Plan In this study the previous assumption (Gharaibeh, 2010) of zero cost of erroneous pay decisions was found to be inco rrect as is shown in Table 65.

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64 Table 65 Difference Between Agency s Desired Pay and Contractor s Expected Pay for n=3 Colorado A.C PWL Colorado Den. PWL Expected Pay Desired Pay Difference between E .P and D .P 30 30 0.46375 0.49440 0.03065 30 5 8 0.63796 0.67984 0.04189 30 75 0.74035 0.77003 0.02968 30 87 0.80059 0.82350 0.02291 30 97 0.83396 0.86162 0.02765 58 30 0.56520 0.60141 0.03621 58 58 0.74728 0.78686 0.03958 58 75 0.84679 0.87704 0.03026 58 87 0.89846 0.93051 0.03206 58 9 7 0.94456 0.96863 0.02407 75 30 0.61977 0.64862 0.02885 75 58 0.80878 0.83407 0.02529 75 75 0.89234 0.92426 0.03192 75 87 0.94434 0.97773 0.03338 75 97 0.98748 1.01584 0.02836 87 30 0.64651 0.67387 0.02736 87 58 0.82445 0.85931 0.03486 87 7 5 0.91494 0.94950 0.03456 87 87 0.97156 1.00297 0.03141 87 97 1.01572 1.04109 0.02537 97 30 0.66922 0.68980 0.02058 97 58 0.84108 0.87525 0.03416 97 75 0.93830 0.96543 0.02714 97 87 0.99119 1.01890 0.02772 97 97 1.03339 1.04822 0.01483 As pay adjustment acceptance plans were not designed to make erroneous pay decision cost consequences cancel each other, it would be coincidental for that to happen. This study showed that the net consequences of erroneous pay decisions not only could be considerable, but that they tended to favor the highway agency, i.e., to underpay contractors. When the agency uses an acceptance plan system that in the

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65 long run underpays contractors, the agencys expected cost of erroneous acceptance decision (hence, the agencys total cost of lot acceptance) is decreased. Figure 6 9 and 610 developed from CDOT data, can be used as examples to illustrate such a situation. In Figure 6 9 the expected cost of erroneous acceptance decisions is negative throughout the rang e of n ; the smaller the n the greater the probability of erroneous acceptance decisions, thus the more negative the expected cost, i.e., the more the contractor is underpaid. These negative costs bring the optimization models total cost of lot acceptance curve below the cost of sampling and testing curve. With respect to optimum n it is n = 3, which is the lowest possible n for the PWL quality measure. ($15) ($10) ($5) $0 $5 $10 $15 3 8 13 18 23 Sample Size, n Expected Cost/Lot (Thousand USD) Expected Cost of Erroneous Acceptance Decision Cost of Sampling and Testing Total Cost of Lot Acceptance F igure 6 9 Application of optimization model with negative agency acceptance costs

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66 Sample Size vs. Cost (With Threshold) ($15) ($10) ($5) $0 $5 $10 $15 3 8 13 18 23 Sample Size, n Expected Cost (Thousand USD) Expected Cost of Erroneous Acceptance Decision Cost of Sampling and Testing Total Cost of Lot Acceptance (With Threshold) F igure 6 10. Application of optimization model with negative agency acceptance costs (with Threshold) Two variables come into play for the situation depicted in Figure 6 9 and 610 to occur the lot quality distribution and the pay adjustment provisions. Figure 6 11 can provide a better understanding. It compares the composite pay equation with its expected pay curve for a twoAQC acceptance system (with the X axis identifying the same PWL for both AQCs (Asphalt Contents and InPlace Density) ). Note that the com posite pay equation has both a minimum acceptable estimated quality level (45 PWL) and a maximum pay factor cap (102.50 PWL ). When the lot quality distribution is such that delivered lots rarely fall below 45 PWL, the portion of the graph to the right of 45 PWL carries a much greater weight. One can thus see that under such conditions, unless one or two specific contractors are consistently delivering the few instances of

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67 below 45 PWL lots, contractors as a whole are in the long run underpaid in compariso n to agency desired pay. Of the two variables, the lot quality distribution appears to be the most influential. Application of the optimization model shows that the better the lot quality distribution, the greater the likelihood of negative costs and ass ociated long run underpayment to the contractors. Further, the lot quality levels need not be all that high for the long run underpayment to occur. It can occur with Gharaibehs regular and good historical quality distribution, whether or not the pay provisions include a maximum pay cap and/or a minimum acceptable estimated quality level (below which lots may be rejected), although use of the latter decreases the long run underpayment. 0.00 20.00 40.00 60.00 80.00 100.00 120.00 0 10 20 30 40 50 60 70 80 90 100 Estimated and True PWL Composite Pay Factor Pay Eq. (Estimated PWL) Expected Pay Eq. (True PWL) F igure 6 11. Comparison of a composite pay equation and its expected pay curve

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68 It is also important to note that sample size is not causing the underpayment. Sample size simply affects the risks associated with contractor being assessed the correct payment. For highquality lots, the risk of an occasional incorrect low quality estimate is greater with small sample sizes, thus resulting in the pay equation bias that leads to a greater expected underpayment. The underpayment should not be an issue, provided the agency has developed, is satisfied with, and has made available to contractors, the expected pay equation. 6.4 C ost of Sampling and Testing The cost of sampling and testing is difficult to model because there are many different possible cost scenarios: a technician could perform two or more AQC tests simultane ously; a technician could perform another function while waiting for test results; one sample could yield several AQC test results; multiple technicians could be employed and be more (or less) efficient than a single technician; etc. In this study, a line ar relationship (Gharaibeh, 2010) between the cost of sampling and testing and sample size n was used This example explains how Figures 6 12 and 613 were developed based on the cost assumptions presented earlier. The cost of sampling and testing of 5 HMA cores for density is computed as follows: Transportation between lab and project site ..................................................$115.1 Coring ($99.6/core).........................................................................................$498 .1 Max. Theoretical Specific Gravity test ($52.3/core) ....................... ...............$261.5 Density test ($26.2/core).................................................................................$130.8 Total (for 5 cores) ......................... ...............................................................$1,005.5

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69 F igure 6 12. National Average Cost of Sampling and Testing for unmolded HMA (Binder Content and Gradation together) (2008 Prices) F igure 6 13. National Average Cost of Sampl ing and Testing for HMA Density Cores (2008 Prices) As these examples If a single assumption has to be made about the cost of sampling and testing element in a generalized optimization model, it would be hard to argue that linearit y should not be the ass umption. The assumption of linearity tends to average all the possible cost scenarios that exist within the state and/or within the acceptance plan system. However, if an agency

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70 believes the cost of sampling and testing element to be other than linear f or its optimization purposes, the agency can substitute its own specific model in performing the necessary calculations. In any case, agencies should also consider stepped functions. A stepped cost of sampling and testing function is appropriate in a sit uation where the cost of performing additional tests is viewed as negligible up to a certain n at which point it suddenly increases (Kopac, 2010) One such example is the cost of nuclear density testing. Under certain situations, the cost of nuclear density testing could stay about the same whether say, 3 tests or 10 tests, are performed in one day or by one technician. The cost could then noticeably increase if n = 11 tests required two technicians or extended the time on the job for one technician from say, 1 day to 2 days. With a stepped function, the best optimization solution is for the agency to accept n = 10 as the optimum in this example rather than the calculated n = 3. Another assumption made by Gharaibeh dealt with the unit costs of sampling and testing. As indicated earlier, the higher the unit costs, the larger the optimum n The unit costs Gharaibeh used represent national averages and are deemed to be sufficiently conservative. In this study, unit bid price was calculated by only CDOT data. State highway agencies that suspect their unit costs are higher can easily input their own costs. 6.5 P ost Construction Costs Previous optimization modeling (Gharaibeh, 2010) assumed the agencys expected cost due to erroneous acceptance decisions i s based solely on bid price and not any other costs such as user costs or maintenance and rehabilitation costs. This

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71 assumption may be valid for accept/reject acceptance plan systems for which there is a clear line between rejectable and acceptable lots a nd no need to distinguish among various levels of acceptable quality. This assumption, however, has no place in pay adjustment acceptance plan systems. In this study, optimization modeling by its nature distinguishes among various levels of acceptable qual ity (and also among various levels of unacceptable quality). It indirectly considers user costs and maintenance and rehabilitation costs. These post construction costs relate to the expected underpayment/overpayment to contractors as a result of erroneous pay decisions. As stated earlier, this study found that the highquality (above AQL) lots typically delivered to state highway agencies along with the caps placed on the maximum pay factor are responsible for the long run underpayment to the contractors. Including user costs and maintenance and rehabilitation costs in the modeling significantly raises the performancerelated pay increase the contractor deserves to have for above AQL lots. PaveSpec performancerelated specifications (PRS) software show s that the inclusion of post construction costs, even when using only 5 percent of the theoretical user cost, can result in very high deserved pay factors ( FHWA RD0 0 131 2000). That is partly why 100 percent of user costs is not used in PRS development, and why a cap is placed on the maximum PRS pay factor (Kopac, 2010) Both have the effect of underpaying the contractors, i.e., of creating negative costs to the agency that in turn decrease the calculated optimum n (below that optimum n calculated wi t hout post construction costs).

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72 6.6 C ost of Contractor Reaction This study considered a third cost element separate from the two elements (Expected Cost of Erroneous Acceptance Decision and Cost of Sampling and Testing) identified in Figure 3 1 It address ed the impact on lot quality and cost if an agency switches to a smaller n than the one it currently uses. Preliminary indications are that with proper precautions there should be little if any difference in delivered (and accepted) quality. Contractors have already delivered many lots to various state highway agencies knowing that the acceptance sample size will be as small as n = 3 or 4 (Kopac, 2010). CDOTs database was used to make a comparison of hot mix asphalt quality delivered when the contractor knows n will be small. The CDOT data base contains lots with n ranging from 3 to well over 100. The comparison consisted of grouping lots by sample size and comparing PWL estimates. To allow a fair comparison, those lots that were meant to have higher n b ut had been prematurely discontinued due to unscheduled mix design changes w ere eliminated from the data. An example comparison, using the n = 3, 9, and 1720 groups, is shown in Figure 6 14. As one would expect, the spread of PWL estimates decreases as n increases; thus, the smaller n distributions contain not only more low quality estimates but also more highquality estimates. This does not mean that the true quality is different, only that the quality estimates have different distributions. The aver age PWL of the distribution is the best measure of true quality in this case, and the three average PWLs are about the same 91.72, 90.23, and 91.87 for n =3, 9, and 17 20 respectively. The nonparametric MannWhitney U test shows no difference among the three population

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73 An argument can even be made that, for some AQCs, quality increases with decreased n A frequently cited example involves concrete compressive strength under an acceptance plan that uses the average as the quality measure (Kopac, 2010) With smaller n a contractor would need to target a higher average compressive strength in order to meet the minimum average estimated strength requirement with the same probability as for larger n This argument holds not only for compressive strength but also other onesided AQCs such as thickness, density and smoothness; and other quality measures such as PWL. In Figure 614 it is the spread of the distributions (the error associated with the estimated PWLs) that w ould motivate a contractor to increase quality for either onesided or twosided AQCs. For twosided AQCs, rather than increasing (improving) the lot average, the contractor would want to decrease variability. Either way, the contractors costs theoretic ally would increase, but the post construction costs would probably decrease. Histogram (n=3) 0 10 20 30 40 50 60 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion N=46 Avg. PWL = 91.72

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74 Histogram (n=9) 0 5 10 15 20 25 30 35 40 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion Histogram (17<=n<=20) 0 5 10 15 20 25 30 35 40 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 PWL Proportion F igure 6 14. Distribution of CDOT asphalt content PWL estimates for lots with n=3, 9, and 1720 N= 32 Avg. PWL = 90.23 N= 45 Avg PWL = 91.87

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75 CHAPTER 7 7 CONCLUSIONS RECOMMENDATIONS 7.1 S ummary and Conclus ions Th is study concluded that previous optimization model (Gharaibeh, 2010) assumptions were indeed conservative. With modified assumptions, based where possible on actual data, the optimum acceptance sample size n is even smaller than that calculated by Gharaibeh. However sample size n cannot be less than 3 for PWL acceptance plans as PWL estimates cannot be made with less than 3 test results. For acceptance plan systems related to pavements, optimum n is primarily a function of incoming lot quality th e higher the quality the lower the n To illustrate with an extreme example, if an agency was certain that incoming lots will be of such high quality that none should be rejected, there would be no need for the agency to do any testing under an accept/rej ect acceptance system. The term acceptance system however, has become somewhat of a misnomer. For many highway agencies, it is more a pay adjustment system, i.e., more for the purpose of making pay adjustment decisions rather than accept/reject decis ions. This study examined pay adjustment acceptance systems and concluded for them too, optimum n is primarily a function of incoming lot quality. When incoming lot quality is anticipated to be high, the expected cost associated with erroneous pay decis ions is frequently negative (i.e., contractors are underpaid in the long run), especially in situations where the agency has no minimum acceptable estimated quality level provision (or it has a provision, but with a low minimum). For acceptance plan systems related to other than pavements (e.g., bridge decks), it should be noted that optimum n may also be influenced considerably by the cost

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76 associated with erroneous accept/reject decisions. If the consequences of erroneous accept/reject decisions are catastrophic and could result in loss of lives, the optimum n could be much higher than 3. Both previous research and this study reported here considered only pavement acceptance plan systems. Optimum n is of course also a function of the AQCs being measuredwhat are they? how many are there? are they correlated? and what is the cost to sample and test them? The number of AQCs is important as they work as a unit to estimate lot quality which then determines the composite pay factor. Because the expected cost of erroneous pay decisions is spread out among the AQCs, each AQCs contribution to the expected cost decreases as the number of AQCs increases, resulting in a smaller optimum n for each AQC. Th is study investigated systems with only two AQCs, and already optimum n was below 3. Correlated AQCs were not found to change the optimum n = 3 conclusion for PWL, provided the agency needed the correlated AQCs within the acceptance system in order to control different key distresses. The unit costs of sampling and testing derived by previous research (Gharaib eh, 2010) were deemed to be conservative and were used in this study as well. Th is study concluded statewide lot quality (and therefore lot performance) is not likely to suffer if agencies that switch to s maller acceptance sample sizes take proper precautions. Some recommendations are provided below in the recommendations section. There is also reason to believe quality might actually increase for some onesided AQCs, as statistical risk aware contractors tend to raise target quality to account for the increased variability of estimates from small sample sizes. In such cases, to keep from having a corresponding increase in the cost of lots, agencies that expect

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77 contractors to raise their quality levels mig ht consider simply lowering the specified quality level. However, it is doubtful that any increase in costs due to such increases in quality would change the optimum n For agencies that believe quality levels will decrease, the simple solution would be to raise the specified quality level. 7.2 R ecommendations It is strongly recommended agencies do their own optimum n determinations. In doing so, they will gain a better understanding of their acceptance plan systems and associated costs, and just as important, of the various underlying assumptions and their effect on optimum n Having this understanding, they will be in a better position to draw their own conclusions from the performed economic decision analysis, which identifies the best decision in the long run and not necessarily the best decision with respect to a specific project or contractor or submitted quality level. The understanding will also provide the agencies greater confidence in applying economic decision analysis principles to minimize expected costs and optimize statistical risks. Once an agency has followed the optimization procedure and determined optimum n for its AQCs, the agency will have simultaneously identified the optimum buyers and sellers risks (as these statistical risks are a function of n ). It is also possible for the agency to determine th e theoretical optimum lot size since lot size affects the expected cost of erroneous acceptance decisions (an element of the optimization model). Assuming the agency is considering switching to a lower n based on its optimization, the following recommendations are offered: In going to less acceptance testing, the agency should consider (a) placing more emphasis on the contractors quality control programs, (b) placing more emphasis on inspection to identify isolated instances of poor quality and/or (c)

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78 replacing programs that use contractor test results for acceptance purposes with programs that use only the agencys test results. The agency should include a strong retest provision i n each acceptance plan to further test lots that yield borderline test results. If the agency can identify specific contractor ( s) with a record of having submitted low quality levels in the past, the agency should consider taking appropriate action. The agency has many options one of which is the use of an acceptance plan system with same low n but greater pay reductions for the specific contractor(s). The agency should monitor how the lower n is working statewide with respect to quality and cost. Here too, the agency has many options that allow it to control overall quality and cost; simply increasing/decreasing specified quality is one such option.

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79 APPENDIX A A PWL QUALITY MEASU RE If the quality characteristic is to be used for payment determination, the quality measure to be related to the payment must be decided upon. There are several quality measures that can be used. In past acceptance plans, the average, or the average deviation from a target value, was often used as the quality measure. However, t he use of the average alone provides no measure of variability, and it is now recognized that variability is often an important predictor of performance. Several quality measures, including percent defective (PD) and percent within limits (PWL), have been preferred in recent years because they simultaneously measure both the average level and the variability in a statistically efficient way. In this appendix, only the PWL quality measure was described for the purpose of this study. The Transportation Resear ch Board ( TRB ) glossary (TRC E C074, 2005) includes the following definition (where LSL and USL represent lower and upper specification limits, respectively): PWL also called percent conforming. The percentage of the lot falling above the LSL, beneath th e USL or between the USL and LSL ( PWL may refer to either the population value or the sample estimate of the population value. PWL = 100 PD. ) This quality measure uses the sample mean and the sample standard deviation to estimate the percentage of the population (lot) that is within the specification limits. This is called the PWL method, and is similar in concept to determining the area under the normal curve.

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80 In theory, the use of the PWL (or PD) method assumes that the population being sampled is nor mally distributed. In practice, it has been found that statistical estimates of quality are reasonably accurate provided the sampled population is at least approximately normal, i.e., reasonably bell shaped an d not bimodal or highly skewed. A .1 Estimating PWL Conceptually, the PWL procedure is based on the normal distribution. The area under the normal curve can be calculated to determine the percentage of the population that is within certain limits. Similarly, the percentage of the lot that is within the specification limits can be estimated. Instead of using the Z value and the standard normal curve, a similar statistic, the quality index, Q, is used to estimate PWL. The Q value is used with a PWL table to determine the estimated PWL for the lot. A sampl e PWL table is shown in T able A 1 A different format for a table relating Q values with the appropriate PWL estimate is shown for a sample size of n = 5 in Table A 2 A more complete set of PWL tables in this format, for sample sizes from n = 3 to n = 30, is available. Another way of relating Q and PWL values is presented in T able A 3 In this table a range of Q values is associated with each PWL value. This table was developed by an agency such that any estimated PWL is rounded up to the next integer PWL value. Other possible rounding rules could be used to develop similar tables. The rounding rule in T able A 3 is the one that is most favorable to the contractor since it rounds any PWL number up to the next whole number. For example, 89.01 is rounded up to 90.00 in T able A 3 A .2 Calculation and Rounding Procedur es As the previous paragraph illustrates, the calculation procedures and rounding rules can influence the estimated PWL value that is obtained. This can become a point

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81 of contention, particularly if the payment determination is based on the estimated PWL value. It is therefore important that the agency stipulate the specific calculation process, including number of decimal places to be carried in the calculations, as well as the exact manner in which the PWL table is to be used. For example, in T able A 1 is the PWL value to be selected by rounding up, rounding down, or by linear interpolation. Each of these will result in a different estimated PWL value. For instance, if the sample size is n = 5, and the calculated Q value is 1.18, the estimated PWL values for rounding up, rounding down, and interpolating would be 89, 88, and 88.5, respectively. A .3 Quality Index and PWL The Z statistic that is used with the standard normal table, an example of which is shown in T able A 4 uses the population mean as the point of reference from which the area under the curve is measured: W here: Z = the Z statistic to be used with a standard normal table (such as T able A 4 ). X = the point within which the area under the curve is desired. = the population mean. s = the popul ation standard deviation. The statistic Z, therefore, measures distance above or below the mean, using the number of standard deviation units, s, as the measurement scale. This is illustrated in F igure A 1.

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82 Table A 1 Quality Index Values for Estimating PWL I PWL n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 to 11 100 1.16 1.5 1.79 2.03 2.23 2.39 2.53 2.65 99 1.47 1.67 1.8 1.89 1.95 2 2.04 98 1.15 1.44 1.6 1.7 1.76 1.81 1.84 1.86 97 1.41 1.54 1.62 1.67 1.7 1.72 1.74 96 1.14 1.38 1 .49 1.55 1.59 1.61 1.63 1.65 95 1.35 1.44 1.49 1.52 1.54 1.55 1.56 94 1.13 1.32 1.39 1.43 1.46 1.47 1.48 1.49 93 1.29 1.35 1.38 1.4 1.41 1.42 1.43 92 1.12 1.26 1.31 1.33 1.35 1.36 1.36 1.37 91 1.11 1.23 1.27 1.29 1.3 1.3 1.31 1.31 90 1.1 1.2 1.23 1.24 1.25 1.25 1.26 1.26 89 1.09 1.17 1.19 1.2 1.2 1.21 1.21 1.21 88 1.07 1.14 1.15 1.16 1.16 1.16 1.16 1.17 87 1.06 1.11 1.12 1.12 1.12 1.12 1.12 1.12 86 1.04 1.08 1.08 1.08 1.08 1.08 1.08 1.08 85 1.03 1.05 1.05 1.04 1.04 1.04 1.04 1.04 84 1.01 1.02 1.01 1.01 1 1 1 1 83 1 0.99 0.98 0.97 0.97 0.96 0.96 0.96 82 0.97 0.96 0.95 0.94 0.93 0.93 0.93 0.92 81 0.96 0.93 0.91 0.9 0.9 0.89 0.89 0.89 80 0.93 0.9 0.88 0.87 0.86 0.86 0.86 0.85 79 0.91 0.87 0.85 0.84 0.83 0.82 0.82 0.82 78 0.89 0.84 0.82 0.8 0.8 0.79 0.79 0.79 77 0.87 0.81 0.78 0.77 0.76 0.76 0.76 0.75 76 0.84 0.78 0.75 0.74 0.73 0.73 0.72 0.72 75 0.82 0.75 0.72 0.71 0.7 0.7 0.69 0.69 74 0.79 0.72 0.69 0.68 0.67 0.66 0.66 0.66 73 0.76 0.69 0.66 0.65 0.64 0.63 0.63 0.63 72 0.74 0.66 0.63 0.62 0.61 0.6 0.6 0.6 71 0.71 0.63 0.6 0.59 0.58 0.57 0.57 0.57 70 0.68 0.6 0.57 0.56 0.55 0.55 0.54 0.54 69 0.65 0.57 0.54 0.53 0.52 0.52 0.51 0.51 68 0.62 0.54 0.51 0.5 0.49 0.49 0.48 0.48 67 0.59 0.51 0.47 0.47 0.46 0.46 0.46 0.45

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83 PWL n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 to 11 65 0.52 0.45 0.43 0.41 0.41 0.4 0.4 0.4 64 0.49 0.42 0.4 0.39 0.38 0.38 0.37 0.37 63 0.46 0.39 0.37 0.36 0.35 0.35 0.35 0.34 62 0.43 0.36 0.34 0.33 0.32 0.32 0.32 0.32 61 0.39 0.33 0.31 0.3 0.3 0.29 0.29 0. 29 60 0.36 0.3 0.28 0.27 0.27 0.27 0.26 0.26 59 0.32 0.27 0.25 0.25 0.24 0.24 0.24 0.24 58 0.29 0.24 0.23 0.22 0.21 0.21 0.21 0.21 56 0.22 0.18 0.17 0.16 0.16 0.16 0.16 0.16 55 0.18 0.15 0.14 0.14 0.13 0.13 0.13 0.13 54 0.14 0.12 0.11 0.11 0.11 0.11 0.1 0.1 53 0.11 0.09 0.08 0.08 0.08 0.08 0.08 0.08 52 0.07 0.06 0.06 0.05 0.05 0.05 0.05 0.05 51 0.04 0.03 0.03 0.03 0.03 0.03 0.03 0.03 50 0 0 0 0 0 0 0 0 Table A 1 Quality Index Values for Estimating PWL I (continued) Table A 1 Quality Index Values for Estimating PWL II PWL n = 12 to 14 n = 15 to 18 n = 19 to 25 n = 26 to 37 n = 38 to 69 n = 70 to 200 n = 201 to 72 0.59 0.59 0.59 0.59 0.59 0.58 0.58 71 0.57 0.56 0.56 0.56 0.56 0.55 0.55 70 0.54 0.53 0.53 0.53 0.53 0.53 0.52 69 0.51 0.5 0.5 0.5 0.5 0.5 0.5 68 0.48 0.48 0.47 0.47 0.47 0.47 0.47 67 0.45 0.45 0.45 0.44 0.44 0.44 0.44 66 0.42 0.42 0.42 0.42 0.41 0.41 0.41 65 0.4 0.39 0.39 0.39 0.39 0.39 0.39 64 0.37 0.36 0.36 0.36 0.36 0.36 0.36 63 0.34 0.34 0.34 0.34 0.33 0.33 0.33 62 0.31 0.31 0.31 0.31 0.31 0.31 0.31 61 0.29 0.29 0.28 0.28 0.28 0.28 0.28 60 0.26 0.26 0.26 0.26 0.26 0.25 0.25 59 0.23 0.23 0.23 0.23 0.23 0.23 0.23 58 0.21 0.21 0.2 0.2 0.2 0.2 0.2 57 0.18 0.18 0.18 0.18 0.18 0.18 0.18

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84 PWL n = 12 to 14 n = 15 to 18 n = 19 to 25 n = 26 to 37 n = 38 to 69 n = 70 to 200 n = 201 to 56 0.16 0.15 0.15 0.15 0.15 0.15 0.15 55 0.13 0.13 0.13 0.13 0.13 0.13 0.13 54 0.1 0.1 0.1 0.1 0.1 0.1 0.1 53 0.08 0.08 0.08 0.08 0.08 0.08 0.08 52 0.05 0.05 0.05 0.05 0.05 0.05 0.05 51 0.03 0.03 0.03 0.03 0.03 0.03 0.02 50 0 0 0 0 0 0 0 Table A 1 Quality Index Values for Estimating PWL II (continued) Table A 2 PWL Estimation Table for Sample Size n = 5 Q 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 50 50.36 50.71 51.07 51.42 51.78 52.13 52.49 52.85 53.2 0.1 53.56 53.91 54.27 54.62 54.98 55.33 55.69 56.04 56.39 56.75 0.2 57.1 57.46 57.81 58.16 58.52 58.87 59.22 59.57 59.92 60.28 0.3 60.63 60.98 61.33 61.68 62.03 62.38 62.72 63.07 63.42 63.77 0.4 64.12 64.46 64.81 65.15 65.5 65.84 66.19 66.53 66.87 67.22 0.5 67.56 67.9 68.24 68.58 68.9 2 69.26 69.6 69.94 70.27 70.61 0.6 70.95 71.28 71.61 71.95 72.28 72.61 72.94 73.27 73.6 73.93 0.7 74.26 74.59 74.91 75.24 75.56 75.89 76.21 76.53 76.85 77.17 0.8 77.49 77.81 78.13 78.44 78.76 79.07 79.38 79.69 80 80.31 0.9 80.62 80.93 81.23 81.54 81.84 82.14 82.45 82.74 83.04 83.34 1 83.64 83.93 84.22 84.52 84.81 85.09 85.38 85.67 85.95 86.24 1.1 86.52 86.8 87.07 87.35 87.63 87.9 88.17 88.44 88.71 88.98 1.2 89.24 89.5 89.77 90.03 90.28 90.54 90.79 91.04 91.29 91.54 1.3 91.79 92.03 92.27 92.51 92.75 92.98 93.21 93.44 93.67 93.9 1.4 94.12 94.34 94.56 94.77 94.98 95.19 95.4 95.61 95.81 96.01 1.5 96.2 96.39 96.58 96.77 96.95 97.13 97.31 97.48 97.65 97.81 1.6 97.97 98.13 98.28 98.43 98.58 98.72 98.85 98.98 99.11 99.23 1.7 99.34 99.45 99.55 99.64 99.73 99.81 99.88 99.94 99.98 100

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85 Table A 3 Another PWL Estimation Table for Sample Size n = 5 QL or QU PWLL or PWLU QL or QU PWLL or PWLU 1.671 or More 100 0.029 to 0.000 50 1.601 to 1.670 99 0.059 to 0.030 49 1.541 to 1.600 98 0.079 to 0.060 48 1.491 to 1.540 97 0.109 to 0.080 47 1.441 to 1.490 96 0.139 to 0.110 46 1.391 to 1.440 95 0.169 to 0.140 45 1.351 to 1.390 94 0.199 to 0.170 44 1.311 to 1.350 93 0.229 to 0.200 43 1.271 to 1.310 92 0.249 to 0.230 42 1.231 to 1. 270 91 0.279 to 0.250 41 1.191 to 1.230 90 0.309 to 0.280 40 1.151 to 1.190 89 0.339 to 0.310 39 1.121 to 1.150 88 0.369 to 0.340 38 1.081 to 1.120 87 0.399 to 0.370 37 1.051 to 1.080 86 0.429 to 0.400 36 1.011 to 1.050 85 0.449 t o 0.430 35 0.981 to 1.010 84 0.469 to 0.450 34 0.951 to 0.980 83 0.509 to 0.470 33 0.911 to 0.950 82 0.539 to 0.510 32 0.881 to 0.910 81 0.569 to 0.540 31 0.851 to 0.880 80 0.599 to 0.570 30 0.821 to 0.850 79 0.629 to 0.600 29 0.781 to 0.820 78 0.659 to 0.630 28

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86 QL or QU PWLL or PWLU QL or QU PWLL or PWLU 0.751 to 0.780 77 0.689 to 0.660 27 0.721 to 0.750 76 0.719 to 0.690 26 0.691 to 0.720 75 0.749 to 0.720 25 0.661 to 0.690 74 0.779 to 0.750 24 0.631 to 0.660 73 0.819 to 0.780 23 0.601 to 0.630 72 0.849 to 0.820 22 0.571 to 0.600 71 0.879 to 0.850 21 0.541 to 0.570 70 0.909 to 0.880 20 0.511 to 0.540 69 0.949 to 0.910 19 0.471 to 0.510 68 0.979 to 0.950 18 0.451 to 0.470 67 1.009 to 0.980 17 0.431 to 0.450 66 1.049 to 1.010 16 0.401 to 0.430 65 1.079 to 1.050 15 0.371 to 0.400 64 1.119 to 1.080 14 0.341 to 0.370 63 1.149 to 1.120 13 0.311 to 0.340 62 1.189 to 1.150 12 0.281 to 0.310 61 1.229 to 1.190 11 0.251 to 0.280 60 1.269 to 1.230 10 0.231 to 0.250 59 1.309 to 1.270 9 0.201 to 0.230 58 1.349 to 1.310 8 0.171 to 0.200 57 1.389 to 1.350 7 0.141 to 0.170 56 1.439 to 1.390 6 0.111 to 0.140 55 1.489 to 1.440 5

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87 QL or QU PWLL or PWLU QL or QU PWLL or PWLU 0.081 to 0.110 54 1.539 to 1.490 4 0.061 to 0.080 53 1.599 to 1.540 3 0.031 to 0.060 52 1.669 to 1.600 2 0.001 to 0.030 51 1.789 to 1.670 1 0.029 to 0.000 50 1.790 or Less 0 Table A 3 Another PWL Estimation Table for Sample Size n = 5 Table A 4 Areas Under the Standard Normal Distribution Z or Z Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 .1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 .2 .0793 .0832 .0871 .0910 .0948 0987 .1026 .1064 .1103 .1141 .3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 .4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 .5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .6 .2257 .2291 .2324 .2357 .2389 2422 .2454 .2486 .2517 .2549 .7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

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88 Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3183 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .336 5 .3389 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4 616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .484 6 .4850 .4854 .4857 2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916 2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936 2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4982 .4982 .4983 .4984 .4 984 .4985 .4985 .4986 .4986 3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990 3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4993 3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4995 3.3 .4995 .4995 .4995 .4996 .499 6 .4996 .4996 .4996 .4996 .4997 3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998 Table A 4 Areas Under the Standard Normal Distribution (continued)

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89 Figure A 1 Illustration of the Calculation of the Z statistic Conceptually, the Q statistic, or quality index performs identically the same func tion as the Z statistic except that now the reference point is the mean of an individual sample, instead of the population mean, and the points of inter est with regard to areas under the curve are the specification limits. and W here: QL = quality index for the lower specification limit. QU = quality index for the upper specification limit. LSL = lower specification limit. USL = upper specification limit.

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90 = the sample m ean for the lot. s = the sample standard deviation for the lot. The Q statistic, therefore, represents the distance in sample standard deviation units that the sample mean is offset from the specification limit. A positive Q statistic represents the number of sample standard deviation units that the sample mean falls inside the specification limit. Conversely, a negative Q statistic represents the number of sample standard deviation units that the sample mean falls outside the specification limit. These ca ses are illustrated in F igures A 2 and A 3 QL is used when there is a onesided lower specification limit, while QU is used when there is a onesided upper specification limit. For twosided specification limits, the PWL value is estimated as: PWLT = PWLU + PWLL 100 W here: PWLU = percent below the upper specification limit (based on QU ). PWLL = percent above the lower specification limit (based on QL ). PWLT = percent within the upper and lower specification limits. A.4 Example An example using a simpli fied P ortland cement concrete specification can be used to explain the PWL concept. In this example, the minimum specification limit for strength is 21,000 kPa. One requirement of the PWL procedure is that the sample size must be greater than n = 2 since both the sample mean and sample standard deviation are necessary to estimate PWL. For this specification, the sample size has been chosen as

PAGE 91

91 n = 4. Furthermore, the specification requires that at least 95 percent of the lot exceed the minimum strength (i.e. PWL > 95). Table A 1 shows that the minimum Q value is 1.35 for 95 PWL and a sample size of n = 4. Whenever the mean is 1.35s above the specification limit, the lot is accepted. However, as used most frequently, the Q value will be used to compute the PW L and that will, in turn, be used to determine a payment factor. For example, suppose that the acceptance tests for a lot have a sample mean of 25,000 kPa and a sample standard deviation of 3,400 kPa. Does this lot meet the specification requirement? The q uality index value is calculated as: Using this Q value, n = 4, and Table A 1 the estimated PWL for the lot is between 89 and 90. This is less than the required 95, so the lot does not meet the specified strength requirement. Figure A 2 Illustration of Positive Quality Index Values

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92 Figure A 3 Illustration of a Negative Quality Index Value Intuitively, PWL is a good measure of quality since it is reasonable to assume that t he more of the material that is within the specification limits, the better the quality of the product should be. A detailed discussion and analysis of the PWL measure of quality is presented in the t echnical report for the project

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93 LIST OF REFER ENCES 1. American Association of State Highway and Transportation Officials (AASHTO), Standard Recommended Practice for Acceptance Sampling Plans for Highway Construction, AASHTO Designation R9 90, Washington, D.C., 1995. 2. American Association of State Highway and Transportation Officials (AASHTO), Quality Assurance Guide Specification, Washington, D.C, 1996. 3. Transportation Research Board, Glossary of Highway Quality Assurance Terms, Transportation Research Board, Elect ronic Circular E C074, Third Update, Washington, D.C., 2005 4. Vidalis, Sofia M., Relation Between Cost, Qulity, and Risk in Portland Cement Concrete Pavement Construction, University of Florida, Ph.D dissertation, 2005. 5. Duncan, A. J., Quality Control a nd Industrial Statistics, R.D. Irwin, ISBN 0256035350, Fifth Edition, 1986. 6. Burati, J.L., Weed, R. M., Hughes, C. S., Hill, H. S., Optimal Procedures for Quality Assurance Specifications, Federal Highway Administration, Publication No. FHWA RD 02095, McLean, VA., 2004. 7. Velivelli, K.L., Gharaibeh, N.G., Nazarian, S., Sample Size Requirements for Seismic and Traditional Testing of Concrete Pavement Strength, Transportation Research Board, Transportation Research Record, No. 1946, pp. 33 38, 2006. 8. We ed, R.M., Quality Assurance Software for the Personal Computer, Federal Highway Administration, Publication No. FHWA SA 96026, McLean, VA., 1996. 9. Kopac, Peter, The Development of Operating Characteristic Curves for PENNDOTs Restricted performance Bit uminous Concrete Specifications, The Pennsylvania Transportation Institute, Project No. 7427, Research Report No.3, 1976. 10. Tagaras George Economic Acceptance Sampling by Variables with Quadratic Quality Costs, IIE Transactions, Vol. 26, No. 6, Novemb er 1994. 11. Von Quintus, H.L., Rauhut, J.B., Kennedy, T.W., and Jordahl, P.R., Cost Effectiveness of Sampling and Testing Programs, Federal Highway Administration, Publication No. FHWA/RD85/030, McLean, VA, 1985. 12. Gharaibeh, N.G., Garber, S.I., and Liu, L., Determining Optimum Sample Sizes for Percent Within Limits Specifications, Transportation Research Board, Transportation Research Record ____, 2010, pp. ____.

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94 13. South, J.B., Selecting an Acceptance Sampling Plan That Minimizes Expected Error and Sam pling Cost, Pittsburg State University, Quality Progress Vol. 15, No. 10, pp. 1822, 1982. 14. Lapin, L. Statistics Meaning and Method. Harcourt Brace Jovanovich, Inc. New York, NY, 1975. 15. Fabrycky, W.J. and Thuesen, G.J., Economic Decision Analysis Prenti ceHall, Inc., Englewood Cliffs, NJ, 1974. 16. Military Standard 414, Sampling Procedures and Tables for Inspection by Variables for Percent Defective, Superintendent of Documents, Government Printing Office, Washington, DC 1957. 17. Miller Warden Associates Development of Guidelines for Practical and Realistic Construction Specifications, Highway Research Board, National Cooperative Highway Research Program Report 17, 1965. 18. National Highway Institute, SpecRisk Quality Assurance Specification Development and Validation Course, Web based course number FHWA NHI 134070, www.nhi.fhwa.dot.gov. 19. Burati, J.L., Risks with Multiple Pay Factor Acceptance Plans, Transportation Research Board, Transportation Research Record No. 1907 Washington, DC, 2005, pp. 3742. 20. Hoerner, T.E. and Darter, M.I., Improved Prediction Models for PCC Performance Related Specifications, Volume II, Federal Highway Administration, Publication No. FHWA RD 00131, McLean, VA, 2000. 21. J. L. Burati, R. M. Weed, C. S. Hughes, H. S. Hill Optimal Procedures for Quality Assurance Specifications, Federal Highway Administration, Publication No. FHWA RD 0295, McLean, VA, 1985.

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95 BIOGRAPHICAL SKETCH Dooyong Cho was born in Seoul Republic of Korea. H e started his college career at S ungKyunKwan University in Seoul. H e earned Bachelor of Science and Master of Science degrees majoring in civil e ngineering in 200 2 H e came over United States of America to study abroad in 2002. At Pennsylvania State Univer sity, he earned Master of Engineering degree in the D epartment of C ivil and E nvironmental engineering He transferred to and entered in a Ph.D program in the C ivil and C oastal E ngineering D epartment at the University of Florida in 2005 in Ph.D program spec ializing in infrastructural engineering and public works