Research Article A Two-Stage Model for Project Optimization in Transportation Infrastructure Management System

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1 Matheatical Probles in Engineering, Article ID , 8 pages Research Article A wo-stage Model for Project Optiization in ransportation Infrastructure Manageent Syste Zhang Chen, 1 Liyuan Liu, 1 Li Li, 1 and Hui Li 2 1 Key Laboratory of Road and raffic Engineering of the Ministry of Education, ongji University, Shanghai , China 2 Departent of Civil and Environental Engineering, University of California, Davis, CA 95616, USA Correspondence should be addressed to Li Li; 622lilian@tongji.edu.cn Received 11 April 2014; Revised 3 June 2014; Accepted 18 June 2014; Published 13 July 2014 AcadeicEditor:X.Zhang Copyright 2014 Zhang Chen et al. his is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Matheatical optiization is very iportant for project decisionin the ransportation Infrastructure Manageent Syste (IMS). However, it has not been widely eployed in IMS due to poor perforance of conventional optiization odels in calculation speed and practical application. herefore, it is necessary to iprove the perforance of optiization odels. According to the process of decision-aking in transportation anageent, a novel two-stage project optiization odel, including budget allocation and project distribution, was proposed in this paper. Moreover, the ethods of dynaic prograing (DP) and genetic algorith (GA) were applied to obtain an effective solution. he findings indicate that the new optiization ethod can provide a satisfactory and reasonable aintenance schedule for transportation infrastructure aintenance agencies whose routine anageent will benefit fro the newly proposed odel. 1. Introduction ransportation infrastructure includes roads, bridges, tunnels, airports, railways, and seaports. As a result, ransportation Infrastructure Manageent Syste (IMS) correspondingly covers any subsystes, aong which Paveent Manageent Syste (PMS) and Bridge Manageent Syste (BMS) are the ost iportant ones. Whatever kind of subsyste it is, project optiization is a key eleent in the process of decision-aking for the infrastructure anageent. Specifically, project optiization [1] refers to finding an optial aintenance strategy with axiized benefit through arranging paveent aintenance reasonably in ters of tie and space in the planning period. Most of existing research efforts on project optiization of IMS focusedonpms. Since decision supporting syste was introduced into the second generation of PMS, project optiization has been paid uch attention by paveent researchers and anageent agencies. Currently, there are ainly two categories of project optiization ethods for network-level paveent anageent syste, naely, prioritization ethod and atheatical optiization [2, 3]. Prioritization ethod is to carry out project selection based on soe principles prescreened and then to deterine the aintenance strategy for each year in the planning period. In 1994, Hass et al. [4] suarized the characteristics of different prioritization ethods, aong which the two ost popular ethods are based on infrastructure perforance paraeters and econoic analysis paraeters. he prioritization ethod based on perforance paraeters is adopted to arrange road aintenance projects by California, US, where road roughness, daaged condition, and average daily traffic volue are considered as influencing factors to develop the prioritization principles. Hudson uses rainfall, nuber of freeze-thaw cycles, and daage severity as paraeters to establish an expression for prioritization through regression or variance analysis so as to deterine the priority index of every section requiring reconstruction in the road network. While prioritization based on perforance paraeters isconvenientforcalculationandofhighpertinence,the results ay be far fro the econoic optiu. hus the prioritization ethod based on econoic analysis paraeters is relatively advanced regarding econoic optiu. In

2 2 Matheatical Probles in Engineering 1980s, the prioritization ethod based on econoic analysis paraeters was very popular [5 8]. he State of Washington, US, used the total cost, including initial project construction cost, aintenance cost, user operating expense, delay cost, and paveent salvage, as the prioritization indicator in their PMS while the perforance index was adopted in UK. At the sae tie, Hass et al. used equivalent annual cost in Canada, which refers to the ratio of the project construction cost and the corresponding expected lifetie to sequence. Aong the PMSs adopting the prioritization ethod, the ost typical one is PAVER in US. PAVER uses the ethod of benefitcost-increent to deterine the priorities of aintenance projects according to budget optiization. Another representative one is the Paveent Maintenance Decision Support Syste of Shanghai (PMDSSS), China. According to different anagerial preferences of road adinistrators and agencies, this PMDSSS has developed eight different prioritization principles taking into account either perforance paraeters or econoic analysis paraeters or both, with different cobinations of traffic volue, paveent daage condition, riding quality, structural capability, and econoic indicators. he principles greatly help policy akers to forulate large or ediu aintenance plans and long-ter rehabilitation strategies in the planning years. However, the tradeoff between the aintenance strategy and the tie has not been included in the prioritization ethod, which leads to huge disparity between the calculated result and the actual optial solution. herefore, any researchers have been, instead, focusing on the atheatical optiization, which refers to considering each project in the planning period, the possible aintenance plan, and the ipleentation schedule through atheatical calculations. he idea of optiization using atheatical ethod was proposed in he atheatical optiization for PMS falls into two categories: static and dynaic. Integer prograing is ostly used in the forer. For instance, PMS in Denark [9], HDM-III in World Bank [10], RAMS of exas, US, and PARS of Ontario, Canada, all adopt 0-1 integer prograing [11, 12].PMSintheStateofIndiana,US[13, 14], usesthenuberofworkdayneededforaintenanceasthe decision variable, and constraints fro budget, anpower, achine, and aterial are also included to build up the integral prograing odel. In the dynaic prograing, PMS of Arizona, US [15], was the first to successfully introduce Markov decision process to the network-level paveent anageent syste. When iproving the PAVER syste, Feighan et al. [16] anaged to develop the strategy for aintenance and reconstruction that iniizes the cost of road network through dynaic prograing. In 1994, Liu and Yao [17] usedmarkovdecisionprocesstoiniize aintenance cost. In 1995, Zou [18] forulated a dynaic odel to predict the paveent perforance and the adaptive iteration algorith. In addition, to solve the probles of paveent aintenance strategy, he adopted the analytical hierarchy process odel and the heuristic optiization technique. In 2001, unoo [19] optiized the aintenance plan of integrated paveent using shuffled coplex evolution algorith. In 2003, Chan et al. [20, 21] atteptedtouse genetic algorith to develop a ultiobjective prograing odel to optiize the strategy for highway paveent aintenance. Meanwhile, they adopted heuristic algorith to yield the result. Ferreira et al. [22] alsodevelopedaprobabilistic segent-linked optiization odel together with a genetic algorith heuristic with the objective of iniizing total discounted cost of M&R actions. While atheatical optiization can produce the optial calculation results, the large nuber of factors to be considered, huge data processing, and barely satisfactory calculation speed greatly liit its practical application. In order to apply atheatic optiization to project optiization, the key point is to find a odel or algorith which can save coputing resources as well as eeting the practical needs. Copared to the existing research studies, the ain contributions of this paper are listed as follows. Firstly, the project optiization is separated into two independent and interrelated processes, naely, budgets allocation and project distribution, and a new two-stage odel is developed. Secondly, the technique of dynaic prograing (DP) and genetic algorith (GA) are applied to solve the odel and yield an effective solution. Finally, the new prograing ethodisverifiedtobeeffectivethroughthecasestudy in Shanghai and the poor calculation speed and the practical application liitations of conventional ethods are iproved. 2. Integer Prograing in Conventional Project Optiization Integer prograing (IP) in the conventional project optiization odel can be stated as follows: Objective j=1 Subject to ax Z= X ijt B ijt (1) X ijt C ijt A (2) X ijt =1 (i=1,2,...,;,2,...,) (3) X ijt ={ 1 if treatent j isappliedinsegenti in year t 0 otherwise (4) B ijt,c ijt,a,s t,v t >0, (5) where Z is total aintenance benefit; B ijt, C ijt are, respectively, the benefit and cost caused by ipleenting treatent j in project i in year t; A is the total budget in planning period; is the length of planning period, norally 5 or 10 years; is the aount of total road units; is the total nuber of treatents for each project. he odel above is a large integer prograing proble in which the set of feasible solutions is very huge. herefore, it is necessary to siplify the odel in order to calculate

3 Matheatical Probles in Engineering 3 conveniently as well as eeting practical needs. Equation (1) is transfored to the following for: X ijt B ijt = X ij1 B ij1 + + X ijk B ijk + + X ij B ij. If the expression f t (Y t )= i=1 j=1 X ijt B ijt (Y t is the budget of year t) is considered as the aintenance benefit of year t, then the total aintenance benefit in planning years should be equal to the suation of aintenance benefit of each year. herefore, X ijt B ijt =f 1 (Y 1 )+ +f t (Y t )+ +f (Y ) = f t (Y t ), where Y t is budget of year t and f t (Y t ) is aintenance benefit of year t. Itisshownin(1) and(7) thatsolutionofprojectoptiization can be approached with the following steps. he first step is budget optiizing to get the budget allocation in each year in the planning period. he second step is project portfolio (including optiization of project schee and schedule) to get the optial benefit aintenance strategy in each year based on the budget allocation of first step. After several iterations of optiization, selection, and coparison through the repetition of above two steps, the optial aintenance strategy under the optial budget allocation canbeachievedfinally.actually,thisethodreflectswellthe actual iterative decision-aking process of the governent agencies, which is budget allocation, project arrangeent, budget adjustent, and project adjustent. hrough this iterative ethod, project optiization is divided into two relatively siple processes, naely, budget optiization and project distribution. 3. wo-stage Optiization Approach According to the above analysis, project optiization can be divided into two stages. he first one is about how to allocate budgets. he second one deterines aintenance projects in each year based on the budget allocation in the first stage. he first stage is defined as budget allocation odel and the second as project distribution odel, which build up the twostage optiization approach Budget Allocation Model. Budget allocation odel is, under certain aount of fund, to find a reasonable budget (6) (7) allocation for each year in planning period in order to axiize the benefit of budget. he odel focuses on the reasonable way of budget allocation. In the process of allocating budgets, paveent anageent agencies need to consider not only the perforance of road network but also the budget liits (axial and inial budgets for each year are expressed with S t and V t, resp.). herefore, the budget allocation odel can be designed as follows: Objective ax f t (Y t )=f 1 (Y 1 )+f 2 (Y 2 ) Subject to + +f t (Y t )+ +f (Y ) (8) Y t A (9) Y t S t (,2,...,) (10) Y t V t (,2,...,). (11) In fact, there are a large nuber of aintenance projects in theroadnetworkfortheplanningperiod,andeachproject has several treatents. Different treatents bring different aintenance benefits. herefore, it is necessary to pose soe liitations to budget allocation considering that paveent anageent agencies ay utilize fuzzy decisions unconsciously in the process of decision-aking which akes their decisions rather reasonable. Firstly, budget change is discontinuous when anageent agencies are allocating or adjusting budgets; therefore the budget change is set as integer ties of the sallest unit. Secondly, axial aintenance benefit in the planning period under certain budget allocation schedule equals the suation of axial benefits in each year: f t (Y t )= ax (f 1 (Y 1 )) + + ax (f t (Y t )) + +ax (f (Y )) = ax (f t (Y t )). (12) 3.2. Project Distribution. Project distribution odel is about how to arrange aintenance projects under given budget allocation schedule (Y 1,Y 2,...,Y ) in each year in order to obtain the axial aintenance benefit. It can be designed as follows: j=1 Subject to Objective n n (ax (X ijt B ijt )) (13) (X ijt C ijt ) Y t, (,2,...,) (14) X ijt 1 (i=1,2,...,n;,2,...,), X ijt =0or 1, (15)

4 4 Matheatical Probles in Engineering Initial species produced by budget optiization Chroosoe represents budget allocation anner Calculation of chroosoe fitness Calculating axial aintenance benefit Inheritance (including copy, intersection, utation) ew species ew budget allocation Meet the rule of optiization Budget allocation axiizes aintenance benefit Final species Results output Figure 1: Calculation process of budget allocation with GA. where n is the total aount of projects in road network; is the total aount of treatents for each project; Boolean variable X ijt =1(or 0) eans treatent j is applied (or not) to project i in year t Relation of Budget Allocation Model and Project Distribution Model. he results of the budget allocation odel and the project distribution odel are interactional. At first, annual budget will be provided through the budget allocation odel. hen, the axial aintenance benefit and the project schedule will be constructed through the project distribution odel according to the result of the first odel and then a feedback will be offered to the first odel to judge whether the solution is the best or not. If not, the budget allocation odel will optiize again and produce a new array of budgets in each year, then the corresponding total aintenance benefit and aintenance schedule will be obtained through the project distribution odel. hrough iterations of the two odels, the optial strategy will be obtained in the planning years. 4. Solving the wo-stage Model he key to applying the atheatic optiization ethod into practice is to find a reasonable solving ethod which ensures certain accuracy and eets practical requireents. In the objective function of (8), it is difficult to be expressed for f i (Y i ) by explicit functions, which causes difficulty in effectively solving it with noral algorith. Given that genetic algorith (GA) is able to search the global optial solution in a coplicated space while its objective functions are not necessary to be explicit functions, GA is chosen to solve the budget allocation odel (8). In the budget allocation odel, decision (aintenance strategy) of each stage (planning year) is the function of paveent network condition given by the decision of last stage. hat is, each stage will influence the next stage through the network condition resulting fro the different treatent strategy. herefore, under a given budget allocation (Y 1,Y 2,...,Y ), the forulation of paveent network aintenancestrategyinplanningyearsisaultistageproblein which all stages have utual connections, naely, a classic dynaic prograing proble. herefore, the ethod of dynaic prograing is eployed to solve the projects distribution odel (13) GA for Budget Allocation Model. Before the calculation with GA, soe paraeters including length of chroosoe, species group scale, intersection rate, and utation rate have to be set in advance according to the scale of paveent network, length of planning period, and budget. hen, the budget allocation odel can be solved in the following process, as shown in Figure 1. (1) Initial species: each chroosoe in the species represents a budget allocation ode. (2) Fitness function: the fitness function of chroosoe is derived fro the axial aintenance benefit under a budget allocation ode. he function is coputed using the project allocation odel. (3) Genetic anipulation of chroosoe: through the genetic anipulation of chroosoe, the budget allocation with superior aintenance benefit can be inherited and those inferior ones will be eliinated. (4) erination of iteration: certain iteration ties can be set as the terination condition of GA. In the above calculation process, the fitness function of chroosoes, which are not subject to the budget constraints, can be liited by penalty function or be given a very sall fitness value. he chroosoe with the biggest fitness value gets to be obtained by GA and the corresponding budget allocation anner is the final solution of the budget allocation odel.

5 Matheatical Probles in Engineering 5 Stage 1 Stage 2 Stage Condition of 1st year Decision of 1st year Condition of 2nd year Decision of 2nd year. Condition of year Decision of year Y 1 U 1 (Y 1 ) Y 2 = F(Y 1,U 1 (Y 1 )) Y 2 U 2 (Y 2 ). Y U (Y ) Y = F(Y 1,U 1 (Y 1 )) Figure 2: DP process of project distribution odel DP for Project Distribution Model. Project distribution odel is ainly used to arrange project ipleentation in each year under a given optiization result of budget allocation. he output results should include a list of aintenance projects in each planning year. Meanwhile, through the project distribution odel, the fitness value can be achieved for the above-entioned chroosoe which can serve as the basis of calculation for further optiization. As discussed above, the deterination of aintenance strategy of paveent network in each stage is a ultistage decision-aking proble in which each stage is interactional. he process has the following characters: at the beginning of each stage, optial decision of each stage is only related to road network condition, but not related to the decision of the previous stage in which the road network condition is known. he cobination of optial decision in each stage is the best strategy for the planning period. On the other hand, the process of decision-aking is a ultistage chain in which each stage does not influence the following stages. herefore, the dynaic prograing is a powerful tool to solve this kind of proble Calculation Process of DP. According to the characteristics of DP and project distribution odel, the process of DP in project distribution can be described as shown in Figure 2. In Figure 2, the whole planning period of years can be divided into stages. Vector Y t (paveent network condition in year t,budget of year t) represents initial condition of each stage. he decision can be ade according to U t (Y t ) = U t (paveent network condition in year t, budget of year t). Finally the axial aintenance benefit, benefit in this stage, can be calculated based on the U t (Y t ) and the condition transition function Y t +1=F(Y t,u t (Y t )) he Calculation of Maxial Maintenance Benefit (Deterination of U t (Y t )). Calculation odel of axial aintenance benefit U t (Y t ) ineachyearcanbedevisedas follows: ax s.t j=1 n (X ij B ij ) (16) n (X ij C ij ) y (17) X ij 1, X ij =0or 1, (18) where n is the nuber of projects; is the nuber of treatents for each project; y is the total aintenance budget; X ij =1(or 0) eans treatent j is selected (or not) in project i. he following is a recursion equation built up by the DP ethod, in which there are 3 treatents for each project. he cost of each treatent is defined as integer and described fro sall to large as w i1, w i2,andw i3. At the sae tie, the corresponding benefit is V i1, V i2,andv i3,respectively: f (i, ) f (i+1,) 0<<w i1 ax (f (i+1,), f(i+1, w i1 )+V i1 ) w i1 <<w i2 { ax (f (i+1,),f(i+1, w = i1 ) +V i1,f(i+1, w i2 )+V i2 ) w i2 <<w i3 ax (f (i+1,),f(i+1, w i1 ) +V { i1,f(i+1, w i2 )+V i2, { f(i+1, w i3 )+V i3 ) X i >>w i3. (19)

6 6 Matheatical Probles in Engineering o. of segents to be treated Length of road to be treated () able 1: Results of new ethod. Area of paveent to be treated ( 2 ) otal aintenance cost ( 10,000) Average PCI of paveent network Average PII of paveent network (inial) (inial) otal hen constraints can be forulated as follows: 0 < w n1 { V f (n, ) = n1 w n2 <<w n1 ax (V { n1, V n2 ) w n2 <<w n3 { ax (V n1, V n2, V n3 ) > w n3. (20) he function of f(i, ) canbesolvedthroughthebackward deduction ethod and the optial solution for each stage can be achieved through the backtracking ethod. 5. Model Verification through Case Study he odel and algorith are tested and verified through a case study with the data collected fro asphalt paveents of 12 districts in Shanghai in he planning period lasts 10 years, the total nuber of sections surveyed is 867, the total road length is 238,332 eters, and the total area is 3,093,293 square eters. (1) Paraeters of funds are provided as follows: (a) total budget: 160 illion RMB, (b) iniu investent for each year: 13 illion RMB, (c) axiu investent for each year: 17 illion RMB, (d) the step change of investent for each year: 100 thousand RMB. (2) Paraeters of GA are provided as follows: (a) reproduction rate: 0.7, (b) utation rate: 0.05, (c) crossover rate: 0.3. he calculation software is prograed with Microsoft Visual Basic 6.0. he odel proposed and developed in this paper (hereafter referred to as new ethod ) is copared Maintenance cost ( 10,000) ew ethod Conventional ethod Figure 3: Maintenance cost coparison of road network. with the current paveent aintenance ethod (hereafter referred to as conventional ethod ), in which aintenance wouldbeconductedoncethepciislowerthan75,currently adopted by paveent anageent agencies in Shanghai. he calculation results are listed in ables 1 and 2. In the tables, the aintenance benefit of the paveent network is indicated with PII (Paveent Iproveent Index). PII ainly considers restoration of paveent indicators, interval tie between two aintenance services, and social ipact. he coparison of the data in ables 1 and 2 is shown in Figures 3 and 4. Based on the case study results, the following observations can be drawn. (1) he total aintenance benefit increases (the PII iproves fro to after optiization with the new ethod), while the average PCI of road network in the planning period decreases after adopting the new ethod. At the sae tie, the total cost in the planning period is significantly reduced (the total cost decreases fro to 15760, a drop

7 Matheatical Probles in Engineering 7 o. of segents to be treated Length of road to be treated () able 2: Results of conventional ethod. Area of paveent to be treated ( 2 ) otal aintenance cost ( 10,000) Average PCI of paveent network Average PII of paveent network (inial) (axial) (axial) (inial) otal (average) Paveent network average PCI ew ethod Conventional ethod Figure 4: Average PCI coparison of road network. PII ( 10,000) ew ethod Conventional ethod Figure 5: PII coparison. of 18%). hat is, paveent anageent agencies will achieve better aintenance benefit through spending fewer resources with the new ethod. (2) As shown in Figure 3, through the conventional ethod, budget deand in each planning year is significantly different (i.e., the axial budget deand is 28,200,000 in the fourth year, and the inial deand is 5,590,000 in the tenth year. he variance of the investent in each year is up to 2447). hat seriously violates the rule of a stable financial plan of local finance departent. hrough the new ethod, the budget of each planning year is well-controlled. he difference of each year s budget effectively decreases (i.e. the axial budget deand is 17,000,000 in the fifth and ninth year, and the inial deand is 13,200,000 in the first year. he variance of the investent in each year is lowered down to 387). It well reflects the rule of a stable financial plan of finance departent. (3) Figures 4 and 5 indicate that the PCI and the PII of road network in each year, siilar to the aintenance budget, change rearkably when adopting the conventional ethod. On the contrary, the new ethod can significantly itigate the variation of PCI and PII. he variance of PCI decreases fro 8.3 to 4.5 and the variance of PII decreases fro to In general, paveent anageent agencies will get ore aintenance benefit with fewer funds through the new odel. Meanwhile, the aount of investent and the variability of paveent network PCI in each planning year can be effectively controlled to eet the rule of a stable project plan for paveent anageent agencies. Consequently, the newodelandthesolvingethodproposedinthispaperare effective and practical for the infrastructure anageent. 6. Conclusion Choosing a proper odel and algorith is very critical for ethods of atheatic optiization. o solve the probles of poor calculation speed and practical application liitation of the conventional atheatic ethod, a two-stage odel consisting of the budget allocation odel and the project distribution odel is developed in this paper. In addition,

8 8 Matheatical Probles in Engineering the odel is solved through dynaic prograing and genetic algorith. Furtherore, it is verified to be effective by practical data in a case study. he findings indicate that the new optiization ethod can provide a satisfactory and reasonable aintenance schedule for transportation infrastructure aintenance agencies whose routine anageent will benefit fro the newly proposed odel. Conflict of Interests he authors declare that there is no conflict of interests regarding the publication of this paper. References [1]. Hegazy, Optiization of resource allocation and leveling using genetic algoriths, Construction Engineering and Manageent,vol.125,no.3,pp ,1999. [2]. Bandara and M. Gunaratne, Current and future paveent aintenance prioritization based on rapid visual condition evaluation, ransportation Engineering, vol.127,no. 2, pp , [3].F.Fwa,K.C.Sinha,andJ.D..Riverson, Highwayroutine aintenance prograing at network level, ransportation Engineering, vol. 114, no. 5, pp , [4] R. Hass, W. R. Hudson, and J. Zaniewski, Modern Paveent Manageent, Krieger, Malabar, Fla, USA, [5] J. J. Hajek and W. A. Phang, Prioritization and optiization paveent of preservation treatents, ransportation Research Record 1216, ational Research Council, Washington, DC, USA, [6] P. B. Still, A Paveent anageent syste for UK roads, Highways and ransportation,vol.37,no.2,pp.6 9,1990. [7] R. Haas, A ethod for integrated priority prograing and budget level analysis for paveent aintenance and rehabilitation, in Proceedings of the 5th International Conference on the Structure Design of Asphalt Paveents,vol.1,pp ,Delft, he etherlands, [8] R. Dekker, R. E. Wildean, and F. A. van der Duyn Schouten, A review of ulti-coponent aintenance odels with econoic dependence, Matheatical Methods of Operations Research,vol.45,no.3,pp ,1997. [9] P. Ullidtz, A danish paveent anageent syste, in Proceedings of the APMC, vol. 3, pp , oronto, Canada, [10] D. Mrawira, W. J. Welch, M. Schonlau, and R. Haas, Sensitivity analysis of coputer odels: World bank HDM-III odel, ransportation Engineering, vol.125,no.5,pp , [11] S. Chi, J. Hwang, M. Arellano, Z. Zhang, and M. Murphy, Developent of network-level project screening ethods supporting the 4-year paveent anageent plan in exas, Manageent in Engineering, vol.29,no.4,pp , [12] R. Kher and W. D. Cook, PARS: the MC ode for prograing and financial planning in paveent rehabilitation, in Proceedingsof the ational Arts Marketing Project Conference (AMPC 85), vol. 3, pp , oronto, Canada, [13] B. Colucci-Rios and K. C. Sinha, An optial paveent anageent syste at the network level, in Proceedings of the 1st orth Aerican Paveent Manageent Conference (APMC 85), vol. 3, pp , oronto, Canada, [14] R. F. Caricheal, Ipleenting paveent anageent systeattheindianadepartentofhighways, inproceedings of the 6th International Conference on Structure Design of Asphalt Paveents, vol. 1, pp , Ann Arbor, Mich, USA, [15] K. Golabi, R. B. Kulkarni, and G. B. Way, A statewide paveent anageent syste, Interfaces,vol.12,no.6,pp.5 21,1982. [16] K. J. Feighan, M. Y. Shahin, and K. C. Sinha, A dynaic prograing approach to optiization for paveent anageent systes, in Proceedings of Second orth Aerican Conference on Managing Paveents (SACMP 87), vol.2,pp , oronto, Canada, [17] B. Liu and Z. Yao, Optiization ethod in network paveent anageent syste, China Highway and ransport, vol. 7, no. 3, pp. 1 9, [18] P. Zou, On optiization ethod of paveent aintenance and rehabilitation prograing, China Highway and ransport,vol.8,no.2,pp.1 8,1995. [19] C. unoo, Optiization of Paveent Maintenance and Rehabilitation prograing Using Shuffled Coplex Evolution Algorith, Florida International Univesrity, Miai, Fla, USA, [20] W.. Chan,. F. Fwa, and J. Y. an, Optial fund-allocation analysis for ultidistrict highway agencies, Infrastructure Systes,vol.9,no.4,pp ,2003. [21]. F. Fwa, W.. Chan, and K. Z. Hoque, Multiobjective optiization for paveent aintenance prograing, ransportation Engineering,vol.126,no.5,pp ,2000. [22] A. Ferreira, A. Antunes, and L. Picado-Santos, Probabilistic segent-linked paveent anageent optiization odel, ASCE ransportation Engineering,vol.128,no.6,pp , 2002.

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