Estimation of Highway Maintenance Marginal Cost under Multiple Maintenance Activities

Transcription

1 Estimatio of Highway Maiteace Margial Cost uder Multiple Maiteace Activities Shadi B. Aai, Ph.D., P.E. 1 ; ad Samer M. Madaat, M.ASCE 2 Abstract: This paper focuses o the estimatio of highway maiteace margial costs. Highway maiteace margial cost has bee estimated i the literature usig the perpetual overlay idirect approach. This approach assumes that pavemet overlay costs domiate maiteace costs ad igores other maiteace activities. This paper focuses o two questios. First, is it acceptable to igore the less costly activities? Secod, if multiple maiteace activities are to be cosidered, is it acceptable to igore their iterdepedece? The results show that less costly maiteace activities caot be igored. Furthermore, if multiple activities are to be cosidered, their iterdepedece should be take ito accout. DOI: /ASCETE CE Database subject headigs: Highways ad roads; Maiteace costs; Estimate. Author keywords: Margial cost; Highway maiteace; Realistic strategy; Multiple activities. Itroductio A typical highway agecy uses multiple types of highway pavemet maiteace, rehabilitatio, ad recostructio MR&R activities. Ofte, highway agecies have MR&R strategies that are coditio resposive; i other words, a give MR&R activity is performed each time a give measure of pavemet coditio reaches a predetermied trigger level. Each type of activity ca be triggered by a differet type of pavemet coditio, such as ruttig, alligator crackig, or roughess. As a result of such a coditio-resposive strategy, a icrease i traffic loadig leads to a idirect icrease i the MR&R total cost icurred by the highway agecy. A icrease i traffic loadig accelerates pavemet deterioratio, which brigs forward all future MR&R activities which, i tur, icreases their preset value. The icrease i the MR&R total cost resultig from a additioal uit of traffic loadig e.g., a additioal equivalet sigle axle load ESAL is the MR&R margial cost. This is oly oe compoet of the margial social cost. Other compoets of the margial social cost iclude the private margial cost the icrease i ow vehicle operatig cost ad the highway user margial cost the icrease i the cost of subsequet vehicles as a result of worse pavemet coditio. This paper oly focuses o the MR&R margial cost compoet. From a equity ad ecoomic efficiecy poit of view, it is 1 Graduate Studet Researcher, Dept. of Civil ad Evirometal Egieerig, Istitute of Trasportatio Studies, Uiv. of Califoria Berkeley, 109 McLaughli Hall #1720, Berkeley, CA correspodig author. shadiaai@gmail.com 2 Professor, Dept. of Civil ad Evirometal Egieerig, Istitute of Trasportatio Studies, Uiv. of Califoria Berkeley, 109 McLaughli Hall #1720, Berkeley, CA madaat@ce. berkeley.edu Note. This mauscript was submitted o February 26, 2008; approved o February 15, 2010; published olie o February 26, Discussio period ope util March 1, 2011; separate discussios must be submitted for idividual papers. This paper is part of the Joural of Trasportatio Egieerig, Vol. 136, No. 10, October 1, ASCE, ISSN X/2010/ /$ desirable that each vehicle pays its margial social cost. There is growig iterest for implemetig margial cost pricig, which is a pricig strategy that sets price equal to the margial social cost, oe compoet of which is MR&R margial cost. The lack of accurate estimates of MR&R margial cost remais a importat obstacle to such implemetatio. Much of this iaccuracy stems from urealistic simplifyig assumptios, such as the assumptio that the oly MR&R activity used by a highway agecy is a overlay of costat itesity. Bruzelius 2004 surveyed the differet approaches used i the literature to estimate MR&R margial cost. Amog these approaches, the perpetual overlay idirect approach is the most detailed because it explicitly models the steps that take place betwee the icrease i traffic loadig ad the icrease i MR&R cost. Bruzelius 2004 referred to it as the idirect approach. This approach assumes that pavemet overlay resurfacig costs domiate MR&R costs, ad it igores all other MR&R costs. It uses a ifiite aalysis horizo ad assumes that a pavemet is overlaid as soo as it deteriorates to a predetermied trigger level Newbery 1988; Small et al It first relates chages i traffic loadig additioal ESAL to chages i overlay frequecy a additioal ESAL brigs forward the future overlays ad possibly chages i the overlay itesity thicker overlays i aticipatio of higher traffic loadigs i the future. The, it relates these chages i overlay frequecy ad itesity to MR&R margial cost. Followig Small et al. 1989, Vitaliao ad Held 1990, ad Lidberg 2002, a additioal ESAL is defied as a evet that recurs aually, ad the MR&R margial cost is defied as the chage i the aualized cost of future overlays, as a result of icreasig the traffic loadig by 1 ESAL this year ad every year i the future. After the icrease, all years have the same aual traffic loadig, which exceeds the curret aual traffic loadig by 1. This is referred to as the recurrig additioal ESAL. The perpetual overlay idirect approach icludes studies by Newbery 1988, Small et al. 1989, Vitaliao ad Held 1990, Trasportatio Research Board 1996, Lidberg 2002, ad Haraldsso 2007a. The basic formulatio i the state-of-the-art proceeds as follows. Cosider oe lae of a flexible pavemet JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010 / 863

2 Fig. 1. Future MR&R costs with oe type of activity sectio of a highway. Let costat L, such that L0, be the aual traffic loadig for this sectio ESAL/year. Also, cosider a highway agecy that uses a simple MR&R strategy with oly oe type of MR&R activity, amely, a overlay of costat itesity that is triggered by a specific pavemet performace measure, M. The pavemet sectio receives a overlay each time M reaches trigger level M f. Assume that pavemet deterioratio ad improvemet are determiistic. Let X, such that X0, be the umber of ESALs to failure for this pavemet sectio. Let T year be the overlay life, i.e., the time betwee two cosecutive overlays T = X L Although Eq. 1 is derived for the case of oe lae, it ca also be applied to the case of a highway sectio with multiple laes, provided that all laes are oly overlaid at the same time, i which case these laes ca be treated as a system of laes. Whe Eq. 1 is used for a system, X ad L should iclude the combied umber of ESALs for all laes. The exact defiitio of failure for this system depeds o the highway agecy for example, the agecy might overlay the system each time ay lae fails, ad it will affect the value of X. Some studies take weatherig or agig ito accout Newbery 1988; Small et al. 1989; Vitaliao ad Held Weatherig is the additioal deterioratio resultig from the passage of time ad the climate. However, we show the basic formulatio used by previous studies. Uder costat aual traffic loadig L, the values of X ad T deped o the uderlyig pavemet structure, the climate, ad the value of the trigger level M f. These three factors are held costat. Let U be the uit cost $/mile for a overlay. The value of U should be cosistet with the values of X ad L; for example, if X ad L are defied for a system of laes, the U should be the uit cost for overlayig all laes. Fig. 1 shows the cash flow diagram for all future overlays. Preset time is time 0. Let r, such that r0, be the discout rate per aum. Let V be the preset value of all future overlays $/mile. Usig cotiuous discoutig V = U exp r T j = U exp r T j exp r T = U 1 exp r T The third equality i Eq. 2 comes from the assumptio that r T is strictly egative ad fiite. As a result of this assumptio, 0exp r T1. Therefore, the ifiite geometric series coverges. Eq. 2 ca be writte as 1 2 U V = 3 expr T 1 Usig cotiuous discoutig, the aualized cost of all future MR&R activities equals e r 1 V. We use cotiuous discoutig for aualizig V i order to be cosistet with its use for expressig V. Note that this approach diverges from some studies that have used a mixture of cotiuous discoutig for expressig V ad aual discoutig for aualizig V, i.e., r V Lidberg 2002; Small et al. 1989; Vitaliao ad Held The additioal ESAL is defied as a recurrig additioal ESAL Lidberg 2002; Small et al. 1989; Vitaliao ad Held Eq. 4 gives the defiitio of C simple, the MR&R margial cost $/ESAL/mile resultig from the MR&R strategy that uses oly oe type of activity C simple ª der 1 V = e r 1 dv dl dt dt 4 dl This expressio for MR&R margial cost implicitly assumes that the icrease i traffic loadig at time 0 takes place immediately followig a MR&R activity. This is clear from Fig. 1 ad Eq. 2. This assumptio is acceptable for comparative aalysis. Usig Eq. 3 Usig Eq. 1 dv dt = r U expr T expr T 1 2 dt dl = X L 2 = T L The, usig Eqs. 4 6 C simple = er 1r U T expr T L expr T 1 2, where T = X/L 7 Eq. 7 is essetially a simplified versio of the equatio of Small et al Eq. 2 9b with m=0 ad Vitaliao ad Held 1990 Eq. 8 with =1 or m=0 whe the effect of weatherig, which is icluded i these works, is igored. Recet research has show that a strategy i which a pavemet receives a overlay of costat itesity every time its coditio deteriorates to a costat, predetermied trigger level is optimal amog strategies for which a sigle activity is used for both the fiite horizo problem Ouyag ad Madaat 2006 ad the ifiite horizo problem Li ad Madaat Still, the assumptio made by the perpetual overlay idirect approach that the oly MR&R activity used by a highway agecy is a overlay of costat itesity is questioable. I reality, a highway agecy uses differet types of MR&R activities, such as pothole repairs, patchig, sealig, thi overlays, regular overlays, ad recostructio, ad it uses differet triggers for differet activities. Sice each highway agecy has its ow MR&R strategy, it is importat to take ito accout this strategy whe determiig MR&R margial cost. It should be oted that some studies have estimated MR&R margial cost uder multiple types of MR&R activities, but they have used approaches other tha the perpetual overlay idirect approach. For example, Li et al. 2001, Lik 2002, ad Haraldsso 2007b used the ecoometric approach. I the ecoometric approach, a pavemet MR&R total cost fuctio is estimated usig ecoometric techiques, ad the the pavemet MR&R margial cost is determied from this total cost fuctio. The MR&R total cost fuctio icludes idepedet variables / JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010

3 such as traffic, road geometry, pavemet structure, ad climate. The ecoometric approach does ot explicitly model MR&R strategies used by a highway agecy. This paper improves the estimatio of MR&R margial cost usig the perpetual overlay idirect approach by relaxig the assumptio that a highway agecy uses oly oe type of MR&R activity. It presets a methodology to estimate MR&R margial cost takig ito accout the pavemet maagemet strategies used i practice. The paper asks two questios. First, is it acceptable to igore the less costly MR&R activities? Secod, if multiple MR&R activities are to be cosidered, is it acceptable to igore their iterdepedece? I order to aswer these questios i a ituitive way, this study makes some simplifyig assumptios of its ow. For example, as the ext sectio explais, oly two activities are cosidered, ad specific pavemet deterioratio models ad pavemet coditio trigger levels are replaced by the umber of ESALs to failure. Methodology Cosider oe lae of a flexible pavemet sectio of a roadway. Flexible pavemets udergo both ruttig ad crackig. Also, cosider a highway agecy that uses a MR&R strategy with two activities C ad R, which are triggered by crackig ad ruttig, respectively. Activity C is performed each time alligator crackig reaches trigger level C f %, ad it cosists of patchig. Activity R is performed each time ruttig reaches trigger level R f mm, ad it cosists of levelig ad overlayig. The iterdepedece betwee the two activities will be take ito accout. Assume that Activity C improves crackig but has egligible effect o ruttig; however, activity R improves both ruttig ad crackig. Assume that pavemet deterioratio ad improvemet are determiistic. Assume that ESAL is the appropriate deterioratio equivalece factor for both activity types C ad R. Let costat L, such that L0, be the aual traffic loadig ESAL/year. Let X C ad X R, both strictly positive, deote the umber of ESALs to failure for Activities C ad R, respectively, for a give pavemet sectio. The values of X C ad X R deped o the pavemet structure, the climate, ad the values of the trigger levels C f ad R f. Assume that Activity C is performed more frequetly tha Activity R, i.e., X C X R. Let T C year be the time betwee two cosecutive type-c activities that do ot have a type-r activity betwee them T C = X C 8 L Let T R year be the time betwee ay two cosecutive type-r activities T R = X R 9 L We igore the effect of weatherig, so X C ad X R are idepedet of L. To accout for weatherig, we would have to use specific deterioratio models. For example, i order to take weatherig ito accout, Small et al used a particular deterioratio model, ad they derived the followig for oe type of MR&R activity: T=X 0 exp m T/L, where m=evirometal coefficiet they used m=0.04, ad X 0 =umber of ESALs to failure uder coditios of egligible weatherig i.e., whe L, so T 0. The actual umber of ESALs to failure that they derived, X=X 0 exp m T, depeds o L the smaller the L, the smaller the X. By igorig weatherig, we keep the methodology geeral ad simple, ad it becomes easier to gai ituitio about the effect of icludig multiple activities. Let be the umber of times type-c activity is performed betwee ay two cosecutive type-r activities. It is a iteger, so floor divisio is used i Eq. 10 = X R/L X C /L = X R 10 = T R T C Eq. 10 shows that does ot deped o L. Assume that T C is strictly less tha T R or, equivaletly, that X R is ot a multiple of X C. This assumptio avoids the situatio where both types of activities occur at the same time. If that situatio were to happe, ay reasoable highway agecy would choose to perform type-r activity oly at that time. More geerally, a highway agecy might decide to skip each type-c activity that precedes a type-r activity by a very short period, e.g., less tha 3 moths. Let U C ad U R be the uit costs $/mile for type-c ad type-r activities, respectively. Fig. 2 shows a possible cash flow diagram for all future activities i this example, =2. Preset time is time 0. It is assumed that a type-r activity took place just before time 0. I order to simplify otatio, defie a cycle as the time period startig at, ad icludig, a type-r activity ad edig at, but ot icludig, the ext type-r activity. Let r, such that r0, be the discout rate per aum. Let U cycle be the equivalet uit cost for a cycle, evaluated at the begiig of the cycle. Eq. 11 gives the expressio for it usig cotiuous discoutig. The example i Fig. 3 is equivalet to the oe i Fig. 2, i.e., they both have the same preset value at time 0 of all future costs U cycle = U R + U C exp r T C i 11 Let V be the preset value of all future type-c ad type-r activities $/mile But Fig. 2. Example of future MR&R costs =2 V = U C X C exp r T C i + U cycle exp r T R j Fig. 3. Equivalet MR&R costs for previous example =2 12 JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010 / 865

4 exp r T R j = exp r T R j = exp r T R 1 exp r T R 1 = 13 expr T R 1 The secod equality i Eq. 13 comes from the assumptio that r T R is strictly egative ad fiite. As a result of this assumptio, 0exp r T R 1. Therefore, the ifiite geometric series coverges. The, usig Eqs. 12 ad 13 U V = U C cycle exp r T C i + 14 expr T R 1 By substitutig the expressio for U cycle from Eq. 11 ito Eq. 14 V = U C exp r T C i + U R + U C exp r T C i expr T R 1 15 Usig the assumptio that T C is strictly less tha T R, a ifiitesimal chage i either T C or T R does ot chage the value of. The, usig Eq. 15, we obtai Simplifyig V V = U C r T C i exp r T C i T C = U C r Also, usig Eq. 15 V T R = Usig Eq. 8 Usig Eq. 9 U C r expr T R i exp r T C i 1 i exp r T C i expr T R expr T R 1 r expr T R U R + U C exp r T C i expr T R 1 2 dt C dl = X C L dt R dl = X R L 2 20 Suppose that the additioal ESAL is defied as a recurrig additioal ESAL Lidberg 2002; Small et al. 1989; Vitaliao ad Held Usig cotiuous discoutig, the aualized cost of all future MR&R activities equals e r 1 V. The, the MR&R margial cost $/ESAL/mile equals Table 1. Default Iput Values Used i Computatios Iput variable Descriptio/uits Default value X C ESALs to failure for activity C 200,000 X R ESALs to failure for activity R 500,000 L Aual traffic loadig ESAL/year 100,000 U C Uit cost for activity C $/mile 20,000 U R Uit cost for activity R $/mile 200,000 r Discout rate per aum 0.05 Usig the chai rule C = der 1 V dl = e r 1 dv dl C = e r 1 V dt C T C dl + V dt R T R d L where the derivatives o the right-had side are give by Eqs Istead of usig particular pavemet structure ad climate data, particular pavemet deterioratio models, ad particular maiteace strategies trigger levels ad itesities for activities C ad R, this study assumes a realistic MR&R strategy ad parametrically varies the iputs X C, X R, L, U C, U R, ad r. For each istace, the realistic margial cost, which takes ito accout both activities ad their iterdepedece, is computed ad compared with the margial cost estimates that take ito accout oly oe type of MR&R activity C orr ad with the margial cost estimate that takes ito accout both activities but igores their iterdepedece. Computatios Table 1 shows the default values for the iput variables used i the computatios. I order to uderstad the effect of the iput variables, each will be varied, while fixig the others to the values show i Table 1 uless otherwise oted. The followig four quatities will be computed: The realistic MR&R margial cost, as give by Eq. 22. The MR&R margial cost that assumes that oly activity C is performed i respose to crackig, as give by Eq. 7 with U=U C ad X=X C. Such a MR&R strategy leads to uacceptable levels of ruttig. The MR&R margial cost that assumes that oly activity R is performed i respose to ruttig, as give by Eq. 7 with U =U R ad X=X R. Such a MR&R strategy leads to uacceptable levels of crackig. The sum of the MR&R margial costs for the two sigleactivity strategies. This sum is expected to be larger tha the realistic MR&R margial cost sice it fails to take ito accout the beeficial effect of activity R o crackig. First, the effect of varyig the frequecies of Activities C ad R will be studied. These frequecies are affected by the values of L, X C, ad X R. Fig. 4 shows that as L icreases, all four margial cost estimates icrease asymptotically to costat values. It is easiest to uderstad this for the case of the MR&R margial cost for a sigle-activity strategy, which is give by Eq. 7. It ca be show that the asymptotic value equals 866 / JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010

5 Fig. 4. Varyig aual traffic loadigs Fig. 6. Varyig ESALs to failure for Activity C higher r=0.40 limc simple L = er 1 U r X 23 For small values of r, this asymptotic value is approximately U/X $/ESAL/mile, which Newbery 1988 called the average maiteace cost, ad Small et al p. 15 called the aïve MR&R margial cost. I Fig. 4, the differece betwee simple MC with R where MC stads for margial cost ad sum of simple MCs represets the importace of takig ito accout the less costly activity C, ad the differece betwee sum of simple MCs ad realistic MC represets the effect of iterdepedece betwee the two activities. Fig. 4 shows that as L icreases, the sizes of these two differeces icrease slightly. I other words, as L icreases, it becomes more importat to take ito accout the less costly activity ad the iterdepedece, but the chage i importace is small. Fig. 5 shows the effects of varyig X C. Whe X C icreases, the frequecy of Activity C decreases but the frequecy of Activity R is ot affected. As a result, the MR&R margial cost for the strategy that uses oly C decreases, the MR&R margial cost for the strategy that uses oly R does ot chage, ad the realistic MR&R margial cost decreases. The realistic MR&R margial cost has drops at the values of X C that are divisors of X R =500,000. For example, a drop occurs at X C =250,000. For values of X C slightly above 250,000, the highway agecy performs oly oe type-c activity betwee each pair of type-r activities. For values of X C slightly below 250,000, the highway agecy performs two type-c activities betwee each pair of type-r activities. I real life, a reasoable highway agecy would ot perform the secod type-c activity; istead, it would wait for the soo-to-come type-r activity. As aforemetioed, the model assumes that X C is ot a divisor of X R. Therefore, the realistic margial cost is ot defied for such values of X C. If we had allowed X C to be a divisor of X R, each type-r activity would coicide with the last type-c activity of the previous cycle, elimiatig the beeficial effect of activity R o crackig. As a result, the value of the realistic MR&R margial cost would coicide with the value of the sum of the simple margial costs the two curves touch, as show i Fig. 5. Although the realistic margial cost curve i Fig. 5 appears to have horizotal segmets, these segmets are, i fact, dowward slopig. The segmets are early horizotal sice each of them correspods to a costat value of i.e., costat umber of type-c activities per cycle. Chagig X C withi each segmet ca oly vary the times of the type-c activities by less tha T R or, to be more exact, by less tha T R / mius T R /+1. With small values of r r=0.05, this has a isigificat effect of the margial cost. However, as the value of r icreases, the slopes of these segmets become more proouced, as show i Fig. 6, which uses a urealistically high r=0.40 i order to clearly show the slope. Whe X C exceeds X R, i.e., whe Activity R becomes more frequet tha Activity C which violates oe of the model assumptios, Activity C is ever triggered ad therefore ever takes place. Fig. 7 shows the effect of varyig X R. Whe X R icreases, the frequecy of Activity R decreases but T C is ot affected. Asa Fig. 5. Varyig ESALs to failure for Activity C Fig. 7. Varyig ESALs to failure for Activity R JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010 / 867

6 Table 2. Proportioately Chagig Activity Uit Costs X C ESAL X R ESAL Iputs L ESAL/year U C $/mile Results Margial cost $/ESAL/mile U R $/mile r C ad R activities Oly C activity Oly R activity 200, , ,000 10, , , , ,000 20, , result, the MR&R margial cost for the strategy that uses oly R decreases, ad the MR&R margial cost for the strategy that uses oly C does ot chage. Furthermore, the realistic MR&R margial cost decreases as log as remais costat. The realistic MR&R margial cost has jumps at the values of X R that are multiples of X C =200,000 sice the umber of type-c activities per cycle icreases by 1 at such values. As aforemetioed, the model assumes that X R is ot a multiple of X C. Therefore, the realistic margial cost is ot defied for such values of X R. Next, the effects of varyig the activity uit costs ad discout rate are examied. Chagig both uit costs, U C ad U R,bythe same positive multiplicative factor simply chages the four margial cost quatities by that same factor. This fact ca be easily see from the equatios for C ad C simple ad is illustrated by a example i Table 2. Therefore, it is ot iterestig to proportioately chage both activity uit costs. Rather, it is more iterestig to vary the relative value of oe of them with respect to the other. Fig. 8 shows the effect of varyig U C. As the value of U C icreases the ratio of U C to U R icreases, the differece betwee the realistic MR&R margial cost ad the sum of the simple margial costs icreases; i other words, it becomes more importat to take ito accout the iterdepedece of the two MR&R activities. Whe U C exceeds U R, a highway agecy might decide to rely solely o type-r activities, which will be performed at itervals of legth T C i other words, crackig triggers activity R. The model does ot capture this possible strategy, which leads to a differet value of margial cost this value is give by the lie labeled MC with R triggered by crackig i Fig. 8. Therefore, the model should ot be used to estimate the realistic margial cost i cases where the highway agecy might use a differet MR&R strategy from what the model assumes. Fig. 9 shows the effect of r o the margial cost values. The differece betwee the realistic MR&R margial cost ad the sum of the simple margial costs is isesitive to r. However, the differece is sesitive to r for lower values of L, as Fig. 10 shows this example correspods to large values of T C ad T R. Coclusios I the highway maiteace cost literature, the perpetual overlay idirect approach is ofte used to estimate maiteace margial cost. This approach is based o the assumptio that a highway agecy oly uses oe type of MR&R activities, amely, overlays. This paper relaxes this assumptio by presetig a methodology for estimatig MR&R margial cost for a strategy with two MR&R activities, C ad R, which are carried out i respose to two differet measures of pavemet coditio. It is assumed that Activity C oly improves oe measure of pavemet coditio, while Activity R improves both. Activity C is assumed to be more frequet tha Activity R. Furthermore, Activity C is assumed to have a lower uit cost tha Activity R. Fig. 9. Varyig discout rates per aum Fig. 8. Varyig uit costs for Activity C Fig. 10. Varyig discout rates per aum lower L=10, / JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010

7 Our computatios show that the realistic MR&R margial cost estimates realistic MC are sigificatly higher tha the MR&R margial cost estimates that accout oly for the domiat Activity R simple MC with R. This differece becomes more sigificat whe Activity C becomes relatively more frequet or relatively more expesive. I other words, simple MC with R is a lower boud for realistic MC, a boud that is typically ot tight. Therefore, both activities should be take ito accout whe estimatig MR&R margial cost. The sum of the two simple estimates of MR&R margial cost that take ito accout oly oe activity sum of simple MCs does ot capture all the pavemet coditio improvemets resultig from Activity R. As a result, the sum of simple MCs is a upper boud for the realistic MC. This ca be uderstood ituitively. Each Activity R, which is performed i order to improve ruttig, has a positive effect o crackig, i that it reduces crackig that has accumulated sice the last type-c activity. As a result, the realistic MC is lower tha the sum of simple MCs. Furthermore, the loger this time period betwee a type-r activity ad the type-c activity immediately precedig it, the larger the differece betwee the sum of simple MCs ad the realistic MC. A factor that affects the tightess of this upper boud is the ratio of uit costs. As U C /U R icreases, the differece betwee the sum of simple MCs ad the realistic MC icreases, ad it becomes more importat to take ito accout the iterdepedece of the two activities. Now, the aswers to the two questios posed i the Itroductio of this paper ca be summarized. First, is it acceptable to igore the less costly MR&R activity? The simple MC with R uderestimates the realistic MC, ad the differece is ofte sigificat. Therefore, the less costly activity C should ot be igored. Secod, if both MR&R activities are to be cosidered, is it acceptable to igore their iterdepedece? Although the sum of simple MCs is ofte close to the realistic MC, it cosistetly overestimates it, ad the differece ca be sigificat i some cases. Therefore, the iterdepedece caot be igored either. Sice each highway agecy has a differet MR&R strategy, MR&R uit costs, ad pavemet deterioratio, it is difficult to geeralize the results. However, we preset a methodology, which ca be modified ad exteded i order to aalyze differet situatios. From a practical poit of view, the highway agecy should first fid the realistic MR&R margial cost, ad the check whether it makes a sigificat differece to igore less costly activity types or igore the iterdepedece betwee differet activity types. I other words, it should ot igore activities or iterdepedece uless it ca justify doig so for that specific situatio. I order to exted this study, future work might look at MR&R strategies that iclude more tha two activities, as well as more complex iterdepedece relatioships, such as partial improvemet. Furthermore, the effect of weatherig might be icluded. Fially, the writers have worked o relaxig two other assumptios made i the perpetual overlay approach. Oe assumptio is that pavemet deterioratio caused by a axle is proportioal to the fourth power of the axle load irrespective of the performace idicator used by the highway agecy to trigger maiteace. The other is that pavemet deterioratio is determiistic, ad as a result, the exact times of all future MR&R activities ca be exactly predicted Aai Ackowledgmets This paper was fuded by the Uiversity of Califoria Trasportatio Ceter UCTC through a faculty research grat to the secod writer. Notatio The followig symbols are used i this paper: C type of MR&R activity which is triggered by crackig; C MR&R margial cost $/ESAL/mile; C simple MR&R margial cost $/ESAL/mile for the case of sigle activity type; L aual traffic loadig ESAL/year; umber of times a type-c activity is performed betwee two cosecutive type-r activities; R type of MR&R activity which is triggered by ruttig; r discout rate per aum; T time i years betwee two cosecutive activities for the case of sigle activity type; T C time i years betwee ay two cosecutive type-c activities that do ot have a type-r activity betwee them; T R time i years betwee ay two cosecutive type-r activities; U uit cost for activity $/mile for the case of sigle activity type; U C uit cost for activity C $/mile; U cycle equivalet uit cost for a cycle evaluated at the begiig of the cycle $/mile; U R uit cost for activity R $/mile; V preset value of all future type-c ad type-r activities $/mile; X umber of ESALs to failure for the case of sigle activity type; X C umber of ESALs to failure for activity C; ad X R umber of ESALs to failure for activity R. Refereces Aai, S. B Revisitig the estimatio of highway maiteace margial cost. Ph.D. thesis, Uiv. of Califoria, Berkeley, Calif. Bruzelius, N Measurig the margial cost of road use A iteratioal survey. Meddelade 963A, Swedish Natioal Road ad Trasport Research Istitute VTI, Liköpig, Swede. Haraldsso, M. 2007a. The margial cost for pavemet reewal A duratio aalysis approach. Workig Papers No. 2007:8, Swedish Natioal Road ad Trasport Research Istitute VTI, Liköpig, Swede. Haraldsso, M. 2007b. Margial costs for road maiteace ad operatio A cost fuctio approach. Workig Papers No. 2007:7, Swedish Natioal Road ad Trasport Research Istitute VTI, Liköpig, Swede. Li, Y., ad Madaat, S. M A steady-state solutio for the optimal pavemet resurfacig problem. Trasp. Res., Part A: Policy Pract., 36, Li, Z., Siha, K. C., ad McCarthy, P. S Methodology to determie load- ad o-load-related shares of highway pavemet rehabilitatio expeditures. Trasp. Res. Rec., 1747, JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010 / 869

8 Lidberg, G Margial cost of road maiteace for heavy goods vehicles o Swedish roads. Aex A2 (versio 0.3) of deliverable 10: Ifrastructure cost case studies, Uificatio of Accouts ad Margial Costs for Trasport Efficiecy UNITE, fuded by 5th Framework RTD Programme, ITS, Uiv. of Leeds, Leeds, U.K. Lik, H Road ecoometrics: Case study o reewal costs of Germa motorways. Aex A1a of Deliverable 10 of UNITE, versio 1.1, fuded by 5th Framework RTD Programme, ITS, Uiv. of Leeds, Leeds, U.K. Newbery, D. M Road damage exteralities ad road user charges. Ecoometrica, 562, Ouyag, Y., ad Madaat, S. M A aalytical solutio for the fiite-horizo pavemet resurfacig plaig problem. Trasp. Res., Part B: Methodol., 409, Small, K. A., Wisto, C., ad Evas, C. A Road work: A ew highway pricig ad ivestmet policy, The Brookigs Istitutio, Washigto, D.C. Trasportatio Research Board Payig our way: Estimatig margial social costs of freight trasportatio. Special Rep. No. 246, Committee for Study of Public Policy for Surface Freight Trasportatio, TRB, Natioal Research Coucil, Washigto, D.C. Vitaliao, D. F., ad Held, J Margial cost road damage ad user charges. Quarterly Review of Ecoomics ad Busiess, 302, / JOURNAL OF TRANSPORTATION ENGINEERING ASCE / OCTOBER 2010