Highway Vertical Alignment Analysis by Dynamic Programming

Transcription

1 TRANSPORTATION RESEARCH RECORD 1239 Highway Vertical Alignment Analysis by Dynamic Programming T. F. FwA In this paper, an optimization program developed to prodce an optimm vertical highway profile for a preselected horizontal alignment is described. The aim of the program was to provide local highway engineers and planners in Singapore with a practical aid for highway geometric design and location analysis. A dynamic programming formlation was adopted to minimize the overall cost, which incldes earthwork, land acqisition, materials, and vehicle operating costs. The program's versatility allows it to model most of the cost items important in highway constrction and can also be sed to provide qick information on the effects of different geometric design parameters on project costs and to determine the relative importance of different cost categories. A real-life nmerical example is presented to illstrate program application. In the process of constrcting new highways and relocating existing highways, a highway location stdy is sally performed to position the proposed highway within a corridor according to engineering, social, economic, and environmental considerations. Sch an analysis typically involves a nmber of conflicting constrction and operational reqirements. A highway designer is reqired to achieve an economical design that is adeqate operationally. Vertical alignment design, a sbproblem of the broad threedimensional highway location problem, is of great practical significance becase its soltion has direct bearing on highway constrction and vehicle operating costs. Vertical alignment design has attracted mch research interest in the last two decades, particlarly in the area of highway vertical alignment optimization. Comparisons with conventional manal designs have shown that an average savings of 15 percent of constrction costs cold be achieved by sing compter-derived optimm alignments (J,2). The optimization techniqes that have been sed for vertical road alignment stdies inclde linear programming (2), qadratic programming (3), dynamic programming (4), and relaxation methods (5). Methods of search-direct search, random search, and gradient search (6,7)-have also been sed. In this paper, an optimization program that was developed to prodce a preliminary vertical highway profile for a preselected horizontal alignment is described. The aim was to provide local highway engineers and planners in Singapore with a practical aid for highway geometric design and location analysis. A dynamic programming formlation was adopted to minimize the overall cost, which inclded earthwork, land acqisition, materials, and vehicle operating costs. Constraints were careflly selected and modeled to sit local conditions and design and constrction practices. Parametric and Department of Civil Engineering, National University of Singapore, Kent Ridge, Singapore 511. sensitivity stdies were condcted, based on a real-life example, to illstrate the salient featres of the program. MODELING OF ALIGNMENT PROBLEM Three basic elements can be identified in modeling the problem of vertical alignment optimization of a highway: (a) representation of inpt grond profile and otpt road alignment, (b) formlation of objective fnction, and ( c) definition of constrction constraints and operational reqirements. Detailed descriptions of these aspects of the optimization program are inclded below. Grond and Road Alignment The natral grond profile is established along the central axis of the proposed highway, the horizontal alignment of which has been predetermined and fixed. This profile is inpt as a series of straight-line segments, each spanning an eqal horizontal distance measred along the central axis of the highway. These eqal horizontal intervals, which are determined by the grid size selected for a particlar problem, also determine the line segment intervals over which the otpt road alignment will be represented. Smoothing of the piecewiselinear alignment by a design engineer is reqired to give a practical crved profile. The dynamic programming optimization process ses a set of vertical data grid lines spaced at the horizontal grid intervals mentioned above, as shown in Figre 1. Both the piecewiselinear inpt grond profile and otpt road alignment pass throgh one grid point each on each vertical grid line. The optimization algorithm determines one point on each grid line to represent the otpt alignment that satisfies specified constraints and yields the lowest overall cost. Objective Fnction The objective fnction is the overall cost that represents the sm of the following for cost components: (a) earthwork cost, (b) pavement cost, (c) land acqisition cost, and (d) vehicle operating cost. Earthwork Cost Earthwork cost may be divided into two major components: (a) cost of ctting when the compted road alignment level

2 2 Start Point x Grid Grid 1 2 l f f k Grond f Along Road Axis x Grid n f FIGURE 1 Data grid for vertical alignment optimization. x End Point is lower than the existing grond level and (b) cost of filling or embankment constrction when the reqired road alignment lies above grond level. These two cost components can be frther sbdivided into excavation or backfilling cost, transport cost, and roadbed preparation cost according to the size of project. Excavation and backfilling costs arc basically fnctions of the volme of earth involved, althogh the nit rate of excavation may vary with the depth of ct and the type of soil. Transport costs can be compted once the haling distance is known. Locations of dmping sites and borrow pits are reqired as inpt to the optimization program. Roadbed preparation costs vary with the type of soil pon which the proposed road is to be constrcted. For the prpose of ct- or fill-earth volme calclation, the cross sections of finished roadbed were simplified (see Figre 2). The cross-sectional area of ct or fill at any grid point is completely defined by three variables: (a) roadbed width, w; (b) ct or embankment slope represented, ; and (c) the difference between compted road alignment level and grond level, h. End area method was sed to compte the volme of ct or fill reqired between any two grid lines. Pavement Cost Pavement cost, compted from pavement thickness and nit costs of pavement materials sed, is basically a fnction of the qality of roadbed soil. In most highway projects, soil type and strength are fond to vary along the length of the rote. Pavement thickness reqirement is also likely to vary between fill and ct sections, as weil as among different depths of ct at a given point. Becase the type and strength of soil is a fnction of grid line location, as well as the relative vertical position of roadbed with respect to grond srface, it is necessary to compte nit pavement cost for each grid point. The pavement cost for each road segment between two vertical grid lines was compted by mltiplying the length of the road segment by the arithmetic mean of the nit pavement costs at the two grid points which defined the road segment. Lana Acqwswn Cose Land acqisition cost was defined as the prodct of nit area land cost mltiplied by the area of land reqirement for highway right-of-way. The nit cost of land may vary along the length of the proposed rote. Depending pon the type of constrction (ct or fill) and the natre of land development on the two sides of road, right-of-way width may also vary along the length of the rote. TRANSPORTATION RESEARCH RECORD 1239 I Road Width WI --rh---road Level _Ae '- I' "-ond,, I "IOiiStrclioo Width h -I--. Lc c f Ve for fill Note : e l -ve for ct FIGURE 2 Simplified cross section for earth volme comptation. When the nit land cost and width of right-of-way reqirement are known, the land acqisition cost at each grid point can be compted. Figre 2 shows that the right-of-way width at a grid point depends on the constrction width, b, at the grond level. The magnitde of width b is, in trn, related to the difference between road alignment and grond level. To obtain the land acqisition cost for any line segment, the arithmetic mean of land acqisition costs at the two corresponding grid points was compted, and the mean vale was mltiplied by the road segment length to calclate the desired land acqisition cost. Vehicle Operating Cost Two elements that inflence the cost incrred by road sers were considered-length and gradients. These factors can be conveniently combined into a single inpt variable expressed in terms of total ser cost/percent gradient/kilometer (or mile) of road length. This cost was compted based on a period eqal to the design service life in the highway. The total highway traffic volme expected over this period mst be estimated for compting the nit ser cost. Constraints The following constraints were inclded in the optimization program: (a) a maximm gradient, (b) crvatre reqirements, and (c) control points with fixed levels. The nmerical control vales for eat:h wnsiraini we1e based on the design standards and reqirements provided by the Singapore Pblic Works Department (8). Gradient control is a traffic operational reqirement to ensre smooth vehicle movement. The maximm allowable gradient is a fnction of design highway speed, as well as the types of vehicles inclded in the design traffic stream. The constraint on gradient can be dealt with in the following manner: IY, - Y,_,1 Gd where i = 1, 2, N Y; = road alignment level at grid line i; G = maximm allowable gradient; d = horizontal distance between two sccessive grid lines; and N = total nmber of grid intervals. Crvatre reqirements were achieved by controlling the magnitde of algebraic change in gradient between two consective line segments. Adopting the recommended practice (1)

3 Fwa of Singapore Pblic Works Department (8), the absolte vale of gradient change, g, which was derived from considerations of highway safety, aesthetics, and comfort of ride, were specified separately for crest and sag vertical crves as follows: For crest vertical crves: g :S: 425/(2S - L) when L :S: S g :S: 425L/S2 when L S For sag vertical crves: g :S: ( S)/(2S - L) when L :S: S g :S: ( S)L/S 2 where when L S g = absolte algebraic difference in gradient ( % ) ; L = length of vertical crve (m); and S = sight distance (m). The sight distance, S, is a fnction of design speed, vehicle type and roadway gradient. The allowable gradient change, g, wold therefore also vary with the geometric parameters selected by the designer. The third constraint, control points with fixed levels, is commonly encontered in actal highway design problems. The levels of the start and end points of a new highway are typically fixed. Intermediate fixed-level control points are needed where a new highway intersects existing roads. DYNAMIC PROGRAMMING FORMULATION The dynamic programming formlation for the vertical alignment problem is smmarized in this section. The objective was to minimize the total sm of selected costs incrred in the constrction of a new highway. The objective fnction was N- 1 Min L [C1(Uk) + C2(Uk) + C3(Uk) + C4(Uk)] (6) k O (2) (3) (4) (5) and vehicle operating costs, respectively, for the line segment k bonded by grid lines k and k + 1. Each of the for cost components is a fnction of the positions of grond levelf(x) and road alignment y(x), where x is the horizontal distance measred from the start point along the roadway central axis. The for cost components can be compted once the relative position of road alignment with respect to grond srface is known, as represented by hk = y (xk) - f (xk) in Figre 3. APPLICATION The work reported in this paper was a response to a need to provide local highway engineers and planners in Singapore with a practical aid for highway location analysis and preliminary geometric design. The program can be sed to provide qick soltions concerning the impacts of different geometric design parameters on project costs and to determine the relative importance of different cost categories. A real-life nmerical example is presented here to illstrate program application. Nmerical Example The optimization program was sed to condct a preliminary constrction cost analysis for a proposed for-lane highway connecting an indstrial town to a residential area with a poplation of 5,. The horizontal alignment of the proposed highway had been fixed. The highway wold shorten the one-way traveling distance between the two locations from 11.5 km to 5.9 km. Figre 4 shows the grond srface profile along the central axis of the proposed rote. The total horizontal length was 5875 m (19,275 ft). The cost data sed in the optimization analysis are smmarized in Table 1. The proposed road was located within government land reserved for highway constrction. Becase sfficient right-of-way width had been reserved, land acqisition constraints of cost and width became irrelevant and were not considered in this example. The following discssion presents the reslts of vertical alignment analysis in the fol- 3 Sbject to the following constraints: Slope k =, 1, "N - 1 (7) Crvatre System eqations k=,1,- -,N-1 (8) k =, 1, -, N - 1 (9) 2 i-1 i i+l N I al Representation of Grond y (x l Road Alignment For bondary conditions, Y and Y N are prescribed. All the variables and symbols in the constraint eqations 7, 8, and 9 have been explained in the preceding section and are graphically represented in Figre 3. C 1 (Uk), Ci( Uk), C/Uk), and C 4 (Uk) represent earthwork, pavement, land acqisition, FIGURE 3 I b J Defi nition of Symbols Dynamic programming model.

4 4 TRANSPORTATION RESEARCH RECORD BO Grond 7 e 6. 5 t:; Ci 4 ;; :.;: FIGURE 4 Grond profile inpt for example problem. Horizontal Distance (x ml TABLE 1 COST DATA AND CONSTRAINT CONDITIONS FOR EXAMPLE PROBLEM Parameter Vertical gradient control Vertical crvatre Filling cost Ctting costs Depth< 1.5 m m m m m 7.5 m Pavement cost Vehicle operating costs Vale 4% maximm (See eqations 2-5) S$1./m 3 S$1./m 3 S$14.11/m 3 S$18.2/m 3 S$25./m 3 S$3./m 3 S$5./m 3 S$8/m 2 Tangent rnning costs on grades recommended by AASHTO (9) for passenger cars on mltilane highways. NOTE: One Singapore dollar (S$) is approximately.5 U.S. dollar. lowing aspects: (a) sensitivity of reslts in terms of total cost with respect to horizontal and vertical grid sizes; (b) effect of maximm e;rnclint constraint on optimm total cost; (c) effect n.f t "":.Y't.,..",,_,,., -.;_ _ - "' - ' -.,, \._ " aa-.a,., Y LLU.l.V ""''-'.l.ll.1. VJ Vl.I. vpijjllujjl LVl,.UJ \.,U;:')l, \UJ effect of earth filling cost on optimm total cost; ( e) effect of earth ctting cost on optimm total cost; and (f) effect of vehicle operating cost on optimm total cost. Following the local practice of project cost comptation, the first five analyses did not consider vehicle operating cost. The sixth analysis was performed to give an indication of the effect of inclding vehicle operating cost. Becase no information was available on the magnitde of vehicle operating costs in Singapore, vales for passenger cars recommended in an AASHTO Manal (9) were sed as examples only. Only tangent rnning costs on grades were inclded in the analysis. Selection of Horizontal and Vertical Grid Intervals The inpt grond profile and the final compted road alignment are each represented by a series of line segments passing throgh one grid point on each vertical grid line. As shown by the data grid in Figre 1, the accracy of the comptation depends to a great extent on the horizontal grid interval, d, as well as the length of each division, v, on the vertical grid lines. A sensitivity analysis of the compted total cost with respect to the grid variables v and d was performed. The reslts of this analysis are presented in Tables 2 and 3. Data in Table 2 indicate that with a horizontal grid interval, d, fixed at 62.5 m, sfficiently accrate soltion cold be obtained with vertical division spacing v eqal to.5 m or less. When fixing v at.25 m (Table 3), acceptable reslts were obtained for vales of d eqal to 62.5 m or less. Analyses condcted on highways in other regions of Singapore also showed similar ranges of acceptability for horizontal and vertical grid intervals. For Singapore terrain it was, 1 (" 1 't, I t t, ' " ' l.ucj.cluj.c, J.C\,;UUlJJlCJJUCU UJdl LUC UUllLUJJlC.U djju Vt;JLH:UJ g11 intervals be not more than 6 m and.5 m, respectively. It is of importance to impose another control on the vales of horizontal and vertical grid intervals selected. The vales of d and v sed mst be sch that their ratio (vld) is less than the maximm gradient constraint specified. The reslts in Tables 2 and 3 indicated that a (vld) ratio of less than 1. percent wold provide an acceptable soltion for a maximm vertical gradient control of 4 percent. Becase the allowable

5 Fwa 5 TABLE 2 EFFECT OF VERTICAL GRID DIVISION ON COST ANALYSIS Vertical Division, Horizontal Interval, Total Cost v(m) d(m) (S$) (vld) x 1% Remarks ,16, No good ,3, Marginal ,244,489.8 Satisfactory ,239,328.4 Satisfactory NOTE: Inpt data and constraint conditions are given in Table 1. One Singapore dollar (S$) is approximately.5 U.S. dollar. TABLE 3 EFFECT OF HORIZONTAL GRID INTERVAL ON COST ANALYSIS Horizontal Interval Vertical Division Total Cost d(m) v(m) (S$) (vld) x 1% Remarks ,97,72.1 No good ,421,626.2 No good ,239,328.4 Satisfactory ,423,647.8 Satisfactory NOTE: Inpt data and constraint conditions are given in Table 1. One Singapore dollar (S$) is approximately.5 U.S. dollar. range of maximm vertical gradient is between 4 and 8 percent in Singapore (8), it has been sggested that a (v/d) ratio of not more than 1 percent be sed. Effect of Maximm Gradient Constraint A highway designed with a stricter gradient control-by imposing smaller vales of maximm vertical gradient constraint on a hilly terrain-allows for a more comfortable ride and a smoother flow of traffic. Savings on vehicle operating costs cold also be achieved by sing stricter gradient control, a direct conseqence of which is the increase in highway constrction cost de primarily to the increased earthworks reqired. Cost comptation of highway constrction for different maximm gradient controls cold therefore provide sefl information for a benefit/cost analysis to aid in the selection of gradient control in highway planning and geometric design. Figre 5 shows the variation of constrction cost (exclding vehicle operating costs) with different vales of maximm allowable vertical gradient. As expected, highway constrction cost decreased as higher maximm gradient was allowed. It shold be noted that the compted road profiles were different for different vales of gradient control imposed. Figre 6 shows the profiles for maximm gradient control of 4 percent and 6 percent. The 6-percent gradient profile conformed more closely to the grond srface, reslting in a saving on the amont of earthwork ctting and filling. Althogh considerable changes in constrction costs were observed when vertical gradient control was changed in this example, it is clear that the sensitivity of compted constrction costs against gradient control is a fnction of grond terrain. The constrction cost of a highway to be bilt on a flat terrain wold not vary mch with changes in vertical gradient control V') = 14 -;;; '--' Note : c: Other problem.!:? 13 data are given in., ilems 2, 3, 4 and 5 c: of Table '--' :!,_ Allowable Max. Vertical Gradient (%) FIGURE 5 Effect of vertical gradient control on highway constrction cost. Effect of Minimm Crvatre Control Crvatre control was achieved by setting an pper limit on the vale of allowable gradient change, g, which is defined in eqations 2 throgh 5. Small g vales allow gradal changes of vertical road alignment, leading to design of highways with smooth riding profile. Constrction costs, however, tend to be higher with smaller g vales becase of the higher qantity of earthwork needed to satisfy the constraint.

6 6 TRANSPORTATION RESEARCH RECORD E c c:,. =.!:::! - c Max. Allowable Vertical Gradient = 6 % Max. Allowable Vertical /, Gradient = 4 "!. I /,.,,, / / 1 D Horizontal Distance Ix ml FIGURE 6 Effect of maximm vertical gradient control on highway alignment in example problem. Figre 8 shows the effect of crvatre control on the compted optimm road alignment. The alignment compted with a higher vale of minimm crvatre vale, g, conformed better to the natral grond profile. -V = '-' c: -:::; 16 "' c: Note: 11 I Other problem data are '-' given in items 1, 2, 3, and 5 of Table 1.!!! I- 121 Allowable gradient change, 15 g, is defined in Eqations 121 throgh 15). i Allowable Gradient Change, g 1%1 FIGURE 7 Effect of crvatre control on highway constrction cost. The relationship between constrction cost and crvatre control was stdied sing the example problem, and the reslts were plotted in Figre 7 for the range of g vales commonly sed in highway design. The changes in constrction costs cased by varying the vale of g were relatively small compared with the changes cased by varying vertical gradient control. This finding has also been fond tre for highways constrcted in other regions of Singapore. Effect of Earth Filling Costs Earth filling cost may vary with the type of filling material and haling distance, when more than one borrow pit is available for a constrction project. Constrction cost analyses for different borrow materials and haling distances are sefl for highway designers and project planners. Figre 9 shows the impact of varying earth filling costs for the example problem. The constrction cost increased with rising earth filling cost. The amont of increase for each nit rise in filling cost became lower at higher filling costs as the optimm alignment moved toward a profile with less and less volme of fill. Figre 1 illstrates this trend by comparing the optimm profiles for two cases of different earth filling costs. Effect of Earth Ctting Costs The effect of varying earth ctting costs on highway constrction cost is similar to the effect of earth filling costs in many aspects, as illstrated by Figres 11 and 12. The crve in Figre 11 tended to level off as ctting costs were increased, reflecting the smaller nit impact on constrction cost when ctting costs became higher. This trend cold be explained by the fact, as depicted in Figre 12, that the volme of ct diminished as ctting costs escalated.

7 Fwa Grond BO 7 6 :; 5 -.!!! = 4 :.e Max. Allowable Gradient Change = 1 % ', / Max. Allowable Gradient "'(' Change = 1 % \ \ \ Horizontal Distance ( x m I FIGURE 8 Effect of gradient change constraint on highway alignment in example problem. -(/) 22 <L = < -:;; 18 L) c:: :::: Note. Other problem data are given In _ - items 1, 2' 4 and 5 of Table 1 "' c:: L) 1 e Earth Filling Cost (S$/m3) FIGURE 9 Effect of earth filling cost on highway constrction cost. Effect of Vehicle Operating Costs Althogh vehicle operating costs have not been commonly considered in cost analyses for highway constrction projects in Singapore, examination of how the inclsion of vehicle operating costs wold affect engineering designs and project costs is instrctive. For the prpose of illstration, the cases analyzed in Figre 5 were recompted with the addition of vehicle operating costs defined in item 6 of Table 1. The reslts of this analysis were plotted in Figre 13. Vehicle operating cost increased as higher vales of vertical gradient were permitted in highway design, which had the effect of offsetting the savings in constrction cost cased by redced earthwork when higher vertical gradients were sed. As a reslt, in contrast to the trend in Figre 5, the total costs shown in Figre 13 were relatively insensitive to the changes in allowable maximm vertical gradient sed in highway geometric design. The impact on total highway cost of constrction prices and vehicle operating costs depends very mch on the relative magnitdes of these two major cost items. In contries where vehicle operating costs are very high, it is possible on certain terrain that the total highway cost might increase as a higher maximm vertical gradient is allowed in design.

8 8 TRANSPORTATION RESEARCH RECORD Grond 7 E. 6 c c. c:::i 5 4 -;;;. - c Earth Filling Cost = 5$ 7 /m3 Earth Filling Cost = S$13 /m Horizontal Distance ( x m I FIGURE 1 Effect of earth filling cost on highway in example problem V "" c:::i < c;; '- lb - 14 Other problem data are given in c. - items 1, 2, 3 and s of Table 1 "' c '- 1 FIGURE 11 cost. 4 (Ctting B Cost Schedle) I I Basic Ctting Cost Schedle in Table 1) Effect of earth ctting cost on highway constrction CONCLUSIONS Highway vertical alignment analysis is sefl for highway designers and planners in the location stdy, benefit/cost analysis, and geometric design of new highways. The dynamic programming formlation developed in Singapore has been fond to be a valable tool for local highway planners and designers. The formlation of the optimization program is simple in concept, yet has the versatility of modeling practically all of the cost items important in highway constrction. The nmerical example presented, based on a real-life grond profile and cost data, illstrated these points. ACKNOWLEDGMENTS This paper is based on research fnded by the National University of Singapore. Assistance received from the Pblic Works Department of Singapore is gratef!ly acknowledged.

9 Fwa Grond E c, -. 4 Cl ;; Earth Ctting Cost = Rates in Table 1 Earth Ctting Cost = 2 x (Rates in Table FIGURE Horizontal Distance (x SB. 75 m) Effect of earth ctting cost on highway alignment in example problem REFERENCES -"" = ; '-' ' , 12 Note: Other items Total Highway Cost problem data are given in 2, 3, ' and 5 of Table 1. OECD Road Research. Optimization of Road Alignment by the Use of Compters. Organisation of Economic Co-operation and Development, Paris, I. Shacke. Optimization of Vertical Alignment-A Case Stdy. Proc. Conference on Compter Systems in Highway Design, Copenhagen, 19n, pp V. Calogew. Compter-Aided Highway Design. Ph.D. thesis. Inslillc of Comp1er Science, Univer. i1y of London, P. A ntoniotti. APOLLON: A New Road Design Optimization Procedre. PTRC Symp. on Cost Models and Optimization in Road Location, Design, and Con!rction, London, 1969, pp J. L. Groboillot and L. Gall as. Optimization d'n Project Retier par Recherche de pls Cort Chemin dans n Graphe a Trois Dimensions. R.I.R.O., Vol. 2, Paris, 1967, pp R. W. Haymon. Optimization of Vertical Alignment for Highways lhrogh Mathemalical Programming. In f'liglzway Research Record36, HRB, National Research Concil, Wa hingion, D.C., 197, pp R. Robinson. A Frther Compter Model for Defining the Vertical Alignment of a Road: Programme VENUS II. TRRL Report LR 469, Department of the Environment, Crowthorne, Berkshire, U.K., Pblic Works Department. Manal for Geometric Design of Highways and Streets. Ministry of National Development, Singapore, A Manal on User Benefit Analyses of Highway and Bs-Transit Improvements. AASHTO, Washington, D.C., 1977, p. 51. ID ' Allowable Max. Vertical 6 Gradient ('/.I FIGURE 13 Effect of vehicle operating cost on total highway cost. The athor is solely responsible for the findings and conclsions of the research. Pblication of this paper sponsored by Committee on Geometric Design.