The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2011 Development and implementation of a networklevel pavement optimization model Shuo Wang The University of Toledo Follow this and additional works at: http://utdr.utoledo.edu/theses-dissertations Recommended Citation Wang, Shuo, "Development and implementation of a network-level pavement optimization model" (2011). Theses and Dissertations. 751. http://utdr.utoledo.edu/theses-dissertations/751 This Thesis is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.
A Thesis entitled Development and Implementation of a Network-Level Pavement Optimization Model by Shuo Wang Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Civil Engineering Dr. Eddie Y. Chou, Committee Chair Dr. George J. Murnen, Committee Member Dr. Liangbo Hu, Committee Member Dr. Patricia Komuniecki, Dean College of Graduate Studies The University of Toledo December 2011
Copyright 2011, Shuo Wang This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.
An Abstract of Development and Implementation of a Network-Level Pavement Optimization Model By Shuo Wang Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Civil Engineering The University of Toledo December 2011 Optimal use of pavement maintenance and rehabilitation dollars is essential in a constrained budget environment such as now. A network-level optimization tool, which could generate the best maintenance and rehabilitation strategies for the entire pavement network, has become necessary for many highway agencies. This thesis presents the development and implementation of a network-level optimization tool within a pavement management information system for the Ohio Department of Transportation (ODOT). Future pavement condition is predicted based on historical pavement data using a Markov transition probability model. Such transition probabilities are updated automatically when new condition data become available each year. The network-level optimization tool integrates a linear programming model and the Markov transition probability model. This optimization tool is capable of (1) calculating the minimum budget required to achieve a desired level of pavement network condition, (2) maximizing the improvements of pavement network condition with a given amount of budget, and (3) determining the corresponding optimal treatment policy and budget iii
allocations. It can be used by highway agencies as a decision support tool for networklevel pavement management. iv
Acknowledgements Many people have assisted me in conducting this study. I would like to thank my advisor Dr. Eddie Y. Chou for providing the opportunity and financial support for me to continue my graduate education. I appreciate his valuable guidance on my thesis. I am also grateful to the members of my thesis committee, Dr. Liangbo Hu and Dr. George J. Murnen, for their helpful suggestions and allowing me to defend this thesis. I would also like to acknowledge the Ohio Department of Transportation for funding this study. Special thanks must be given to Debargha Datta, Jun Zhang and Haobo Zhen, for their previous research efforts for the pavement management information system. Without their assistance, this study could not have been completed. v
Table of Contents Abstract... iii Acknowledgements...v Table of Contents... vi List of Tables... ix List of Figures... xi 1 Introduction...1 1.1 Background...1 1.2 Statement of Problem...3 1.3 Objective of Study...4 2 Literature Review...5 2.1 Optimization Algorithms...5 2.2 Pavement Performance Prediction Models...6 2.3 Previously Developed Optimization Models...7 3 Methodology...15 3.1 Introduction...15 3.2 Development of Markov Transition Probability Model...16 3.3 Formulation of Network-Level Optimization Model...18 3.4 Implementation...24 vi
4 Example Problems...26 4.1 Overview...26 4.2 Problem 1: Minimum Budget to Achieve a Desired Condition Level...27 4.2.1 Problem Statement...27 4.2.2 Optimized Results without Budget Constraints...28 4.2.3 Optimized Results with Budget Constraints...33 4.2.4 Comparison between Two Solutions...36 4.3 Problem 2: Allowable Treatments Effects on Annual Budget Requirements...39 4.3.1 Problem Statement...39 4.3.2 Results and Discussions...40 4.4 Problem 3: Budget Allocation among Different Treatments...41 4.4.1 Problem Statement...41 4.4.2 Results and Discussions...41 4.5 Problem 4: Budget Allocation among Different Districts...45 4.5.1 Problem Statement...45 4.5.2 Applying Statewide Policy to Each District...45 4.5.3 Optimizing Pavement Expenditure for Each District...48 4.5.4 Comparison between Two Results...50 5 Conclusions...52 5.1 Summary of the Study...52 5.2 Conclusions...52 References...54 Appendices...56 vii
A Optimization Model Formulation Flowchart...56 B Network-Level Pavement Optimization Model...57 viii
List of Tables 3.1 Example of Allowable Treatments... 23 4.1 Unit Cost of Maintenance and Rehabilitation Treatments... 26 4.2 Pavement Condition Classification... 27 4.3 Current Pavement Condition Distribution... 27 4.4 Allowable Treatments for Problem 1... 28 4.5 Recommended Treatment Budget for Problem 1 without Budget Constraints... 29 4.6 Projected Pavement Condition Distribution for Problem 1 (without Budget Constraints)... 31 4.7 Recommended Treatment Budget for Problem 1 with Budget Constraints... 33 4.8 Projected Pavement Condition Distribution for Problem 1 (with Budget Constraints)... 35 4.9 Allowable Treatments for Problem 2 (A)... 39 4.10 Allowable Treatments for Problem 2 (B)... 39 4.11 Flexible Pavements Treatment Policy for Year 2011 Obtained from Model A in Problem 1... 46 4.12 Required Budget Obtained by Applying Statewide Policy to Each District... 47 4.13 Predicted Deficiency Level Obtained by Applying Statewide Policy to Each District... 48 ix
4.14 Required Budget Obtained by Optimizing Pavement Expenditure for Each District... 49 4.15 Predicted Deficiency Level Obtained by Optimizing Pavement Expenditure for Each District... 50 x
List of Figures 4-1 Recommended Treatment Budget for Problem 1 (without Budget Constraints)... 30 4-2 Projected Pavement Condition Distribution for Problem 1 (without Budget Constraints)... 32 4-3 Recommended Treatment Budget for Problem 1 (with Budget Constraints)... 34 4-4 Pavement Condition Distribution for Problem 1 (with Budget Constraints)... 36 4-5 Recommended Budget versus Available Budget for Problem 1... 37 4-6 Comparison of Deficiency Level for Problem 1... 38 4-7 Impact of PM on Required Average Annual Budget... 40 4-8 Recommended Treatment Budget for Problem 3... 42 4-9 Pavement Condition Distribution for Problem 3... 43 4-10 Comparison of Deficiency Level Trends between Problem 1 (Model A) and Problem 3... 44 4-11 Comparison of Average Annual Budget for Each District... 51 xi
Chapter 1 Introduction 1.1 Background Transportation is essential for a nation s development and growth (Garber and Hoel 2001). Pavement networks are a key component of the transportation infrastructure system, especially in developed countries such as the United States, where a large number of pavement networks have been constructed (Li et al. 1998). As a result of the aging pavement network compounded by budget cuts at most agencies, maximizing the benefits of available maintenance and rehabilitation dollars has become necessary for many highway agencies (Akyildiz 2008; Li et al. 1998). Generally, pavement conditions can be classified into several categories, such as good, fair or poor and maintenance and rehabilitation treatments can also be grouped into a few types, such as preventive maintenance, minor rehabilitation, and reconstruction. The unit costs of those treatments vary significantly, and their effect on pavement condition improvement also varies (Abaza 2007). It is often a complicated problem for highway agencies to determine the optimal treatment policy and budget allocation to maintain the overall pavement condition above an acceptable level with the least agency cost. Some 1
highway agencies determine the annual budget level and allocate the available funds among different repair treatments based upon experience or engineering judgment, which is not always an efficient way of managing pavement networks, especially in a constrained budget environment such as now. Therefore, an effective pavement management system (PMS) that can find the optimal policy has become a necessity for highway agencies to determine the best maintenance and rehabilitation strategies. The optimization tool should be able to address two critical issues facing the decision-maker: (1) determining the minimum budget required to achieve a desired level of pavement network condition, and (2) maximizing the improvements of pavement network condition with a given amount of maintenance and rehabilitation dollars. A pavement management system is defined by AASHTO as a set of tools or methods that assist decision-makers in finding optimum strategies for providing, evaluating, and maintaining pavements in a serviceable condition over a period of time (Huang 2004). The pavement optimization tool, which is capable of generating the best maintenance and rehabilitation strategies for the entire pavement network, is a critical component of a PMS. The financial benefits of optimizing pavement expenditures can be very significant. For instance, the optimization system developed for the state of Arizona saved almost 1/3 of Arizona s pavement preservation budget during the first year of implementation (Golabi et al. 1982). Optimization of pavement expenditures can be conducted at either the network- or the project-level. The network-level optimization takes a global view of the entire pavement network, and focuses on the overall condition distribution and budget allocation problems (Huang 2004). The network-level 2
optimization is capable of estimating the total mileage of pavements to be repaired by the applicable treatments, and determining the amount of budget required to maintain the whole pavement network above a certain acceptable condition level (Bako et al. 1995). On the other hand, the project-level optimization concentrates on a specified subset of the whole network and generates maintenance and rehabilitation plans for each pavement section (Huang 2004). The network-level optimization is usually implemented by a macroscopic approach, in which the repair variables are introduced for each pavement condition category and they represent the proportions of pavement that should be treated by the applicable treatments (Abaza 2007). The project-level optimization is generally conducted by a microscopic approach, in which each pavement section is assigned a repair variable for each repair treatment and the value of this variable is 1 if the repair treatment is recommended for this pavement section, otherwise it is 0 (Abaza 2007; Bako et al. 1995). Generally, the linear programming model is used at the network-level, and the integer programming model is applied at the project-level. 1.2 Statement of Problem A pavement management information system (PMIS) has been developed to assist ODOT to manage the Ohio pavement network, which includes more than 40,000 lane miles of highways. The current PMIS is capable of generating various reports regarding the pavement condition, performance, project history, and so on. In order to generate pavement maintenance and rehabilitation plans, a set of decision trees has been 3
developed by ODOT and implemented using the PMIS. However, the work plan generated by the decision tree would recommend a rather large number of pavements be repaired and the funding required is far beyond the maximum available budget. Therefore, a more targeted decision support tool is necessary to help ODOT determine the most cost-effective budget allocation and select the best pavement treatment policy under the stringent budget environment. The main purpose of this research is to develop and implement an optimization tool within the current PMIS to support the decision making process in pavement management at the network-level. 1.3 Objective of Study The main objectives of this study are: 1. To review prior research in pavement network-level optimization. 2. To develop a Markov prediction model for pavement condition deterioration and effects of repair treatments based on historical pavement condition data. 3. To develop a linear optimization model for the network-level pavement optimization. 4. To develop a decision support tool by implementing the network-level optimization model using Microsoft Visual Basic.NET (2008) and IBM ILOG CPLEX 12.1. 4
Chapter 2 Literature Review 2.1 Optimization Algorithms Linear and integer programming are two optimization algorithms utilized by most developed pavement optimization models. Selecting an appropriate algorithm is important in establishing an efficient optimization tool. Linear programming is a powerful mathematical technique for dealing with the problem of allocating limited resources among competing activities in a best possible way (Hillier and Lieberman 2010). All functions and constraints of a linear programming model are required to be linear functions. A linear programming model is generally utilized in a macroscopic approach for pavement optimization at the network-level (Abaza 2007). Thanks to the efficient solution algorithms and the rapid progress in computation power, linear programming models can be solved within an acceptable time period even if the problem size is quite large (Hillier and Lieberman 2010). Therefore, many researchers, such as Golabi et al. (1982), Bako et al. (1995), Chen et al. (1998), and Abaza (2007), have developed network-level optimization models using linear programming. 5
The more complete name for integer programming is integer linear programming, which indicates that the integer programming model is derived from the linear programming model by adding a restriction that all variables must be integers (Hillier and Lieberman 2010). Solving an integer programming model is much more difficult than solving a linear programming model, especially when the problem size is large or the constraints are complicated (Hillier and Lieberman 2010). Li et al. (1998) and Ferreira et al. (2002) use integer programming models, in which each pavement section is assigned a set of repair variables and a specific maintenance and rehabilitation plan can be generated for each pavement section. However, this approach results in a very large number of variables and makes the optimization model extremely difficult to solve, especially when it is used for a large pavement network (Abaza 2007). Therefore, it is often used in the project-level optimization, where the number of pavement sections is much less than that of the entire network (Ferreira et al. 2002; Li et al. 1998). 2.2 Pavement Performance Prediction Models An accurate and reliable pavement condition prediction model is essential for a pavement optimization model (Akyildiz 2008). There are two main types of prediction models, namely deterministic models and probabilistic models. De la Garza et al. (2010) developed a regression prediction model by deterministically computing pavement deterioration rates based on historical data. However, the pavement deterioration rates are often uncertain (Butt et al. 1994). Therefore, the probabilistic 6
model based on the Markov process is the most frequently used approach (Bako et al. 1995; Chen et al. 1996; Golabi et al. 1982; Abaza 2007). A critical component of this model is the Markov transition probability matrix. Most developed models use two transition matrices for each repair treatment: one for condition improvements in the first year the treatment is conducted, and the other for the deterioration trend after the treatment (Chen et al. 1996). Generally, the elements of the transition probability matrices are calculated based on historical pavement condition data, or are assumed when historical data are insufficient or not available (Bako et al. 1995). 2.3 Previously Developed Optimization Models Two optimization models utilizing the linear programming algorithm and the Markov prediction model are Arizona s model developed by Golabi et al. (1982), and Oklahoma s model developed by Chen et al. (1996). The first modern network-level pavement management system was developed by Golabi et al. (1982) for Arizona Department of Transportation (ADOT) (Ferreira et al. 2002). In Golabi et al. s optimization model, a total of 120 pavement conditions states are defined by the variables including present amount of cracking, change in amount of cracking during the previous year, the present roughness, and index to the first crack. The statewide pavement network is divided into nine road categories (sub-networks) based on traffic volume and a regional environmental factor. The maintenance actions are grouped into 17 types ranging from routine maintenance to substantial corrective measures. 7
Golabi et al. (1982) developed a Markov transition probability prediction model using historical pavement condition data to address the probabilistic aspect of pavement deterioration. A single Markov transition probability matrix is used to estimate the deterioration trend of pavements receiving routine maintenance, which is equivalent to Do Nothing in other researcher s models, no matter what repair action they receive before the routine maintenance (Chen et al. 1996). As a result, pavements with different repair treatments, such as reconstruction and thin overlay, are assumed to deteriorate at the same rate after the treatments are conducted, which is considered by Chen et al. (1996) as one of the major limitations of Golabi et al. s model. The network-level optimization model for Arizona is composed of a long-term model and a short-term model. The objective functions of the two models are to minimize the total expected cost. The long-term model calculates a maintenance policy that minimizes the expected long-term average cost to keep the pavement network condition at a desired level. The short-term model then seeks a maintenance policy over an analysis period T that minimizes the total expected cost to achieve the long-term standard within the first T years. (Golabi et al. 1982) The outcome of this optimization model includes the optimized maintenance policy, the expected minimum budget required, and the predicted pavement condition (Golabi et al. 1982). 8
Another network-level optimization model is established by Chen et al. (1996) for the Oklahoma Department of Transportation with the application of linear programming and the Markov decision process. Pavement conditions are divided into five states, namely excellent, good, fair, poor, and bad, in terms of the overall pavement condition index. Nine treatments are defined: thin, medium, thick overlay on both asphalt and concrete pavements, medium and thick asphalt reconstruction, and concrete reconstruction. Chen et al. (1996) uses a global optimization model which seeks the optimal solution for the entire network, although the network is divided into six pavement groups by traffic volume and pavement types. The main improvement of this optimization model is that it uses two Markov transition matrices for each repair treatment. One is for the immediate impact of the treatment on the pavement condition improvement when it is conducted in the first year. The other is for the deterioration trend after the treatment, which is also known as a Do Nothing matrix. In other words, the deterioration trends for different repair treatments are estimated separately. Therefore, this prediction model is more realistic and accurate than previous ones in that pavements with different last treatments tend to deteriorate at different rates (Chen et al. 1996). Both cost minimization and benefit maximization approaches are implemented in Chen et al. s optimization model. Two methods for estimating the benefits of pavement maintenance and rehabilitation are developed for the benefit maximization model. One method is to convert pavement conditions into benefit indexes. The benefit index is 9
determined subjectively by engineering judgment considering traffic volume and pavement condition (Chen et al. 1996). The other method is to estimate benefits on the basis of the area under the performance curve after a treatment is applied. The benefit index of conducting treatment a on a pavement in state i with last treatment b can be calculated by: N 1 N 1 B mibat WmjADTmjlaj WmjADTmjlbj / C j 2 j 1 miat (2.1) where the Wmj is the weight for pavement group m in state j; ADTmj is the average daily traffic of pavement group m in state j; l aj is the number of years that a pavement section with last treatment a staying in state j; and at time t on pavement group m in state i (Chen et al. 1996). Cmiatis the unit cost of conducting treatment a Other methodologies except linear programming and Markov prediction model, such as integer programming and deterministic prediction models, have also been utilized previously by other researchers. Li et al. (1998) presented an integer programming optimization model for pavement network maintenance and rehabilitation. A time-related Markov probabilistic model is established for pavement condition prediction considering both the immediate treatment effects and the potential impact on the rate of future condition deterioration, which is similar to the prediction model developed by Chen et al. (1996). The major difference 10
between the two Markov models is that Li et al. s model predicts the exact pavement condition state (PCS) score, such as pavement condition index (PCI) or pavement serviceability index (PSI), rather than the pavement condition category, such as excellent or poor. This approach facilitates the establishment of the cost-effectiveness-based integer programming optimization model, as the predicted PCS score can be used directly to estimate the benefit of a treatment in the following objective function: Maximize s s M PCS stm Ast Lst ESALs Dst X stm t 1 m 1 Lst Wst Cstm 1 R (2.2) where X stm is the decision variable which is equal to 1 if treatment m is recommended for Section s at Year t and is equal to 0 otherwise; PCS st is the generalized Pavement Condition State (such as PCI or PSI) for Pavement Section s at Year t; minimum acceptable level of PCS required for Pavement Section s at Year t; Ast is the Lst is the length of pavement Section s at Year t; ESALs is ESAL applications on Section s at Year t; D st is the number of service days for traffic flow by Section s at Year t if treatment m is selected; Wst is the width of Section s at Year t; C stm is the unit cost of treatment m; R is the discount rate for calculating the present value of future cost (Li et al. 1998). The network optimization model developed by Li et al. (1998) uses a multiyear integer programming model on a year-by-year basis. The objective of the optimization model is 11
to maximize the total value of cost-effectiveness in each analysis year, given the available budget constraints and other applicable constraints. The main output of this program consists of the optimal maintenance and rehabilitation treatment strategy and the predicted condition state for each pavement section in each analysis year (Li et al. 1998). However, integer programming models are much more difficult to solve than linear programming models especially when the problem size is large or the constraints are complicated (Hillier and Lieberman 2010). Therefore, integer programming is not appropriate for pavement maintenance and rehabilitation optimization at the networklevel. De la Garza et al. (2010) developed a network-level linear programming optimization model, in which a deterministic prediction model is utilized for pavement condition deterioration. Five pavement condition states are defined based on the Combined Condition Index (CCI) values. Nine maintenance and rehabilitation treatments, ranging from ordinary maintenance to reconstruction, are identified. Each treatment is allowed to be conducted on only one pavement condition category. De la Garza et al. s model assumes that the deterioration rates are fixed for each pavement condition state and that pavements only deteriorate from an upstream condition to the next downstream condition. The pavement deterioration rates are calculated deterministically from historical data. The objective function of the optimization model is stated as follows: Minimize w N1 i w2 N2i w3 N3 i w4 N4i w5 N5i i P 1 (2.3) 12
where P is a set of funding periods (1, 2, 3,, i,, 15); w 1, w 2, w 3, w 4, and w 5 are weighting coefficients for each condition state; Nki is the number of lane-miles in condition k at the end of period i (de la Garza et al. 2010). The model is subject to several sets of constraints such as performance targets and budget limitation. One important constraint representing pavement condition state transition is stated as follows: N5( i 1) X i1 X i2 X i3 X i4 N 5i N5( i 1) 0 (2.4) D U U U U 54 i1 i2 i3 i4 where D 54 is the deterioration rate from condition state 5 (excellent) to condition state 4 (good); X ij is the amount of money spent on treatment j within period i; U ij is the unit cost of treatment j within period i (de la Garza et al. 2010). This constraint indicates that the number of lane-miles in excellent condition at period i is equal to the sum of the following components: the number of lane-miles in excellent condition at period (i-1); minus the number of lane-miles deteriorating from excellent condition to good condition; plus the number of lane-miles in excellent condition restored from other conditions via corresponding treatments (de la Garza et al. 2010). Similar constraints are developed for the other four condition states. 13
The optimization model developed by de la Garza et al. (2010) can be used as a powerful decision support tool in pavement management at the network-level. The objective function can be modified to solve different problems. However, there are two limitations in the deterioration model: (1) the same deterioration rates are used for all pavements no matter whether the last treatment is reconstruction or thin overlay (Chen et al. 1996); (2) the deterministic prediction model cannot consider the uncertain aspect of pavement deterioration (Butt et al. 1994). 14
Chapter 3 Methodology 3.1 Introduction As discussed in the previous chapter, the Markov transition probability prediction model and linear programming algorithm are the most popular techniques utilized in networklevel optimization. The development of this optimization model is mainly based on the methodologies adapted from Arizona s model (Golabi et al. 1982) and Oklahoma s model (Chen et al. 1996). A Markov prediction model is developed using historical pavement condition data. Then a network-level optimization model is formulated by integrating a linear programming model with the Markov prediction model. Appendix A shows a flowchart of the formulation process of this optimization model. The main feature of this decision support tool is that it provides more flexibility in setting optimization objectives and defining constraints, resulting in more capability in analyzing pavement management problems at the network-level. 15
3.2 Development of Markov Transition Probability Model The Markov transition probability model assumes that the probabilities that a pavement deteriorates from a given condition state to other condition states are stationary transition probabilities (Hillier and Lieberman 2010; Chou et al. 2008). These transition probabilities can form a Markov transition matrix, expressed by Equation 3.1: p p P p 11 21 n1 p p p 12 22 n2 p 1n p p 2n nn (3.1) where pij is the probability that the a pavement section moves from state i in the current year to state j in the next year, and n is the total number of condition states (Chou et al. 2008). In this research, pavement conditions are categorized into five states, namely Excellent, Good, Fair, Poor and Very Poor, based on the pavement condition rating (PCR) score; pavement repair treatments are grouped into four types, respectively Preventive Maintenance, Thin Overlay, Minor Rehabilitation and Major Rehabilitation. Pavement deterioration rates can be influenced by many factors, such as pavement type, last repair treatment, traffic loading, construction quality, climate, underlying soil characteristics, and system priorities (Yu 2005). Markov transition probabilities should be estimated for each pavement group with similar characteristics. However, a pavement group must have a significant number of pavement sections at various condition states to develop a reliable 16
prediction model (Chou et al. 2008). Therefore, three critical factors, namely pavement type, system priorities and last repair treatment, are used as parameters to form pavement groups. Two transition probability matrices, namely treatment matrix and Do Nothing matrix, are developed for each repair treatment in each pavement group. The treatment matrix is for the condition improvement the first year the treatment is applied and the Do Nothing matrix is for the deterioration trend after the treatment. The elements of the transition probability matrices are derived from historical pavement condition data, and are updated automatically when new condition data become available each year. There are three challenges in estimating the Markov transition matrices from actual historical data. First, noises or outliers in the data need to be excluded to improve the accuracy of the estimation. An example of the noises is that a pavement section in poor condition may become in good condition the next year without any record of repair treatment. Such pavement sections are removed from the calculation process in this research. Therefore, the Do Nothing matrices are upper triangular matrices and treatment matrices are lower triangular matrices. Second, pavement condition data are often subject to attrition, also referred to as dropouts (Laird and Ware 2004). Overtime, only good performing pavements remain, while poor performing pavements are more likely to receive treatments and drop out ; therefore, prediction models that do not consider dropouts tend to overestimate future pavement conditions, particularly at the later stage of pavement life span (Chou et al. 2008). In this research, imputation is conducted for those drop out pavement sections to avoid overestimation (Laird and Ware 2004). Third, some pavement groups do not have a sufficient number of pavement 17
sections, which makes the transition matrices tend to be inaccurate and unrealistic. For this research, the total mileage of a pavement group should be at least 300 miles; otherwise, the transition probabilities are derived from other similar groups. 3.3 Formulation of Network-Level Optimization Model This section presents the development of a linear programming model for network-level pavement optimization based on the Markov transition probability model. In this research, the pavement network is divided into three sub-networks according to the pavement types (1, Concrete; 2, Flexible; 3, Composite). Each sub-network is divided into four groups according to the last repair treatments (1, Preventive Maintenance (PM); 2, Thin Overlay; 3, Minor Rehabilitation; 4, Major Rehabilitation). Each group is further divided into five pavement condition states (1, Excellent; 2, Good; 3, Fair; 4, Poor; 5, Very Poor) based on the PCR score. Each pavement condition class may be recommended for one of the five repair treatments (0, Do Nothing; 1, Preventive Maintenance; 2, Thin Overlay; 3, Minor Rehabilitation; 4, Major Rehabilitation). In the optimization model described in this chapter: N is the number of pavement types, K is the number of repair treatment types, I is the number of pavement condition states and T is the number of analysis years. Y ntk ' ik is the decision variable representing the proportion of pavement type n in condition state i with last treatment k receiving recommended repair treatment k in year t. Two assumptions are: (1) the total mileage of the pavement network remains constant and (2) the pavement types do not change for any pavement section during the analysis period. 18
Two objective functions are developed. The first one is to minimize the total repair cost of the pavement network to achieve a certain condition level goal (Equation 3.2): Minimize N T K n 1 t 1 k' 1 i 1 k 0 I K Y ntk ik C (3.2) ' ntk' ik where C ntk ' ik is the unit cost of applying treatment k in year t to pavement type n in state i with last treatment k. The second one is to maximize the proportion of pavements in Excellent, Good, and Fair condition over the analysis period with given budget constraints (Equation 3.3): Maximize N T K 3 K Y ntk ik n 1 t 1 k' 1 i 1 k 0 ' (3.3) There are four sets of required constraints namely non-negativity constraints, sum-to-one constraints, initial condition constraints, and state transition constraints. The nonnegativity constraints (Equation 3.4) ensure that all variables in the optimization model are non-negative. Y ntk' ik 0 for all n = 1,, N; t = 1,, T; k = 1,, K; i = 1,, I; k = 0,, K (3.4) 19
The sum-to-one constraints (Equation 3.5) ensure that the entire pavement network is divided into many proportions and each proportion is represented by a decision variable. N K n 1 k' 1 i 1 k 0 I K Y 1 for all t = 1,, T (3.5) ntk' ik The initial condition constraints (Equation 3.6) pass the values representing current pavement condition state distribution for each pavement group to the optimization model. K k 0 Y n1k ' ik Q nk' i for all n = 1,, N; k = 1,, K; i = 1,, I (3.6) where year. Q nk ' i is the proportion of pavement type n in state i with last treatment k in initial The state transition constraints (Equation 3.7) integrate the Markov transition probability model with the linear programming model. From the second analysis year on, the proportion of pavement type n in condition state j with last treatment k in year t is derived from two parts of pavement in various condition states in year t-1: one part with last treatment k receiving no new treatment (Do Nothing) and the other part receiving new treatment k. 20
K Y ntk' jk k 0 i 0 k 1 I K Y n( t 1) kik' P nk' ij I i 0 Y n( t 1) k' i0 DN for all n = 1,, N; t = 2,, T; k = 1,, K; j = 1,, I; (3.7) nk' ij where P nk ' ij is the probability that pavement type n receiving new treatment k transit from state i to state j and DN nk ' ij is the probability that pavement type n with last treatment k receiving no new treatment (Do Nothing) moves from state i to state j. In order to make the optimization model more practical, several sets of optional constraints are also introduced. The condition constraints (Equation 3.8 and Equation 3.9) ensure that the proportion of pavement in certain condition states is in a prescribed range. N K K n 1 k' 1 k 0 Y ntk ik for all t = 2,, T; selected i (3.8) ' it N K K n 1 k' 1 k 0 Y ntk ik for all t = 2,, T; selected i (3.9) ' it where it is the upper limit of proportion of pavement in condition i in year t and it is the lower limit of proportion of pavement in condition i in year t. For instance, pavements in Poor and Very Poor condition are considered as deficient. It may be desirable to limit the total amount of deficient pavements (or deficiency level) to a given percentage, say, 5%, of the entire network. If the desirable deficiency level is 21
much lower than the existing deficiency level, a significant amount of rehabilitation would be required to achieve the desired condition target immediately. Therefore, it is more reasonable to allow the condition target (in term of desired deficiency level) to be achieved gradually by linearly reducing the proportion of deficient pavements using Equation 3.10: i1 i i1 t 1 2 t t' it t' 1 (3.10) i t' t T where i is the desired proportion of condition state i; it is the upper limit of proportion of pavement in condition i in year t; t is the year to achieve condition target specified by the user and T is the number of analysis years. The allowable treatment constraints (Equation 3.11) ensure that certain treatments can only be applied to pavements in certain condition states or with certain last treatments. Yntk' ik 0 for all t = 1,, T; selected n, k, i, k (3.11) Experience reveals that some treatments are cost effective only when pavements are in certain condition states and with appropriate last treatments. For example, Thin Overlay is only cost effective on pavements in Fair and Poor condition as shown in Table 3.1, so the corresponding decision variables are set to zero to disallow Thin Overlay on pavements in other condition states. 22
Table 3.1: Example of Allowable Treatments Condition Do Nothing PM Thin Overlay Minor Rehab Major Rehab Excellent Yes Good Yes Yes Fair Yes Yes Yes Poor Yes Yes Yes Yes Very Poor Yes Yes The effectiveness of some treatments is also associated with the last treatment. For instance, if PM is conducted on pavements with last treatments of PM, the underlying distress of the pavement can only be masked for a short period of time and the distress may resurface quickly within a few years after treatment. However, PM is a lower cost treatment, which may cause the optimized solution to recommend PM treatments to be applied repeatedly. Therefore, it is necessary to add a set of constraints to disallow PM treatment on pavements with last treatment of PM. The budget constraints (Equation 3.12) ensure that the required budgets recommended by the optimized solution do not exceed the maximum available budget for each year. N T K n 1 t 1 k' 1 i 1 k 0 I K Y ntk ik Cntk ik L B for all t = 1,, T (3.12) ' ' t where L is the total length of entire pavement network and Bt is the maximum available budget in year t. 23
It is possible that the optimized repair policy obtained from the mathematical model would recommend a large number of pavements to be repaired in the first couple of years in order to minimize the total cost over the analysis period. However, the recommended budget may be far beyond the maximum available budget of the highway agency, which makes the optimized repair strategy unsuitable for practical use. For that reason, the budget constraints are included in the model. On the basis of the above objective functions and constraints, a linear programming model for pavement maintenance and rehabilitation optimization at the network-level is formulated as described in Appendix B. 3.4 Implementation The network-level optimization model is implemented using Microsoft Visual Basic.NET (2008) and IBM ILOG CPLEX 12.1. The optimization tool is composed of four parts: pavement database, data preparation, optimization analysis and results output. The pavement database stores current and historical pavement conditions, project history, and road inventory data for analysis. The data preparation part enables the user to define pavement condition states (Excellent, Good, Fair, Poor, and Very Poor) by selecting the corresponding PCR thresholds; to generate the current pavement condition distribution table for further analysis; and to determine the year from which historical condition data are used to generate the Markov transition probability matrices. The optimization analysis part allows the user to select the pavement network for optimization; to input unit cost for each type of repair treatment; to choose appropriate objective functions; to 24
set pavement condition constraints; to select allowable treatments for pavements in different condition states; and to enter the maximum available budget for each year in the analysis period. The results output part enables the user to view the projected pavement condition distribution, the optimized recommended treatment policy, and the corresponding budget allocation for the analysis period of up to 30 years. 25
Chapter 4 Example Problems 4.1 Overview This chapter presents four examples problems solved by the optimization tool developed in this study. For the example runs, ODOT s priority system pavement network which consists of 11,941 lane miles of interstate highways, U.S. routes, and state routes is analyzed over the next 20 years. The unit costs of four types of maintenance and rehabilitation treatments are shown in Table 4.1. Table 4.1: Unit Cost of Maintenance and Rehabilitation Treatments Preventive Thin Minor Major Treatment Maintenance Overlay Rehab Rehab Cost ($1,000 per lane-mile) 40 100 200 1,000 Pavement conditions are classified into five categories based on PCR scores as shown in Table 4.2. 26
Table 4.2: Pavement Condition Classification Pavement Condition PCR score range Excellent PCR >= 85 Good 75 =< PCR < 85 Fair 65 =< PCR < 75 Poor 55 =< PCR < 65 Very Poor PCR < 55 Table 4.3 presents the current overall pavement condition distribution. Since pavements in poor and very poor conditions are considered to be deficient, the current network deficiency level is 2.7%. Table 4.3: Current Pavement Condition Distribution Pavement Condition Category Excellent Good Fair Poor Very Poor Proportion (%) 66.1 22.8 8.4 2.4 0.3 4.2 Problem 1: Minimum Budget to Achieve a Desired Condition Level 4.2.1 Problem Statement This problem is to calculate the minimum budget required to improve the overall pavement network condition by reducing the deficiency level from 2.7% to 1% within three years and to determine the corresponding fund allocation among different maintenance and rehabilitation treatments. Both the optimized results with and without budget constraints are analyzed and compared. Table 4.4 shows the allowable treatments for Problem 1. 27
Table 4.4: Allowable Treatments for Problem 1 Condition Do Nothing PM Thin Overlay Minor Rehab Major Rehab Excellent Yes Good Yes Yes Fair Yes Yes Yes Poor Yes Yes Yes Very Poor Yes Yes The minimization of total cost is used as the objective function for this problem. 4.2.2 Optimized Results without Budget Constraints The optimization model without budget constraints (Model A) yields a theoretical optimized solution for the problem. Since no maximum available annual maintenance and rehabilitation budget is defined, the mathematical optimization model could recommend any amount of pavement mileage to be repaired in each year in order to minimize the total cost over the analysis period, which is 20 years in this case. Table 4.5 and Figure 4-1 show the recommended budget allocation for each type of treatment. Table 4.6 and Figure 4-2 show the corresponding projected pavement condition distribution. 28
Table 4.5: Recommended Treatment Budget for Problem 1 without Budget Constraints Year Recommended Budget ($ Million) PM Thin Overlay Minor Rehab Major Rehab Total Budget 2011 0.7 0.0 78.7 0.0 79.4 2012 19.9 0.0 93.3 30.7 143.9 2013 11.6 0.0 195.1 0.0 206.7 2014 13.2 0.0 143.7 0.1 156.9 2015 14.5 0.0 141.0 0.0 155.6 2016 15.9 0.0 136.9 0.0 152.8 2017 17.5 0.0 129.7 0.1 147.2 2018 18.9 0.0 121.7 0.1 140.6 2019 20.1 0.0 116.9 0.1 137.0 2020 20.8 0.0 114.8 0.1 135.6 2021 21.5 0.0 112.7 0.1 134.2 2022 22.1 0.0 111.7 0.1 133.8 2023 22.5 0.0 111.5 0.1 134.0 2024 21.8 0.0 111.9 0.1 133.8 2025 22.2 0.0 112.6 0.1 134.9 2026 22.1 0.0 115.6 0.1 137.7 2027 23.2 0.0 112.4 0.1 135.7 2028 22.3 0.0 117.1 0.1 139.5 2029 16.1 0.0 143.9 0.1 160.0 2030 33.9 0.0 78.7 0.1 112.6 29
Budget ($ Million) 250 Recommended Treatment Budget for Problem 1 (without Budget Constraints) 200 150 100 50 0 Year PM Thin Overlay Minor Rehab Major Rehab Figure 4-1: Recommended Treatment Budget for Problem 1 (without Budget Constraints) 30
Table 4.6: Projected Pavement Condition Distribution for Problem 1 (without Budget Constraints) Year Condition Distribution (%) Excellent Good Fair Poor Very Poor 2011 66.1 22.8 8.4 2.4 0.3 2012 58.6 27.1 12.2 1.8 0.3 2013 56.1 29.7 12.7 1.5 0.0 2014 57.1 30.5 11.5 1.0 0.0 2015 56.2 31.2 11.6 1.0 0.0 2016 55.7 31.6 11.7 1.0 0.0 2017 55.4 31.8 11.8 1.0 0.0 2018 55.1 32.1 11.8 1.0 0.0 2019 54.8 32.3 11.9 1.0 0.0 2020 54.4 32.6 12.1 1.0 0.0 2021 54.0 32.8 12.2 1.0 0.0 2022 53.7 33.0 12.4 1.0 0.0 2023 53.4 33.1 12.5 1.0 0.0 2024 53.2 33.2 12.6 1.0 0.0 2025 52.8 33.4 12.8 1.0 0.0 2026 52.5 33.6 13.0 1.0 0.0 2027 52.3 33.6 13.1 1.0 0.0 2028 52.2 33.7 13.1 1.0 0.0 2029 52.1 33.7 13.2 1.0 0.0 2030 52.1 33.7 13.2 1.0 0.0 31
100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Pavement Condition Distribution for Problem 1 (without Budget Constraints) Year Very Poor Poor Fair Good Excellent Figure 4-2: Projected Pavement Condition Distribution for Problem 1 (without Budget Constraints) From Table 4.5 and Figure 4-1, it can be seen that the required budget for the year 2013 is $206.7 million, much higher than the other years. Table 4.6 and Figure 4-2 indicate that the deficiency level is reduced gradually from 2.7% to 1%. However, this result is not suitable for practical use, since the recommended budget for the third year may be far beyond the available maximum annual budget. Besides, the recommended annual budget varies significantly in the first several years, which makes the treatment strategy difficult to be implemented by highway agencies. It should be noted that the funds for years after 2014 are used to maintain the deficiency level at 1%, since pavements tend to deteriorate over years. 32
4.2.3 Optimized Results with Budget Constraints The optimization model with budget constraints (Model B) provides an optimal solution under the constraint that recommended budgets do not exceed the maximum available budget for each year. In this example run, it is assumed that the annual budget limitation is $150 million. All other constraints and objective functions are the same with the Model A described in 4.2.2. Table 4.7 and Figure 4-3 show the recommended budget allocation for each type of treatment. Table 4.8 and Figure 4-4 show the corresponding projected pavement condition distribution over the next 20 years. Table 4.7: Recommended Treatment Budget for Problem 1 with Budget Constraints Year Recommended Budget ($ Million) PM Thin Overlay Minor Rehab Major Rehab Total Budget 2011 1.2 0.0 111.2 30.6 143.0 2012 1.2 0.0 148.8 0.0 150.0 2013 24.5 0.0 125.4 0.0 150.0 2014 12.4 0.0 137.5 0.0 150.0 2015 14.9 0.0 135.1 0.0 150.0 2016 17.2 0.0 132.8 0.1 150.0 2017 19.6 0.0 130.3 0.1 150.0 2018 19.1 0.0 122.8 0.1 141.9 2019 20.2 0.0 117.4 0.1 137.7 2020 20.8 0.0 115.6 0.1 136.5 2021 21.5 0.0 113.6 0.1 135.1 2022 22.0 0.0 112.6 0.1 134.7 2023 22.4 0.0 112.3 0.1 134.7 2024 21.8 0.0 112.5 0.1 134.3 2025 22.2 0.0 113.0 0.1 135.3 2026 22.0 0.0 115.8 0.1 137.9 2027 23.1 0.0 112.7 0.1 135.9 2028 22.3 0.0 127.2 0.1 149.6 2029 16.1 0.0 133.9 0.1 150.0 2030 33.9 0.0 78.6 0.1 112.6 33
Budget ($ Million) 160 Recommended Treatment Budget for Problem 1 (with Budget Constraints) 140 120 100 80 60 40 20 0 Year PM Thin Overlay Minor Rehab Major Rehab Figure 4-3: Recommended Treatment Budget for Problem 1 (with Budget Constraints) 34
Table 4.8: Projected Pavement Condition Distribution for Problem 1 (with Budget Constraints) Year Condition Distribution (%) Excellent Good Fair Poor Very Poor 2011 66.1 22.8 8.4 2.4 0.3 2012 60.4 27.1 11.1 1.5 0.0 2013 57.4 28.8 12.3 1.5 0.0 2014 57.1 30.6 11.4 0.9 0.0 2015 55.9 31.4 11.7 1.0 0.0 2016 55.2 31.9 12.0 1.0 0.0 2017 54.8 32.3 12.0 1.0 0.0 2018 54.9 32.2 11.9 1.0 0.0 2019 54.6 32.5 12.0 1.0 0.0 2020 54.3 32.7 12.1 1.0 0.0 2021 53.9 32.8 12.2 1.0 0.0 2022 53.6 33.0 12.4 1.0 0.0 2023 53.4 33.1 12.5 1.0 0.0 2024 53.2 33.2 12.6 1.0 0.0 2025 52.8 33.4 12.8 1.0 0.0 2026 52.5 33.6 13.0 1.0 0.0 2027 52.3 33.6 13.1 1.0 0.0 2028 52.2 33.7 13.1 1.0 0.0 2029 52.6 33.7 12.9 0.9 0.0 2030 52.1 33.7 13.2 1.0 0.0 35
100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Pavement Condition Distribution for Problem 1 (with Budget Constraints) Year Very Poor Poor Fair Good Excellent Figure 4-4: Pavement Condition Distribution for Problem 1 (with Budget Constraints) It can be seen from Table 4.7 and Figure 4-3 that the recommended annual budgets are all within the limit of $150 million during the analysis period. Table 4.8 and Figure 4-4 indicate that the deficiency level is reduced gradually from 2.7% to 1% in three years. Although the average annual pavement expenditure is $141 million, which is slightly higher than the theoretical optimized result ($140.6 million) obtained from Model A, this model yields a more practical and stable solution especially for the first several years. 4.2.4 Comparison between Two Solutions As shown in Figure 4-5, Model A (without budget constraints) recommends a large amount of pavement mileage be repaired in the third year, which may exceed the 36
$ Million available budget limitation; whereas the annual treatment budget recommended by Model B (with budget constraints) is more stable and practical. 250 Recommended Treatment Budget versus Available Budget for Problem 1 200 150 100 50 0 Year Without Budget Constraint With Budget Constraints Available Budget Figure 4-5: Recommended Budget versus Available Budget for Problem 1 As shown in Figure 4-6, both Model A and Model B reach the goal to reduce the deficiency level from 2.7% to 1% within three years as described in the problem statement. 37
Deficiency Level 3.0% Comparison of Deficiency Level Trends for Problem 1 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 2011 2013 2015 2017 2019 2021 2023 2025 2027 2029 Year Without Budget Constraints With Budget Constraints Figure 4-6: Comparison of Deficiency Level for Problem 1 Model B yields a better condition level than Model A in the first several years, mainly because the total recommended budget in the first three years of Model B is $13.6 million higher than that of Model A. In conclusion, Model A provides a maintenance and rehabilitation strategy to minimize the total cost in the 20 years without considering the budget limitation; whereas Model B has one more set of constraints to ensure the recommended annual budgets do not exceed the maximum available budget limitation. The average annual pavement expenditure obtained from Model A is slightly lower than that of Model B, which means Model A yields a better solution than Model B if the total cost in the analysis period is the only 38