Safety Science 50 (2012) 1513–1521

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Safety Science

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Risk-optimal highway design: Methodology and case studies

Karim Ismail a,⇑, Tarek Sayed b

a Department of Civil and Environmental Engineering, Carleton University, 1125-3432 Colonel By Drive, Ottawa, ON, Canada K1S 5B6b Dept. of Civil Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4

a r t i c l e i n f o

Article history:Received 24 December 2010Received in revised form 23 December 2011Accepted 5 February 2012Available online 22 March 2012

Keywords:Highway geometric designReliability analysisOptimizationCross-section elements

0925-7535/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.ssci.2012.02.001

⇑ Corresponding author. Tel.: +1 613 520 2600 170E-mail addresses: karim_ismail@carleton.ca (K.

(T. Sayed).

a b s t r a c t

Highway geometric design in mountainous areas has been a typical challenge. The combination of shorthorizontal curves and restricted right-of-way is a common ground for contemplating design exception inBritish Columbia, Canada. In practice, collision modification factors (CMFs) are advocated as quantitativemeasures of changes in road features on safety. However, in many situations, there are no CMFs in theliterature to predict the safety impact of changing particular road features. An important example ofthese road features is sight distance restriction on horizontal curves. A mechanism for risk measurementhas been proposed in earlier work to assist designers in comparing the safety impact of different devia-tions from sight distance requirements. This paper attempts to answer the questions as to whether it ispossible to reduce overall risk and achieve consistency in such reduction without demanding wider right-of-way. This problem was formulated in a multi-objective optimization framework. Following this meth-odology, it was possible to achieve an average reduction in risk of 25% on the nine critical cross-sections.This reduction in risk was achieved without demanding wider right-of-way and without creating mea-surable increase in expected collision frequency due to independent re-dimensioning of different geo-metric elements. On theoretical grounds, this paper represents another step into the direction ofdeveloping fully probabilistic geometric design standards. On practical grounds, this paper provides animportant decision mechanism that enables the efficient use of available right-of-way for new highwayconstruction. Case studies in this paper have been applied on a major highway development in BritishColumba, Canada.

� 2012 Elsevier Ltd. All rights reserved.

1. Background

Many of the major highway developments in British Columbiaare in mountainous terrain. These developments involve new high-way construction and improvement to existing infrastructure. Inthis roadside environment, most highway developments are lo-cated in a constricted right-of-way. Therefore, the designer andthe decision maker are faced with the dilemma of meeting budgetconstraints or approving of geometric designs that involve, to var-ious degrees, some violation of, or exception to, standard require-ments. The most common contemplated design in these mountainterrain, and the main concerned exception of this paper, is avail-ability of sight distance. The feasible alternative to avoid this di-lemma is to impose some mobility restrictions. The latteralternative however does not typically meet the expectations ofthe road users, who aspire to use new infrastructure with en-hanced mobility as well as safety.

Geometric design guides, such as The American Association ofState Highway and Transportation Officials (AASHTO) Green Book

ll rights reserved.

9.Ismail), tsayed@civil.ubc.ca

(American Association of State Highway and Transportation Offi-cials, 2004) and the Canadian design guide by the TransportationAssociation of Canada (TAC,1999), provide deterministic standardsfor design requirements. Deterministic design is based on adoptingconservative percentile values for design parameters to compen-sate for any uncertainty of their values. Deterministic design stan-dards lack a quantitative measure of uncertainty or degrees ofbelief. For example, with respect to stopping sight distance (SSD)requirements, design requirements are mainly based on nearworst-case scenarios of stochastic design parameters. Furthermore,available sight distance (ASD), is calculated at the location alongthe highway alignment of minimum value, with no regard to theprecise failure mechanism that leads to collision. According todeterministic design, the domain of all possible designs is split intotwo discrete sets: acceptable (e.g. ASD � SSD P 0) and unacceptable(e.g. ASD � SSD < 0). The deterministic approach to highway geo-metric design suffers from two main shortcomings (Ismail andSayed, 2010):

1. Many design parameters, such as perception and brake reactiontime (PRT), driver eye height, maximum deceleration rate, andobject as well as design variables such as operating speed arestochastic in nature. According to current deterministic design

1514 K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521

standards, design parameters are typically selected at conserva-tive percentile values drawn from their respective distributions.The safety margin of the design output (e.g. percentage of alldesign outcomes with ASD � SSD < 0) is unknown. Further-more, minimum standard requirements are not known to targetan explicit safety margin.

2. When the granting of design exceptions is contemplated, geo-metric design guides provide little knowledge on the safetyimplications of deviating from standard requirements. Forexample, according to deterministic design standards, a slightviolation to sight distance requirement is tantamount to themost egregious visual obstruction – both are unacceptable.

This study suggests a methodology to address the two short-coming, The first shortcoming is addressed by representing designparameters as random variables with pre-specified probability dis-tributions. The methodology proposed in this study can also beadopted to address the second shortcoming. When roadside spatiallimitations restrict the ability to meet design requirements, thedecision-maker may rely on the probabilistic risk indicator andthe optimization methodology presented in this study to quantifyand mitigate the consequences of violating design requirements.This was also the practical motivation behind the case studies pre-sented in this paper. An alternative traditional approach is to relyon empirical relationships between dimensions of road elementsand collision frequency.

In order to predict the safety consequences of geometric dimen-sioning decision, collision modification factors (CMFs) have beenadvocated as quantitative measures of the safety impact associatedwith changes in road features or traffic control. A sizeable body ofliterature exists on quantifying safety implications of dimensionsof highway geometric elements, e.g. (Hadi et al., 1995; Miltonand Mannering, 1998; Abdel-Aty and Radwan, 2000; Harwoodet al., 2003; Bonneson et al., 2005; Sayed and de Leur, 2008).Despite the extensive development of CMFs in the literature, notall dimensioning decisions are known for fact to influence roadsafety. The lack of knowledge on the safety implication of dimen-sioning decisions may be attributable to data idiosyncrasies usedfor CMF development, limitations in collision data availability,compound effect of dimensioning scenarios, the previously men-tioned inadequacy in accounting for design uncertainty, or genuineabsence of significant safety impact. An important example ofthese road features is sight distance restriction on horizontalcurves. Despite the empirical evidence in favor of granting designexceptions, e.g. (Malyshkina and Mannering, 2010), issues of liabil-ity and lack of a unified decision support mechanism have beenunsettling for many practitioners and decision makers.

An earlier work related to this paper presented a methodologyfor measuring the risk of various dimensioning scenarios for anumber of case studies of new major highway development inBritish Columbia, Canada (Ismail and Sayed, 2010). Each case studyincluded a road segment with constrained roadside environmentthat restrict sight distance below standard stopping sight distancerequirements. The absence of CMF’s for restricted sight distance onhorizontal curves required a solution outside the literature ofCMFs.

This paper applies a probabilistic framework for highway geo-metric design. The question of whether a dimensioning designmeets the requirements is reformulated from a crisp separationbetween acceptable and unacceptable designs into a matter of de-gree of probability of meeting, or complying to, the design require-ments. The measurement of such probability of non-complying todesign requirements, Pnc, is performed using reliability theory.Reliability theory provides the analytical tools necessary for trac-ing the propagation of uncertainty throughout the design process,

beginning with probability distributions of design inputs until finaldesign outputs (Harr, 1987).

The probability term in the previous alternative formulationdoes not measure the safety implication in absolute terms, butrather in a relative term. Probabilistic estimation of system perfor-mance has been advocated as nominal value for comparing differ-ent design alternatives. It should be noted that the issue of whetherthe probabilistic measures of design limitation are related toempirical safety measures such as collision frequency, was not at-tempted in this study. It is the subject of an ongoing research at theUniversity of British Columbia as a continuation of this study.However, such relationship was explored by Faghri and Demetsky(1988) at rail-highway grade crossings. They found a positive cor-relation between probabilistic measures of design limitation andcollision frequency. In companion work submitted by authors(Ibrahim and Sayed, 2011), a positive empirical evidence wasfound for the relationship between Pnc and collision frequency. Thisclearly proves the practical significance of the proposed methodol-ogy for risk-based optimization.

Earlier work on the subject of this paper has provided a meansfor risk measurement for a number of critical design cases in a ma-jor highway development in British Columbia, Canada. However,earlier work does not provide guidance regarding the particulardimensioning of critical cross-sections. This paper presents anextension to the consultation work presented in earlier work(Ismail and Sayed, 2010). The objective of this paper is to presenta methodology for selecting the dimensions of cross-section ele-ments on highway segments with restricted sight distance. Thegeneral premise underlying the proposed dimensioning methodol-ogy is to reduce average risk and balance the risk in design for bothcarriageways while considering the safety impact in terms of colli-sion frequency. The proposed methodology was applied to the ear-lier case studies in order to produce re-dimensioned cross-sectionswith reduced and consistent risk levels. The first section of thispaper discusses previous work. Later sections discuss the dimen-sioning methodology, application on case studies of recent high-way development in British Columbia, summary of the findingsand a brief on future work.

2. Previous work

The development of probabilistic highway design has beenadvocated in several studies, notable of which are (Faghri andDemetsky, 1988; Ibrahim and Sayed, 2011; Ben-Akiva et al.,1985; Hirsh et al., 1986; Easa, 2000; Navin, 1990; Richl and Sayed,2006; El Khoury and Hobeika, 2007; Sarhan and Hassan, 2008,2011; Ismail and Sayed, 2009). Among the previous studies, Faghriand Demetsky (1988) was the first study to show that there is alink between Pnc and collisions at rail-highway grade crossings.Studies (Ismail and Sayed, 2010; Faghri and Demetsky, 1988; Ibra-him and Sayed, 2011; Richl and Sayed, 2006) and this paper arebased on real-world data. Previous work has been largely moti-vated by theoretical shortcomings of deterministic highway geo-metric design as well as absence of relevant CMFs.

This study extends on previous work by taking a step from themere measurement of risk in terms of uncertainty of design param-eters and variables into the realm of design optimization. Earlierattempts for optimization of highway design have been proposedbased on construction cost (Jha and Schonfeld, 2000) and railwaycrossings (Park and Saccomanno, 2005). Aside from the originalwork by Ben-Akiva et al. (1985), little work has been conductedon risk-based optimization of highway geometric design. To thebest of the authors’ knowledge, the present study presents the firstrisk-based design optimization application in highway geometric

K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521 1515

design using reliability theory. While a developed topic of researchand practice in other disciplines, e.g. structural engineering(Haukaas, 2008), mechanical design (Youn et al., 2003; Grujicicet al., 2010), the adoption of this design concept in the disciplineof highway geometric design is very limited. This paper is a furtherstep toward: the formal development of probabilistic standards forhighway geometric design and the adoption of risk-based designoptimization.

3. Methodology

The methodology sections include: (1) definition of designrequirement as a limit state function, (2) a brief description of reli-ability analysis and (3) the specifics of the design optimizationproblem.

3.1. Limit state representation of design requirement

In the context of highway geometric design, the purpose of reli-ability analysis is to calculate the probability of meeting standarddesign requirements. In this paper, the design requirement is thatavailable sight distance (ASD) is greater than stopping sight dis-tance (SSD). ASD on horizontal curves is calculated as follows:

ASD ¼ 2 � R � cos�1 1�wlane

2 þwclearance

R

� �ð1Þ

where ASD is available sight distance (m), wlane is lane width (m),wclearance is the width of lateral clearance (m), and R is the horizontalcurve radius of the centerline of the lane closer to the restrictive ele-ment (m). SSD is calculated based on two models: friction model(TAC, 1999) and deceleration model (AASHTO, 2004) as follows:

SSD ¼ Voperating � PRTþV2

operating

2gðfTþlÞ Friction model

V2operating

2ðaþglÞ Deceleration model

8><>: ð2Þ

where Voperating is operating speed (km/h), PRT is perception andbrake reaction time (s), g is gravitational acceleration (m/s2), fT islongitudinal friction available for braking action (unitless – see Eq.(13)), l is longitudinal grade (unitless – positive if uphill), and a isthe deceleration rate (m/s2).

Design requirement are met when ASD exceeds the designrequirement represented by SSD. Formally, the criterion for meet-ing design requirement is represented by a limit state function g(X)(also called performance function as follows (Melchers, 1999):

gðX; hÞ � 0 indicates unacceptable designgðX; hÞ > 0 indicates acceptable design

ð3Þ

where X is an n-dimensional vector of design inputs X = x1, x2, . . .,xn, h is a vector of deterministic design parameters, and a limit statefunction or performance function g(X). The value of the limit statefunction is positive when the system performance is acceptable orsafe, and its value is negative otherwise. In this paper, g(X) repre-sents the difference between ASD and design requirementsrepresented by SSD as follows:

gðXÞ ¼ ASD� SSD ð4Þ

According to deterministic design, each design case is consti-tuted by a set of design inputs, i.e. a set of design variables suchas lateral clearance and parameters such as road friction, will beassociated with a single value for g(X). In reality, both ASD andSSD as well as their constituent design variables are stochastic innature and so is g(X). Furthermore, there is no single value forg(X) but rather a distribution of its own with a corresponding

probability of assuming negative values. The goal of reliabilityanalysis is to compute the probability of acceptable design, asshown in Eq. (3).

3.2. First-Order Reliability Analysis (FORM)

Due to the stochastic nature of many design inputs, the proba-bility of a vector of design inputs f(X) is denoted by the joint prob-ability of its constituent variables and parameters. Probability ofnon-compliance Pnc is therefore the integration of f(X) over thedomain in which g(X) 6 0:

Pnc ¼Z

gðXÞ�0f ðXÞdx1dx2 . . . dxn ð5Þ

A number of numerical solutions exist for Eq. (5), e.g. MonteCarlo simulation, First-Order Reliability Method (FORM). MonteCarlo simulation is based on numerical sampling of g(X) at suitabledesign points such as the mean of the design inputs X. Monte Carlosimulation is generally more computationally expensive than othermethods that involve some approximation of g(X) such as FORManalysis. For design optimization, which involves many iterationsand gradient computations, the use of Monte Carlo simulation be-comes significantly less appealing. Other approximations existsuch as First Order Second Moment (FOSM) reliability method,the simplest and most common reliability method in the majorityof previous work. FOSM analysis however suffers from to theinvariance problem, in which alternative but equally representativeformulations of g(X), e.g. g X ¼ ln ASD

SSD

� �� �, will yield different Pnc

estimate.FORM analysis overcomes all previous limitations. It can repre-

sent non-Normal input distributions, it does not suffer from invari-ance problem, and efficient algorithms exist for Pnc evaluation.Therefore, FORM analysis was used for conducting reliability anal-ysis in this study. FORM analysis is conducted in a transformedspace in which all variables are mapped from their original poten-tially non-Normal and correlated space to an independent standardnormal distribution using the Rosenblatt TransformationT : Rn ! Rn (Rosenblatt, 1952), such that:

UðY ¼ UðTðXÞÞ ¼ FðXÞ ð6Þ

where U(.) is the standard normal cumulative density function(CDF), Y is the corresponding vector of design inputs transformedfrom the problem space, and F(.) is the CDF of the joint probabilityof the design inputs f(X) which can be represent any continuousrandom variable. FORM analysis is different from FOSM in thatthe first-order expansion is performed at a more accurate point,the design point Y�. The design point is the point on the hyperplane:

gðXÞ ¼ GðYÞ ¼ 0 ð7Þ

with the highest probability of occurrence. Geometrically, the de-sign point is the closest point on the hyperplane G(Y) = 0 to the ori-gin of the standard independent Normal space. The design point Y�

is found by solving the following optimization problem:

Y� ¼ argminfkYk; such that GðYÞ ¼ 0g ð8Þ

where G(Y) is the limit state function in the standard normal space.After the design point is found, Pnc can be evaluated simply as;Pnc = U(�Y�). For smooth and differentiable limit state functions,such as the limit state function constituted by Eqs. ((1), (2) and(4)), FORM analysis is relatively efficient (Der Kiureghian et al.,2006). On average the number of evaluations of the limit state func-tion evaluations in the optimization problems to follow is 86 con-ducted over approximately seven iterations before converging toan optimal solution. This is a significantly smaller number com-pared to the number of function evaluations for an efficient Monte

1516 K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521

Carlo simulation (importance sampling �15,000 function evalua-tions for a single Pnc evaluation).

3.3. Cross-section risk optimization

This section presents a methodology for risk optimization forhighway segments constructed in restricted right-of-way. Previouswork by the authors (Ismail and Sayed, 2010) presented a method-ology for risk measurement. A logical question follows as to howcross-section elements can be re-dimensioned to mitigate risk.Risk optimization in this context potentially involves multipleobjectives. Probably the most appealing objective is the reductionof average risk for both carriageways. Another objective can be toproduce a cross-section with balanced risk for both carriageways.Finally, it is also relevant to achieve previous objectives using spe-cific re-dimensioning of various cross-section elements withoutconsequent increase in expected collision. Other objectives existand their relative priority or importance may vary. This study how-ever was limited only to the previous three objectives, mainly as aninitial step into a relatively new research direction.

3.3.1. Objective(s) functionThe following objective function enables the optimization for

the previous objectives and also provides some control over therelative importance of each objective:

cðIÞ ¼ a1maxðPnco ; Pnci

ÞminðPnco ; Pnci

Þ þ a2CMFðIÞCMFðIoÞ

þ a3Vo�Pnco þ V i

�Pnci

V i þ Voð9Þ

where

c(.) objective or cost function that is inversely proportional to the design desirabilityI input vector composed of six elements that represents a dimensioning scenarioI1,2,3 the first three elements of I, the outer shoulder width, lane width of both traffic lanes, and inner shoulder width of the inside

carriageway. Refer to Fig. 1 for an illustration of the inside/outside inner/outer terminologyI4,5,6 the second three elements of I, the inner shoulder width, lane width of both traffic lanes, and outer shoulder width of the

outside carriagewayVi expected traffic volume on the inside carriagewayVo expected traffic volume on the outside carriagewayCMF(.) the weighted average of the compound collision modification factors calculated for both carriageways. The functional details of

collision modification factors are based on the work by Harwood et al. (2003). For the inside carriageway, the compoundcollision modification factor is calculated as follows:CMFiðI1;2;3Þ ¼ exp½�0:021ð3:28I1 � 10Þ � 0:047ð3:28I2 � 12Þ � 0:021ð3:28I3 � 4Þ�CMFo is calculated identical to CMFi exception that I1,2,3 are replaced with I6,5,4 the weighted average of both carriageways is:

CMFðIÞ ¼ ViCMFiðI1;2;3ÞþVoCMFoðI4;5;6ÞV1þVo

a1 weight factor assigned for the first cost function component; risk balance between both carriagewaysa2 weight factor assigned for the second cost function component; increase in expected collision risk balance between both

carriagewaysa3 weight factor assigned for the third cost function component; weighted average risk for both carriagewaysIo input vector composed of six elements that represents a cross-section before optimizationPnco probability of non-compliance for the outside carriagewayPnci probability of non-compliance for the inside carriageway

Since median width used before optimization was the minimumallowable width, no further attempts have been made to reduce it.In addition, all possibilities to afford longer horizontal curve radiiwere tried out before the commencement of this optimization work.Therefore, optimization was performed on the following six cross-section elements: outer shoulders, inner shoulders and lane widthsof both carriageways (as represented earlier by the vector I).

3.3.2. ConstraintsCross-section optimization must be performed in respect of the

permitted right-of-way. All attempts have been made before thisanalysis to acquire additional roadside room. Therefore, the firstconstraint limits the right-of-way of the optimized section withinthe right-of-way available before optimization. Furthermore, lanewidth and shoulder widths were restricted to minima (and/ormaxima) in order to avoid unrealistic optimized cross-sectiondimensions. Formally the constraints can be stated as follows:

X6

i¼1

Ii ¼X6

j¼1

Ioj

I2;5 2 ½3:05;3:65�I1;3;4;6 2 ½0;3:05�

ð10Þ

3.3.3. Optimization algorithmSince the objective function is obtained through an iterative process,

FORM analysis, the function derivative was modified to be obtained bylonger step length. The minimum step length for finite difference calcu-lations was 1e�02 of current input vector values and maximum steplength was 1e�01. Three optimization algorithms were used: ActiveSet (Gill et al., 1991), Sequential Quadratic Programming (SQP, Nocedaland Wright, 2006), and Interior-point Method (Waltz et al., 2006).Slightly better results were obtained using SQP as will be demonstratedin later sections. Therefore, optimization was conducted using SQP forall cases. The reader is referred to a dedicated chapter in Nocedal andWright (2006) for implementation details.

4. Case studies

The case studies analyzed below are parts of two major high-way developments in British Columbia, Canada. All nine cross-sec-tions belong to horizontal curves with restricted sight distance. Therestrictive elements (i.e. block the line of sight connecting the dri-ver eye and an object on the occupied lane) are roadside concrete

Table 3Distribution parameters.

Parameter Mean Standarddeviation

Distribution Designvalues

Percentilevalue

Perception andbrake-reaction time

1.5 s 0.4 s Log normala 2.5 sc 98

Driverdeceleration

4.2 m/s2

0.6 m/s2 Normalb 3.4 m/s2

9

Total friction 0.38 0.09 Normald 0.30 16(Voperating ffi 90) 0.35 0.09 Normal 0.30 16Total friction Model Model Normal – –((Voperating ffi 95)Operating speed

a Obtained from Lerner et al. (1995).b Obtained from Fambro et al. (1997).c AASHTO, Green Book (2004).d Richl and Sayed (2006).

Table 4Mean and variance of operating speed on horizontal curves (Richland Sayed, 2005).

Radius (m) Mean (km/h) Standard deviation (km/h)

300 86.8 5.6350 88.6 5.1400 90.0 4.8450 91.1 4.6500 91.9 4.6550 92.7 4.6600 93.3 4.6650 93.8 4.7700 94.2 4.7750 94.6 4.8800 94.9 4.9

Table 2Summary of scenarios.

Case SSD model a1 a2 a3

1 Friction 1 1 12 Friction 1 1 03 Friction 0 0 14 Deceleration 1 1 15 Deceleration 1 1 06 Deceleration 0 0 1

Table 1Summary of cross-section dimensions of different case studies. Dimensions in meters.

Case Radius Inside carriageway Median Outside carriageway Longitudinal slope(%)

Right-handcurve = 1

Outershoulder

Lanewidth

Innershoulder

Medianwidth

Innershoulder

Lanewidth

Outershoulder

Left-handcurve = 0

1.1 440 2.5 3.7 1.15 0.6 1.15 3.7 2.5 �0.7 11.2 440 2.5 3.7 1.15 0.6 1.15 3.7 2.5 1.5 01.3 440 2.5 3.7 1.15 0.6 1.15 3.7 2.5 �1.3 11.4 440 2.5 3.7 1.15 0.6 1.15 3.7 2.5 �0.4 01.5 440 2.5 3.7 1.15 0.6 1.15 3.7 2.5 0.1 1

2.1 450 2.5 3.6 1.7 0.6 1.7 3.6 2.5 �3.9 12.2 320 2.5 3.6 1.7 0.6 1.7 3.6 2.5 0.6 12.3 350 2.5 3.6 1.7 0.6 1.7 3.6 2.5 �3.3 12.4 350 2.5 3.6 0.5 0.6 0.5 3.6 2.5 �2.3 1

K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521 1517

barriers, median barriers, roadside structures, and bridge parapet.The question was raised as to how to utilize available right-of-way. The methodology presented in the previous section wasapplied to these cross-sections. Table 1 shows a summary ofcross-section elements. Table 2 lists all the optimization scenariosdiscussed in later sections. Following sections are (1) presentationof input variables and distribution assumptions (2) presentation ofoptimization results (3) discussion of results.

4.1. Input variables

The stochastic input variables in this paper are: perception andbrake reaction time (PRT), operating speed (Voperating), pavement to-tal friction factor (fT), and deceleration rate (a). A summary of theprobability distributions of the design inputs used is presented inTable 3. In the reliability analysis presented in this paper, operatingspeed was used, opposed to adoption of design speed in previouswork. The significance of using operating speed instead of designspeed in reliability analysis was highlighted in previous work(Ismail and Sayed, 2010; Sarhan and Hassan, 2011). Design speedis a fundamental input to standard design models in order toachieve balanced dimensioning of various geometric elements.The inputs to reliability analysis are the actual distributions thatrepresent real-world variation. Design speed is an assumed speedthat may not exhibit strong relationship with actual operatingspeed (Fitzpatrick et al., 1996). For example, a review of operatingspeed models on horizontal curves shows that for a design speed of100 km/h or less, operating speed exceeds design speed on hori-zontal curves (Fitzpatrick et al., 1996). Exceeding design speed isan unacceptable design outcome that should be reflected by higherPnc values. Obtaining high Pnc in case of excessive driving wouldnever be possible if design speed is used in reliability analysis.Operating speed in this study is assumed to follow Normal distri-bution with mean and variance obtained from the meta-analysisconducted in Richl and Sayed (2005). A summary of these valuesis presented in Table 4.

Two stopping sight distance models were used in this paper.The first model, based on TAC design guide (Association of Canada.

Geometric Design Guide for Canadian Roads. Ottawa, 1999),assumes that deceleration during braking action depends on thepavement surface friction. The total tangential friction in this studyis assumed to follow Normal distribution with mean and varianceobtained from Table 3; obtained from Richl and Sayed (2006). Themaximum radial friction fRmax is assumed to be 0:925 � fTmax basedon (Lamm et al., 1999, p. 10.21).

The centrifugal force induced on a vehicle is balanced by itsweight component pointing to the center as well as the lateralfriction. The amount of tangential friction available for braking ac-tion is the total friction (fRmax Þ reduced by the amount required forlateral stability on a horizontal curve. The friction term used forbraking action fT is obtained as follows (Lamm et al., 1999, p.10.23):

Table 5Summary of optimization results for friction model using Sequential Quadratic Programming (SQP).

Scenario (cf.Table 2)

Case Input: Inside carriageway Input: Outside carriageway Objectivefunction

Pnc forcurrentdesign

Pnc optimaldesign

Objective functioncomponents

Innershoulder

Lanewidth

Outershoulder

Innershoulder

Lanewidth

Outershoulder

Inner Outer Inner Outer pnc

RatioCMFratio

AveragePnc

1 1.1 3.05 3.66 0.75 2.68 3.61 0.79 2.44 0.53 0.80 0.42 0.42 1.00 1.01 0.421.2 2.66 3.60 0.80 3.05 3.66 0.78 2.44 0.45 0.85 0.43 0.43 1.00 1.01 0.431.3 3.05 3.66 0.89 2.50 3.56 0.92 2.46 0.55 0.79 0.45 0.45 1.00 1.01 0.451.4 3.05 3.66 0.68 2.78 3.64 0.71 2.42 0.52 0.81 0.41 0.41 1.00 1.01 0.411.5 3.05 3.66 0.58 2.94 3.66 0.60 2.41 0.50 0.82 0.40 0.40 1.00 1.01 0.40

2.1 3.05 3.66 1.79 1.79 3.50 1.86 2.54 0.65 0.55 0.54 0.54 1.00 1.00 0.542.2 3.05 3.66 1.04 3.05 3.66 1.04 2.57 0.68 0.83 0.57 0.57 1.00 1.00 0.572.3 3.05 3.66 1.74 1.99 3.46 1.80 2.65 0.75 0.71 0.65 0.65 1.00 1.00 0.652.4 3.05 3.66 1.55 2.20 3.53 1.62 2.54 0.72 0.96 0.62 0.62 1.00 0.92 0.62

2 1.1 3.04 3.66 0.78 2.68 3.60 0.80 2.01 0.53 0.80 0.43 0.43 1.00 1.01 0.431.2 2.65 3.60 0.81 3.04 3.66 0.79 2.01 0.45 0.85 0.43 0.43 1.00 1.01 0.431.3 3.03 3.66 0.90 2.48 3.57 0.92 2.01 0.55 0.79 0.45 0.45 1.00 1.01 0.451.4 3.05 3.66 0.69 2.78 3.63 0.71 2.01 0.52 0.81 0.41 0.41 1.00 1.01 0.411.5 3.05 3.66 0.58 2.94 3.66 0.59 2.01 0.50 0.82 0.40 0.40 1.00 1.01 0.40

2.1 3.02 3.66 1.83 1.77 3.50 1.88 2.00 0.65 0.55 0.54 0.54 1.00 1.00 0.542.2 3.05 3.66 1.04 3.05 3.66 1.04 2.00 0.68 0.83 0.57 0.57 1.00 1.00 0.572.3 3.00 3.65 1.79 1.95 3.46 1.83 2.00 0.75 0.71 0.66 0.66 1.00 1.00 0.662.4 3.03 3.66 1.62 2.21 3.49 1.65 1.92 0.72 0.96 0.62 0.62 1.00 0.92 0.62

3 1.1 3.05 3.66 0.00 3.05 3.66 1.07 0.39 0.53 0.80 0.42 0.35 1.21 1.01 0.391.2 3.05 3.66 0.00 3.05 3.66 1.07 0.39 0.45 0.85 0.34 0.43 1.24 1.01 0.391.3 3.05 3.66 0.00 3.05 3.66 1.07 0.39 0.55 0.79 0.44 0.33 1.36 1.01 0.391.4 3.05 3.66 0.00 3.05 3.66 1.07 0.39 0.52 0.81 0.41 0.36 1.14 1.01 0.391.5 3.05 3.66 0.00 3.05 3.66 1.07 0.39 0.50 0.82 0.39 0.38 1.05 1.01 0.39

2.1 3.05 3.66 0.00 3.05 3.66 2.07 0.38 0.65 0.55 0.53 0.23 2.29 1.00 0.382.2 3.05 3.66 0.00 3.05 3.66 2.07 0.57 0.68 0.83 0.57 0.57 1.01 1.00 0.572.3 3.05 3.66 0.00 3.05 3.66 2.07 0.52 0.75 0.71 0.65 0.39 1.66 1.00 0.522.4 3.05 3.66 0.00 3.05 3.66 2.07 0.52 0.72 0.96 0.61 0.42 1.45 0.92 0.52

1518 K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521

ðfT=fTmax Þ2 þ ðfR=fRmaxÞ

26 1 ð11Þ

which can be re-arranged as follows:

fT � fTmax

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðfR=fRmax Þ

2q

ð12Þ

but from lateral stability between centrifugal force and centripetalforce provided by friction and superelevation we have: fR ¼ v2

gR � e.Substituting into Eq. (12), we can estimate the maximum availablelongitudinal friction after reducing friction due to balance of forcesin the lateral direction:

fT ¼ fTmax

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

V2operating

gRfRmax

� efRmax

!2vuut ð13Þ

where g is gravitational acceleration (m/s2), R is the horizontalcurve radius of the centerline of the lane closer to the restrictive ele-ment (meter), and e is cross-slope (unitless). For the second model,based on AASHTO Green Book (American Association of StateHighway and Transportation Officials, 2004), the distribution ofthe driver deceleration was assumed Normal, with mean and vari-ance obtained from Fambro et al. (1997). A summary of the inputdistributions for braking action is provided in Table 3.

4.2. Cross-section optimization results

Optimization was conducted for the nine case studies (1.1–1.5and 2.1–2.4) presented in Table 1. The average number of iterationsper cross-section was seven iterations. In general, the computational

cost for obtaining optimal results was negligible. Different scenar-ios were considered as presented in Table 2. SQP was used forcross-section optimization for all case studies. Different combina-tions were used for weighing cost function components presentedin Eq. (9). The first combination (a1 = 1, a2 = 1, a3 = 1) reflects anequal importance of the three objectives that constitute the costfunction; risk balance, change in CMF, and average risk. The secondcombination, (a1 = 1, a2 = 1, a3 = 0), eliminates the importance ofaverage risk. As stated, this objective strives to achieve equitablerisk for traffic on both carriageways at the expense of having rela-tively higher total risk for all traffic navigating the concerned high-way section. The third combination (a1 = 0, a2 = 0, a3 = 1)represents the designer’s preference for reducing average risk atthe expense of having unbalanced risk for both carriageways. Bal-ance in risk is assumed to be measured against traffic volume oneach carriageway. In the case studies presented in this study, trafficvolume was expected to be equal for all pairs of carriageways. Ta-bles 5 and 6 show the dimensions of optimized cross-sections ofthe nine case studies using both friction and deceleration models.In these case studies, it was assumed that the net available right-of-way after optimization, not including median width, was21.8m for 1.1-5 and 22.8m for 2.1-4.

4.3. Discussion

In general, there appears to be a noticeable difference betweendimensions of geometric elements before and after optimization.With respect to average risk for both carriageways (also total risk),there was a consistent reduction in risk after optimization for allcase studies and for all combinations. The following patterns ofchange can be noticed from the analysis results:

Table 6Summary of optimization results for deceleration model using Sequential Quadratic Programming (SQP).

Scenario (cf.Table 1)

Case Input: Inside carriageway Input: Outside carriageway Objectivefunction

Pnc forcurrentdesign

Pnc optimaldesign

Objective functioncomponents

Innershoulder

Lanewidth

Outershoulder

Innershoulder

Lanewidth

Outershoulder

Inner Outer Inner Outer Pnc

RatioCMFratio

AveragePnc

4 1.1 3.05 3.66 0.69 2.71 3.66 0.71 2.21 0.32 0.72 0.20 0.20 1.00 1.01 0.201.2 2.69 3.66 0.72 3.05 3.66 0.71 2.22 0.24 0.79 0.21 0.21 1.00 1.01 0.211.3 3.05 3.66 0.78 2.53 3.66 0.80 2.23 0.35 0.70 0.22 0.22 1.00 1.01 0.221.4 3.05 3.66 0.65 2.81 3.66 0.67 2.20 0.31 0.73 0.19 0.19 1.00 1.01 0.191.5 3.05 3.66 0.57 2.96 3.66 0.59 2.19 0.29 0.75 0.18 0.18 1.00 1.01 0.18

2.1 3.05 3.66 1.62 1.84 3.66 1.66 2.31 0.47 0.37 0.31 0.31 1.00 1.00 0.312.2 3.04 3.66 1.04 3.05 3.66 1.04 2.35 0.50 0.74 0.35 0.35 1.00 1.00 0.352.3 3.05 3.66 1.59 2.05 3.58 1.63 2.45 0.60 0.57 0.45 0.45 1.00 1.00 0.452.4 3.05 3.66 1.45 2.28 3.61 1.49 2.33 0.56 0.95 0.41 0.41 1.00 0.92 0.41

5 1.1 2.94 3.66 0.80 2.61 3.66 0.82 2.01 0.32 0.72 0.22 0.22 1.00 1.01 0.221.2 2.58 3.66 0.84 2.92 3.66 0.82 2.01 0.24 0.79 0.23 0.23 1.00 1.01 0.231.3 2.91 3.66 0.91 2.41 3.66 0.93 2.01 0.35 0.70 0.25 0.25 1.00 1.01 0.251.4 2.93 3.66 0.76 2.70 3.66 0.78 2.01 0.31 0.73 0.21 0.22 1.00 1.01 0.211.5 2.90 3.66 0.72 2.81 3.66 0.73 2.01 0.29 0.75 0.21 0.21 1.00 1.01 0.21

2.1 1.41 3.66 3.05 0.67 3.66 3.05 2.00 0.47 0.37 0.81 0.81 1.00 1.00 0.812.2 3.04 3.66 1.05 3.05 3.66 1.04 2.00 0.50 0.74 0.35 0.35 1.00 1.00 0.352.3 3.00 3.65 1.62 2.01 3.61 1.65 2.00 0.60 0.57 0.46 0.46 1.00 1.00 0.462.4 2.98 3.65 1.53 2.22 3.61 1.56 1.92 0.56 0.95 0.42 0.42 1.00 0.92 0.42

6 1.1 3.05 3.66 0.00 3.05 3.66 1.07 0.17 0.32 0.72 0.20 0.14 1.41 1.01 0.171.2 3.05 3.66 0.00 3.05 3.66 1.07 0.17 0.24 0.79 0.14 0.21 1.47 1.01 0.171.3 3.05 3.66 0.00 3.05 3.66 1.07 0.17 0.35 0.70 0.22 0.13 1.73 1.01 0.171.4 3.05 3.66 0.00 3.05 3.66 1.07 0.17 0.31 0.73 0.19 0.15 1.27 1.01 0.171.5 3.05 3.66 0.00 3.05 3.66 1.07 0.17 0.29 0.75 0.18 0.16 1.09 1.01 0.17

2.1 3.05 3.66 0.00 3.05 3.66 2.07 0.19 0.47 0.37 0.31 0.07 4.34 1.00 0.192.2 3.05 3.66 0.00 3.05 3.66 2.07 0.35 0.50 0.74 0.35 0.35 1.02 1.00 0.352.3 3.05 3.66 0.00 3.05 3.66 2.07 0.31 0.60 0.57 0.44 0.18 2.40 1.00 0.312.4 3.05 3.66 0.00 3.05 3.66 2.07 0.30 0.56 0.95 0.40 0.21 1.91 0.92 0.30

C

Outside Lane

3.70 3.70

1.15 InsideLane

2.50

Outer Shoulder

2.50

OuterShoulder

Conc. Barrier0.6m Median Barrier

Outside Lane InsideLane

Right-hand-side horizontal curve

1.15

3.70

Inner Shoulders

Inside CarriagewayOutside Carriageway

3.70

Cross section: Case 1.1

All dimensions in meters.

Fig. 1. Typical cross-section in a right-hand-side horizontal curve. The restrictive element for the inside direction is the inner, i.e. closer to the center, barrier. The lateralclearance for the inside direction is its outer shoulder width. The restrictive element for the outside direction is the median barrier. The lateral clearance for the outsidedirection is its inner shoulder width.

K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521 1519

1. For both the friction and deceleration SSD models, the highestreduction in total risk was achieved for the third weight factorcombination (scenarios 3 and 6 in Table 2). The absolute aver-age reduction in Pnc, from current to optimal design, for all casesin scenario 3 is 0.25 (0.18–0.32). This reduction is calculated forthe average of Pnc for both directions (inner and outer lanes) forevery case. For example, absolute reduction for case 2.4 in sce-nario 3 should be (0.72 + 0.96)/2 � 0.52.The average reductionin risk for scenario 6 is 0.33 (0.23–0.45).

2. Consistently, the cross-section with the most risk reduction was2.4. This can be explained by the fact that the outside carriagewayat this cross-section suffered from the highest pre-optimizationrisk of approximately 0.95. This was achieved by maximizing pos-sible sight distance clearance for both carriageways.

3. Focusing optimization on the reduction of average risk onlyproduces on a consistent basis unbalance in risk for both car-riageways. The average risk ratio for both carriageways of eachcross-section is 1.37 for scenario 3 and 1.85 for scenario 6.

1520 K. Ismail, T. Sayed / Safety Science 50 (2012) 1513–1521

Therefore, it is remarkable that striking a balance in risk forboth carriageways and reducing their average risk are two com-peting objectives.

4. For most cases, it is more desirable to afford wider clearancefrom restrictive elements, in concrete barriers in the presentedcase studies, by reducing the shoulder width on the oppositeside. This approach for width compensation is desirable forreducing lane width. This is can be explained by the fact thathalf the lane width is included in the clearance calculation.

5. Scenarios 3 and 6 bring what appears once to be a surprisingoptimization outcome. The weight factor that represents inclu-sion of expected collision in the objective function appears tohave little effect on the CMF of the optimized section. For exam-ple, there is no noticeable difference in the ratio between theCMFs of cross-sections before and after optimization even whena2 = 0 in scenarios 3 and 6. This could be explained by the pres-ence of a link between average risk and expected collisionwhereby minimizing one of them indirectly results in minimiz-ing the other. Whether such link exists is subject of future work.However, these findings are consistent with results that prove apositive relationship between Pnc and collision frequency (Ibra-him and Sayed, 2011).

6. Despite the evident difference in Pnc values obtained from thetwo SSD models proposed used in this study, SSD model selec-tion appears to have little effect on optimal cross-sectiondimension. The average difference in width between corre-sponding cross-section elements obtained using both SSD mod-els is only 6.5% for scenarios 1, 2, 4, and 5 and 0% for scenarios 3and 6. This is generally a desirable robustness characteristic ofthe proposed optimization methodology. Despite the differentdeparture points for both SSD models, obtained from differentliteratures and for two different design guides, they appear toconverge to a unique risk-optimal design.

7. The design models adopted in this study are themselves ofuncertain nature. Design models typically involve a level of ide-alization of real-world driving conditions. For example, the SSDmodel adopted in Eq. (2) implies that a single distribution forPRT may be used irrespective of the driving conditions duringthe perception and reaction to the danger on the road. It canbe argued that light conditions, object shape, road geometry,traffic conditions, among other factors, may influence PRT.Therefore, this SSD model may have omitted variables. More-over, the friction factor values suggested in Table 3 are limitedto specific pavement conditions, namely wet pavement. Otherweather conditions, e.g., dry or icy, may exist that may influ-ence positively or negatively braking distance. Therefore, theweather condition should be represented by a random dummyvariable in the SSD model. Furthermore, the type of evasiveaction may involve braking and swerving. The assumption ofa braking course along the lane centerline may not be com-pletely realistic.

8. Note that lane and shoulder widths are represented in Eq. (9) bycontinuous variables not discrete variable at increments of 5 cmas is common in practice. This simplifies the optimization prob-lem as the algorithms adopted in this study may not be suitablefor discrete variables. The decision maker however may selectto adopt widths slightly different from those obtained from thisoptimization problem in order to meet specific categoricalrequirements. The latter decision may be made with the expec-tation that slightly suboptimal dimensions are selected in orderto meet an external categorical dimensioning policy.

5. Summary and Future Work

This study presented a novel methodology for re-dimensioningcross-sections located at highway segments with restricted sight

distance. The risk entailed by different dimensioning decisions ismeasured in terms of the uncertainty in final design output inmeeting design requirements. Uncertainty measurement is con-ducted using reliability analysis. Re-dimensioning of cross-sectionswas formulated as a multi-objective optimization problem. The fol-lowing objectives were minimized in different priorities, averagerisk and imbalance in risk for both carriageways as well as the in-crease in expected collisions consequent to cross-section re-dimensioning. Sequential Quadratic Programming proved a reli-able optimization approach for this problem. A number of findingswere drawn from the analysis results. Most notably, the findingthat for the case studies presented in this paper, following eitherAASHTO (2004) or TAC (1999), leads to almost the same optimalcross-section.

Note that, while this study advocated the use of an indicator ofthe risk of failure to meet design requirements, i.e., normativesafety, the use of other indicators for the risk of collisions, e.g.,CMFs, whenever available can be readily integrated into the deci-sion making criterion. This is a key advantage of the proposedmethodology which integrates different safety cues, normativeand empirical, in order to aid the re-dimensioning of cross-sectionelements.

Several future directions can be derived from this work. First isthe composition of a multi-objective cost function that more accu-rately reflects both practical needs and safety consequences. Thevariant effect of risk imbalance for carriageways and change inaverage risk is unknown. The cost function proposed in Eq. (9)can be constructed to reflect the variant safety consequences of dif-ferent cost function components. Another future direction is theinvestigation of the compound effect of simultaneous change indimension of different cross-section elements. It is assumed thatthe total effect on expected collision is the product of differentCMFs calculated independently for every cross-section element.It is not known however whether the mathematical product isthe precise representation of the concerned safety implications. Fi-nally, the issue of construction cost minimization can be re-visitedin conjunction with the optimization methodology proposed inthis paper within what could be a unified framework for highwaygeometric design.

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