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Christian, Steve Clarence.
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A sensitivity analysis of a heuristic model used for the placement allocation of utilities in transportation rightofway corridors
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by Steve Clarence Christian.
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Thesis (M.S.I.E.)University of South Florida, 2004.
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ABSTRACT: The requirements for public utility systems in the United States of America have grown enormously over the years triggering a tremendous shortage for space available to public utilities on and within transportation rightofways (ROW). Overcrowding and improper location of utilities has resulted in problems such as, damage to infrastructure, traffic accidents and, interruption of service to customers. The project titled, "Optimal Placement of Utilities within FDOT RightofWay", sponsored by the Florida Department of Transportation (FDOT), and currently being investigated at the University of South Florida, presents a decisionmaking heuristic aimed at developing a better utility placement allocation system (Kranc et. al) [6].Working in accordance with the guidelines of safety, relocation, and clearance for utility placement set by the American Association of State Highway and Transportation Officials organization (AASHTO), the heuristic finds suitable locations for the utilities in ROW corridors. However, a model being used to advocate a practice having large social and economical impacts is more likely to play the role of generic evidence in a trial, whose weight must ultimately be established by a 'jury'. The question being addressed to the system must be scrutinized carefully, and the formal structure updated iteratively until it proves capable of providing an answer to the given question. A good sensitivity analysis can provide this generic quality assurance to the model and help demonstrate the worthiness of the model itself. This thesis is a quantitative and qualitative sensitivity analysis of the abovementioned heuristic.The analysis is conducted in two parts, 1. The 'Model Factor Sensitivity Analysis', with the objective of assessing the uncertainties associated with the modeling of the heuristic. This analysis focuses primarily on providing an evaluation of the confidence in the heuristic and its predictions by analyzing the influences that variations in the input factors have on the outputs of the utility cost assessment models and the final output of the heuristic itself. Variance based sensitivity indices derived from Sobol' sensitivity indices [42] are used here for this purpose. 2. The 'Model Output Evaluation and Enhancement' study, which initially focuses on understanding / evaluating the complexities of the discrete step, cost optimization procedure used in the heuristic and later, based on certain observed shortcomings and problems develops an enhancement, the Ideal Configuration Selector (ICS) to be implemented with the heuristic.The ICS addresses all the problems of the heuristic with the help of experimental speedup, positional sensitivity and refinement tools and employs a multi criterion evaluation technique for utility configuration assessment to provide substantiation to the outputs determined by the heuristic.
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Coadviser: Miller, William A.
Coadviser: Kranc, Stanley C.
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Sobols' sensitivity indices.
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multicriteria decision making.
transportation corridor planning.
utility placement allocation.
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A Sensitivity Analysis of a Heuristic Model used for the Placement Allocation of Utilities in Transportation RightofWay Corridors by Steve Clarence Christian A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Ma nagement Systems Engineering College of Engineering University of South Florida CoMajor Professor: W illiam A. Miller, Ph.D., P.E. CoMajor Professor: Stanley C. Kranc, Ph.D., P.E. Jos Zayas Castro, Ph. D. Date of Approval: November 8, 2004 Keywords: Sobol Sensitivity Indices, Factor Analysis, MultiCrit eria Decision Making, Transportation Corridor Planning, Utility Placement Allocation Copyright 2004, Steve Clarence Christian
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That is what we mean by science. That both question and answer are tied up with uncertainty, and that they are painful. But there is no way ar ound them and that you hide nothing, instead everything is brought out into the open. Peter Heg (1995)
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ACKNOWLEDGEMENTS Gratitude is the least of the virtues, but ingratitude is th e worst of vices. I would like to express my gr atitude to the following peopl e and organizations for their help and support during the formatio n and completion of this thesis. To Dr. William A. Miller, for your consiste nt guidance that has molded me into the Industrial Engineer that I am today. To Dr. Stanley C. Kranc, for presenting me with the opportunity to work on the Utility Placement project for the Florida Department of Transportation. Your vast knowledge of research procedures is truly amazing. You exemplify everything a researcher should be. To Dr. Zayas Castro and the staff of the I ndustrial and Management Systems Engineering Department, for their generosity, time and re sources allotted to me for this thesis. To my friend and mentor Nathaniel Collier, for helping me through every step of this thesis. Thank you for all the many hours. To my family (Mom, Dad, Mark and Mayura) thank you for all your support and love. And finally, but most importantly, thank you Lord Jesus, for everything in my life.
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i TABLE OF CONTENTS LIST OF TABLES iv LIST OF FIGURES vi ABSTRACT ix CHAPTER 1 INTRODUCTION 1 1.1 Thesis Focus: Sensitivity Analysis & Model Enhancement 4 1.2 Thesis Outline 7 CHAPTER 2 LITERATURE REVIEW 10 2.1 Sensitivity Analysis 10 2.1.1 Why Carry Out A Sensitivity Analysis 11 2.1.2 Types Of Sensitivity Analysis 13 2.1.2.1 Screening Designs 15 2.1.3.2 Local Sensitivity Analysis 16 2.1.3.3 Global Sensitivity Analysis 18 2.1.3 Application Examples Of Sensitivity Analysis 20 CHAPTER 3 THE HEURISTIC 22 3.1 Mathematical Representation Of The Heuristic 23 3.1.1 Model Objective Function 23 3.1.2 Cost Models 24 3.1.2.1 Installation Cost Model 25 3.1.2.2 Access Cost Model 29 3.1.2.3 Damage Cost Model 30 3.1.2.4 Accident Cost Model 32 3.1.3 AASHTO Utility Placement Constraints 38 3.2 Model Structure And Working 43
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ii CHAPTER 4 PROBLEM STATEMENT 47 4.1 The General Problem 48 4.2 The Thesis Problem 49 CHAPTER 5 MODEL FACTOR SENSITIVITY ANALYSIS 53 5.1 Sensitivity Analysis Of The Heuristic 53 5.1.1 Sensitivity Indices 55 5.1.2 Factor Sensitivity Studies 56 5.1.2.1 Sensitivity Analysis Of Accident Model Factors 57 5.1.2.2 Sensitivity Analysis Of Damage Model Factors 60 5.1.2.3 Sensitivity Analysis Of Installation Surcharge Models 61 CHAPTER 6 RESULTS AND CONCLUSIONS OF MODEL FACTOR SENSITIVITY ANALYSIS 63 6.1 Results Of The Sensitivity Analysis Of The Accident Model Factors 63 6.1.1 Main Effects Of Accident Factors 66 6.1.2 Accident Model Factor Interactions 74 6.2 Results Of The Sensitivity Analysis Of The Dama ge Model Factors 76 6.2.1 Main Effects Of Damage Model Factors 77 6.2.2 Damage Model Factor Interactions 81 6.3 Results Of The Sensitivity Analysis Of The Installation Surcharge Model Factors 83 6.3.1 Main Effects Of Insta llation Surcharge Model Factors 84 6.3.2 Installation Surcharge Models Factor Interactions 90 6.4 General Conclusions 91 CHAPTER 7 MODEL OUTPUT EVALUATION AND ENHANCEMENT STUDY 92 7.1 Problems With The Present Working Procedure 92 7.2 The Ideal Configuration Selector 97 CHAPTER 8 RESULTS AND CONCLUSIONS OF MODEL OUTPUT EVALUATION AND ENHANCEMEN T STUDY 109 8.1 Advantages Of Using The Ideal Configuration Selector 109 8.2 General Conclusions 118
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iii CHAPTER 9 FUTURE WORK 119 9.1 Sensitivity Analysis Of The AASHTO Utility Placement Rules 120 9.2 Development Of The Damage Model 121 REFERENCES 122 APPENDICES 129 Appendix A: Variance Sensitivity Indi ces (Sobol' 1990b) 130 Appendix B: Standard Utility Placement Experiments 133 Appendix C: Analysis of Variances Tables 135 Appendix D: Configuration Di fferentiation And Clustering Techniques 139 D.1 Cost Dot Technique (CDT) 139 D.2 The Metric 140 Appendix E: Jiggle Sensitivity Tool 144
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iv LIST OF TABLES Table 5.1: Levels Of Factors Varied For The Acci dent Model Factor Analysis 59 Table 5.2: Levels Of Factors Varied For The Da mage Model Factor Analysis 61 Table 5.3: Levels Of Factors Varied For The In stallation Surcharge Mode l Factor Analysis 62 Table 6.1: First Order Sensitivity I ndices For Accident Model Factors 63 Table 6.2: Second Order Sensitivity Indices Fo r Accident Model Factors 64 Table 6.3: Major Factor Interactions Influe ncing The Accident Costs 75 Table 6.4: First Order Sensitivity Indices For Damage Model Factors 76 Table 6.5: Second Order Sensitivity Indices For Damage Model Factors 76 Table 6.6: Major Factor Interactions Influe ncing The Damage Costs 82 Table 6.7: First Order Sensitivity Indices For Installation Surcharge Model Factors 83 Table 6.8: Second Order Sensitivity Indices For Installation Surcharge Model Factors 84 Table 7.1: Nine Point Scale For Characteristic Im portance 102 Table 7.2: Decision Matrix For Th e Weighted Product Model 102 Table 8.1: Optimal Solution Determined By The He uristic 111 Table 8.2: Optimal Solution Determined By The ICS 111 Table 8.3: List Of 10 Optimal Solutions Determined By The ICS 112 Table 8.4: Solution Determined By The ICS For The Best Corridor Space Utilization 113
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v Table 8.5: Solution Determined By The ICS For Flexibility In Utility Positioning 114 Table 8.6: Solution determined By The ICS For Fa irness In Individual Utility Costs 115 Table 8.7: Top 10 Configuration Obtained With The ICS 116 Table 8.8: Sensitivity / Criticality Of The Resu lts 117 Table B.1: Standard Utility Placement Experiments Initial Setup 133 Table B.2: Standard Utility Placement Experiment 1 134 Table B.3: Standard Utility Placement Experiment 2 134 Table C.1: Analysis Of Variance s (ANOVA) Of Accident Model Fact ors 135 Table C.2: Analysis Of Va riances (ANOVA) Of Damage Model Factors 137 Table C.3: Analysis Of Va riances (ANOVA) Of Installati on Surcharge Model Factors 137
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vi LIST OF FIGURES Figure 1.1: Diagram Of U tilities Placed In Transportation ROW 2 Figure 3.1: Installation Cost Function Of A Utility 26 Figure 3.2: Inconvenience Surcharge Re gion And Model 27 Figure 3.3: Shoring Surcharge Region And Model 28 Figure 3.4: Access Costs Function Of A Utility 30 Figure 3.5: Damage Cost Model 31 Figure 3.6: Damage Cost Function Of A Utility 32 Figure 3.7: Encroachment Angle, Swath Wi dth And Impact Zones 34 Figure 3.8: Accident Cost Function Of A Utility 37 Figure 3.9: Cumulative Cost Function Of A Utility 38 Figure 3.10: Clearance Constraints 40 Figure 3.11: Safety Constraints 42 Figure 3.12: Stacking Constraints 43 Figure 3.13: Working Structure Of The Heuristic 45 Figure 5.1: Growth In Traffic Over Th e Project Life 58 Figure 6.1: Main Effect Of Design Year On The Accident Costs 66 Figure 6.2: Main Effect Of Design Speed Of The Road On The Accident Costs 67 Figure 6.3: Main Effect Of Average Daily Tra ffic On The Accident Costs 68 Figure 6.4: Main Effect Of Tr affic Growth Rate On The Accident Costs 69
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vii Figure 6.5: Highway Diagram Explaining A ccident Factors 70 Figure 6.6: Main Effect Of Nu mber Of Lanes On The Accident Costs 71 Figure 6.7: Main Effect Of Lane Width On The Accident Costs 72 Figure 6.8: Main Effect Of Number Of Aboveground Facilities On The Accident Costs 73 Figure 6.9: Main Effect Of Project Life On The Accident Costs 74 Figure 6.10: Interaction E ffects Of Accident Factors 75 Figure 6.11: Main Effect Of Maximum Damage On The Damage Costs 78 Figure 6.12: Main Effect Of Default Cover On The Damage Costs 79 Figure 6.13: Main Effect Of Maximum Depth On The Damage Costs 80 Figure 6.14: Main Effect Of Damage Fracti on On The Damage Costs 81 Figure 6.15: Interaction Eff ects Of Damage Model Factors 82 Figure 6.16: Main Effect Of Shoring Surcharge On The Total Optimal Costs 85 Figure 6.17: Initial Optimal Configuration Determined (Shoring) 86 Figure 6.18: Optimal Configuration Determ ined After Increasing The Shoring Surcharge 86 Figure 6.19: Main Effect Of Inconvenience Surcharge On The Total Optimal Costs 87 Figure 6.20: Initial Optimal Configuration Determined (Inconvenience) 88 Figure 6.21: Optimal Configuration Determ ined After Increasing Inconvenience Surcharge 88 Figure 6.21: Optimal Configuration Determined W ith Larger Increase In Inconvenience Surcharge 89 Figure 6.22: Main Effect Of Inconvenien ce Surcharge Region On The Total Optimal Costs 90 Figure 6.23: Interaction Effect Of Installation Surcha rge Models Factors 91
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viii Figure 7.1: Corridor Search Coverage At Step Size 1 94 Figure 7.2: Corridor Search Coverage At Step Size 0.5 94 Figure 7.3: Variation In The Total Cost s Of Optimal Solutions For 3 Utility Experiment Using Varied Search Step Sizes 95 Figure 7.4: Variation In The Total Cost s Of Optimal Solutions For 4 Utility Experiment Using Varied Search Step Sizes 95 Figure 7.5: Variation In The Total Cost s Of Optimal Solutions For 5 Utility Experiment Using Varied Search Step Sizes 95 Figure 7.6: Working Structure Of The Heuristic With The Ideal Configuration 98 Selector Figure 8.1: Optimal Confi guration Determined Using The Heuristic 110 Figure 8.2: Optimal Configuration Determined By The ICS 111 Figure 8.3: Configuration Determined By The IC S For The Best Corridor Space Utilization 114 Figure 8.4: Configuration Determined By The ICS Fo r Flexibility In Utility Positioning 115 Figure 8.5: Configuration Determined By The ICS Fo r Fairness In Individual Utility Costs 116 Figure D.1: Quantifying Configuratio nal Differences Using The Metric 141 Figure D.2: Cost Dot And Metric Value Plots For Differentiating Between Configurations In A 3 Utility Step Size Sweep 142 Figure D.3: Cost Dot And Metric Value Plots Fo r Differentiating Between Configurations In A 5 Utility Step Size Sweep 143 Figure E.1: Jiggling of Utility for Configuration Sensitivity Study 144
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ix A SENSITIVITY ANALYSIS OF A HE URISTIC MODEL USED FOR THE PLACEMENT ALLOCATION OF UTILITIES IN TRANSPORTATION RIGHTOFWAY CORRIDORS Steve Clarence Christian ABSTRACT The requirements for public utility system s in the United States of America have grown enormously over the years triggering a tremendous shortage fo r space available to public utilities on and within transportati on rightofways (ROW). Overcrowding and improper location of utilities ha s resulted in problems such as, damage to infrastructure, traffic accidents and, interrup tion of service to customers. The project titled, Optimal Placement of Utilities within FDOT RightofWay, sponsored by the Florida Department of Transportation (FDOT), and curr ently being investigated at the University of South Florida, presents a decisionmaking he uristic aimed at developing a better utility placement allocation system (Kranc et. al) [6]. Working in accordance with the guidelines of safety, relocation, and clea rance for utility placement set by the American Association of State Highway and Transportation Offici als organization (AASHTO), the heuristic finds suitable locations for the utilities in ROW corridors. However, a model being used to advocate a practice having large social and economical impacts is more likely to play
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x the role of generic evidence in a trial, whos e weight must ultimately be established by a jury. The question being addressed to the sy stem must be scrutinized carefully, and the formal structure updated iteratively until it proves capable of provi ding an answer to the given question. A good sensitivity analysis can provide this generic quality assurance to the model and help demonstrate the worthiness of the model itself. This thesis is a quantitative and qualitative sensitivity analysis of the abovementioned heuristic. The analys is is conducted in two parts, 1. The Model Factor Sensitivity Analysis, with the objective of assessing the uncertainties associated with the modeling of the heuristic. This analysis focuses primarily on providing an evaluation of the confidence in the heuristic and its predictions by analyzing the in fluences that variations in the input factors have on the outputs of the utility cost assessment m odels and the final output of the heuristic itself. Variance based sensitivity indices deri ved from Sobol sensi tivity indices [42] are used here for this purpose. 2. The Model Output Evaluation and Enhancem ent study, which initially focuses on understanding / evaluating the complexities of the discrete step, cost optimization procedure used in the heuristic and later, based on certain observed shortcomings and problems develops an enhancement, the Id eal Configuration Sel ector (ICS) to be implemented with the heuristic. The ICS a ddresses all the problems of the heuristic with the help of experimental speedup, positional sensitivity and refinement tools and employs a multi criterion evaluation technique for utility configuration assessment to provide substantiation to th e outputs determined by the heuristic.
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1 CHAPTER 1 INTRODUCTION Roads and highways are the backbone of our transportation system. But apart from their obvious application, they also play the impor tant role of accommodating utilities in their rightofway (R OW). A utility is defined as a privately, publicly or cooperatively owned line, facility or system, for producing, tr ansmitting or distributing communications, cable television, power, electri city, light, heat, gas, oil, crude products, water, steam, waste and storm water, not connected with highway drainage or any other similar commodity including any fire or poli ce signal system or street lighting system, which directly or indirectly serves the public [2]. Utility lines can be subsurface lines (like water or sewer lines) or above the ground aerial structur es (like telephone or electric lines). Around 1916, the United States of Ameri ca embraced the concept of utility transportation corridors [1]. Since then, uti lities have been located within the ROW of transportation roads and highways (F igure 1.1 is an illustration of a Highway with Utilities placed within the ROW). A rightofway is defined as any part or access to a public agencys transportation facility abov e, at the surface or below the ground [3]. State Departments of Transportation (DOT) are public agencies th at have regulatory responsibility for the maintenan ce and operations of the roads and highways in a state. It is their duty to carry out these functions in an efficient ma nner, ensuring the safety, traffic
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carrying ability, and physical integrity of the facilities within and along the ROW. A utilitys presence in the ROW affects these functions and hence the DOT is in part responsible for its location. Figure 1.1: Diagram Of Utilities Placed In Transportation ROW 2
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3 The previous systems used by the DOTs for allocating placement locations to utilities within the ROW were based on a first come first served method with no governing rules or regulations. Such evolut ionary systems were neither safe nor economic solutions to the problem. In 1956, when the national system of interstate highway program was created it became appare nt that the control of access by utility firms to the ROW was essential to ensure th e safe operations of the highway systems. The AASHTO prepared the Policy on the Accommodation of Utilities on the National System of Interstate and Defense Highways [4] in 1959, and in 1966 it was made mandatory for all DOTs incharge of their states roadways and highways to follow the regulations given by the AASHTO. The Fede ral Government required each State to develop and maintain a Utility Accomm odation Manual (UAM) to summarize policies regarding location and reloca tion of facilities within transportation corridors [4]. Since then, there has been a rapid gr owth in vehicular volumes, speeds and weights resulting in a larger network of ro ads and highways. Recent years have also witnessed a tremendous growth in tra ffic and customers for companies like telecommunication, cable television and internet providers. This has created a demand for increased access to various utility lines and, a much bigger distribution of utility systems. Considering the present number of utilitie s and forecasting a probable requirement for new ones, in the future, a wide range of utilities will have to share the already crowded ROWs. Many of the present roads have na rrow ROWs or are running through crowded urban areas. It has become increasingly difficu lt for the DOTs to upgrade older roads to provide the necessary capacity for placement of new utilities, and also ensure the safety of motorists using them. Crowding and increa se in the demand for space has resulted in
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4 problems of damage to infrastructure, public safety, interruption of service to customers and traffic disruptions. Obviousl y, there exists a very urgent need for a bette r solution to the utility placement problem than that pr ovided in the utility accommodation manuals alone. The ongoing research project titled, Optim al Placement of Utili ties within FDOT rightofway, sponsored by the FDOT and curren tly being investigated at the University of South Florida (Kranc et. al.) [6], presen ts a decisionmaking heuristic, the goal of which is to build a better utility placement a llocation system. The heuristic, described as a discrete step, cost optimization model, numer ically simulates the shape and dimensions of the transportation co rridor, and physical information of the utilities to be located within it. Working in accordan ce with the rules and regulati ons of safety, relocation, and clearance for utility placement set by AASHTO, and utilizing positional cost assessment models, the heuristic finds suitable (near optim al cost) locations for the utilities in the ROW corridors. 1.1 Thesis Focus : Sensitivity Analysis & Model Enhancement When a model is used for making decisi ons that could have large social and economical impacts, verification analysis is naturally invoked for the corroboration, quality assurance, and defensib ility of its output. Issues of relevance and transparency become critical in this context. This thes is is a quantitative a nd qualitative sensitivity study of the abovementioned heuristic. Th e study primarily aims at providing an evaluation of the confidence in the model by assessing the uncer tainties associated with the modeling process and the out come of the model itself.
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5 The heuristic finds suitable placement lo cations for utilities within the ROW corridors by optimizing the total costs of en tire utility systems. Positional costs of individual utilities of a configuration (a pos itional arrangement of utilities in the cross section view of the corridor) are determined from respective overall cost functions, estimated from various global model factors and smaller cost models integral to the main model. The first part of the model analysis, that is, the factor se nsitivity study, addresses the issue of assessing the uncertainties asso ciated with the modeling process by analyzing the influences that variations in the input factors (both global and intra modular) have on the outputs of the uti lity cost assessment models and th e heuristic itself. The approach adopted for this purpose is a combination of the design of experi ments (DOE) technique and sensitivity analysis perfor med in a specific manner to determine variance sensitivity indices (based on Sobol sensitivity indi ces) [42], a measure used commonly for quantifying the effect of input factors on the output of complex models. The factors considered in this study are categorized and analyzed separately based on their specific application and area of influence in the mode l. The different categories considered are, 1. Accident Model Factors 2. Damage Model Factors 3. and, Installation Surcharge Models Factors. The second part of the model analysis is an enhancement study (an assessment of the quality) of the final output of the heuristic. This study initially focuses on better understanding the complexities of the discrete step, cost optimization procedure used in the heuristic, and later, based on certain observed shortcomings and problems in the
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6 output determination technique, suggests an enhancement, that is, the Ideal Configuration Selector (ICS) to be implemented with the heuristic. During conference presentations, it was no ticed that besides the DOT, a diverse group of stakeholders such as, the public (consumers), utility owners (public and private,) and other corporate parties (c ontractors, services etc.) expressed interests in the development of a utility corridor orga nization scheme. A study was conducted to determine a set of criteria to be used fo r the assessment of u tility configurations. Considering the requirements of each party (stakeholder), the following characteristics (qualities) of an ideal utility c onfiguration was finally decided on. 1. Optimality in the total cost of the configurational solution. 2. Fairness in location for util ities of the configuration. 3. Flexibility in the positioning of utilities of the configuration. 4. Low usage of corridor space by the configuration. The next steps involved defining and calcu lating quantifying measures for these ideal configurational characteristics. Experi mental tools and techniques like, the Jiggle Sensitivity Tool (JST), for determining the positional sensitivities and positional flexibilities of utilities in a configuration, the Cost Dot Technique (CDT), and the Metric used in conjunction to identify and quantify differences between output configurations were developed and put to use in the proposed Ideal Confi guration Selector (ICS). The ICS can be described as an e xperimental utility c onfiguration assessmen t tool which uses a multicriterion decision making procedure called the Weighted Product Model (WPM) to assess and rank configura tions according to their c onformity to the desired configurational characteristics. The utility configurati on embodying most of the ideal
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7 configurational characteristic s is selected as the best solution. The configurational rankings obtained from the ICS depend heav ily on the weights (importance measures) assigned by the decision maker to each qua lity characteristic. The sensitivity and criticality of the results (rankings) to the we ights assigned is also determined to provide the decision maker with further insight into the configuration se lection procedure. 1.2 Thesis Outline This thesis underscores through a realw orld application, the usefulness of sensitivity analysis and the scientific chal lenges faced in model development and model corroboration. This thesis is organized as follows: 1. Chapter 2 discusses the literature review of sensitivity analysis and describes the present techniques that are being employed for sensitiv ity studies. Examples are presented to illustrate the use of sensitivity analysis in a wide variety of application areas. 2. Chapter 3, titled The Heuristic, describes the formulation and the working structure of the heuristic. The cost factors, cost assessment models and, rules (constraints) of safety and clearance set by the AASHTO for placement of utilities are also explained here. 3. Chapter 4, The Problem Stat ement, describes the reason for this research and details the proposed studies and work to be performed in the chapters ahead. 4. Chapter 5 is the first part of the analysis titled, Model Factor Se nsitivity Analysis. It is aimed at increasing the confidence in the heuristic and its predictions by assessing the uncertainties associated with the certain input fact ors (global and intra
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8 modular) of the model. The chapter introdu ces variance based sensitivity indices (Sobol sensitivity indices) [42] for quantifying the effect of the input factors on the output of the heuristic. 5. Chapter 6 presents the results of the factor sensitivity studies conducted in chapter 5. 6. Chapter 7, the second part of the model anal ysis titled, Model Output Evaluation & Enhancement, is an evaluation (assessment) study of the final output of the heuristic. This chapter defines characteristics / criterion for an ideal utility configuration, the quantifying measures for wh ich are then used in an experimental utility configuration assessment tool, the Id eal Configuration Selector (ICS) designed to work in conjunction with the previously developed heuristic. 7. An example illustrating the working of the ICS is included in Chapter 8. Output substantiation advantages of using the ICS are highlighted here. 8. Chapter 9 provides recommendations for futu re work and interesting topics for further research / development of the heuristic. 9. The Appendices of the thesis is organized as follows a. The proof for the variance based sensitivity indices developed by Sobol (1990b) [42] is included in Appendix A. b. Appendix B describes the standard utility placement problem considered for most of the analysis conducted on the heuristic. c. Appendix C contains the analysis of vari ances (ANOVA) results for the design of experiment tests conducted on the cost models.
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9 d. Appendix D describes the experimental Cost Dot Technique and the Metric used together in the Ideal Configuration Selector for differentiating between, and clustering common orientati on utility configurations. e. Another experimental tool called the Jiggl e Sensitivity Tool used in the ICS to determine positional sensitivities and utility jiggle (positional flexibility) capabilities is described in Appendix E.
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10 CHAPTER 2 LITERATURE REVIEW This thesis is a quantitative and qualitat ive sensitivity analysis of the heuristic being developed with the intension of building a good utility placement allocation system. Sensitivity analysis is being used he re for the corroboration, quality assurance, and defensibility of this model. In this chap ter, sensitivity analysis and present techniques that are being employed for sensitivity studi es are introduced. Prac tical hints about the associated advantages and efforts needed to effectively select a t echnique and perform a functional sensitivity analysis of a numerical model are in cluded. As a final point, the discussions are illustrated into concrete examples showing the power of sensitivity analysis in a wide variety of application areas. 2.1 Sensitivity Analysis Sensitivity analysis is defined as the study of how the variation in the output of a model (numerical or otherwis e) can be apportioned, qualitat ively or quantitatively, to different sources of variati on, and of how the given model depends upon the information fed into it [60]. Sensitivity analysis is, in the opinion of most scientists, an important element of modeling. Kolb, quoted in Rabitz [36], states that theoretical methods are sufficiently advanced, a nd, it is intellectually dishonest to perform modeling without sensitivity analysis , while Furbringer [24] argues in Sensitivity analysis for modelers,
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11 Would you go to an orthopedist that didnt use Xrays? . Rabitz [36] presented sensitivity analysis as a funda mental ingredient for model bu ilding and a key tool in the understanding of complex physical processes. According to him, sensitivity analysis helps analyze the contents of the model and interface it with observational data. It helps to identify, which factors are critically im portant, how they are interrelated, and especially how they at a given level of descri ption of the system infl uence the behavior of the model. 2.1.1 Why Carry Out A Sensitivity Analysis? Many processes are so complex that physical experimentation is too time consuming, too expensive, or even impossi ble. As a result, to explore systems and processes, investigators ofte n turn to mathematical or computational models. When models are used for making decisions, having a large social and economical impact it is not surprising to meet cynic opinions about them. Accordi ng to Hornberger and Spear [28], .most heuristics will be complex, with many parameters, statevariables and non linear relations. Under the best circumstanc es, such models have many degrees of freedom, and with judicious fiddling, can be made to produce virtually any desired behavior, often with both plausible structure and parameter values. This problem highlighted by Hornberger is acutely felt in the modeling community. The awareness of the danger implicit in selecting a model struct ure as true and working happily thereafter leads to the attempts to map rigorously altern ative model structures into the space of the model predictions [60]. The natural extension of which is the analysis of how much each source of uncertainty weighs on the model prediction.
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12 Thus, almost all models have use for sens itivity analysis. Applications worked by the Joint Research Centre group for Applie d Statistics include: Atmospheric Chemistry [13], transport emission modeling, fish popul ation dynamics [60], co mposite indicators [60], portfolios, oil basin models [60], capital adequ acy modeling, macroeconomic modeling, radioactive waste management [60]. The EC handbook for extended impact assessment, a working document by the European Commission, 2002, states A good sensitivity analysis s hould conduct analyses o ver the full range of pl ausible values of key factors and their interactions, to access how impact change in res ponse to change in key factors . Similar recommendations are found in the United States EPAs White Paper on model acceptability, 1999. In the context of numerical modeling, sens itivity analysis means very different things to different people. For a reliability engineer, sensitivity analysis could be the process of moving or changi ng components in the design. For a chemist, sensitivity analysis could be the analysis of the strength of the relation between kinetic or thermodynamic inputs and measurable outputs of a reaction system, and for a software engineer, sensitivity analysis could be related to the robustness and reliability of the software with respect to diffe rent assumptions. These different types of analyses have in common the aim to investigate how a given co mputational model responds to variations in its input. Modelers generally conduc t sensitivity analys is to determine: 1. If a model resembles the system or pro cess under study. This pr ocess is also known as the validation of the model. 2. Which factors contribute largely to the output variability and require additional research. This process is conducted primarily to streng then the modelers knowledge
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13 base and is known as the calibration pro cess. Part of the calibration study would involve determining the optimum regions w ithin the space of the influential model factors. 3. If certain model factors (or parts of the model) are sign ificant, and if not can be eliminated from the final model. This process is known as the mechanism reduction which enables building a simpler model from a more complex (lumped) model. 4. If there is some region in the space of the input factors for which the model variation is maximum. 5. If and which group of model factors intera ct with each other enough to effect the output of the model. 2.1.2 Types Of Sensitivity Analysis This section, gives an overview of the va rious methods that are currently used in sensitivity studies. The choice of which sensiti vity analysis method is a difficult, since each technique has strengths and weaknesse s and would depend on the problem the investigator is trying to address and the characterist ics of the model under study. Let us assume that we are studying a system of k input factors x = (x 1 ,x 2 ,.,x k ) and an output variable y. In practice, the inpu t factors are affected by several kinds of heterogeneous uncertainties that reflect our imperfect knowledge of the system. Hence it is convenient for the purpose of sensitivity analysis to tr eat them as random variables with assumed probability distributions. The vector x can be seen as a realization of a random vector X characterized by a joint probability density function P(X ) = P(X 1
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X 2 ,.., X k ) assumed to be known. The output variable y can then also be seen as a realization of a random variable Y, and the relationship between the input factors and the output under study can be represented by a mathematical construction f(.) such that, f(X)).....XX,f(XYk21 Different sensitivity analysis strategies may be applied, depending on settings. The three main settings identified are, 1. Factor Screening: Where the task is to identify influential factors in a system with many factors. This method is used in dealing with models that are computationally expensive to evaluate and have a large number of input factors. As a drawback these economical methods tend to provide only qualitative sensitivity measures i.e. they rank the input factors in importance but do not quantify how much more important a given factor is than another. 2. Local Sensitivity Analysis: Where the emphasis is on the local impact of the factors on the model. Local sensitivity analysis involves computing partial derivatives of the output functions with respect to input factors. 3. Global Sensitivity Analysis: Where the emphasis is on apportioning the output uncertainty to the uncertainty in the input factors described typically by probability distribution functions or range of factor existence. Global sensitivity analysis typically takes a sampling approach, and the uncertainty range given in the input reflects the imperfect knowledge of the model factors and parameterization. A global method evaluates the effect of input factor x i while all other x j ji, are varied as well. In contrast, the local perturbative approach is based on partial derivatives, the 14
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15 effect of the variation of the input factor x i when all other x j j i, are kept constant at their central value [32]. 2.1.2.1 Screening Designs Screening designs are preliminary numerical experiments whose purpose is to isolate the most important factors from amongs t a large number that may affect the model response. Typical screening designs are OneAtaTime (OAT) experiments in which, the impact of changing the values of each factor is evaluated in turns (Daniel, [19], [20]) The experiment which uses the standard values is defined as the control experiment. For each factor, two extreme values are selected and then the analyst decides the control value (normally, midway between the two extremes ). The magnitude of residuals, defined as the difference between the perturbed experimental results and the control, are compared in order to evaluate factors to which the model is significantly sensitive. One major limitation of the OAT experi ments is that they allow only the evaluation of the main effects (the effects of the input factors w ithout including their mutual interactions). The use of factorial experimentation (Box et al., [12]) allows not only for the evaluation of the main effects, but also that of the inter actions. In a factorial experiment approach, all fact ors are perturbed simultaneously to one of their possible values called levels and all possible comb inations of values are covered. When the number of factors is too large, or the mode l evaluation is very time consuming, a useful alternative is given by the fractional factoria l experiment (Box et al., [12]). Andres developed the Iterated Fractional Factoria l Design (IFFD) (Andres and Hajas, [10]),
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16 which required fewer runs than there were fa ctors. IFFD estimated the main effects, quadratic effects and two factor interactions of influential factors. Many of the screening methods described rely on strict assumptions about the nature or absence of interactions between factors. One exception however, is that of Cotter [16]. Cotters method does not require pr ior assumptions about interactions, and its results are hence easier to interpret. This design is also called the systematic fractional factorial design. 2.1.2.2 Local Sensitivity Analysis Local sensitivity analysis concentrates on the local impact of the factors on the model. Local sensitivity analysis is usually carried out by computi ng partial derivatives of the output function with respect to the in put variables, that is, local sensitivities provide the slope of the calcu lated model output in the fact or space at a given set of values. A differential sensitivity analysis involves the following four steps. In the first step, base values and ranges are selected for each input factor. In the second step, a Taylor series approximation of the output is developed aro und the base values of the input. In the third step, va riance propagation techniques are used to estimate the uncertainty in the output in terms of its expe cted value and its variances. In the final step, the Taylor series approximations are used to estimate the importance of individual input factors [32].
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17 The greatest effort in a differential sensitiv ity analysis is the determination of the partial derivatives in the Taylor seri es approximation. A number of specialized techniques have been developed to facilitate the calculation of thes e derivatives, namely, 1. The Brute force method which uses the finite difference approximations. 2. The method of Miller and Frenklach [34], based on approximations by empirical models of the solution of the system in a parameter region. 3. The Green function method also called the variation method. 4. The polynomial approximation method elaborated by Hwang [29], which transforms the sensitivity differential equations into a set of algebraic ones. Usually only the first order partial derivatives called the first order local sensitivity coefficients are computed and studied. They constitute the sensitivity matrix S which represents a linear approximation of the dependence of the solutions on factor changes. The order of importance that can be deduced from local sensitivities is called order of tuning importance (Turanyi [43]). If the system under consideration is not spatially homogeneous constant factor system (factors are also a function of time and space), sensitivity analysis is based on their perturbation by another function usi ng the principles of non linear functional analysis. Dickinson and Gelinas [22] were the first to tackle the problem of factor function, and introduced a sensitivity meas ure depending on the perturbing function (Turanyi [43]). The sensitivity measure was named sensitivity density (Demiralp and Rabitz [21]). For all models of real systems, the valu es of the factors are subject to some uncertainty. In most cases, such uncertainties can be very high, and sometimes when the
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18 factors are changed within the range of uncer tainty, a qualitatively different model is obtained. Local sensitivities however, are totally incapable of providing information on the effect of significant factor changes. Lo cal sensitivities are really local, and the information provided is related to a si ngle point in the space of factors. 2.1.2.3 Global Sensitivity Analysis Global sensitivity analysis techniques have been discussed by C ukier et al. [18], Iman and Helton [30], Sobol [ 42] and Saltelli and Homma [26] Global sensitivity analysis apportions the output uncertainty to the uncertain ty in the input factors, described typically by probability distributive f unctions that cover the factors ranges of existence. The ranges are valuable since they represent our knowledge or lack of it with respect to the model and its parameterization. Global sensitiv ity analysis methods can be characterized by the following two properties: 1. The inclusion of influen ce of scale and shape : The sensitivity estimates of individual factors incorporate the effect of the range and the shape of their probability density functions. 2. Multidimensional averaging : The sensitivity estimates of individual factors are evaluated varying all ot her factors as well. Global sensitivity analysis techniques are known as variance based methods. Variance based techniques such as the stan dardized regression coefficients (SRC), correlation coefficients (Pearso n) and partial correlation coefficients (PCC) rely on the assumption that the output and th e input factors are near linearly related, and their rank equivalents such as the standardized ra nk regression coefficients (SRRC), Spearman
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19 correlation and partial rank corr elation coefficients (PRCC) rely on the assumption that the output and input are n ear monotonically related. Correlation ratios and importance measur es (Hora and Iman [27]) are derived from a simple description of uncertainty using probability distributions and are based on the conditional variance of the model output. The Fourier amplitude sensitivity test (FAST), created in the 1970s by Cukier, Schaibly [17] and others and further developed by Koda and McRae [32], offers a sensitivity analysis method that is indepe ndent of any assumptions about the model structure, and works for monotonic and nonmo notonic models. The core feature of the FAST is that it explores the multidimensiona l space of the input factors by a search curve that scans the entire input space. Some varia tions of the basic scheme of the FAST are also known an example is given by the Wals h amplitude sensitivity procedure (WASP) (Pierce and Cukier [35]). Salte lli et al.[38] proposed a new FAST technique which uses a new Fourier transform function and a resampling plan. The Sobol sensitivity indices [42], an orig inal extension of design of experiments (DOE) to the world of numerical experime nts first published in 1990, are similar to FAST in the sense that the total variance of the model output is assu med to be made up of terms of increasing dimensionality. Sobol indices are superior to the original FAST in that the computation of the higher interaction terms is very natural and is similar to the computation of the main effects. In recent years, global quantitative sensitivity analysis techniques have received considerable attention in the lite rature (RESS 1997; JSCS 1997; CPC 1999; JMCDA 1999).
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20 2.1.3 Application Examples Of Sensitivity Analysis The following illustrations are examples of the applicability of sensitivity analysis in a wide variety of functional areas. 1. Scenario and Parametric Sensitivity and Uncertainty Analysis in Nuclear Waste Disposal Risk Management : The case of GESAMAC. Sensitivity analysis was used here in the process of model audit, studyi ng the scenario and parametric uncertainty in nuclear waste disposal risk assessment [23]. 2. Sensitivity Analysis for Signal Extraction in Economic Time Series : Sensitivity analysis was used here to answer the question of how sensitive the unobserved components in a time series are to a m odel and the parameter choice within the chosen model. Bayesian techniques and im portance measures were used to explore the effect of different model assumptions and to direct the model choice [60]. 3. Analysis and Interpretati on of Sensitivity Measures related to Ground water Pressure Decrease and Resulting Ground Subsidence : Application of First order FORM and second order (SORM) reliabil ity methods were used to determine reliability measures to study sensitivity measures for ground subsidence in an engineering context [60]. 4. OneataTime and Mini Global Analysis for Characterizing Model Sensitivity in the Nonlinear Ozone Predictions from the US EPA Regional Acid Deposition Model (RADM) : This analysis involved applying sens itivity analysis to a large, complex Eulerian air quality model. Both Oneata Time and global techniques for a restricted set of model inputs under two scenarios of emission [60].
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21 5. An Application of Sensitivity Anal ysis to Fish Population Dynamics : Sensitivity Analysis was applied to an ecological mode l used to explore the dynamics of fish ecosystems, particularly the collapse a nd regeneration of fish species. Morris screening techniques were applied to identify factors that required further investigation [60]. 6. Global Sensitivity Analysis : A Quality Assurance Tool in Environmental Policy Modeling. This study was a policy problem, how to dis pose of solid waste and explore an incineration versus landfill option for solid waste using different sets of indicators. The FAST method was used he re to quantitatively rank the group of factors according to their influen ce on the output uncertainty [60].
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22 CHAPTER 3 THE HEURISTIC The network of roads and highways in th e United States of America has grown enormously over the years, and with it, the need for public uti lities. Crowding and improper location of utilities in public tran sportation rightofways (ROW) has resulted in problems such as, damage to infrastruc ture, traffic accident s and, interruption of service to customers. The present system adopted by the State Departments of Transportation (DOT) for allocating placemen t locations to utilities within ROW corridors is based on a first co me first served method with certain governing rules set by the AASHTO way back in 1959. This regulatory system however is ne ither a safe nor an efficient (economically or space utilizati on wise) solution to th e utility placement allocation problem. The research project title d, Optimum Placement of U tilities within FDOT RightofWay, (Kranc et. al.) [6], sponsored by the Florida Department of Transportation (FDOT) and currently being investigated at th e University of South Florida, aims at building a better utility placement allocation sy stem. The formulation and working of the heuristic being developed as part of this investigation to provide a ba sis for making rational decisions regarding the organizati on of utilities within transportation ROW corridors is described in the following sections.
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3.1 Mathematical Representation Of The Heuristic A mathematical model is defined as a series of equations, input factors, parameters, and variables aimed at characterizing the process being investigated or simulated. The utility placement allocation heuristic related to this research is characterized as a discrete step, cost optimization mathematical model. It numerically simulates the shape and dimensions of the transportation ROW corridor, and the physical information of the utilities to be located within the corridor. Guided by the constraining rules and regulations of safety, relocation, and clearance for utility placement set by the AASHTO, and with the help of four positional utility cost assessment models the heuristic finds optimal cost locations for the utilities in the ROW corridors. The models objective function, the formulation of its constituent cost models and the AASHTO utility placement guidelines (constraints) under which it operates are explained in the following sections. 3.1.1 Model Objective Function The objective of the heuristic is to determine the most economically advantageous configuration of the utilities selected for installation in a transportation ROW corridor. The total cost of a configuration is the sum of the individual position sensitive cost of each of its constituent utilities. The best utility configuration is determined by optimizing the total cost of all feasible configurations determined for that ROW corridor. Mathematically this objective function is represented as, N1jN21iy)(x,C....y)(x,Cy)(x,CTCMin 23
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Where, iTC = Total cost of the utility configuration i and, y)(x,Cj = Positional cost of utility j. ( j= 1 to N ) The individual positional cost of a utility j located at (x, y) is the sum of the four position sensitive cost components. That is, y)(x,cy)(x,cy)(x,cy)(x,cy)(x,CACCIDENT jDAMAGE jACCESS jONINSTALLATI jj where, y)(x,cONINSTALLATI j = Positional Installation cost of utility j. y)(x,cACCESS j = Positional Access cost of utility j. y)(x,cDAMAGE j = Positional Damage cost of utility j. 24 y)(x,cACCIDENT j = Positional Accident cost of utility j. ( j= 1 to N ) 3.1.2 Cost Models A principal requirement for corridor optimization is the understanding and quantification of the position sensitive costs (initial and recurring) associated with individual utilities installed in the ROW corridor. The cost of the j th utility of a configuration located at position (x, y) in the ROW corridor is given as, the sum of four
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25 position sensitive components c j namely, installation, access, damage and accident costs. These costs are determined from resp ective cost models described below. 3.1.2.1 Installation Cost Model The installation cost of a utility is define d as, the initial (nonrecurring) cost of placing the utility within a ROW corridor. This includes the costs of excavation, maintenance of traffic, conflict accommodati on, and shoring but excludes the material costs of the utility conduit itself. The inst allation cost model assu mes that all utilities have approximately the same position sensitiv e installation costs which are determined by the following. 1. Depth of Installation : Installation costs of a utility increases with increase in the installation depth because of added diggi ng, burying, reinforcing (shoring), and soil treatment costs at deeper locations. 2. Horizontal Positioning : Installation costs of a utility vary horizontally based on the placement region in the ROW. The tw o basic regions defined are, a. Paved Region : The part of the ROW that is below the pavement (road). b. Unpaved area : The part of the ROW that is not presently paved over. Installation costs for utilities placed below the pavement are generally greater than for those placed in the unpaved region for obvious reasons. Figure 3.1 shows a typica l installation cost func tion plot obtained from information collected by a survey of utilit y companies. The plot shows the cost of installation of a utility in K$/Mile with respect to the depth of installation in inches. For
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this plot, the paved region installation costs were considered to be twice that of the unpaved region, shown as two different installation cost function curves. INST ALLA T I ON COST S (k$/MILE) 70 80 10 20 30 40 50 60 0 0.0 20.0 40.0 D E P T H 60.0 80.0 100.0 120.0 140.0 UNPAVED PAVED Figure 3.1: Installation Cost Function Of A Utility Besides the def a ult co sts, a utility m i ght also have ad dition a l ins t allation surcharg es applied, cond ition a l to it being loca ted in certain undesira ble regions within the ROW corrido r. Thes e surcharges are used p r im arily as deterr ents in th e heuris tic. The surcharg es are summ arized as, 1. Inconven ien ce Surcharg e : This is an additiona l installation c h arge applie d to a utility when it has to be placed within th e ROW in close proxim ity to the p a vem e nt. Since installation and access events to this utility will cause dis r up tion of traffic plying th e road, the inconvenience caused is factored in as a surcharge to the utility for installation at that partic ular location. The inconvenien ce surcharge model adds a surcharg e that is m a ximum startin g from the edge of the pavem e nt and reduces 26 linearly to zero at the end of the surc harge region. Figure 3.2 shows the surcharg e region and the associated inconvenience surcharge m odel.
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Figure 3.2: Inconvenience Surcharge Region And Model 2. Shoring Surcharge: This additional installation charge is applied to a utility that has to be placed close to the extreme most position (easement) of the ROW corridor. Shoring costs are used to factor in, the difficulties involved, additional labor and extra materials required for locating utilities at this undesirable location. The shoring surcharge model assumes the region starting from the edge of the ROW extending 3 feet inward as the shoring region. A flat cost is applied to all utilities to be placed in this region. Figure 3.3 depicts the shoring region and the associated shoring surcharge model. 27
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Figure 3.3: Shoring Surcharge Region And Model Mathematically, as shown by equation (1), the installation costs of a utility is typically modeled as a vertical function g(y), modified by a multiplicative factor, represented as a(x) (a function of horizontal position), to account for under pavement installation and, an additive cost b(x, y) to account for additional charges like shoring surcharge, inconvenience surcharge, or material costs associated with deep installations. P j is the probability of installation of utility j in year Y inst and to cover cases involving damage incidents during deferred installation or relocation, an additional additive damage factor c j dam is included in the installation cost model. (y)]cy)(x,b(y)g[a(x)Py) (x,cdam jjinstjjONINSTALLATI j (1) 28
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29 3.1.2.2 Access Cost Model For any facility placed with in the ROW corridor there exist needs to access the subsurface utility installation, perhaps for a new connection or for routine maintenance. The costs incurred per year over the entire project life for providing this kind of access to a utility is known as the access cost of the utility. This cost is determined by the following. 1. Depth of Installation : Access costs of a utility increases with increase in the installation depth because of added diggi ng, burying, reinforcing (shoring), and soil treatment costs at deeper locations. 2. Horizontal Positioning : Access costs of a utility vary horizontally based on the placement region in the ROW. The tw o basic regions defined are, a. Paved Region : The part of the ROW that is below the pavement (road). b. Unpaved area : The part of the ROW that is presently not paved over. Access costs of utilities placed below the pavement are generally greater than for those placed in the unpaved region. 3. Frequency of access : Is the number of times a year the utility will be accessed for maintenance. 4. Length of excavation : Access to a subsurface utility re quires only certain parts of the entire line to be exposed. The ratio of the trench length excavated to the length of the entire utility line is kno wn as the equivalent length of excavation. Figure 3.4 shows a typical utility access cost function plot with the access costs of a utility in K$/Mile with respect to the depth of installation in inches. For this plot, the
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paved region access costs were considered to be twice that of the unpaved region, shown as two different access cost function curves. Figure 3.4: Access Costs Function Of A Utility Mathematically, as shown by equation (2), the same functional dependence used to model the initial installation function with three a multiplicative factors, the equivalent length of excavation L eq the rate of access f acc (the number of events / year / distance along corridor) and the number of years of service is adopted to determine the actual access costs. ACCESS MAINTENA NCE COSTS (k$/MILE) 140 160 20 40 60 80 100 120 0 0.0 20.0 40.0 D E P T H 60.0 80.0 100.0 120.0 140.0 UNPAVED PAVED acc inst sl eq dam j j inst j j ACCESS j )f Y (Y (y)]L c y) (x, b (y) g [a(x) P y) (x, c (2) 3.1.2.3 Damage Cost Model During routine excavations (new installati ons o r access ev ents) in the corridor there exis ts som e probability of acciden t al dam a ge to the u tility itse l f or to f acilities already located in the co rrido r. 30
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The data on damage events is not very accurate and hence a simple linear damage model is used to determine damage costs. The model assumes that the number of accidental damage incidents is proportional to the expected number of access events and that excavating to conduits buried deep within a corridor will more likely result in damage to the utility itself and other utilities in the corridor (Depicted in Figure 3.5). Figure 3.5: Damage Cost Model Mathematically, as assumed in the damage model, the cost per damage incident is primarily a function of depth g dam modified by multiplicative factors such as, the rate of access f acc the fraction of events resulting in damage incidents f dam (taken arbitrarily as 1%) and a maximum cost per incident c j max at the maximum depth that reduces linearly to the highest possible location for the utility. 31
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damaccinstsldammax jjDAMAGE jf)fY(y)](Yg[cPy) (x,c (3) The plot in Figure 3.6 shows the access damage costs in K$/Mile versus installation depth in inches. A CCESS DAM A GE COS T S (k$/MILE) 32 Figure 3.6: Da m a ge Cost Functio n Of A Utility 3.1.2.4 Accident Cost Model The cost of traffic accidents with the above ground com ponent of a utility is an im portant p a rt of a utility s co st f unction. Th is cos t is p r im arily dependent on the horizon tal p o sition i ng o f the utility in the ROW corridor. A procedure to estim a te the econom ic values for traffic accidents with sta tio nary objects at the side of the roadway was developed by the F e deral Highway Adm i ni stration and is used for developing the acciden t m odel. The con s tructi on of the acciden t function an d m odel is based on Figure 3.7. Consider the traffic travel ing in one direction along th e roadway in adjacent lanes (i.e. the lanes closest to an above ground object). A certain f r act ion of these vehicles will leave the p a vem e nt and trav el for som e distance beyond the pavem e nt edge. The 0.0 0.00 3.00 3.50 0.50 1.00 1.50 2.00 2.50 20.0 40.0 60.0 80.0 100.0 120.0 140.0 D E P T H
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33 approach used in the accident model is to calculate the probability that a vehicle leaving the roadway within an interval along the paveme nt, travels sufficiently far to collide with some portion of the object. For an approxima te mix of vehicula r traffic, a single encroachment angle e is defined, characterized as a function of the roadway design speed. P(x), the probability of an encroaching vehicle traveling a perpendicular distance x os from the pavement (encroachment distance) for a set of typical design speed is also tabulated. The above ground object is partitioned into several zones, each with different likelihood for impact. For a rectangular object, collisions with th e face perpendicular (Zone 1) and the face parallel (Zone 3) to the roadway are possible, as is a collision with the corner of an object facing the traffic (Z one 2). Round objects are tr eated in a slightly different manner and are represented in term s of reduced diameter. To account for the possibility of skid with rota tion, the vehicle path width is taken to be a swath of 3.6 meters. The encroachment factor EF, which is the dimensionless ratio between the distance along the pavement, and the distance along the line perpendicular to the pavement, defines the impact zone of interest The number of impacts with a particular zone occurring as a result of vehicles leavi ng the pavement within the boundaries of the path leading to the zone is defined as the impact factor IF, and is given by the product of the encroachment factor and the integrated pr obability that a vehicl e will travel to the offset distance of the zone. Th is distance corresponds to th e distance along the pavement equivalent to a particular component of the object times the ratio of impacts per encroachment.
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Figure 3.7: Encroachment Angle, Swath Width And Impact Zones For Zone 1, EF 1 is the distance along the traveled way corresponding to a unit length along the perpendicular face of the object equal to 1/tan e To obtain the number 34
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of impacts with this face resulting from encroachments from the corresponding interval along the pavement AB requires an integration of the probability of impact over the offset of the face (from XA to XB) then multiplication by the encroachment factor to give, 'A'BX0X0e1P(x)dx)P(x)dx(tan 1IF (4) To obtain the encroachment factor for Zone 2, an integrated probability is again required between the offsets for C and D to account for the variable offset across the swath path. Calculation of the encroachment factor for this zone requires the length along the normal distance across the swath that project to give a unit length along the perpendicular (1/Cos e ). This dimension corresponds to a length along the traveled way so that EF 2 = (1/Sin e )/Cos e Thus the impact factor for Zone 2 is, 'C'BX0X0ee2P(x)dx)P(x)dx( CosSin 1IF (5) For Zone 3 the encroachment factor EF 3 = 1, unit length along the traveled way/unit length along the face (since the parallel face has a constant offset) so that the number of impacts with this face along the pavement is, IF 3 = P(x os ) (6) 35 A severity index is utilized to describe the nature of possible accidents by the type of object involved and the design speed of the roadway. To establish a cost per impact a relationship between accident costs and the severity index is established. Consistent with the partitioning of the object into separate accident zones different severity indices c coll (SI) are employed for each impact factor defined above. The product of ER, IF and
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the cost of a single accident is the total cost of accidents expected annually per traffic volume due to a single object at nominal offset x os The cost of an impact with a specific object at x os is then given in units of cost / annual traffic volume, 31iicolliimp)(SIcIFERc (7) Where, the summation is over all impact zones considered. For traffic on one side of roadway, going in one direction the annual encroachment rate (annual encroachments per unit distance along pavement per vehicular volume) is taken as constant ER=0.0003 enc/km/y/vehicles/day. The average daily traffic (total traffic count, independent of direction or number of lanes) for the roadway in year i, ADT i can be expressed as, iTdyiTGR)(1ADTADT (8) Where, ADT dy is the design value for average daily traffic. T i is the number of years from i to the design year and TGR is the traffic growth rate expressed as a decimal fraction. If traffic is two way, the total volume in one direction is one half the ADT. The model assumes that the traffic volume is the same in both directions. Since costs vary with the changing traffic volume a summation over years is conducted. Ypl0iiosimpospl/2]ADT)(x[c)(xc (9) Thus for utility j (having an above ground facility), the accident cost component is the sum of the terms accounting for traffic flow in the adjacent lanes, and those accounting for encroachments from the opposite direction, striking the above ground object. This 36
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latter component is also calculated in the same manner as previously described, except that the adjacent lane width is added to increase the effective offset difference which changes the encroachment probabilities. )](x[cPN)](x[cPNy) (x,cospljojospljajACCIDENT j (10) N j represents the number of objects per unit distance along the roadway, P ja and P jo represent the encroachment probabilities for adjacent and opposite lanes of traffic respectively. Accident costs generated by the accident model is a function for the cost per impact in K$/Mile as a function of offset (Ft) as seen in the Figure 3.8. 37 ACCIDENT COST FUNCTION A CC (k$/MILE) 250 200 I Figure 3.8: Accident Cost Function Of A Utility The cumulative individual cost function C j (x, y) shown in equation (11) of a utility j is, the sum of all position sensitive cost functions c j (installation, access, damage and accident) for that utility. That is a sum of equations (1), (2), (3) and (10). 0 50 100 0 5 10 15 20 25 HORIZONTAL (FT) DENT COST 150
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y)(x,cy)(x,cy)(x,cy)(x,cy)(x,CACCIDENT jDAMAGE jACCESS jONINSTALLATI jj (11) The plot in Figure 3.9 shows the overall cost function gradient for a typical utility over the cross section of a standard ROW corridor. Figure 3.9: Cumulative Cost Function Of A Utility 3.1.3 AASHTO Utility Placement Constraints The term ROW corridor refers to a profile view of the cross section of the subterranean area adjacent to and underneath the pavement available for the placement of 38
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utilities. The horizontal extent is the joint use ROW from the center of the pavement to the outer edge of the easement. The vertical extent of the corridor is governed by practical considerations (water table, shoring requirements). Constraints are rules and regulations set by the AASHTO to ensure overall safety of the utilities placed within the ROW corridor. These constraints are summarized as follows. 1. Clearance Constraints also understood as proximity constraints are imposed on utilities to prevent interference leading to accidental damage. Clearance is defined as the space around a utility, which should not be occupied by another utility. A utilitys clearance requirements are relative, that is, it depends on the type of the other utility being considered for proximal placement. The heuristic considers 10 different types of utilities and clearance requirements as specified by AASHTO are tabulated and utilized. The model demarcates utility clearance boundaries by two techniques (Bounding box and Radial boundaries) as shown in Figure 3.10. Mathematically, the clearance required between two utilities i and j is as, Bounding box: ijiijjX)r(x)r(x where, X ij = Horizontal clearance ijiijjY)r(y)r(y Y ij = Vertical clearance Radial: ijiijjR}ry){(x,}ry){(x, R ij = Radial clearance Where, (x i y i ) and (x j y j ) are the placement positions of, and r i and r j are the radii of utilities i and j respectively. 39
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Figure 3.10: Clearance Constraints 2. Safety Constraints are placement constraints (depicted in Figure 3.11) that are imposed on utilities in the interest of overall safety. These constraints are, a. Minimum Cover is the minimum depth below the surface of the ground, above which a utility should not be placed. This constraint is imposed on the placement of utilities to prevent damage caused due to superficial location. In the heuristic, the cover requirements are unique (specified by the user) for every utility type and the required cover adapts to the ground profile of the ROW to maintain a constant minimum earth cover over the utility. Mathematically, the cover constraint for a utility j with a radius of r j is specified as, 40
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0)r(y)(yjcovermin j Where, y min cover is the minimum cover specified for that utility. b. Maximum Depth is the maximum allowed depth for placement a utility within the ROW corridor. This constraint governed by practical considerations of safety (presence of water tables, application of high pressures) prevents very deep placement of utilities. The heuristic considers a unique maximum depth constraint (specified by the user) for every utility type. Mathematically for a utility j with radius r j the maximum depth constraint is specified as, 0)r(y)(yjjdepthmax Where, y maxdepth is the maximum allowed positional depth for that utility. c. Under Pavement: Utilities with above ground components for obvious reasons can not be placed below the pavement but besides these, certain other utilities for technical reasons and reasons of safety are not allowed placement below the pavement. The heuristic, uses the under pavement placement constraint to prevent restricted utilities from being placed below the pavement. Mathematically, this constraint for a utility j with radius r j is specified as, 0)r(x)(xjidthpavement wj Where, x pavement width = Horizontal width of the pavement 41
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d. Clear Zone is the recovery area; the region starting from the edge of the pavement that should be free of utilities. This placement constraint can be imposed instead of the inconvenience surcharge (additional installation costs) to h prevent traffic disruptions and accidents. Figure 3.11: Safety Constraints 3. Stacking Constraints: Stacking in terms of utility placements is defined as the positioning of one utility above or below another in the ROW corridor. Inconvenience for accessing, interference, increased probability of accidental damage and overall safety, are some of the reasons why certain utilities are not allowed stack positioning. In the heuristic, utilities with above ground components 42
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have an automatic no stack constraint applied to them (shown in Figure 3.12). Mathematically, the stacking constraint applied to utilities is specified as, ijiijjX)r(x)r(x where, X ij = Horizontal safety clearance Figure 3.12: Stacking Constraints 3.2 Model Structure And Working The heuristic is characterized as a discrete step, cost optimization model, which determines economically advantageous utility configurations for transportation ROW corridors by optimizing the estimated total costs of entire utility systems (configurations). 43
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44 The step by step working procedure of the model (shown in Figure 3.13) is explained as follows. 1. Project analysis setup : An analysis is initiated with problem defining inputs to the model such as, a. Information on the utilities to be placed : Number, and type of utilities to be placed (with or without above ground component) and, Utility parameters (probability of pl acement, diameter, minimum safety cover required etc.), b. Project duration and eva luation parameters : Project life and, Project design year, c. Traffic details : Design year traffic and Traffic growth rate, Number of lanes of traffic Lane width Pavement Design Speed etc d. Rightofway corridor specifications : Max depth, ROW width and, Ground profile.
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Figure 3.13: Working Structure Of The Heuristic 2. Configurations Search: Next, a search for all possible positional configurations for the utilities within the corridor is conducted using the mover program. The number of configurations obtained is a function of the user defined search step size used. 45
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46 3. Application of Filters : Filters are basically rules and regulations of clearance, safety and, stacking, set by the AASHTO for placement of utilities in ROW corridors. Utility configurations obtained from the pr evious step are tested for acceptability (feasibility) by the application of filters. Configurations th at violate filtering rules are eliminated at this stage. 4. Configuration Costing: The next step, that is, the valuation / costing of acceptable configurations is very important to the working of th e model. The model is based on the premise, that every utility to be placed in the ROW corridor has certain position sensitive costs (both initial and recurring) associated with it. Individual costs of utilities are estimated from relevant cost functions, generated by four integral cost models (installation, access, damage and accident cost models). The summation of the individual costs of each of the constituent utilities of a configuration yields the total societal cost of that configuration. 5. Optimize Total Costs : The final operation in the wo rking of the model is the optimization of the total costs of the ut ility configurations. The configuration associated with the least total societal cost is selected as the optimal.
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47 CHAPTER 4 PROBLEM STATEMENT The present system being used by the State Departments of Transportation (DOT) for allocating placement locations to ut ilities within transportation rightofways (ROW) is based on a first come first served method with certain governing rules provided by the AASHTO in 1959. Unplanned installations and excessive crowdi ng of utilities in ROW corridors has resulted in problems of da mage to infrastructure, interruption of service to customers and traffic disruptions / accidents. It has become increasingly difficult for the DOTs to upgrade older roads for placement of new utilities, and also ensure the safety of motorists using them. Obviously, there exis ts a very urgent need for a solution to the utility placement problem. The project Optimal Placement of Utilities within FDOT RightofWay, sponsored by the Florida Department of Transportation (FDOT) and currently being investigated at the University of South Florida (Kranc et. at .)[6] is aimed at addressing this need. It presents a decisionmaking heur istic designed to be a safe and economically efficient utility placement allo cation system. The model numerically simulates the shape and dimensions of the ROW corridor, and physic al information of the utilities to be located within it. Working in accordance wi th the rules of safety, relocation, and clearance for utility placement set by AASHTO, and utilizing positional cost assessment
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48 models, the heuristic finds suitable (optimal cost) locations for the utilities in the ROW corridors. 4.1 The General Problem Heuristics are models / tools designed fo r a scientific task and must be proven capable of dealing with uncertainty. A model such as this heuristic, being used to advocate a practice having large social and economical impacts is more likely to play the role of generic evidence in a trial, whos e weight must ultimatel y be established by a jury. Not only must the model be shown not to contradict the evidence, but it must do so when all driving forces relevant to the probl em have been incorporated in a way that is plausible to the jury. During the formulati on of a model, the questions being addressed to the system must be scrutinized carefull y, and the formal structure possibly updated iteratively until it proves capable of providing an answer given the question. A good sensitivity analysis can provide the generic quality assurance desired to the model and help demonstrate the worthiness of the model itself. According to Rabitz [36] a sensitivity analysis will help: 1. Analyze the contents of the model and in terface it with the ob servational data. 2. Identify which factors are critically important, how they are interrelated, and especially how they influence the behavior of the model. 3. Serve as a guide to any further use of the model by effectively communicating the modelers confidence in the model, its properties and his understanding of the sources of uncertainties to the decision maker.
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49 4.2 The Thesis Problem This thesis is a quantitative and qualitative sensitivity analysis of the abovementioned heuristic conduc ted in two parts namely, 1. Model Factor Sensitivity Analysis 2. Model Output Evaluation & Enhancement Part 1: Model Factor Sensitivity Analysis Objective : is to assess (quantify) the uncertainties associated with the modeling of this heuristic. Reason: As explained in chapter 3, the heuris tic finds economically advantageous placement locations for utilities within transportation ROW corri dors in accordance to the utility placement rules (constraints) set by the AASHTO, by optimizing the total costs of entire utility systems (configurations). The tota l cost of a configuration is the sum of the individual positional costs of each of its cons tituent utilities, dete rmined from respective cumulative cost functions generated by util ity cost assessment models (accident, installation access and damage) integral to th e main heuristic. Each cost model is influenced by input factors (global and m odel specific) which determine the shape and value of the cost function generated by them Since the output of the heuristic relies heavily on the cost models and their functi ons, it becomes imperative to fully understand the uncertainties associated with their input factor influences (dir ect and interaction). Preliminary analysis of the cost models revealed the following requirements for factor sensitivity analysis (i.e. for factor influen ce determination, factor calibration and further model development):
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50 1. Damage Model : Accurate data on damage costs and events is not available, hence a make do linear damage model is utilized in the heuristic to estimate the damage costs associated with a utility. 2. Accident Model : The accident model is derived from the procedure developed by the Federal Highway Administration to estimat e the economic value of traffic accidents with stationary objects at the side of the roadway. 3. Installation Surcharge Models : The installation cost model has experimental surcharge models (i.e. the inconvenience surcharge and the shoring surcharge model) which add to the installation cost functions only in certain regions of the ROW. Analysis : The analysis focuses on providing an evaluation of the confidence in the heuristic and its pr edictions by analyzing the influences that variati ons in the input factors have on the cost models and, the final out put of the heuristic itself. The following sensitivity studies are conducted: 1. A study of the local influence of the accident and damage cost model factors on their respective individual cost functions and, 2. A study to determine the global influence of untested installation surcharge models on the final output of the heuristic. The sensitivity studies addr ess the following questions: 1. Which factors contribute most to the output variability a nd require additional research? 2. Which model factors arent significant, a nd can be eliminated from the model?
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51 3. Is there some region in the space of the input factors for which the model output variation is maximum? 4. And finally, which group of factors if any, interact with each other? Part 2: Model Output Evaluation & Enhancement Objective : An evaluation and enhancement study of the final output of the heuristic. Reason: The working structure of the heuristic, though well defined ha s certain inherent problems that are highlighted when implemented as a program code, such as: 1. The model employs a mover pr ogram which moves ea ch utility to be placed, one at a time, with a user specified search step size within the ROW corridor boundaries to find possible placement locations (configuratio ns) for them. However, this discretized search is conducted over continuous cumulativ e individual cost functions generated for the utilities selected. Variability in the step size chosen causes unpredictable variability in the outputs determined (configurational and total costs). 2. Executing the program at the lowest possible st ep size (for the move r, 0.1 of a foot) to obtain the best possible refine ment on the output solves the problem of variability but is computationally very e xpensive (time consuming). 3. The heuristic compares the estimated total co sts of all feasible utility configurations determined for a ROW corridor to select th e configuration associated with the least total cost as the optimal. Very often the analysis determines many configurations (somewhat similar or totally different) with the same least total cost. The program code in such a case selects either the first (if < is used) or the last configuration (if
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52 is used) from the set of optimal configur ations which does not always present the best solution. During conference presentations it was noticed that besides departme nts of transportation (DOT), a diverse group of stakeholders such as, the public (consumers), utility owners (public and private,) and other corporate parties (contracto rs, services etc.) expressed interests in the development of this utility corridor organization scheme. Each stakeholder expressed requirements that the present singl e objective heuristic doe s not address, like, 1. The issue of locational fairness for all utilitys in the corridor. 2. Flexibility in the positioning accuracy require d for installation of the utilities in the corridor and, 3. Renovation capabilities of the configuration (i.e. the scope for addition of more utilities, and pavement extensions). Analysis : This analysis focuses initially on understanding (evaluating) the complexities of the discrete step, cost optimization procedure used in the heuristic. Based on the observed shortcomings and problems (implementation, speed, output identification and verification), develop an enhancement to be implemented with the heuristic. The enhancement will address all the problems of the heuristic by employing experimental speedup tools for refining the solutions (c onfigurations) obtained from coarse step configuration searches with the mover program and also by implementing a multi objective / criterion evaluation technique for utility configur ation selection to provide substantiation to the outputs determined by the heuristic.
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53 CHAPTER 5 MODEL FACTOR SENSITIVITY ANALYSIS This chapter constitutes the first part of the analysis of the heuristic. Its objective is to serve as a guide for any future use and development of the model. The main focus of this study will be on providing an evaluation of the confidence in th e heuristic and its predictions by analyzing the influences that va riations in the input factors (global and intra modular) have on the utility cost a ssessment models (i.e. the cost functions generated by them) and the final output of th e heuristic itself. Variance based sensitivity indices derived from Sobol [42] sensitivity indices are used here for this purpose. 5.1 Sensitivity Analysis Of The Heuristic Model development consists of several logical steps, one of which is the determination and analysis of the input f actors which influence the model output. An input factor is defined as any quantity that can be changed in the model prior to its running. This quantity can be a parameter (to be estimated), an input variable (directly observable in the real system), or a module of the model. The heuristic has four integral utility cost assessment models (i.e. the in stallation, access, damage and accident cost model, explained in chapter 3), each having i nput factors (global and model specific) that determine the shape and value of the cost functions generated by them. Preliminary
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54 examinations / observations made on the cost models and their input factor influences revealed the following. 1. The installation and access cost functions deri ved from the data collected by a survey of utility companies, show a vertical tendency (i.e. they vary with depth). The factors influencing theses models have a uniform multiplicative or additive effect all through their cost functions. The installation cost model however, has additional surcharge models (i.e. the inconvenience surcharge and the shoring surcharge model) which add to the installation cost functions onl y in certain regions of the ROW. Both surcharge models are experime ntal and further investigation into their effect on the output of the heuristic is required for calibration and fu ture model developments. 2. The data available on damage events is not very accurate and hence, a simple linear make shift damage model is used in the heuristic to determine damage costs associated with a utility. Since most of th e factors in the damage model are assumed, their influences need to be assessed fo r calibration and further model development purposes. 3. The accident model employs the proce dure developed by the Federal Highway Administration to estimate the economic valu e of traffic accidents with stationary objects at the side of the roadway. The accident model has several factors (model and problem specific) whose influences on the accident function have yet to be determined (quantified). Based on these observations, the following fact or sensitivity studies are conducted on the heuristic:
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1. A study of the local influence of the accident and damage cost model factors on their respective individual cost functions and, 2. A study determining the global influence of untested installation surcharge models on the final output of the heuristic. The sensitivity studies are guided by and answer the following questions in regards to model factors and their influences. 1. Which are the factors that mostly contribute to the output variability and require additional research? 2. The model factors that arent significant, and can be eliminated from the model. 3. Is there some region in the space of the input factors for which the model variation is maximum? 4. And finally, If and which group of factors interact with each other? 5.1.1 Sensitivity Indices The method adopted here for determining factor sensitivity indices is a variance based technique, also called ANOVA (analysis of variances) like sensitivity method, used generally for estimating the influences of individual factors or a group of factors on the output of complex models. The technique is based on the fact that, the sensitivity index for a given input factor X i represents the fractional contribution to the total variance observed in the model output. In order to calculate the sensitivity indices, the total variance V of the model output Y is apportioned to all the input factors X i as, mjik1,2,..,ijmjiijiiV.....VVVV (1) 55
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where, ]xXE(Y[VV*iii and, (2) ]xXE(Y[V]xXE(Y[V]xX,xXE(Y[VV*jj*ii*jj*iiij and so on. ]xXE(Y[*ii denotes the expectation of Y conditional on X i having a fixed value x i and the operator V[.] denotes conditional variance. The first order sensitivity index S i for the factor X i is defined as, VVSii (3) Higher order sensitivity indices responsible for interaction effects among input factors can also be determined. The sensitivity indices are nonnegative and their cumulative sum is 1. 1S...SSn1inji1n1,2,..,iji The entire proof for Sobol sensitivity indices [42] is included in the appendix A. 5.1.2 Factor Sensitivity Studies The approach adopted for factor sensitivity studies is a combination of the design of experiments (DOE) technique and sensitivity analysis performed in a specific manner to determine variance based sensitivity indices. DOE is a statistical technique that involves running a series of experiments in which purposeful changes are made to the 56
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57 input variables of a process or system to provide an objective measure of how a given change in the output might be dependent upon the change in values of its input variables. 5.1.2.1 Sensitivity Analysis Of Accident Model Factors The accident cost per impact with a utilitys above gr ound facility in the heuristic is estimated from the accident cost func tion generated by the accident model. The intention of this sensitivity analysis is to determine the influence that certain factors (accident model related factor s and problem, corridor specific parameters) have on the accident cost function generated by the acciden t model. The factors considered for this analysis are, 1. Design Year : Since the present values for factor s are not always known, the accident model allows for the use of predicted data fo r a future period (i.e. the design year). 2. Design Speed of the road : The vehicular speed for which the road is designed. 3. Design Year Average Daily Traffic (ADT dy ) : The average daily traffic predicted for the design year. It is also the capacity traffic for which the road is designed. 4. Traffic Growth Rate (TGR) : The rate at which the av erage daily traffic (ADT) increases every year over the project life. Traffic volume is calculated backwards from the design year traffic to the presen t value, decreasing w ith the TGR explained by equation 8, in chapter 3. Traffic volume be yond the design year remains constant at the design year traffic for the rest of the project life as shown in Figure 5.1 (Design year = 10 and ADT dy = 10 K Cars /day).
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CHANGES IN ADT/LANE OVER THE PROJECT LIFE TRAFF 58 Figure 5.1: Growth In Traffic Over The Project Life 5. Number of Above Ground Facilities (AGF): The measure of the number of above ground components that a utility has per mile of ROW. 6. Number of Lanes: The measure of the lanes of traffic in either direction. 7. Lane Width: The width of a traffic lane on the pavement. 8. Project Life: is the time interval from the original installation of the utility within the ROW until some time in the future when the roadway would be replaced or abandoned. 9. Size of the AGF: Is the size of the facility originating from the utility line below. The size of the component affects the possibility of impact and most importantly the severity of the impact in an accident. The size (diameter / dimensions) of the AGF is not considered for study in this analysis since the minimum diameter for severity index in the accident model is 0.5 meters or 19.685 inches and the utilities in this analysis are assumed not to have diameters greater than that. 0.000 2.000 4.000 8.000 10.000 12.000 0 5 10 15 20 25 TIME YR I C VOLUME K/D/LN 6.000
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59 10. ROW Width : The horizontal space available for th e location of the utility in the ROW corridor. The rightofway wi dth for this analysis is fi xed at the maximum possible value (40 Ft) to ensure accommodation space for lane addition and increase in the lane widths. The accident model generates a function fo r the accident cost associated with a utility varying horizontally over the ROW wi dth. Theoretically, unless forced by certain placement constraints, the heuristic would se lect an optimal configuration having the utility with an above ground f acility at a position in the RO W where its accident cost contribution to the total cost of the configuration is minimal. For this analysis however, to study the effect of the abovementioned f actors on the accident costs of a utility, the average of the accident function generated by the accident model is used as the response variable. A total of 90720 experimental runs of the accident model are made varying the accident factors mentioned above at various le vels within their possible ranges, shown in Table 5.1. Table 5.1: Levels Of Factors Varied Fo r The Accident Model Factor Analysis FACTORS UNITS RANGE FACTOR LEVELS # Design Year Yrs 5 20 5, 10, 15, 20 4 Design Speed MPH 35 70 35, 40, 45, 50, 55, 60, 65, 70 8 Average Daily Traffic(DY) K Cars/Day 5 40 10, 20, 30, 40 4 Traffic Growth Rate % 0 20 5, 10, 15, 20 4 Number of Lanes # 2 6 2, 3, 4 3 Lane Width Ft. 12 15 11, 12, 13 3
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60 Table 5.1 (Continued) Number of ABGF # 1 30 1,10, 20, 30 4 Project Life Yrs 20 40 20, 25, 30, 35, 40 5 Total Number of Runs 90720 5.1.2.2 Sensitivity Analysis Of Damage Model Factors The data on damage events is not very accurate and hence a simple linear damage model is used in the heuristic to determin e damage costs associated with a utility. Mathematically, as assumed in the damage model, the cost per damage incident is primarily a function of depth, modified by factors such as, the frequency of access, the fraction of events resulting in damage incident s (taken arbitrarily in the model as 1%) and a maximum cost per incident (specified by th e user) at the maximum depth that reduces linearly to the highest possible location for the utility (default cover). The following factors are considered for sensitivity studies on the damage cost model. 1. Maximum Damage and, 2. Damage Fraction for factor calibration purposes. 3. Default Cover and, 4. Maximum Depth for function shape and va lue influence analysis. The damage model generates a linear func tion for the damage costs of a utility varying vertically through the depth of the ROW. For this analysis, that is to study the effect of the abovementioned factors on th e damage function generated by the damage model, the average value of the damage functi on is used as the response variable. A total
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61 of 8470 experimental runs of the damage model are made varying the abovementioned factors at various levels within thei r possible ranges, shown in Table 5.2. Table 5.2: Levels Of Factors Varied Fo r The Damage Model Factor Analysis FACTORS UNITS RANGE FACTOR LEVELS # Maximum Damage K$ / Mile 0 1000 0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 11 Default Cover Inches 0 36 0, 6, 12, 18, 24, 30, 36 7 Maximum Depth Inches 60 120 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120 11 Damage Fraction % 0.5 5 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5 10 Total Number of Runs 8470 5.1.2.3 Sensitivity Analysis Of Inst allation Surcharge Models The heuristic has additional surcharge mode ls included in its utility installation cost assessment model, used primarily as dete rrents for utility placements in undesirable regions of the ROW. The inconvenience su rcharge model adds a surcharge to the installation costs of a utility in the regi on in close proximity to the pavement. The surcharge is maximum starting from the edge of the paveme nt and reduces linearly to zero at the end of the specified region. The shoring surcharge model adds a surcharge to the installation costs of a utility in the region close to the extreme most position (easement) of the ROW corridor. A flat cost is applied to all utilitie s to be placed in the shoring region (3 ft inwards from the easement). The experimental surcharge models while rather simple, in crowded rightofway situations are capable of in fluencing the model output si gnificantly. This sensitivity
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62 analysis aims at exploring the influences th at these surcharge models have on the output of the model with the intension of calibrati ng the models and providing guidelines for the correct use of their model f actors. The analysis involves making a total of 1452 runs (3 replicates of 484 runs each) of the stan dard experiment 1 while varying the abovementioned factors at various levels w ithin their possible ranges, shown in Table 5.3. The initial setup factors and the inform ation of the utilities of the standard experiment 1 are shown in Tables AB.1, AB.2 in appendix B at the end of the thesis. The total cost of the optimal solution arrived at in the analysis is used as the output variable. Table 5.3: Levels Of Factors Varied For The Installation Surcharge Model Factor Analysis FACTORS UNITS RANGE FACTOR LEVELS # Inconvenience Surcharge K$ / Mile 0 1000 0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 11 Surcharge Region Ft. 0 3 0, 1, 2, 3 4 Shoring Surcharge K$ / Mile 0 100 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 11 Replicates 3 Total Number of Runs 1452
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63 CHAPTER 6 RESULTS AND CONCLUSIONS OF MODEL FACTOR ANALYSIS The results of the sensitivity studies conducted on the heuristic are, 6.1 Results Of The Sensitivity Analysis Of The Accident Model Factors The sensitivity analysis of the accident model factors involved making a total of 90720 experimental runs of the accident mode l, varying 8 selected factors at various levels within their suggested ranges to dete rmine their influences on the accident cost function generated for a utility (with above ground facilities). The average value of the accident function was used as the response variable for this analysis. The analysis of variances (ANOVA) output determined using Mi nitab Release 14 (Statistical Software) is shown in Table C.1 in appendix C. The test was conducted at a 5% le vel of significance ( = 0.05). First order and second order sensitivity indices derived from the output variances from the ANOVA results are shown in Table 6.1 and Table 6.2 respectively, Table 6.1: First Order Sensitivity In dices for Accident Model Factors ACCIDENT MODEL FACTORS FIRST ORDER S.I. PERCENTAGE VARIATION Design Year 0.00827 0.89% Design Speed 0.31723 33.95% Average Daily Traffic (DY) 0.07188 7.69%
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64 Table 6.1 (Continued) Traffic Growth Rate 0.00202 0.22% Number of Lanes 0.00042 0.05% Lane Width 0.00004 0.00% Number of ABGF 0.18188 19.47% Project Life 0.02882 3.08% Total 0.6106 65.35% Table 6.2: Second Order Sensitivity I ndices For Accident Model Factors ACCIDENT MODEL FACTORS SECOND OREDR S.I. PERCENTAGE VARIATION Design Year & Design Speed 0.00730 0.78% Design Year & Design Year Average Daily Traffic 0.00165 0.18% Design Year & Traffic Growth Rate 0.00059 0.06% Design Year & Number of Lanes 0.00001 0.00% Design Year & Lane Width 0.00000 0.00% Design Year & Number of ABGF 0.00419 0.45% Design Year & Project Life 0.00000 0.00% Design Speed & Design Year Average Daily Traffic 0.06345 6.79% Design Speed & Traffic Growth Rate 0.00179 0.19% Design Speed & Number of Lanes 0.00021 0.02% Design Speed & Lane Width 0.00002 0.00% Design Speed & Number of ABGF 0.16054 17.18% Design Speed & Project Life 0.02544 2.72%
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65 Table 6.2 (Continued) Average Daily Traffic Design Year & Traffic Growth Rate 0.00040 0.04% Average Daily Traffic Design Year & Number of Lanes 0.00008 0.01% Average Daily Traffic Design Year & Lane Width 0.00001 0.00% Average Daily Traffic Design Year & Number of ABGF 0.03638 3.89% Average Daily Traffic Design Year & Project Life 0.00576 0.62% Traffic Growth Rate & Number of Lanes 0.00000 0.00% Traffic Growth Rate & Lane Width 0.00000 0.00% Traffic Growth Rate & Number of ABGF 0.00102 0.11% Traffic Growth Rate & Project Life 0.00000 0.00% Number of Lanes & Lane Width 0.00002 0.00% Number of Lanes & Number of ABGF 0.00021 0.02% Number of Lanes & Project Life 0.00003 0.00% Lane Width & Number of ABGF 0.00002 0.00% Lane Width & Project Life 0.00000 0.00% Number of ABGF & Project Life 0.01458 1.56% Total 0.32371 34.65% The main factor (first order) influe nces account for 65.35%, and the factor interaction (second order) influences account for 34.65% of the total variations in the accident model output (i.e. the average accident cost). The following inferences are made about the accident factor in fluences on the accident cost of a utility based on the sensitivity indices calculated.
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6.1.1 Main Effects Of Accident Model Factors 1. Design Year: The present values for factors such as the average daily traffic (ADT) are not always known, hence the accident model allows for the use of predicted data for a future period (i.e. the design year). The traffic volume for every year of the project life is then calculated using a compounding formula (shown as cost equation 8, in chapter 3). Since the traffic volume plying the road directly affects the accident probabilities, the design year is influential in determining the accident costs of a utility, as seen in Figure 6.1. Design YearAverage Accident Cost (K$/Mile) 2015105 2400230022002100200019001800170016001500 Main Effects Plot for Design Year Figure 6.1: Main Effect Of Design Year On The Accident Costs Change in Design Year: From 5 to 20 (yrs) Change in Average Accident Costs: Decrease from 2311.8 to 1526 (K$ / Mile) The reason for this decrease is a direct effect of the method used for calculating the average daily traffic for every year of the project life using the traffic growth rate. 66
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The Ftest value (3852.0) and the Pvalue (0.00) from the ANOVA results verifies this factors mild influence on the accident cost of a utility. Sensitivity Index: is 0.00827, which accounts for about 0.89% of the variation in the average accident costs. 2. Design Speed of the road is the vehicular speed for which the road is designed. It has a strong influence on the value and shape of the accident cost function (as depicted in Figure 6.2), because it influences the following: a. the lateral encroachment probabilities which is used to determine the number of accidents per year, b. the finite lateral extent of encroachment into the rightofway for a vehicle, c. the length of the road which contributes towards impacts with the facility and, d. the severity of accidental impacts. Design Speed (Miles /hr)Average Accident Cost( K$/Mile) 7065605550454035 6000500040003000200010000 Main Effects Plot for Design Speed Figure 6.2: Main Effect Of Design Speed Of The Road On The Accident Costs 67
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Change in Design Speed of Road: From 35 to 70 (M / hr) Change in Average Accident Costs: Increases exponentially from 192.2 to 5673.2 (K$/ Mile) The Ftest value (63324.9) and the Pvalue (0.00) from the ANOVA results table verifies this factors very strong influence on the accident cost of a utility. Sensitivity Index: is 0.31723, which accounts for about 33.95% of the variation in the average accident costs. 3. Design Year Average Daily Traffic (ADT dy ) is the average daily traffic predicted for the design year. It is also the capacity traffic for which the road is designed. The accident model estimates a present day value for future accident costs associated with a utility by summing the ADT calculated over all the years of the project life. Thus the predicted ADT dy value is influential to the accident costs of a utility, shown in Figure 6.3. Design Year Average Daily Traffic (KCars / day)Average Accident Cost (K$/Mile) 40302010 30002500200015001000 Main Effects Plot for Design Year Average Daily Traffic Figure 6.3: Main Effect Of Average Daily Traffic On The Accident Costs 68
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Change in the Design Year Average Daily Traffic: From 10 to 40 (KCars / day) Change in Average Accident Costs: Increases from 777.3 to 3109.2 (K$ / Mile) The Ftest value (33481.7) and the Pvalue (0.00) from the ANOVA results table verifies this factors moderately strong influence on the accident cost of a utility. Sensitivity Index: is 0.07188, which accounts for about 7.69% of the variation in the average accident costs. 4. Traffic Growth Rate (TGR) is the rate at which the average daily traffic (ADT) increases every year over the project life. Thus the TGR is important to determining the accident cost of a utility, illustrated by Figure 6.4. Traffic Growth Rate (% / yr)Average Accident Cost (K$/Mile) 2015105 22002100200019001800 Main Effects Plot for Traffic Growth Rate Figure 6.4: Main Effect Of Traffic Growth Rate On The Accident Costs Change in the Traffic Growth Rate: From 5 to 20 (% / yr) Change in Average Accident Costs: Decreases from 2164.7 to 1777.6 (K$ / Mile) 69
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The reason for this decrease is explained by the fact that a higher rate of growth in traffic means a smaller number of vehicles plying the roads initially, building up to the design year traffic. The Ftest value (942.4) and the Pvalue (0.00) from the ANOVA results table verifies this factors weak influence on the accident cost of a utility. Sensitivity Index: 0.00202, which accounts for about 0.22% of the variation in the average accident costs. 5. Number of Lanes is the measure of the lanes of traffic in either direction. Vehicular traffic from both the directions have lateral encroachment possibilities. Encroachment probabilities for the adjacent lanes are smaller because of the additional offset (i.e. the pavement width), shown in Figure 6.5. Figure 6.5: Highway Diagram Explaining Accident Factors 70
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Number of LanesAverage Accident Cost (K$/Mile) 432 2050200019501900 Main Effects Plot for Number of Lanes Figure 6.6: Main Effect Of Number Of Lanes On The Accident Costs As seen in Figure 6.6, Change in the Number of Lanes: From 2 to 4 Change in Average Accident Costs: Increases from 1874.5 to 2033.7 (K$ / Mile) Increase in the number of lanes reduces the offset distance of the utility from the traffic thus increasing the possibilities of accidents and the associated accident costs. The Ftest value (296.6) and the Pvalue (0.00) from the ANOVA results table verifies this factors weak influence on the accident cost of a utility. Sensitivity Index: 0.00042, which accounts for about 0.05% of the variation in the average accident costs. 6. Lane Width is the width of a traffic lane on the pavement. Lane width affects the lateral encroachment probability values in the accident model. The main effect of variation in lane width on the accident cost of a utility is illustrated in Figure 6.7. 71
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Lane Width (Ft.)Average Accident Cost (K$/Mile) 131211 197019601950194019301920 Main Effects Plot for Lane Width Figure 6.7: Main Effect Of Lane Width On The Accident Costs Change in the Lane Width: From 11 to 13 (Ft.) Change in Average Accident Costs: Increases from 1920.7 to 1967.7 (K$ / Mile) The Ftest value (24.6) and the Pvalue (0.00) from the ANOVA results table verifies this factors very weak influence on the accident cost of a utility. Sensitivity Index: 0.00004, which accounts for about 0.004% of the variation in the average accident costs. 7. Number of Aboveground Facilities (AGF) The measure of the number of above ground components that a utility has per mile of ROW. The number of above ground facilities per mile affects the probability of accidents and thus the accident costs, shown in Figure 6.8. 72
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Number of ABGFAverage Accident Cost (K$/Mile) 3020101 40003000200010000 Main Effects Plot for Number of Above Ground Facilities Figure 6.8: Main Effect Of Number Of Aboveground Facilities On The Accident Costs Change in the Number of AGF: From 1 to 30 Change in Average Accident Costs: Increases from 127.4 to 2033.7 (K$ / Mile) The Ftest value (84716.2) and the Pvalue (0.00) from the ANOVA results table verifies this factors strong influence on the accident costs of a utility. Sensitivity Index: 0.18188, which accounts for about 19.47% of the variation in the average accident costs. 8. Project Life is the time interval from the original installation of the utility within the ROW until some time in the future when the roadway would be replaced or abandoned. The project life is used to calculate the total traffic plying the road over all the years under consideration, thus determining the total number of possible accidents over the entire life of the utility. The Main effect plot for project life on the accident cost of a utility is shown in Figure 6.9. 73
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Project Life (Yrs)Average Accident Cost (K$/Mile) 4035302520 2800260024002200200018001600140012001000 Main Effects Plot for Project Life Figure 6.9: Main Effect Of Project Life On The Accident Costs Change in the Project Life: From 20 to 40 Change in Average Accident Costs: Increases from 1165.1 to 2721.4 (K$ / Mile) The Ftest value (10066.8) and the Pvalue (0.00) from the ANOVA results table verifies this factors strong influence on the accident costs of a utility. Sensitivity Index: 0.02882, which accounts for about 3.08% of the variation in the average accident costs. 6.1.2 Accident Model Factors Interactions Certain accident model factors interact with each other to produce variation in the cost function generated by the accident model. Table 6.3 and Figure 6.10 detail the major factor interactions contributing towards the variations in the accident cost of a utility. 74
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Table 6.3: Major Factor Interactions Influencing The Accident Costs FACTOR INTERACTIONS CHANGE IN ACCIDENT COST (K$/ Mile) SENSITIVITY INDEX CONTRIBUTION TO VARIATION IN OUTPUT Design Speed & Design Year Average Daily Traffic 924.7 to 2441.6 0.06345 6.79% Design Speed & Number of ABGF 151.6 to 3002 0.16054 17.18% Design Speed & Project Life 1533.6 to 2304.2 0.02544 2.72% Design Year Average Daily Traffic & Number of ABGF 51.0 to 6116.4 0.03638 3.89% Number of ABGF & Project Life 76.4 to 5353.57 0.01458 1.56% Design YearDesign YearAverage Daily TrafficAverage Daily TrafficTraffic Growth RateTraffic Growth RateNumber of LanesNumber of LanesLane WidthLane WidthNumber of ABGFNumber of ABGFProject LifeProject LifeDesign SpeedDesign Speed 7065605550454035 40302010 2015105 432 131211 3020101 4035302520 1000050000 1000050000 1000050000 1000050000 1000050000 1000050000 1000050000 Design1520Year510 Design455055Speed6065703540 Average3040DailyTraffic1020 Traffic1520GrowthRate510 Number4ofLanes23 Lane13Width1112 Number2030ofABGF110Interaction Plot for Accident Model Factors Figure 6.10: Interaction Effects Of Accident Factors 75
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76 6.2 Results Of The Sensitivity Analysis Of The Damage Model Factors The sensitivity analysis of the damage model factors involved making a total of 8470 experimental runs of the damage model, va rying 4 selected factors at various levels within their suggested ranges to determine their influences on the damage cost function generated for a utility. The average value of the damage function generated was used as the response variable. The analysis of vari ances output (ANOVA) that were determined using Minitab Release 14 (Statistical Software) is shown in Table C.2 in appendix C. The test was conducted at a 5% level of significance ( = 0.05). First and second order sensitivity indices derived from the output variances from the ANOVA results are shown in Table 6.4 and Table 6.5 respectively, Table 6.4: First Order Sensitivity Indices For Damage Model Factors DAMAGE MODEL FACTORS FIRST ORDER S.I. PERCENTAGE VARIATION Maximum Damage 0.42087 42.66% Default Cover 0.00070 0.07% Maximum Depth 0.09496 9.63% Damage Fraction 0.28696 29.09% Total 0.8035 81.45% Table 6.5: Second Order Sensitivity I ndices For Damage Model Factors DAMAGE MODEL FACTORS SECOD ORDER S.I. PERCENTAGE VARIATION Maximum Damage & Default Cover 0.00028 0.03%
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77 Table 6.5 (Continued) Maximum Damage & Maximum Depth 0.03798 3.85% Maximum Damage & Damage Fraction 0.11478 11.64% Default Cover & Maximum Depth 0.00389 0.39% Default Cover & Damage Fraction 0.00019 0.02% Maximum Depth & Damage Fraction 0.02590 2.63% Total 0.18303 18.55% The main factor (first or der) influences account for 81.45% and the factor interaction (second order) influences account for 18.55% of the total variations in the damage model output (i.e. the average damage cost). The following inferences are made about the damage factor influences on the damage costs of a utility, based on the sensitivity indices calculated. 6.2.1 Main Effects Of Damage Model Factors The data on damage events is not very accurate and hence a simple linear damage model is used in the heuristic to determine damage costs associated with a utility. The damage model is based on the assumption that th e cost per damage incident is primarily a function of depth, modified by factors such as, 1. Maximum Damage : a maximum cost per incident, specified by the user at the maximum depth. The damage cost of the ut ility reduces linearly from this maximum value at the deepest possible position to th e highest possible location for the utility (i.e. the default cover). Main effect of variation in ma ximum damage specified by the user is depicted in Figure 6.11.
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Maximum Damage (K$ / event)Average Damage Costs (K$/Mile) 10009008007006005004003002001000 100806040200 Main Effects Plot for Maximum Damage Figure 6.11: Main Effect Of Maximum Damage On The Damage Costs Change in the Maximum Damage: From 0 to 1000 (K$/ event) Change in Average Damage Costs: Increases from 0 to 97.67 (K$ / Mile) The Ftest value (24918.3) and the Pvalue (0.00) from the ANOVA results table verifies this factors strong influence on the damage cost associated with a utility. Sensitivity Index: 0.42087, which accounts for about 42.67% of the variation in the average damage costs. 2. Default Cover: the minimum depth below the surface of the ground, above which the utility should not be placed. This constraint is imposed on the placement of utilities to prevent damage caused due to superficial location. Main effect of variation in default cover required for a utility on the associated damage costs is shown in Figure 6.12. 78
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Default Cover (Inches)Average Damage Costs (K$/ Mile) 363024181260 5150494847 Main Effects Plot for Default Cover Figure 6.12: Main Effect Of Default Cover On The Damage Costs Change in the Default Cover: From 0 to 30 (inches) Change in Average Damage Costs: Decreases from 51.17 to 47.53 (K$ / Mile) After which any increase in a mandatory cover imposed causes the damage cost associated with a utility to increase. This is explained by the fact that the linear damage function tends to flattens out as the corridor height is reduced. The Ftest value (69.1) and the Pvalue (0.00) from the ANOVA results table verifies this factors extremely weak influence on the damage cost of a utility. Sensitivity Index: 0.00070, which accounts for about 0.07% of the variation in the average damage costs. 3. Maximum Depth is the maximum allowed depth for placement a utility within the ROW corridor. This constraint governed by practical considerations of safety (presence of water tables, application of high pressures) prevents very deep 79
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placement of utilities. Main effect of variation in maximum depth for positioning of the utility on the damage costs is illustrated in Figure 6.11. Maximum Depth (Inches)Average Damage Costs (K$/ Mile) 12011410810296908478726660 807060504030 Main Effects Plot for Maximum Depth Figure 6.13: Main Effect Of Maximum Depth On The Damage Costs Change in the Maximum Depth: From 60 to 120 (Inches) Change in Average Damage Costs: Decreases from 79.40 to 31.58 (K$ / Mile) The Ftest value (5622.2) and the Pvalue (0.00) from the ANOVA results table verifies this factors mild influence on the damage costs of a utility. Sensitivity Index: 0.09496, which accounts for about 9.63% of the variation in the average damage costs. 4. Damage Fraction is the fraction of events (access or installation) assumed to result in damage incidents. The heuristic arbitrarily takes the value 1%. This analysis experiments with different values for this fraction starting from 0.05 % until 5%. As seen from Figure 6.14, 80
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Damage Fraction (%)Average Damage Costs (K$/ Mile) 5.04.54.03.53.02.52.01.51.00.5 9080706050403020100 Main Effects Plot for Damage Fraction Figure 6.14: Main Effect Of Damage Fraction On The Damage Costs Change in the Damage Fraction: From 0.5 to 5 (%) Change in Average Damage Costs: Increases from 8.88 to 88.78 (K$ / Mile) The Ftest value (18877.5) and the Pvalue (0.00) from the ANOVA results table verifies this factors strong influence on the accident costs of a utility. Sensitivity Index: 0.28696, which accounts for about 29.09% of the variation in the average damage costs. 6.2.2 Damage Model Factor Interactions Certain damage model factors interact with each other to produce variation in the output of the damage model. Table 6.6 and Figure 6.10 detail the major factor interactions contributing towards the variations in the damage cost of a utility. 81
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Table 6.6: Major Factor Interactions Influencing The Damage Costs FACTOR INTERACTIONS CHANGE IN AVERAGE DAMAGE COSTS (K$/ Mile) SENSITIVITY INDEX CONTRIBUTION TO VARIATION IN OUTPUT Maximum Damage & Maximum Depth 0 to 63.16 0.03798 3.85% Maximum Damage & Damage Fraction 0 to 177.57 0.11478 11.64% Maximum Depth & Damage Fraction 14.44 to 57.42 0.02590 2.63% MAXIMUM DAMAGEMAXIMUM DAMAGEMAXIMUM DEPTHMAXIMUM DEPTHDAMAGE FRACTIONDAMAGE FRACTIONDEFAULT COVERDEFAULT COVER 363024181260 12011410810296908478726660 5.04.54.03.53.02.52.01.51.00.5 2001000 2001000 2001000 MAXIMUM200300400DAMAGE50060070080090010000100 DEFAULT121824COVER303606 MAXIMUM727884DEPTH90961021081141206066Interaction Plot for Damage Model Factors Figure 6.15: Interaction Effects Of Damage Model Factors 82
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83 6.3 Results Of The Sensitivity Analysis Of The Installation Surcharge Models Factors The heuristic has additional surcharge models (inconvenience and shoring) included in its utility installation cost assessm ent model, used primarily as deterrents for utility placements in undesira ble regions of the ROW. The sensitivity analysis of the installation surcharge models involved making a total of 1452 experimental runs of the heuristic (experiment 1, appendi x B), varying 3 factors at va rious levels within their suggested ranges to determine their influe nces on the total co sts of the optimal configuration and the positioning of the utilitie s of the optimal solution. 3 replicates of the experiment were made, varying the ROW width on each occasion to eliminate (block) the effect of the corridor and problem set up. The analysis of vari ances output (ANOVA) generated using Minitab Release 14 (Statistic al Software) is show n in Table C.3 in appendix C. The test was conducted at a 5% level of significance ( = 0.05). First and second order sensitivity indices derived fr om the output variances from the ANOVA results are shown in Table 6.7 and Table 6.8 respectively, Table 6.7: First Order Sensitivity Indices For Installation Surc harge Model Factors INSTALLATION SURCHARGE FACTORS FIRST ORDER S.I. PERCENTAGE VARIATION Shoring Surcharge 0.03329 4.05% Inconvenience Surcharge Region 0.23320 28.37% Inconvenience Surcharge 0.24862 30.25% Total 0.5151 62.67%
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84 Table 6.8: Second Order Sensitivity Indices For Installation Surcharge Model Factors DAMAGE COST FACTORS SECOND ORDER S.I. PERCENTAGE VARIATION Shoring Surcharge & Inconvenience Surcharge Region 0.03648 4.44% Shoring Surcharge & Inconvenience Surcharge 0.00550 0.67% Inconvenience Surcharge Region & Inconvenience Surcharge 0.02960 3.60% Total 0.07158 8.71% The ANOVA results determined a 28.62% e ffect of the blocks and 71.38 % effect of the factors. The main fact or (first order) influences account for 81.45% and, the factor interaction (second order) influences account fo r 18.55% of the total variations in optimal total costs due to factor effects. Based on the sensitivity indices calculated, the following inferences are made about the instal lation surcharge fact or influences. 6.3.1 Main Effects Of Installation Surcharge Model Factors 1. Shoring Surcharge is applied to a utility that has to be placed close to the extreme most position (easement) of the ROW corridor. Shoring costs are used to factor in, the difficulties involved, additional labor and extra materials required for locating utilities at this undesira ble location. The shoring su rcharge model assumes the region starting from the edge of the ROW extending 3 feet inward as the shoring region and applies a flat cost to all utilities placed there. The effect of varying the maximum shoring charge associated w ith a utility is shown in Figure 6.16.
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Shoring Surcharge (K$/ Mile)Optimal Total Cost (K$/Mile) 1009080706050403020100 1220120011801160114011201100 Main Effects Plot for Inconvenience Surcharge Figure 6.16: Main Effect Of Shoring Surcharge On The Total Optimal Costs Change in the Shoring Surcharge: From 0 to 1000 (K$ / Mile) Change in Optimal Total Costs: Increases from 1134 to 1179 (K$ / Mile) The following observations were made in regards to the positional changes of the utilities of the optimal configurations determined with changes in the shoring surcharge applied (Figures 6.17 and 6.18) a. Initial application and increase in shoring surcharge moves the utilities of the optimal configuration to the left (if there is space available to do so). b. Further increase in the shoring surcharge just increases the optimal cost determined. c. The maximum shoring surcharge (range is 0 to 100 K$ / Mile) never gets large enough to move a utility to a deeper position in the corridor. 85
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86 Figure 6.17: Initial Optim al Configuration Determ ined (Shoring) Figure 6.18: Optim al Configuration Dete rm ined After Increasing The Shoring Surcharge The Ftest value (23.7) and the Pvalue ( 0.00) from the ANOVA results table verifies this facto r s weak influence on the output of the heuristic. Sensitivity Index : 0.033 29, which accounts for about 4.05 % of the variation in th e optim al tota l costs. 2. Inconven ien ce Surcharg e is an additional installa tion ch arg e applied to a utility when it has to be placed within the ROW in close proxim ity to the pavem e nt. Since installation and access events to this utility will cause dis r up tion of traffic plying th e 36 144 154 164 174 184 194 204 214 41 46 Series1 51 Series2 56 Series3 61 66 71 36 144 154 164 174 184 194 204 214 41 46 Series1 51 Series2 56 61 66 71 Series3
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road, the inconvenience caused is factored in as a surcharge to the utility for installation at that particular location. The effect of varying the maximum inconvenience charge associated with undesirable positioning of a utility is illustrated in Figure 6.17. Inconvenience Surcharge (K$ / Mile)Optimal Total Cost (K$/Mile) 10009008007006005004003002001000 118011701160115011401130 Main Effects Plot for Shoring Surcharge Figure 6.19: Main Effect Of Inconvenience Surcharge On The Total Optimal Costs Change in the Inconvenience Surcharge: From 0 to 40 Change in Optimal Total Cost: Increases from 1134 to 1224 (K$ / Mile) The optimal cost increases rapidly with initial increase in the inconvenience surcharge but flattens out with further increase. The following observations were made in regards to the positional changes of the utilities of the optimal configurations determined with changes in the shoring surcharge applied (Figures 6.20, 6.21, 6.22). 87
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a. Initial application and increase in inconvenience surcharge moves the utilities of the optimal configuration to the right (if there is space available to do so). b. Further increase in the inconvenience surcharge just increases the optimal cost determined. c. At some value of maximum inconvenience surcharge (200 to 500 K$ / Mile depending on the ROW width available), the surcharge gets large enough to change the orientation of the optimal solution by moving a utility to a deeper position in the corridor. 144 149 154 159 164 169 174 179 184 189 194 199 204 209 214 36 41 46 Util 1 51 Util 2 56 Util 3 61 66 71 Figure 6.20: Initial Optim al Configuration Determ ined (Inconvenience) 88 Figure 6.21: Optim al Configuration Determ ined After Increasing Inconvenience Surcharge 36 144 154 149 159 164 169 174 179 184 189 194 199 204 209 214 41 46 Util 1 51 Util 2 56 61 66 71 Util 3
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149 159 144 154 164 169 174 179 184 189 194 199 204 209 214 36 41 46 Util 1 51 Util 2 56 61 66 71 Util 3 Figure 6.21: Optim al Configuration Dete rm ined W ith Larger Increase In Inconvenience Surcharge The Ftest value (176.7) and the Pvalue ( 0.00) f r om the ANOVA results table verify this factors strong influence on the output of the heuristic. Sensitivity Index : 0.24862, which accounts for about 30.25% of the vari ation in the optim al tota l cost. 3. Shoring Surcharge Region is the re gion star tin g f r om the edge of the pavem e nt extending o u twards ( e x t ent sp ecif i e d by the us er) within which a utility h a s an inconvenience surcharge associ ated with it. The inconveni ence surcharge m odel adds a surcharge that is m a xim u m starting from the edge of the pavem e nt and reduces linearly to zero at the en d of the surcharge region. As seen in Figure 6.22, Change in the Inconvenience Surcharge Region : From 0 to 3 Change in Total Optimal Cost : Increases from 1134 to 1224 (K$ / Mile) The Ftest value (552.5) and the Pva lue (0.00) from the ANOVA results table verif i es this f actors m o derately s t ron g influence on the output of the heuristic. 89
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Sensitivity Index: 0.23320, which accounts for about 28.37% of the variation in the optimal total costs. Incovenience Surcharge Region (Ft.)Optimal Total Cost (K$/Mile) 3210 12201200118011601140 Main Effects Plot for Inconvenience Surcharge Region Figure 6.22: Main Effect Of Inconvenience Surcharge Region On The Total Optimal Costs 6.3.2 Installation Surcharge Models Factor Interactions The only second order that is factor interaction influence noticed was the interaction between the inconvenience surcharge region and the shoring surcharge. As seen in Figure 6.23, Change in Total Optimal Cost: Increases from 1134 to 1252 (K$ / Mile) The Ftest value (8.6) and the Pvalue (0.00) from the ANOVA results table verifies this factors moderate influence on the output of the heuristic. Sensitivity Index: 0.03648, which accounts for about 4.44% of the variation in the optimal total costs. 90
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SHORING SURCHARGESHORING SURCHARGEINCONVENIENCE SURCHARGE REGIONINCONVENIENCE SURCHARGE REGIONINCONVENIENCE SURCHARGEINCONVENIENCE SURCHARGE 3210 1009080706050403020100 210018001500 210018001500 SHORING200300400SURCHARGE50060070080090010000100 INCONVENIENCE23SURCHARGEREGION01Interaction Plot for Installation Surcharge Model Factors Figure 6.23: Interaction Effect Of Installation Surcharge Models Factors 6.4 General Conclusions In his article Verification, validation and confirmation of numerical models in the earth sciences Oreskes [65] described Sensitivity Analysis as a tool to improve, verify, validate and corroborate a model by demonstration of agreement between observation and prediction. Sobol variance based sensitivity analysis used here is a global method in which the entire space of existence of the input factors is covered and all factors are varied simultaneously for analysis. The results derived (factor sensitivity indices) are informative (including both main and factor interaction effects), the computation is relatively inexpensive and the method is model independent (can be used in monotonic and nonmonotonic models). 91
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92 CHAPTER 7 MODEL OUTPUT EVALUATION AND ENHANCEMENT STUDY The second part of the model analysis is an evaluation study (i .e. an assessment of the quality) of the final output of the heuristic. This chapter delves into the complexities of the present output determin ation technique of the heur istic, and based on certain observed shortcomings suggests an enhancement to be implemented with it. The enhancement called the Ideal Configuration Selector addre sses all the problems of the heuristic and implements a multi objective / criterion evaluation technique for utility configuration assessment and selection. 7.1 Problems With The Present Working Procedure The output of the heuristic is a configuration of the utilities selected for placement in the ROW corridor having the least estimated total cost associated with it. The working structure of the heuristic, st arting with the identificatio n of configurations, their feasibility assessment, cost evaluation, and finally, select ion of the best based on optimality explained in chapter 3 is very func tional. However, a verification analysis of this working structure revealed the following problems. 1. Problems With The Configura tion Identification Process The heuristic is sometimes referred to as a brute force cost optimization model because of the discrete step operation of its mover program. The mover program moves
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93 each utility, one at a time, by a specified step size within the corridor boundaries to find possible placement locations (configurations) fo r them. The movement step size, that is, the refinement for configurational search is specified in fractions of a foot (step size range is 0.1 to 1). Tests conducted on the heuristic however revealed the following implementation problems associated with the mover. a. If the user decides on a very refined search (step size 0.2 or 0.1), the mover determines a very large number of conf igurations and, takes a long time to do so. The subsequent steps until the determina tion of an optimal solution are also computationally very expensive. An analysis with 3 utilities to be located in a ROW corridor having a crosssection of 6 x 6 feet employing a very refined search can take anywhere between 12 to 72 hours of processing time on a 2.8 GHz. Pentium 4 processor to determine an optimal solution. The use of a coarse step size for the configurational search is not a solution to the problem either. Figure 7.1 shows the positions assessed as feasible for utility placement by the mover in the ROW corridor at step size 1 and, Figure 7.2 shows the placement positions assessed while using a more refine d step size of 0.5. It is obvious from these figures that the use of a coarse se arch step size results in an incomplete coverage of the available ROW corridor space thus eliminating possible good solutions.
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94 Utilit y P o siti ons from Feasible Confi gurati ons in ROW Corrid o r RO W Width Figure 7.1: Corridor Search Coverage At Step Size 1 Figure 7.2: Corridor Search Coverage At Step Size 0.5 b. Another problem with the use of the m ove r in the heu r istic is, th e variability observed in the final (optim al) outputs determ ined with different s earch step sizes. Experim e ntal sweeps with reducing search st ep s i zes showed an erratic variation in the total costs of the optim a l solutions determ ined as illu strated in Figure 7.3 for an analysis with 3 utilities, Figure 7.4 (4 utilities) and F i gur e 7.5 (5 utilities) respectively. 36 144 164 184 204 41 D Utility 1 46 e p 51 Utility 2 t h 56 Utility 3 61 66 71 Utilit y P o siti ons from Feasible Confi gurati ons in ROW Corrid o r RO W Width 36 144 164 184 204 41 D e Utility 1 46 p 51 t Utility 2 h 56 61 66 71 Utility 3
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3 UTILITY EXPERIMENT $1,160.00 TOT $1,155.00 $1,150.00 $1,145.00 A L 95 Figure 7.3: Variation In The Total Costs Of Optimal Solutions For 3 Utility Experiment Using Varied Search Step Sizes Figure 7.4: Variation In The Total Costs Of Optimal Solutions For 4 Utility Experiment Using Varied Search Step Sizes Figure 7.5: Variation In The Total Costs Of Optimal Solutions For 5 Utility Experiment Using Varied Search Step Sizes 5 UTILITY EXPERIMENT $1,840.00 $1,850.00 $1,860.00 $1,870.00 $1,880.00 $1,890.00 $1,900.00 $1,910.00 $1,920.00 $1,930.00 0 0.2 0.4 0.6 0.8 1 1.2 STEP SIZE T O T A L C O S T 4 UTILITY EXPERIMENT $1,450.00 $1,460.00 $1,470.00 $1,480.00 $1,490.00 $1,500.00 $1,510.00 $1,520.00 $1,530.00 0 0.2 0.4 0.6 0.8 1 1.2 STEP SIZE TOT A L COS T $1,115.00 $1,120.00 $1,125.00 $1,130.00 $1,135.00 $1,140.00 COS 0 0.2 0.4 0.6 0.8 1 1.2 STEP SIZE T
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96 The cause of this variability is obvious. Th e mover uses discrete steps for utility movements in the corridor while finding possible placement configurations. However, this discretized search is being conducted over continuous cumulative cost functions generated by the cost models for each utility. The problem is this indicates that the best estimate for an optimal solution can be determined only by using the finest sear ch step possible with the move r (step size 0.1) which poses problems of excessive computati onal time and large data files. c. The final step in the working of the heuristic is the optimization of the estimated total costs of all the feasible utility confi gurations to determine the configuration associated with the least total cost. The problem arises when the analysis determines many configurations (somewhat similar or to tally different) with the same least total costs (optimal solutions). If a < (less than) is used in the code for comparing total cost, the first configurati on amongst the many with the sa me least total cost is selected and, if the (less than equal to) is used, the last configuration with the least total cost is chosen. This however doe s not always present th e best solution, but only one amongst many possible optimal solutions. 2. Problems With The Heuristics Output Quality The purpose of the heuris tic is to develop a good utility conf iguration assessment tool to help the Department of Transportation (DOT) make rational decisions on the placement a llocation of utilities in RO W corridors. During conference presentations however, it was noticed that besides the department of transportation (DOT), a diverse group of stakeholders such as, the public (consumers), utility owners
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97 (public and private,) and othe r corporate parties (contractor s, services etc.) expressed interests in the development of a utility co rridor organization scheme. Each stakeholder expressed certain requirements that the present single objective simulation does not address. For example, a. Economic fairness for all utility comp anies. The displayed optimal solution (configuration) does not guarantee all th e utilities being placed at inexpensive positions in the ROW. b. Present utility installation techniques and procedures are not accurate and the solution does not provide information on the positioning flexibilities of the utilities in the selected configuration. c. With the ever increasing demand for corridor space, for the placement of new utilities in the ROW or for extensions in the road ways, the present method does not evaluate configurations for renovation adaptability (i.e. the measure of the scope for addition of more utilitie s, and pavement extensions). The proposed Ideal Configuration Select or (ICS) is designed to remedy the problems and shortcomings of the current output methodology used by the heuristic and also present a method for producing substa ntiated results (outputs) from it. 7.2 The Ideal Configuration Selector The ICS is a utility configuration asse ssment tool which uses a multicriterion decision making procedure called the Weight ed Product Model (WPM) to assess and rank configurations according to their c onformity to the desired configurational characteristics. The ICS uses a similar assessment procedur e as the original heuristic
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aided by a few experimental tools and techniques like, the Jiggle Sensitivity Tool (JST), the Cost Dot Technique (CDT) and the Metric. The working structure of the ICS is as shown in Figure 7.6 and explained in the following steps. Figure 7.6: Working Structure Of The Heuristic With The Ideal Configuration Selector 98
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99 Step 1: Identification Of Configuration Shape Sets The ICS employs the original mover program to initially identify configurations using a moderately course step size (sugge sted range 0.6 to 0.4 from search pattern observation studies to ensure proper coverage of the ROW co rridor space). Rather than assess all the configurations obtained, the ICS uses two experimental techniques namely the Cost Dot Technique (CDT) and the Me tric to identify configuration shape (orientation) sets from the c onfigurations determined. The working of the CDT is based on the fact that, the individual cost of a utility is a direct function of its location within the ROW. It uses this interaction between th e utility cost functions and the constrained positioning possibilities of utilitie s in the ROW to group the configurations into sets of similar orientation as follows. 1. The CDT utilizes the individua l costs of the utilities in a configuration as vector coefficients to determine the correlati on between two conf igurations. (The correlation between two vectors is obtained by taking the dot produc t of the two cost vectors). 2. The correlation value is then used as a measure of the difference between the two configurations. (The correla tion values lie between 0 and 1. Similar orientation configuration will have equal cost dot values). In certain cases, like those involving large ROW or few utilities to be placed, it is possible for very different configurations to have the similar costs estimated for each utility. To determine and separately group these configurations the Metric is used in conjunction with the CDT. The Metric quantif ies the difference between configurations with the help of the positional coordinates of the utilities that is, by the conventional
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100 sum of the square of differences method. Detailed explanations of the CDT and the Metric are included in appendix D. Step 2: Optimization Of Shape Sets Once the configurational shape sets have been identified, another experimental tool called the Jiggle Sensitivity Tool (JST ) is used to determine a configuration to represents the best possible (optimal) position for utilities in each shape set. The JST is a program that jiggles (moves) the utilities of a configuration by finite steps in specified directions (up, down, to the left and to the right) while monitoring, 1. The percentage change in the individual cost of the utility and, the percentage change in the total cost of the configuration, 2. The possibility for movement of a utility in a particular direction without violations to other utility clearances, corridor boundaries and utility stacking rules. The detailed working of the JS T is explained in appendix E. The optimization of a shape sets is achieved with the following steps. 1. A configuration is selected from each shape set. 2. All the utilities in a configura tion are jiggled (by 1 step = 1/12 th of a foot) in all specified directions. 3. The configurational sensitivity for each of the 4n movements is analyzed and a positional change for a utility is accepted only if: a. it improves (reduces) the total co st of the configuration and, b. does not violate any rules (utility clea rance, stacking and corridor boundary).
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101 4. Steps 2 and 3 are repeated iteratively until: a. No movement is possible for any utility. (E very utility is allowed a maximum of 6 steps in each direction to maintain configurational orientation and ensure complete coverage of ROW corridor space). b. Jiggling of the utilities does not improve the total cost of the configuration. Step 3: Setup Of The Wei ghted Product Model (WPM) The ICS is formulated on a multicriterion decision making procedure also known as the Weighted Product Model. The WPM is based on a numerical technique developed by Bridgman [58] and used later by Miller and Starr [61]. It is used here to select the shape configuration embodying most of the ideal configurational char acteristics as the best solution. The WPM has the following components. 1. Alternatives : Alternatives represents the different options available for assessment. The alternatives in the ICS are the sh ape configurations to be assessed. 2. Attributes : Attributes are referred to as goals or decision criteria. The decision criteria in the ICS are the desired characte ristics of an ideal utility configuration (defined and determined in the next step) with respect to which the shape configurations will be assessed. 3. Decision Weights : The weights of importance of the decision criteria decided by the decision maker. The ICS suggest a nine poin t scale shown in Tabl e 7.1 to the user for weighing the importance of each ideal configuration characteristic. The weights assigned are then normalized to sum up to 1 before being used in the WPM.
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102 Table 7.1: Nine Point Scale For Characteristic Importance INTENSITY OF IMPORTANCE DEFINATION 1 Very Weak Importance 3 Moderate Importance 5 Strong Importance 7 Demonstrated Importance 9 Absolute Importance 2, 4, 6, 8 Intermediate values between two judgments 4. Decision Matrix : A decision matrix as shown in Table 7.2 is an (m x n) matrix in which element c ij indicates the performance of shape configuration C i when evaluated in terms of ideal utili ty configuration characteristic Ch j Table 7.2: Decision Matrix Fo r The Weighted Product Model ATTRIBUTES (Characteristics) Ch 1 Ch 2 Ch n WEIGHTS (Importance) w 1 w 2 w n C 1 c 11 c 12 c 1n C 2 c 21 c 22 c 2n ALTERNATIVES (Set Configurations) C m c m1 c m2 c mn
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103 Step 4: Quantifying Ideal C onfiguration Characteristics A study was conducted to determine a se t of ideal utility configuration characteristics to be used in the ICS for a ssessing utility configur ations. Considering the requirements of the different parties concerned, th e following characteristics were finally decided on. 1. Optimality in the total cost of the configuration. 2. Economic fairness for the uti lities of the configuration. 3. Flexibility in the positioning of utilities of the configuration. 4. Low usage of corridor space by the configuration. The explanations and quantifying measures for these ideal utility configuration characteristics are, 1. Optimality in the total cost of the configuration The total societal cost of the configurati on selected should be at or close to the lowest possible value for the placement of utilities in the ROW. The optimal costs determined for each shape configuration is used directly in the WPM as performance measures for this characteristic. 2. Economic fairness for the uti lities of the configuration. Utility companies required that the configuration selection procedure ensure economic fairness to all the utilities in the ROW corridor. The co efficient used to represent economic fairness for the utilities of a configuration in the WPM is called the Balance Coefficient (BC). The BC is based on the premise, that if all utilities in the configuration were at or close to their individual minimum cost values, they would
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definitely be located in equally fair (less expensive) positions. The BC for a configuration is determined as the maximum of the normalized differences from individual minimum costs of the utilities in a configuration. That is, min jmin jjICIC ICmax (BC)t Coefficien Balance for j = 1 to n Since the WPM works on a minimization principle the shape configuration having the minimum of the maximum deviations of individual utility costs will be favored. This technique is derived from Chebychevs Min Max Normalization Theory [64]. 3. Flexibility in the positioning of utilities of the configuration. The output of the heuristic is a positional configuration for the utilities to be placed within the ROW corridor. Utility installation procedures in use today are not very accurate and in most cases placement precision to the very last inch for all practical purposes can not be achieved. Thus it is very important to determine the positioning flexibility associated with each utility of a configuration before selecting it for implementation in a ROW corridor. The flexibility of a configuration is the measure of the capability of the utilities in a configuration to be positioned finite distances away from their optimal position without violating placement rules (corridor boundaries and clearance constraints). The coefficient used to represent the flexibility of the utilities in a configuration in the WPM is called the Average Flexibility Coefficient (AFC), which is the average of the flexibility coefficients for all the utilities of a configuration. 104
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(n) UtilitiesofNumber (FC) tCoefficieny Flexibilit(AFC)t Coefficieny Flexibilit Averagejn1 j The coefficient for flexibility of a utility in a configuration that is, the Flexibility Coefficient (FC) is defined as the number of valid positions for the utility (in the specified directions) around its position in the configuration. The JST is utilized to determine the validity of a utilitys position 6 mm in each direction (up, down, to the left and the right in 1 mm steps). A position is considered valid only if, a. The rules for utility placement are not violated and, b. The percentage change in the individual cost of a jiggled utility, that is, the positional sensitivity of that utility within the configuration does not exceed 10%. 4. Low usage of corridor space by the configuration. With the ever increasing demand for space, be it for the placement of new utilities in the ROW or for extensions in the road ways, the measure of the scope for renovations that is, the addition of more utilities is a very important characteristic. The coefficient used to quantify this characteristic is the Corridor Space Usage Coefficient (CSUC), which is based on the premise that the measure of the utility addition capability of a configuration is a direct measure of the space available. The CSUC is calculated as the ratio of the area covered by the clearance boundaries of the utilities in a configuration to the total corridor space. AreaCorridor TotalClearances by Utility covered Area (CSUC)t Coefficien UsageSpaceCorridor 105
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Step 5: Ranking the Shape Configurations The ranking of the alternatives (shape configurations) in the Weighted Product Model (WPM) involves comparing each shape configuration with the others by multiplying a number of ratios, one for each ideal utility configurational characteristic. Each ratio is raised to the power equivalent to the relative weight of the corresponding characteristic, that is, to compare two configurations C K and C L the following product (Bridgman [58] and Miller and Starr [61]) has to be calculated n1jwLKLKjjj)/c(c)/CR(C Where, n is the number of characteristics, c ij is the performance value of the i th configuration in terms of the j th characteristic, and w j is the weight of importance of the j th characteristic. If the term R(C K /C L ) is less than one, then it indicates the shape configuration C K is more desirable than shape configuration C L (minimization problem). The best alternative is the one better than all other alternatives, that is, the utility configuration embodying most of the ideal configurational characteristics is selected as the best solution. Step 6: Sensitivity / Criticality Of The Weights The results obtained from the Ideal Configuration Selector are based entirely on the weights assigned by the user (decision maker) to each characteristic of the ideal 106
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configuration in the WPM. To provide the decision maker with further insight into the selection procedure, the ICS provides a sensitivity / criticality analysis of the results to the weight decided on for each characteristic. The following procedure is followed for this purpose. Suppose (for 1 i < j m and 1 k n) denotes the minimum change in the current weight w ji,k, k of characteristic Ch k such that the ranking of configurations C i and C j are reversed. ji,k, > K if K 0 and, ji,k, < K otherwise. Where, kjkikn1ywjyiyw100cclogcclogKy and 100 ji,k, A critical degree of ideal utility configuration characteristic Ch k denoted as can be determined, which is, the smallest percent amount by which the current value of w 'kD k must change, such that the existing ranking of the configurations will change. 107
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1kn allfor Dji,k,m j i 1'kmin From this, a Sensitivity Coefficient of ideal configuration characteristic Ch k denoted as sens (Ch k ) which is the reciprocal of the critical degree is determined. 1knany for D1)sens(Ch'kk If the critical degree is infeasible (i.e., impossible to change any configuration rankings with any weight change), then the sensitivity coefficient is set equal to zero. 108
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109 CHAPTER 8 RESULTS AND CONCLUSIONS OF MO DEL OUTPUT EVALUATION AND ENHANCEMENT STUDY Chapter 7, Model Output Evaluation & E nhancement Study hi ghlighted certain problems associated with the working (imp lementation) and output determination methodology of the heuristic. Based on these sh ortcomings, it suggested an enhancement, the Ideal Configuration Selector (ICS) to be implemented with the heuristic. The ICS was specifically designed to tackle the problem s of the heuristic and implement a multi criterion configuration assessm ent procedure to substantiate the results presented by it. This chapter demonstrates the advantages of using the Ideal Configuration Selector with the heuristic. 8.1 Advantages Of Using The Ideal Configuration Selector To demonstrate the functioning and advantages of the ICS, the following tests were conducted on the Standard Utility Placement E xperiment 2 (Table B.3, appendix B) using the Standard Setup Parameters (Tables B.1) at step size 0.6 (moderately refined) as suggested in the ICS. Test runs were ma de on a Pentium IV, 2.8 GHz. 512 MB computer. 1. Speed : One of the problems highlighted with the use of the heuristic, was the computational time required for refined an alysis. The ICS solves this problem by clustering (grouping) similar orientation conf igurations into sets and analyzing only
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one optimal configuration from each shape set, thus reducing the number of configurations assessed and decreasing computational time. The speeding up of the heuristic is demonstrated from the timing shown below. Analysis time using only the heuristic = 8:00:33 mins. Analysis time using the ICS with the heuristic = 7:11:07 mins. The important point to be noted here is that the heuristic was run at step size 0.6, where as the ICS refined the solutions obtained from runs at step size 0.6 by using the Jiggle Sensitivity Tool at jiggle size 0.1. The refinement in the solution is evident from results shown in Tables 8.1 (only heuristic) and Table 8.2 (heuristic with ICS). 2. Refinement in Output: Using different step sizes in configuration searches with the mover program in the heuristic resulted in, unpredictable variability in the total costs of the optimal solutions determined and in certain cases failure to identify possible good solutions. The ICS solves this problem by extracting one configuration from each shape (orientation) set identified and optimizing the positions of its utilities using the Jiggle Sensitivity Tool at jiggle size 0.1 before assessment. This procedure guarantees always determining the best possible solution. Figure 8.1: Optim al Configuration Determ ined Using The Heuristic 110
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Table 8.1: Optimal Solution Determined By The Heuristic UTILITY TYPE HORIZ [in] DEPTH [in] COST [k$/mi] POWER DIST 212 40 $463 RECLAIMED 189 41 $288 GAS DIST 155 39 $336 TELECOM 149 67 $455 TOTAL $1,541 Tables and Figures 8.1 and 8.2, detail the configuration determined as optimal by the heuristic the ICS respectively. Table 8.2: Optimal Solution Determined By The ICS UTILITY # HORIZ [in] DEPTH [in] COST [k$/mi] POWER DIST 212 40 $463 RECLAIMED 153 41 $258 GAS DIST 177 62 $425 TELECOM 178 38 $335 TOTAL $1,480 Figure 8.2: Optim al Configurati on Determ ined By The ICS 111
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112 Table 8.3 shows the top 10 near optimal solutions determined by the ICS (all cheaper than that determined by the heuris tic), highlighting the problem of lack of refinement in the heuristics results and the associated refinement benefits of using the ICS. Table 8.3: List Of 10 Optimal Solutions Determined By The ICS OPTIMAL TOTAL COSTS CONFIGURATION RANKING CONFIGURATION NUMBER 1 1 964 1480.42 2 967 1481.08 3 3697 1481.09 4 3699 1481.75 5 18967 1481.75 6 18969 1482.41 7 12644 1484.23 8 12645 1484.89 9 5695 1485.11 10 5696 1485.77 3. Customization of Output : The optimization routine in the heuristic compares the total costs of all the feasible configurations to determine an optimal solution. However when several configurations have the same total costs the routine selects either the first or the last configuration dependi ng on the program code. The single objective nature of the heuristic produ ces outputs (utility configurations) which arent very
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113 flexible, that is, they can not be adapted to specific requirements. The ICS implements a multi objective utility c onfiguration assessment and selection procedure which firstly eliminates the am biguity from the output determination and presents the user (decision maker) with the option of customiz ing the outputs. The user can select and weigh the characteristi cs that he or she is looking for in a configuration for a particular ROW corridor. For example: a. Better Utilization of Corridor Space : If the user (decision maker) is designing a ROW corridor which will be upgraded by addition of new utilities, he will obviously want to implement the best po ssible (safe and economically efficient) utility configuration which ut ilizes the least amount of co rridor space to facilitate future expansions. With the ICS, the user can select and emphasize the importance of this characteristic, to customize the heuristics output. Table 8.4 and Figure 8.3 details the configuration determined by the ICS for best corridor space utilization. The space utilized by this c onfiguration is just 20.99% of the total avai lable corridor space. Table 8.4: Solution Determined By The IC S For The Best Corridor Space Utilization UTILITY # HORIZ [in] DEPTH [in] COST [k$/mi] POWER DIST 212 40 $463 RECLAIMED 189 41 $288 GAS DIST 213 68 $492 TELECOM 192 67 $485 TOTAL $1,727
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Figure 8.3: Configuration Determ ined By The ICS For The Best Corridor Space Utilization b. Better Positioning Fle x ib ility for Utilities : If corridor space is not a constraint, and the user wants to redu ce the ins t allati on costs and avoid the hass les o f accurate position i ng of utilitie s in the c o rrido r, he has the option of selec ting a conf iguratio n which has high positio ning f l ex ibilities f o r its c onstitu ent u t ilitie s by weighing th e utility flex ibility op tio n accord ingly. Table 8.5 and Figure 8.4 details the configuration determ ined by the ICS for highest flexibility in utility positioning. Th e average flexibility coefficient for this configuration was 0.24 which indicates an average of 6 steps of flexibility for each utility with less that 10 % in crease in individual cos t s. Table 8.5: S o lution Determ ined By The IC S For Flexibility In Utility Pos ition i ng UTILITY # HORIZ [ i n ] DEPTH [ i n ] COST [ k$/ m i ] POWER DIST 204 40 $467 R E C L A I M E D 1 5 2 5 7 $ 3 0 0 GAS DIST 213 62 $455 T E L E C O M 1 7 8 3 9 $ 3 3 6 T O T A L $ 1 5 5 9 114
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Figure 8.4: Configuratio n Determ ined By The ICS For Flexib ility In Utility Position i n g c. Balance / F a irness in U tility Costs : If the us er requires a co nf iguration which is econom ically fair to all ut ility companies (a m a jor requirem ent with utility com p anies), selecting and weighing th e balance cost option ass e sses and determ ines the best so lution with th e most balance in individual costs. Table 8.6 and Figure 8.5 details the configuration determ ined by the ICS for econom ic fairness to all utility. T h e ba lance coefficient determ ined for this configuration is 0.77 which i ndicates that the m a xim u m va riation of the individual cost of the utilities of this conf igur ation is 77% from their m i ni m u m possible individual costs. Table 8.6: S o lution Determ ined By The ICS For Fairness In Individual Utility Costs UTILITY # HORIZ [ i n ] DEPTH [ i n ] COST [ k$/ m i ] POWER DIST 211 59 $532 R E C L A I M E D 1 5 3 4 1 $ 2 5 8 GAS DIST 213 39 $366 T E L E C O M 1 7 8 5 7 $ 3 9 3 T O T A L $ 1 5 4 9 115
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Figure 8.5: Configuratio n Determ ined By Th e ICS For Fairness In Ind i vid u al Utility Costs 4. Substantiation of Results : The ICS perf orm s a sensitivity / c r itica lity ana l ysis of th e im portance weights assigned by the us er (dec ision m a ker) to the desired configuration characteris tics. This analysis is conducted on the 10 top ranked solution s to provid e th e user with usef ul inform ation on other configurations that nearly m eet his requ irem ents. Table 8.7: T op 10 Confi guration Obtained W ith The ICS OPT I M A L TOT A L COSTS BALANCED IN DI VI D U A L COSTS PERCENTAGE SPACE UTI L IZE D UTI L IT Y PO SI TION AL FLEX IBIL IT Y CONF I G RANKING CONF I G NUMBER 5 5 5 5 1 18 8 65 15 1 2 41 0. 91 32 .7 2 0. 27 2 1 9 7 2 0 14 8 6 4 3 0. 9 1 35 .1 9 0. 2 6 3 2 8 6 8 0 15 8 7 8 1 0. 8 9 38 .0 7 0. 2 4 4 5 6 9 6 14 8 5 7 7 0. 9 1 35 .1 9 0. 2 7 5 1 9 7 1 9 14 8 5 7 7 0. 9 1 35 .1 9 0. 2 7 6 2 3 2 2 5 15 4 6 4 0 0. 9 1 32 .7 2 0. 2 8 7 2 8 6 7 9 15 6 2 1 8 0. 7 7 38 .4 8 0. 2 8 8 2 8 9 0 7 16 2 1 2 3 0. 8 9 38 .0 7 0. 2 4 116
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117 Table 8.7 (Continued) 9 28906 1595.60 0.77 38.48 0.28 10 16535 1611.05 1.05 33.13 0.24 Table 8.8: Sensitivity / Criticality Of The Results CRITICALITY BETWEEN WEIGHT FOR OPTIMAL TOTAL COSTS WEIGHT FOR BALANCED INDIVIDUAL COSTS WEIGHT FOR PERCENTAGE SPACE UTILIZED WEIGHT FOR UTILITY POSITIONAL FLEXIBILITY 1 AND 2 93.56 NF NF NF 1 AND 3 98.14 NF NF NF 1 AND 4 NF NF NF NF 1 AND 5 NF NF NF NF 1 AND 6 NF NF NF NF 1 AND 7 NF NF NF NF 1 AND 8 98.70 NF NF NF 1 AND 9 NF NF NF NF 1 AND 10 NF NF NF NF SENSITIVITY 0.010688 0 0 0 Table 8.8 details the criticality between th e output configurations detailed in Table 8.7. An increase of 93.56 % in the weight s assigned to the optim ality criterion will cause the rankings between confi gurations 1 and 2 to change. Th e sensitivity of the result to the optimality characteristic is 0.01. Th e sensitivity of the output to the other characteristics is zero which indicates th at changing the importance weights for these characteristics will not change the result.
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118 8.2 General Conclusions Multi Criteria Decision Making has been one of the fast est growing problem areas during the last two decades. In business, d ecision making has cha nged from a single (the Boss!) and single criteria (profit), deci sion environment to a multi person and multi criteria situation. For problem s with discrete decision spac es, i.e. with countable few decision alternatives, the Weighted Product Model (WPM) is very useful for making justifiable decisions. What makes this techni que so valuable is that even though the analyses are very rigorous, th e results are described very clearly and are understandable even to non specialists.
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119 CHAPTER 9 FUTURE WORK Uncertainty is not an accident of th e scientific method, but its substance. The ongoing research project titled, Optim al Placement of Utili ties within FDOT RightofWay, sponsored by the Florida Depa rtment of Transportation (FDOT), and currently being investigated at the University of South Florida [6], presents a decisionmaking heuristic aimed at developing a safe and economically effici ent utility placement allocation system for transportation ROW corridors. When a model is used to drive a choice or a decision, it becomes imperative to assess the importance of its associated uncerta inties to ensure its relevance and guarantee the validity of its outputs. Th e above mentioned heuristic fi nds suitable (optimal cost) locations for the utilities in the ROW corridor s with the help of utility cost assessment models while adhering to the rules and regula tions of safety, reloca tion, and clearance for utility placement set by AASHTO. From this it is obvious that the cost assessment models and the AASHTO utility placement ru les heavily influence the outcome of the heuristic. This thesis, has partly analyzed the uncertainties associated with the input factors affecting the cost assessment models of th e heuristic. The follo wing uncertainties and questions still need to be evaluated to complete the analysis of the heuristic.
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120 9.1 Sensitivity Analysis Of The AASHTO Utility Placement Rules The rules for utility placement (utility clearance, stacking, and safety) are set by the AASHTO to ensure overall safety of the utilities placed within the ROW corridor. While the rules are well defined, thei r applicability is subject to a variety of interpretations, giving rise to doubt s and uncertainties. For example, a. Mandatory clearance required between utilities (varying with types) is defined in terms of inches, horizontally and vertically. However how this clearance is to be implemented is subject to interpretation. Question like, Do you consider a rectangular, circ ular or elliptical boundary? and What are the cost ramifications of co nsidering different types of boundaries? need to be answered. b. Placement of utilities very close to the pa vement poses problems of disruption to traffic and increased possibility of accidents. The AASHTO utility placement rules defines a clear zone starting from th e edge of the pavement within which no utility can be placed. However it woul d be interesting to determine: The cost ramifications of im plementing such a constraint. The optimal extent for a clear zone. c. Mandatory no stacking rules ar e applied to certain utilities. The rule for stacking again is open for interpretation. Questions like: How do you define a no stacking boundary? What is the cost ramifications of a no stacking constraint applied to a utility? need to be assessed.
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121 9.2 Development Of The Damage Model The data available on damage events is not very accurate and hence a simple linear damage model is used in the heuristic to estimate damage costs associated with a utility. The model assumes that the number of accidental damage incidents is proportional to the expected number of access events and that excavating to conduits buried deep within a corridor will more likel y result in damage to the utility itself and other utilities in the corridor. While these are all valid assumptions the following issues raise serious doubts about the validity of the model. a. The probability of damage not only depends on the depth of location and frequency of access to a utility but also on the presence, nature (t ype) and location (proximity) of other utilities within the corridor. b. A linear model varying with depth might not fully represent the damage cost of a utility because damaging util ity line at any depth should essentially cost the same. c. The assumption of fraction of events resulti ng in damage incidents, arbitrarily taken as 1% in the damage model would be better modeled as distribution derived from better data.
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122 REFERENCES Civil & Transportation [1] Brydia, Kuhn, Jasek, Parham and Blaschke Feasibility of Utility Corridors in TXDOT Right of Way, Texas Transportation Institute. [2] Federal Highway Administration, Hi ghway guide, U.S Department of Transportation, 1993. [3] Florida Department of Transportati on, Utility Accommodation Manual, 1999. [4] Federal Aid and Design Division, Util ity Adjustment and Accommodation on FederalAid Highway Projects, 4th Edition, Federal Highway Administration, Washington DC, 1998. [5] Iseley, T., Trenchless Technologies, Proceedings of the Fourth National Highway/Utility Conference, 1994. [6] Kranc, S. C., Miller, W. A., The Op timum Placement of Utilities within FDOT RightofWay, Sponsored by Florida DOT, 2002. [7] Mckin, R. A., Selection Method for Trenchless Technologies, Journal of Infrastructure Systems, 1997. [8] Scott, P., Subsurface Utility Engineering, Proceedings of the Fourth National Highway / Utility Conference, 1994.
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123 Sensitivity Analysis [9] Andres, T. H. Sampling Methods and Se nsitivity Analysis for Large Parameter Sets, Journal of Statistical Computation and Simulation, 57, 1997. [10] Andres, T. H., Hajas, W. C., Using Iterated Fractional Factorial Design to Screen Parameters in Sensitivity Analysis of a Probabilitic Risk Assessment Model, Proceedings of the Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications, Germany, 1993. [11] Archer, G., Saltelli, A., Sobol, I. M., Sensitivity Measures, ANOVA like Techniques and the use of Bootstrap, J ournal of Statistical Computation and Simulation, 58, 1997. [12] Box, G. E. P., Meyer, R. D., An Analysis for Unreplicated Fractional Factorials, Technometrics, 28, 1986. [13] Camponolongo, F., Kleijnen, J., Andres, T., Mathematical and Statistical Methods for Sensitivity Analysis of Model Output, 1999. [14] Camponolongo, F., Saltelli, A., Sensitivity An alysis of an Environmental Model: A worked Application of different Analysis Methods, Rehabilitation Engineering and Systems Safety, 52, 1997. [15] Chan, K., Saltelli, A., Tarantola, S., Sensitivity Analysis of Model Output; Variance based Methods make the Differe nce, Proceedings of the 1997 Winter Simulation Conference, 1997. [16] Cotter, S. C., A Screening Design for F actorial Experiments with Interactions, Biometrika, 66, 1979. [17] Cukier, R. L., Schaibly, J. H., Shuler, K. E., Study of the Sensitivity of Coupled Reaction Systems to Uncertainties in Rate Coefficients, Chemistry and Physics, 63, 1975.
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124 [18] Cukier, R. I., Levine, H. B., Shuler, K. E., Nonlinear Sensitivity Analysis of Multiparameter Model Systems, Journa l of Computational Physics, 26, 1978. [19] Daniel, C., On Varying One Factor at a Time, Biometrics, 14, 1958. [20] Daniel, C., One at a Time Plans, Am. Statistics Association, 68, 1973. [21] Demiralp, M., Rabitz, H., Chemical Kinetic Functional Sensitivity Analysis: Derived Sensitivities and General Applic ations, Chemistry and Physics, 75, 1981. [22] Dickinson, R.P., Gelinas, R. J., Sensitiv ity Analysis of Ordinary Differential Equation Systems A Direct Metho d, Computational Physics, 21, 1976. [23] Draper, D., Pereira, A., Pr ado, P. Saltelli, A., Cheal, R ., Eguilior, S., Mendes, B., Tarantola, S. Scenario and Parame tric Uncertainty in GESAMAC: A Methodologic Study in Nuclear Waste Di sposal Risk Assessment, Computer Physics Communications, 117, 1999. [24] Furbringer, J. M., Sensitivity Analysis for Modelers, Air Infiltration Rev., 17, 1996. [25] Helton, J.C., Uncertainty and Sensitivity Analysis Techniques for use in Performance Assessment for Radioactive Waste Disposal, Reliability Engineering and System Safety, 42, 1993. [26] Homma, T., Saltelli, A., Importance Meas ures in Global Sensitivity Analysis of Model Output, Reliability Engine ering and System Safety, 52, 1996. [27] Hora, S.C., Iman, R. L., Expert Opin ion in Risk Analysis: The NUREG1150 methodology, Nuclear Scien ce and Engineering, 60, 1989. [28] Hornberg, G. M., and Spear, R. C., An a pproach to the Preliminary Analysis of Environmental Systems, Envi ronment Management 12(1), 1981.
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125 [29] Hwang, J. T., A Computat ional Algorithm for the Polynomial Approximation Method of Sensitivity Analysis in Chemi cal Kinetics, Chinese Chemical Society, 32, 1985. [30] Iman, R. L., Helton, J. C., Campbell, J. E. An Approach to Sensitivity Analysis of Computer Models, Part I, Part II, Journal of Quality Technology, 1981. [31] JSCS, Journal of Statistical Comput ation and Simulation, Special issue on Sensitivity Analysis, 57, 1997. [32] Koda, M., McRae, G. J., Seinfeld, J. H., Automatic Sensitivity Analysis of Kinetic Mechanisms, International Jour nal of Chemistry and Kinetics, 11, 1979. [33] McKay, M. D., Morrison, J. D., Evaluati ng Prediction Uncertainty in Simulation Models, Computer Physics Communications, 117, 1999. [34] Miller, D., Frenklach, M., Sensitivity Analysis and Parameter Estimation in Dynamic Modeling of Chemical Kinetics, International Journal of Chemisty and Kinetics, 15, 1983. [35] Pierce, T. H., Cukier, R. I., Global N onLinear Sensitivity Analysis using Walsh Functions, Journal of Com putational Physics, 41, 1981. [36] Rabitz, H., Ali F., in Saltelli A. et al. (Eds.), Managing the Tyranny of Parameters in Mathematical Modeling of Physical Systems, 2000. [37] Saltelli, A., Tarantola, S., Planas, C., Sensi tivity Analysis and Official Statistics, Institute for Systems, Informatics and Sa fety, Joint Research Center, European Commission. [38] Saltelli, A., Global Sensitivity Analysis an Introduction, Tutorial Lecture for the International Conference on Sens itivity Analysis, March 2004. [39] Saltelli, A., Sensitivity Analysis for Im portance Assessment, Joint Research Center of the European Communities in Ispra.
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126 [40] Saltelli, A., Andres, T. h., Homma, T., Some new techniques in Sensitivity Analysis of Model Output, Computati onal Statistics and Data Analysis, 15, 1993. [41] Sobol, I. M., Quasi Monte Carlo Methods, Prog. Nuclear Energy, 24, 1990a. [42] Sobol, I. M., Sensitivity Estimates for NonLinear Mathematical Models, Math. Modeling Comput. Exp., 1, 1990b. [43] Turanyi, T., Sensitivity Analysis of Complex Kinetic Systems: Tools and Applications, Mathematics and Chemistry, 5, 1990a. [44] Turanyi, T., Kinal A Program Package for Kinetic Analysis of Reaction Mechanisms, Computers Chemistry, 14, 1990b. MultiObjective / Criteria Decision Making [45] ChungJen Chen, ChinChen Huang, A Multip le Criteria Evaluation of Hightech Industries for the Sciencebased Industria l Park in Taiwan, Information and Management 41, 2004. [46] de Graan, J. G., Extension of the Multiple Criteria Analysis Methods of T. L. Saaty, National Institute for Water Supply, 1980. [47] Hwang, F., An Expert Decision making Support system for Multiple Attribute decision making, Ph.D. Thesis, Dept. of Industrial Engineering, Kansas State University, 1987. [48] Hwang C. L., Yoon, K., Multi Attribut e Decision Making: Methods and Applications, SpringerVerlag, 1981. [49] Insua, R. D., Sensitivity Analysis in MultiObjective Decision Making, Lecture notes in Economics and Mathematical Systems Series, Sp ringerVerlag, 1990.
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127 [50] JMCDA, Journal of Multicriteria Decision Analysis, Special issue on Sensitivity Analysis, D. Rios Insua Vol. 8, 1999. [51] S. Cheng, C. W. Chan, G. H. Huang, A n Integrated MultiC riteria Decision Analysis and Inexact Mixed Integer Li near Programming approach for Solid Waste Management, Engineering Applic ations of Artificia l Intelligence 16, 2003. [52] Simon Mardle, Sean Pascoe, Ines Herrero, Management Objective Importance in Fisheries: An Evaluation using the Anal ytic Hierarchy Process, Environmental Management, 2004. [53] Stelios H. Zanakis, Anthony Solomon, Nico le Wishart, Sandipa Dublish, MultiAttribute Decision making: A Simulati on Comparison of selected Methods, European Journal of Operation Research 107, 1998. [54] S. Cheng, C. W. Chan, G. H. Huang., U sing Multiple Criteria Decision Analysis for supporting decisions of Solid Waste Ma nagement, Journal of Environmental Science and Health. Part A, Toxic / H azardous Substances and Environmental Engineering. [55] Saltenis, V. Dzemyda, G., Structure anal ysis of external problems using an approximation of characteristics, Op timal Decision Theory, Vilnius 8, 1982. [56] Triantaphyllou, E., Mann, S. H., An Exam ination of the Effectiveness of Multi Dimensional Decision Making Met hods: A Decision Making Paradox, International Journal of D ecision Support Systems, 5, 1989. Bibliography [57] Box, G. E. P., Hunter, W. G., Hunter, J. S., Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building, Wiley, 1978. [58] Bridgman, P. W., Dimensionless Analys is, Yale University Press, 1992.
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128 [59] Design and Analysis of Experiments, 5 th Edition, Douglas C. Montgomery, John Wiley & Sons Ltd., 2001. [60] Mathematical and Statistical Methods: Sensitivity Analysis, edited by Andrea Saltelli, Karen Chan, E. Marian Scott, John Wiley & Sons Ltd., 2000. [61] Miller, D. W., Starr, M. K., Executiv e Decisions and Operations Research, Prentice Hall Inc., 1969. [62] MultiCriteria Decision Making Me thods: A Comparative Study, Evangelos Triantaphyllou, Kluwer A cademic Publishers, 2000. [63] Statistics for Experimenters: An Intr oduction to Design, Data Analysis, and Model Building, George E. P. Box, Willia m G. Hunter, J. Stuart Hunter, John Wiley & Sons Ltd., 1978. General [64] Chatterjee, G., Chebychev Approximati on methods for Evaluating Conicity, Measurement, 23, 1998. [65] Oreskes, N., Evaluation of quantitative models for assessing the effects of environmental lead exposure, E nvironment Health Perspect, 106, 1998.
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129 APPENDICES
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Appendix A: Variance Sensitivity Indices (SOBOL' 1990b) The method adopted for the sensitivity studies on the factors of the heuristic is a variance based technique, also called ANOVA (analysis of variances) like sensitivity method. Let f(x) denote the model function where x = (x 1 ., x n ) is the set of input variables, and, let I denote the unit interval [0,1], I n the input factor space as a ndimensional unit hypercube and x I n The integrable function can be defined as, n1sni...iii......ii0s1s1s1)x,......,(xfff(x) (1) Where, the interior sum is over all sets of s integers i 1 ,..i s that satisfy 1 i 1 <..< i s n. Formula (1) means that nji1n2112...njiijn1iii0)x,....,x,(xf...)x,(xf)(xfff(x) The idea used by SOBOL was to decompose the function f(x) into summands of increasing dimensionality. The general decomposition of equation (1) is non informative, and for equation (1) to hold, f 0 must be constant and the integrals of every summands over any of its own variables must be zero. 0dx)x,......,(xfns1s1i10ii......ii for k = (2) s1i,......,i 130
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Appendix A (Continued) Equation (1) satisfying equation (2) is called decomposition into summands of different dimensions. In this case each member is responsible for the joint distribution of the variables to the variability of f(x) in I s1......iif s1iix,......,x n The integrals below are as a rule from 0 to 1 for each variable and dx = dx 1 dx n Integrating equation (1) over I n we obtain 0ff(x)dx Integrating equation (1) over all variables except x i we obtain )(xffdxf(x)ii0ikk thus define Similarly, integrating (1) over all variables except x )(xfii i and x j we obtain )x,(xf)(xf)(xffdxf(x)jiijjjii0ji,kk and define We continue the procedure until all (n1) dimensional summands are defined, and then the last member is found from identity (1). )x,(xfjiij )......xx,(xfn2112....n Since f(x) is a square integral, so are all the, therefore constants s1....iif s1s1s1s1iiii2....ii....ii.....dx)dxx,......,(xfV called partial variances can be introduced as well as the total variance V of f(x) 131
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Appendix A (Continued) 202f(x)dxfV Squaring equation (1) and integrating over I n we obtain n1sni...i...iis1s1VV This means, n1inji1n1,2,..,ijiV...VVV (3) The origin of this term is clear if x were a random point uniformly distributed I n then f(x) and all would be random variables, and V and their variances. The term ANOVA comes from Analysis of Variances. s1....iif ),.....x(xs1ii s1...iiV The ratios VVSs1s1...ii...ii are called sensitivity indices for 1 i 1 < < i s k. The indices are nonnegative and their sum is 1. 1S...SSn1inji1n1,2,..,iji S i is called the first order sensitivity index for factor x i which measures the main effect of x i on the output. S ij for i j is called the second order sensitivity index which measure the interaction effect of the variation in f(x) due to x i and x j 132
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133 Appendix B: Standard Utility Placement Experiments The sensitivity studies c onducted on the heuristic i nvolve running the standard utility placement experiments, using nominal values for the setup factors. Two extreme values are proposed to represent the range of likely values for each setup factor and the nominal value is taken midway between the tw o extremes values. The initial setup for the standard experiments is shown in Tables B.1. Table B.1: Standard Utility Placement Experiments Initial Setup INPUT PARAMTERS / FACTORS UNITS NOMINAL VALUE RANGE Right of Way Width Ft. 18 12 40 Maximum Depth Inches 72 120 Number of Initial Lanes # 2 2 6 Lane Width Ft. 12 12 15 Project Life Years 20 10 50 Design Year Average Daily Traffic K Cars / Day 20 5 40 Design Year Years 10 1 20 Design Speed MPH 50 30 75 Default Cover Inches 36 24 48 Traffic Growth Rate % 10 0 20
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134 Appendix B (Continued) The utilities considered for placement in the standard experiment 1 are: Table B.2: Standard Utility Placement Experiment 1 UTILITY TYPE DIA. STACK AG DIA. AG FAC. #/MILE POWER DIST 6 NO 6 CYLINDER 1 POTABLE 10 YES 0 NO 0 TELECOM 4 NO 0 NO 0 The utilities considered for placement in the standard experiment 2 are: Table B.3: Standard Utility Placement Experiment 2 UTILITY TYPE DIA. STACK AG DIA. AG FAC. #/MILE POWER DIST 8 NO 4 CYLINDER 2 RECLAIMED 10 YES 0 NO 0 GAS DIST 6 YES 0 NO 0 TELECOM 4 NO 0 NO 0
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135 Appendix C: Analysis Of Variances Tables Table C.1: Analysis Of Variances (ANOVA) Of Accide nt Model Factors SOURCE OF VARIATION DF ADJ. S.S. ADJ. M.S. F P Design Year 3 8007531431 2669177144 3852.0 0.00 Design Speed 7 307163000000 43880428571 63324.9 0.00 Average Daily Traffic (DY) 3 69602451976 23200817325 33481.7 0.00 Traffic Growth Rate 3 1959155084 653051695 942.4 0.00 Number of Lanes 2 411098572 205549286 296.6 0.00 Lane Width 2 34113204 17056602 24.6 0.00 Number of ABGF 3 176110000000 58703333333 84716.2 0.00 Project Life 4 27902923228 6975730807 10066.8 0.00 Design Year & Design Speed 21 7067618940 336553283 485.7 0.00 Design Year & Design Year Average Daily Traffic 9 1601506302 177945145 256.8 0.00 Design Year & Traffic Growth Rate 9 571237010 63470779 91.6 0.00 Design Year & N umber of Lanes 6 9459098 1576516 2.3 0.09 Design Year & Lane Width 6 784923 130821 0.2 0.99 Design Year & N umber of ABGF 9 4052185335 450242815 649.8 0.00 Design Year & Project Life 12 0 0 0.0 1.00 Design Speed & Design Year Average Daily Traffic 21 61432617340 2925362730 4221.7 0.00 Design Speed Traffic Growth Rate 21 1729191809 82342467 118.8 0.00 Design Speed & N umber of Lanes 14 205456777 14675484 21.2 0.00 Design Speed & Lane Width 14 16984838 1213203 1.8 0.14 Design Speed & N umber of ABGF 21 155439000000 7401857143 10681.8 0.00
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136 Appendix C (Continued) Table C.1 (Continued) Design Speed & Project Life 28 24627718532 879561376 1269.3 0.00 Design Year Average Daily Traffic & Traffic Growth Rate 9 391830976 43536775 62.8 0.00 Design Year Average Daily Traffic & Number of Lanes 6 82219711 13703285 19.8 0.00 Design Year Average Daily Traffic & Lane Width 6 6822640 1137107 1.6 0.24 Design Year Average Daily Traffic & Number of ABGF 9 35222094774 3913566086 5647.8 0.00 Design Year Average Daily Traffic & Project Life 12 5580585115 465048760 671.1 0.00 Traffic Growth Rate & Number of Lanes 6 2314304 385717 0.6 0.99 Traffic Growth Rate & Lane Width 6 192045 32008 0.0 1.00 Traffic Growth Rate & Number of ABGF 9 991423970 110158219 159.0 0.00 Traffic Growth Rate & Project Life 12 0 0 0.0 1.00 Number of Lanes & Lane Width 4 21086818 5271705 7.6 0.25 Number of Lanes & Number of ABGF 6 208035079 34672513 50.0 0.00 Number of Lanes & Project Life 8 32961056 4120132 5.9 0.39 Lane Width & Number of ABGF 6 17262905 2877151 4.2 0.63 Lane Width & Project Life 8 2735129 341891 0.5 1.00 Number of ABGF & Project Life 12 14120184425 1176682035 1698.1 0.00 Error 91822 63627216654 692941 Total 92159 968251000000
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137 Appendix C (Continued) Table C.2: Analysis Of Variances (ANOVA) Of Damage Model Factors SOURCE OF VARIATION DF ADJ. S.S. ADJ. M.S. F P Maximum Damage 10 8079341 807934 24918.3 0.00 Default Cover 6 13447 2241 69.1 0.00 Maximum Depth 10 1822891 182289 5622.2 0.00 Damage Fraction 9 5508641 612071 18877.5 0.00 Maximum Damage & Default Cover 60 5379 90 2.8 0.00 Maximum Damage & Maximum Depth 100 729156 7292 224.9 0.00 Maximum Damage & Damage Fraction 90 2203457 24483 755.1 0.00 Default Cover & Maximum Depth 60 74711 1245 38.4 0.00 Default Cover & Damage Fraction 54 3667 68 2.1 0.00 Maximum Depth & Damage Fraction 90 497152 5524 170.4 0.00 Error 7980 258738 32 Total 8469 19196580 Table C.3: Analysis Of Vari ances (ANOVA) Of Installati on Surcharge Model Factors SOURCE OF VARIATION DF ADJ. S.S. ADJ. M.S. F P Blocks 2 1746988 873494 835.8 0.000 Shoring Surcharge 10 247250 24725 23.7 0.000 Inconvenience Surcharge Region 3 1732269 577423 552.5 0.000 Inconvenience Surcharge 10 1846753 184675 176.7 0.000
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138 Appendix C (Continued) Table C.3 (Continued) Shoring Surcharge & Inconvenience Surcharge Region 30 270952 9032 8.6 0.000 Shoring Surcharge & Inconvenience Surcharge 100 40863 409 0.4 1.000 Inconvenience Surcharge Region & Inconvenience Surcharge 30 219873 7329 7.0 0.000 Error 1266 1323160 1045 Total 1451 7428108
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Appendix D: Configuration Differentiation And Clustering Techniques The Ideal Configuration Selector uses two experimental techniques namely the Cost Dot Technique (CDT) and the Metric to differentiate between and cluster (group) configurations into shape sets based on similarity in orientation. D.1 Cost Dot Technique (CDT) The Cost Dot Technique uses the individual cost of the utilities for quantifying the difference between configurations. The idea used in the CDT is that any feasible configuration has N utilities with individual costs. Since the individual cost of a utility is a direct function of its location within the ROW, the individual cost of utilities can be used to differentiate between two configurations. The CDT employs the individual costs of the utilities in a configuration as vector coefficients. The correlation between the vectors of two configurations is taken as the measure of the difference between those two configurations. The correlation is calculated as the dot product of those two vectors. Consider an example with 3 utilities, if the individual utility costs of the o th configuration are (C o1 C o2 C o3 ) and the individual utility costs of the i th configuration are (C i1 C i2 C i3 ), the coefficients for the first configurational vector will be, 2o32o22o1o32o32o22o1o22o32o22o1o1CCCC,CCCC,CCCC and the vector will be represented as, 139
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Appendix D (Continued) kCCCCjCCCCiCCCCC2o32o22o1o32o32o22o1o22o32o22o1o1o The coefficient for the i th configurational vector will be, 2i32i22i1i32i32i22i1i22i32i22i1i1CCCC,CCCC,CCCC and the vector will be, kCCCCjCCCCiCCCCC2i32i22i1i32i32i22i1i22i32i22i1i1i The correlation coefficient or Cost Dot Coefficient (CDC) is calculated as the dot product of the two vectors which is, 2i32i22i13i2o32o22o13o2i32i22i12i2o32o22o12o2i32i22i1i12o32o22o1o1ioCCCCCCCCCCCCCCCCCCCCCCCCCCCDC The range of the Cost Dot Coefficient is between 0 and 1. Similar orientation configurations have equal cost dot coefficients. 140
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Appendix D (Continued) D.2 The Metric The metric quantifies the difference between two configurations with the help of the positional coordinates of the utilities. The idea is that any feasible solution can be identified as a 2N vector, describing the configuration of N utilities with x and y coordinates. The difference between two configurations (i.e. the Metric value M oi ) is quantified by the sum of the square of differences method, represented in the equation below and depicted in Figure D.1. 2ojij2ojijN1joi)y(y)x(xM Figure D.1: Quantifying Configurational Differences Using The Metric 141
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Appendix D (Continued) The Ideal Configuration Selector (ICS) applies the Metric to the orientation clustering process after shape (orientation) set have been determined by the Cost Dot Technique. The Metric helps determine configurations of different orientation having similar individual costs for their constituent utilities, a rare occurrence which is not identified by the CDT. Configurations varying by more than a 1000 metric value points are considered to be configurationally different. The functioning of the Cost Dot Technique and the Metric for differentiating between configurations is demonstrated for the configurational sweep search results shown in Figures D.2 and Figure D.3. 3 UTILITY SWEEP $1,160.00 T $1,150.00 O T $1,140.00 A L $1,130.00 COS $1,120.00 T $1,110.00 0 0.2 0.4 0.6 0.8 1 1.2 STEP SIZE 3 UTILITY SWEEP0.9850000.9900000.9950001.0000001.00500000.511. 5 3 UTILITY SWEE P 02004006008001000120000.511STEP SIZEMET .5 R STEP SIZ E COST D O Figure D.2: Cost Dot And Metric Value Plots For Differentiating Between Configurations In A 3 Utility Step Size Sweep 142
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Appendix D (Continued) 143 5 UTILITY SWEEP $1,940.00 TOT A $1,920.00 $1,900.00 L $1,880.00 $1,860.00 $1,840.00 1.2 0.2 0.4 0.6 0.8 0 1 STEP SIZE COST 5 UTILITY SWEEP0.980000000.985000000.990000000.995000001.000000001.0050000000.20.40.60.811.2STEP SIZECOST D O 5 UTILITY SWEEP0200040006000800010000120001400000.20.40.60.811.2STEP SIZEMET R Figure D.3: Cost Dot And Metric Value Plots For Differentiating Between Configurations In A 5 Utility Step Size Sweep Figure D.2 illustrates the configurational difference in the optimal solutions obtained using different step sizes in the analysis. Optimal configuration obtained for step sizes 0.3 and 0.5 are similar, so are the configurations for step size 0.4 and 0.8, 0.2 and 0.7, 0.6 and 0.9. Figure D.3 shows configurations with step size 0.3, 0.5 and 0.8 are similar using the CDT. The need for the Metric in the ICSs clustering process is highlighted with the identification of an orientationally different configuration for step size 0.5, not detected by the CDT.
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Appendix E: Jiggle Sensitivity Tool (JST) The Jiggle Sensitivity Tool is a program employed in the Ideal Configuration Selector to jiggle (move) the utilities of a configuration by finite steps in specified directions (up, down, to the left and to the right) as shown in Figure E.1 while monitoring the following, Figure E.1: Jiggling Of Utility For Configuration Sensitivity Study 1. The percentage change in the individual cost of a jiggled utility represents the positional sensitivity of that utility within the configuration and, )(yor )(xIC)%(Change PositionalUtility ChangeCost Individual Percentagey)Sensitivit l(Positionajjj 144
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Appendix E (Continued) 2. The percentage change in the total cost of the configuration with the jiggling of a utility is the configurational sensitivity of the configuration with respect to that particular utility. )(yor )(xTC)%(Change PositionalUtility ChangeCost Total Percentagey)Sensitivit tional(Configurajjj 3. The validity of the movement of each utility at every jiggled step for, a. Violations to the clearance boundaries of other utilities. b. Violations to the ROW corridor boundaries (i.e. ROW width, maximum depth, and default cover). c. Violations to utility placement constraints (clear zone, below pavement, and stacking). 145
