Optimization of Highway Safety and Operation by Using Crash Prediction Models with Accident...

108

way segments, homogenous segments are highway sections withinwhich the ADT and AMF values are constant. For intersections, dif-ferent models may be developed for different types of intersections.The general forms of crash prediction models with productive AMFsare shown below:

where

N = total expected number of crashes,Ns = expected number of crashes for highway seg-

ments,NI = expected number of crashes for intersections,

nhwy = number of highways,nsh

= number of homogenous segments for highway h,ADTh,i = ADT for homogenous segment i of highway h,

Lh,i = length of homogenous segment i of highway h,fs (ADTh,i, Lh,i) = function of ADTh,i and Lh,i of segment i of high-

way h,a = number of highway segment features for which

there are AMFs in model,AMFSh,i = AMF number S for homogenous segment i of

highway h,Sh,i = index associating AMFs to highway features of

segment i of highway h (from 1 to a),nint = number of intersections,

ADTi,b = ADT for approach b of intersection i,fi (ADTi,b) = function of ADTs of approaches of intersection i,

c = number of intersection features for which thereare AMFs in model,

AMFSi = AMF number S for intersection i, andSi = index associating AMFs to features of inter-

section i (from 1 to c).

Crash prediction models can be used to compare the safety per-formance of different combinations of safety improvements for

N fI i i b SS

c

i

n

i

i

ADT AMF= ( )⎡

⎣⎢

⎤

⎦⎥

==∏∑ ,

int

( )11

3

N f LS s h i h i SS

a

ih i

h i

ADT AMF= ( )⎡

⎣⎢

⎤

⎦⎥

==∏, ,,

,

, 1111

2n

h

n Sh

∑∑=

hwy

( )

N N NS I= + ( )1

Optimization of Highway Safety and Operation by Using Crash Prediction Models with AccidentModification Factors

Mohamadreza Banihashemi

The linear programming model that is presented optimizes the crashand delay costs of highway networks by selecting safety and operationalimprovements with implementation costs within a constrained budget.The costs that are minimized in this study are those of crashes plus thecosts of delay times. This optimization model works with crash predic-tion models that estimate the expected number of crashes by using basemodels and accident modification factors (AMFs). In these models thebase model predicts the expected number of crashes for a base highwayor intersection. The AMFs modify the prediction on the basis of thespecific characteristics of each highway segment or intersection. Sev-eral improvement alternatives for different highways and intersectionswithin a network would mean many improvement combinations, eachwith a different cost for implementation, which would cause certaincrashes and delay. The proposed model uses linear optimization to findthe combination of improvements with the lowest total of crash and delaycosts within an acceptable range of implementation costs. The optimiza-tion model is presented in conjunction with crash prediction modelsborrowed from the Interactive Highway Safety Design Model software.The optimization model could be modified to work with other types ofcrash prediction models. A test case study containing a network of threehighways and 13 intersections is presented. The corresponding opti-mization problem is solved with the CPLEX software. The variations ofcrash and delay cost savings versus improvement costs are studied, andthe results are discussed.

Safety improvement benefits and delay time reductions are often themajor criteria for budget allocation to different safety improvementprojects within a network of highways and intersections. This is acomplicated task because there could be many different combinationsof solutions that need to be compared.

Different types of crash prediction models are used for the safetyevaluation of highways and intersections. One type includes modelswith a base component as a function of average daily traffic (ADT)and highway length for highway segments combined with produc-tive accident modification factors (AMFs). In the models for high-

LENDIS Corporation, c/o FHWA, GDL, 6300 Georgetown Pike, Mail Stop HRDS-05,McLean, VA 22101. [email protected].

Transportation Research Record: Journal of the Transportation Research Board,No. 2019, Transportation Research Board of the National Academies, Washington,D.C., 2007, pp. 108–118.DOI: 10.3141/2019-14

highway networks. This comparison could be conducted by esti-mating the total expected crash and delay costs for each combi-nation. Without an automated approach for such comparisons, the number of combinations that could be studied by a designer is limited. A linear optimization model is presented here that, inconjunction with crash prediction models for highway segmentsand intersections, evaluates all possible combinations of improve-ments to minimize the total of crash and delay costs given a specificbudget.

The crash prediction models used in this effort to show the prac-ticality of this approach are from the Crash Prediction Module (CPM)of the Interactive Highway Safety Design Model (IHSDM) (1, 2).The linear optimization model is described here, a brief overview ofthe IHSDM crash prediction models and related AMFs is given, anda case study is presented to illustrate how the optimization modelcan be used to minimize total crash and delay costs of a highway net-work given various improvement cost constraints. This optimizationmodel is not part of the IHSDM software. The IHSDM models areused only as examples in this research.

RELATED RESEARCH

This research is the continuation of work conducted by Banihashemiand Dimaiuta (3), who presented a mixed integer linear programming(MILP) model for maximizing the safety benefits of the availableimprovement options on a single highway within a certain budgetlimit. Harwood et al. (4) introduced an optimization procedure tomaximize the safety improvements for a series of resurfacing, reha-bilitation, or restoration projects within a certain budget. FHWAcurrently is developing Safety Analyst software, which will helpusers to optimize the safety benefits for improvements within aseries of sites or projects. This software will have four modules:network screening, diagnosis and countermeasure selection, appraisaland priority ranking, and evaluation. The third module has an opti-mization component that maximizes the safety benefits of preselectedimprovements on preselected roadway sections and intersections (5).

The Safety Analyst optimization model is

subject to

where

TB = total benefits from all selected improvements,y = number of sites,z = number of improvements,

NBjk = present value of safety benefits of improvement k at site jminus construction cost for improvement k at site j,

Xjk = indicator whose value is 1 if improvement k at site j isselected as part of optimum allocation of funds and whose

CC jk jkk

z

j

y

X B==

∑∑ ≤11

6( )

X j yjkk

z

=∑ = ∀ =

1

1 1 2 5, , . . . , ( )

maximize TB NB===

∑∑ jk jkk

z

j

y

X11

4( )

Banihashemi 109

value is 0 if improvement k at site j is not selected as partof optimum allocation of funds (for each site exactly oneimprovement or combination of improvements should beselected),

CCjk = construction cost for improvement k at site j, andB = improvement budget or maximum funding available for

improvement of sites under consideration.

For more information readers are referred to the Safety Analystwebsite (5).

LINEAR OPTIMIZATION MODEL

Alternative project designs may result in changes to AMF valuesfor some of the homogenous segments or for some intersections orchanges in intersection type. These changes could represent designimprovements from a safety point of view that may also changesome user costs such as operating or delay costs. Table 1 showssome common highway and intersection features for which safetycould be improved, as well as whether the feature is represented inthe IHSDM CPM.

Crash and Delay Costs

On the basis of a specific crash severity distribution and crash cost foreach crash type, an average crash cost is calculated. The crash cost foreach crash type is from White Paper for Module 3—EconomicAppraisal and Priority Ranking (5). FHWA’s 2002 estimates are$3,000,000, $208,000, $42,000, $22,000, and $2,300 for each fatal,incapacitating injury, serious injury, minor injury, and property-damage-only (PDO) crash, respectively. By using the IHSDM defaultcrash severity distributions the average crash costs for highway seg-ments and three-leg stop-controlled, four-leg stop-controlled, and four-leg signalized intersections are $59,562, $55,239, $81,375, and$31,665, respectively. Details of the IHSDM default crash severitydistributions are found in Report FHWA-RD-99-207 (1).

A unit cost of $12.50 per hour per vehicle is considered for in-vehicle travel time or delay cost for highway users. This unit cost iscalculated by multiplying a $10 per hour in-vehicle travel time cost bya simplified vehicle occupancy rate of 1.25. Delay times are calculatedfor highway segments and intersections separately.

The only highway segment improvement option whose effect ondelay time is considered is the addition of passing or climbing lanes.Other safety improvements may have an effect on the travel timeson highway segments as well, but the author believes that the mar-gin of error in estimating travel times on homogenous segments forthe other improvements is so high that using such estimations in thisoptimization is not appropriate.

The types of intersection improvements whose effects on inter-section delay time are considered in this research are

• Changing the traffic control from minor stop to all stop,• Changing the traffic control from minor stop or all stop to

signalized, and• Adding a left-turn lane.

For a network of highways with some alternatives for each high-way or intersection, it is not practical to manually establish and

evaluate all possible combinations. This model finds the improve-ment combinations with the lowest user crash and delay costs withinan available budget. The linear optimization model statements areas follows:

subject to

Y ii s ts t i

, , ,..., ,...

, , . . . ,( )( )∑ = ∀ =

all for

1 1 2 nnint ( )9

X hh i j kj k h i

, , , ,..., ,...

, ,( )( )

∑ = ∀ =all for ,

1 1 2 .. . . , , , . . . , ( )n i nShhwy ∀ = 1 2 8

min

ADT

AMF

scC f Ls h i h i

SS

h i j k

h i

, ,,

, ,( , ,. . .)

,

( )aa

h i j kj k

X∏⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥( )

( ), , , ,...

, ,...all foor

hwy

A

h i

i

n

h

n

i ic i

Sh

C f

,

,

∑∑∑⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

+

== 11

DDT AMFi b SS

c

i s ti s t

i

Y, , , ,..., , ,. . .( ) ⎛

⎝⎜⎞⎠⎟( )∏ (( )

( )=

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

+

∑∑all fors t ii

n

, ,...

int

1

CC D XD h i j k h i j ki

n

h

Sh

, , , ,... , , , ,...( ) ( )=

⎡⎣ ⎤⎦∑1==

( ) ( )

∑

+ ⎡⎣ ⎤⎦

1

n

D i s t i s ts

C D Y

hwy

all, , ,... , , ,...

,, ,...

int

t ii

n

( )=∑∑

for

(7)1

110 Transportation Research Record 2019

where

Csc = unit cost for segment crashes;Ci,ic = unit cost for crashes for intersection i;

( j, k, . . .) = improvement combination for all highway featuresrepresented in safety model through AMFs, where( j, k, . . .) are improvement-type indices associatingimprovement types ( j, k, . . .) to one of the alterna-tives; existing highway segments have improve-ment combination (0, 0, . . . , 0);

all variables integer (12)=

. . .

. . . . . .

11 0k( )

X Xh i j k h i j kk

, , , ,... , , , ,...,...

0 01( ) + ( )− ∑allaall

hwy and and

k

h n i i

,...

, , . . . ,

∑ =

∀ = ∀ +

0

1 2 1(( ) = −( )( )

1 2 1

11 0

, , . . . , n

j

S

S

h

with the same MIL

L C Xh i h i j k h i j kj k

, , , , ,... , , , ,..., ,...

( ) ( )all(( )==

( )

∑∑∑

+

for

,

hwy

h ii

n

h

n

i s t i

Sh

C Y

,

, ,...

11

,, , ,..., ,...

int

s ts t ii

n

( )( )=∑∑ ≤

all for

tota1

ll budget ( )10

TABLE 1 Highway and Intersection Features and Safety Improvements

Highway and Intersection Feature Safety Improvement Represented in IHSDM

Lane width Widening Yes

Shoulder width Widening Yes

Horizontal curve Flattening Yes

Superelevation Removing deficiencies Yes

Vertical grade Reducing the grade Yes

Driveway density Decreasing Yes

Passing lanes and short Adding Yesfour-lane sections

Two-way left-turn lanes Adding Yes

Roadside hazard rating Decreasing Yes

Median Adding/modifying No

Rumble strip Adding No

Posted speed Reducing No

Edge line Adding No

Guardrail Adding No

Intersection skew angle Reducing/removing Yes

Intersection traffic control Modifying Partially yes (there is no three-leggedsignalized model in IHSDM)

Left-turn lane at intersection Adding Yes

Right-turn lane at intersection Adding Yes

Intersection sight distance Removing deficiency Yes

Intersection traffic control Adding Nodevices visibility

Shoulder by-pass at intersection Adding No

Intersection on curve Moving No

Intersection corner radius Increase No

Intersection/general Change to roundabout No

(s, t, . . .) = improvement combination for all intersection fea-tures represented in safety model through AMFs,where (s, t, . . .) are improvement-type indices asso-ciating improvement types (s, t, . . .) to one of thealternatives; existing intersections have improvementcombination (0, 0, . . . 0);

AMFSh,i,( j, k,...)= AMFs for homogenous segment i of highway h

and highway segment improvement combination( j, k, . . .);

Xh, i,(j, k,...) = variable associated with homogenous segment iof highway h and improvement combination (j,k, . . .), which would be 1 if this combination waschosen and 0 otherwise;

AMFSi,(s, t,...)= AMFs for intersection i and intersection improve-

ment combination (s, t, . . .);Yi,(s, t,...) = variable associated with intersection i and improve-

ment combination (s, t, . . .), which would be 1 ifthis combination was chosen and 0 otherwise;

CD = unit cost for delay time;Dh, i,(j,k,...) = delay time for highway segment i of highway h for

improvement combination ( j, k, . . .);Di,(s,t,...) = delay time for intersection i for improvement com-

bination (s, t, . . .);Ch,i,(j,k,...) = improvement cost associated with implementing

improvement combination ( j, k, . . .) for homoge-nous segment i of highway h (Ch,i,(0,0,...) = 0);

Ci,(s,t,...) = improvement cost associated with implementingimprovement combination (s, t, . . .), for intersec-tion i (Ci,(0,0,...) = 0); and

MILs = minimum improvement length for highway seg-ments along which improvement type s should bemaintained [more explanation and some examplesmay be found elsewhere (3)].

The rest of the items are as defined for Equations 1, 2, and 3.The components of the model are defined in the following with

the mathematical statement numbers in parentheses:

• Objective function (7). The objective is to minimize the totalcost associated with crashes and delays.

• Segment covering constraints (8). These constraints ensure thatone and only one improvement combination is selected for eachhomogenous segment of each highway.

• Intersection covering constraints (9). These constraints ensurethat one and only one improvement combination is selected for eachintersection.

• Budget constraint (10). This constraint limits the total budgetassociated with all selected improvement combinations for highwaysegments and intersections.

• Improvement length limitation constraints (11j0), (11k0), . . . .These constraints are based on the practicality of implementing dif-ferent types of improvements along different segments of each high-way. In this research, limitations based on minimum length, horizontaldesign element length, vertical design element length, or a combi-nation of these are considered. For example, if a section of the high-way is improved by widening the through lanes, it is not practical todo it for only a short homogenous segment. Instead, there might bea minimum length on which this type of improvement makes sensebecause of practical and operational considerations. Also, for sometypes of improvements, the improvement may need to be continu-

Banihashemi 111

ous on certain types of highway segments (e.g., grade reductionsthat should be maintained on vertical design elements). These con-straints are referred to as improvement length limitation constraintsfor each type of improvement. The length of highway for which animprovement must be maintained is called the MIL in the foregoingformulation.

• Integrality constraints (12). These constraints ensure that thesolutions to the problem are integers.

Other Assumptions

Other assumptions and rules for this linear optimization model areas follows:

1. No length change is considered because of changes in horizontalalignment;

2. Superelevation improvement accompanies horizontal curvatureimprovement;

3. Superelevation improvement cost is mandatory with horizontalcurvature improvements to prevent the superelevation improvementcost from being double-counted;

4. Points of vertical intersection locations do not change; only thegrades may change;

5. No length change is considered because of a change in verticalgrade; and

6. Improvement, crash, and delay costs are all for the same periodof time (usually known as the project lifetime).

More explanation about some of these concepts and assumptionsmay be found elsewhere (3).

IMPROVEMENT COST CONSIDERATIONS

For highway segments a cost per unit length is associated with eachimprovement. This cost could be different for each homogenous seg-ment because of many factors, such as land and construction costs.An assumption is that the unit cost for each improvement is constantfor the whole length of each homogenous segment. More consider-ations about the improvements of highway segments may be foundelsewhere (3).

For each intersection, improvements, their costs, and the asso-ciated intersection delay increase or reduction are considered separately. Improvement costs and delay times need to be esti-mated in advance. The major sources for intersection improvementcosts are

• Construction cost (including the delay cost during theconstruction),

• Land cost,• Environmental cost, and• Cost of obstacle removal.

Also, it is assumed that the costs for different improvements forthe same intersection are independent from each other. Thisassumption means that the cost for improvement (s, t, . . . , 0) isequal to the summation of the costs for improvements (s, 0, . . . , 0)and (0, t, . . . , 0).

ESTIMATION OF HIGHWAY SEGMENT DELAY TIMES

Highway segment delay cost for each improvement combination isthe unit cost for delay time, CD, multiplied by the delay time for thatsegment for the improvement combination. TWOPAS software wasused to estimate the delay time for the existing highways as well asfor the highways with passing and climbing lanes. These lanes aredesigned along the high grades. The following are some assumptionsin the calculation of the segment delay times:

• The total delay along a passing or climbing lane is uniformlydistributed along the lane,

• There are 6 h of peak-period traffic with a flow rate of 12% ofthe ADT and 18 h of off-peak period with a flow rate of 1.556% ofthe ADT,

• There are 7% trucks and 3% recreational vehicles (RVs) in thetraffic flow, and

• The desired speed for passenger cars, trucks, and RVs was100 km/h, 95 km/h, and 95 km/h, respectively.

TWOPAS simulations for peak and off-peak periods were run for60 min, and the delay times were used for calculations of total delayfor the entire project period for all segments that were affected by theimprovement. These total delays were then split among the segmentsin proportion to their lengths.

ESTIMATION OF INTERSECTION DELAY TIMES

Intersection delay cost for each improvement combination is the unitcost for delay time, CD, multiplied by the delay time for that inter-section for the improvement combination. It is necessary to considerthis cost because there might be intersection improvements thatimprove the safety but increase the intersection delay time. Chang-ing a minor-stop intersection to all-stop or to a signalized one is anexample.

Equations and graphs in Appendix A (Chapter 30) of the High-way Capacity Manual (HCM) 2000 (6) were used to estimate theintersection delay times. Equations A30-11 and A30-15 were usedfor signalized and minor-stop intersections, and Exhibit A30-6 wasused for all-stop intersections.

STRENGTHS AND WEAKNESSES OF OPTIMIZATION MODEL

The major strengths of this model that do not exist in other modelsare as follows:

• Use of the MIL concept, which adds to the model the capabilityof autogenerating all improvement options;

• Consideration of the delay cost in the objective function pre-venting the selection of improvements, which adds to the safety witha high additional user cost; and

• Absence of need to conduct any preprocessing to eliminate theimprovements that increase network costs more than the benefits thatthey offer to the network. These improvements would automaticallybe eliminated from the solution. Any improvement that is chosen inthe solution would at least be better than the do-nothing option.


The major deficiency of the model, which also appears in othersimilar models, is that it is based on the assumption that the volumewill not change after the safety improvements are implemented.Improving the safety of a highway element may change the trafficpattern and reroute some traffic that was previously using a “safer”route. This situation is a safety dilemma in which an improvementin the safety of a highway element creates a reverse effect for networksafety as a whole.

IHSDM CRASH PREDICTION MODELS

The IHSDM CPM has one highway segment model and three inter-section models. For highway segments, the estimation is based onthe ADT, highway length, and highway segment productive AMFs.For intersections the estimation is based on the approach ADTsand the intersection productive AMFs. Currently, nine highway-segment AMFs and five intersection AMFs are defined in IHSDM.Improvements that decrease the value of one or more AMFs wouldresult in a safety improvement, measured by a decrease in the costof crashes.

The AMFs and base conditions for highway segments used in theCPM model are

• AMF1, lane width: 3.6 m (12 ft);• AMF2, shoulder width and type: 1.8 m (6 ft) paved;• AMF3, horizontal curvature: no curves;• AMF4, superelevation deficiency: none;• AMF5, grades: level (0%);• AMF6, driveway density: three driveways per kilometer (five

driveways per mile);• AMF7, passing lanes and short four-lane sections: none;• AMF8, two-way left-turn lanes: none; and• AMF9, roadside hazard rating: 3.

The AMFs and base conditions for intersections used in the CPMmodels are as follows:

• AMF10, skew angle: 0;• AMF11, traffic control (all-way stop versus two-way stop): two-

way stop;• AMF12, left-turn lane: none;• AMF13, right-turn lane: none; and• AMF14, intersection sight distance limitation: none.

The CPM estimation of the expected number of crashes starts withthe calculation of AMFs for different highway segments and inter-sections. For highway segments it continues with establishing thehomogenous segment. Homogenous segments are segments withinwhich ADT and AMF values are constant. New segments beginwhere ADT or one or more AMF values change. The CPM calculatesand adjusts the expected number of crashes for each homogenoussegment and each intersection by using base models and AMFs. Therate of the expected number of crashes per length along a homogenoussegment is constant. The expected number of crashes for all homoge-nous segments and intersections is summed to make the expectednumber of crashes for any given highway.

The CPM base models are described by Vogt and Bared (7, 8).Because these models are developed and validated by using data from

one or two states, the IHSDM includes calibration factors that scale theestimated number of crashes to the data of other states. The calibrationprocess is explained by Harwood et al. (1).

The CPM models (Equations 2 and 3) for highway segments andintersections are as follows:

where Cr and Ci are the state calibration factors for road segmentsand intersections, respectively. For this research, the values chosenfor Cr and Ci were the ones developed by a state in the northeasternUnited States, but any state’s factors could be substituted. The valuefor Cr for this state was 2.376. The Ci-values for this state were 0.79,0.98, and 0.94 for three-leg and four-leg stop-control and four-legsignalized intersections, respectively.

In Equation 15, α, β, and γ are −10.9, 0.79, and 0.49, respectively,for three-leg stop-control; −9.34, 0.60, and 0.61, respectively, forfour-leg stop-control; and −5.73, 0.60, and 0.20, respectively, forfour-leg signalized intersections.

In this paper, the proposed model is described in the context ofthe IHSDM CPM. The optimization model could be tailored to usealternative crash prediction models with productive AMFs by mod-ifying model components and assumptions. The IHSDM CPM mod-els could be used in conjunction with an empirical Bayes (EB)procedure in which the available crash history data would affect thepredicted number of crashes. This optimization model is for casesin which the EB procedure is not appropriate or there are not enoughcrash data to support this process. Further study is needed for caseswith the EB procedure.

LINEAR PROGRAMMING MATHEMATICAL MODEL

The mathematical optimization model for the CPM is as follows:

min 529.6 10 ADT6sc

all

× × × ×( )− C Lh i h ij k l

, ,( , , ,mm n o p q r h ii

n

h

n sh

, , , , , ) ,for

hwy

AMF

∑∑∑ ⎡

⎣⎢

== 11

SS h i j k l m n oh i j k l m n o p q rX

, ,( , , , , , , , , ) , ,( , , , , , ,pp q rS

a

h i

e

, , )

)

,

∏⎛

⎝⎜⎞

⎠⎟⎤

⎦⎥

+ + +( LnADT LnADT1α β γ 2 ×× ×C Ci i ici

n

s t u v w i

,

( , , , , )

int ⎧⎨⎩

×

=∑

∑

1

all for

AAMFS i s t u v wS

i s t u v w

i

Y,( , , , , )

) ,( , , , , )=

∏⎛⎝⎜

⎞

1

5

⎠⎠⎟⎡

⎣⎢

⎤

⎦⎥⎫⎬⎭

+ C D XD h i j k l m n o p q r, ,( , , , , , , , , ) hh i j k l m n o p q ri

n

h

n shhwy

, ,( , , , , , , , , )⎡⎣ ⎤⎦==

∑∑11

++ [ ]C D YD i s t u v w i s t u v ws t u

,( , , , , ) ,( , , , , )( , ,all ,, , )

int

v w ii

n

for

(15)∑∑=1

N e CI i SS

i

i

= × ×⎡

⎣⎢

⎤

⎦+ +

=∏( LnADT LnADT1 AMFα β γ 2

1

5)

⎥⎥ (14)

N e C LS r i i SS

h i

h

= × × × × ×( )− −365 10 6 0 4865.

,

,

ADT AMFii

sb

i

n

==∏∑

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪1

9

1

(13)

Banihashemi 113

subject to

where ( j, k, l, m, n, o, p, q, r) is the improvement combination forhighway features represented in the safety model through AMFs(i.e., lane width, shoulder width, horizontal curve, superelevation,vertical grade, driveway density, passing lane/climbing lane/shortfour-lane section, two way left-turn (TWLT) lane, and roadsidehazard rating, respectively).

Other parameters and variables as well as the description of thestatements are as previously defined for the general model. The MILs

for different types of improvements are as follows:

• For lane width, combination of minimum length and horizontaldesign element length;

• For shoulder width, combination of minimum length andhorizontal design element length;

• For horizontal curvature, corresponding horizontal curve;• For superelevation, corresponding horizontal curve;• For vertical grade, corresponding grade;• For driveway density, combination of minimum length and

horizontal design element length;• For passing lane/climbing lane/short four-lane section, corre-

sponding grade for which these lanes are designed;• For TWLTL, combination of minimum length and horizontal

design element length; and• For roadside hazard rating, combination of minimum length

and horizontal design element length.• Statement 21: integrality constraints ensuring that all variables

are integers.

all variables integer (2 )= 0

. . . (19 )0r

Xh i j k l m n o p q rj l m n o p q

, ,( , , , , , , , , ), , , , , , ,

0all rr

h i j k l m n o p q rj l

X...

, ,( , , , , , , , , ),

∑ − =+1 00

all ,, , , , , , ...

, , . . . ,

m n o p q r

h n i

∑

∀ = ∀1 2 hwy and annd 1 1, 2 1

with the same MIL

i nSh+( ) = −( ), . . . ,

ss k(19 )0

. . . . . .

Xh i j k l m n o p q rk l m n o p q

, ,( , , , , , , , , ), , , , , , ,

0all rr

h i j k l m n o p q rk l

X...

, ,( , , , , , , , , ),

∑ − =+1 00

all ,, , , , , ,

, , . . . ,

m n o p q r

h n i

∑∀ = ∀1 2 hwy and and ii nS

s

h+( ) = −( )1 1, 2 1

with the same MIL (1

, . . . ,

88 )0j

L C Xh i h i j k l m n o p q r h i j k l m, , , , , , , , , , , , ,( , , , ,( ) nn o p q rj k l m n o p q r h ii

, , , , )( , , , , , , , , ) ,all for

∑=111

n

h

n

i s t u v w i s t u v

Shhwy

C Y

∑∑=

+ ,( ), , , , ,( , , , ,wws t u v w i

n

)( , , , , )

≤∑∑ total costall for1i=

int

((17)

Y ii s t u v ws t u v w i

( , , , , )( , , , , )

, ,all for

∑ = ∀ =1 1 2 .. . . ,nint (17)

Xh i j k l m n o p q rj k l m n o p q

, ,( , , , , , , , , )( , , , , , , ,all ,, )

, , . . . ,

, , .

r h i

h n

i

for ,hwy and∑ = ∀ =

∀ =

1 1 2

1 2 .. . ,nSh(16)

TABLE 2 Highway Segment Features for Existing and Alternative Designs

RoadsideLane Shoulder Horizontal Climbing Driveway Hazard

Alternative Width (m) Width (m) Curves Superelevation Lane Density Rating ADT

Highway 1Existing

Highway 1Alternative 1



Highway 2Existing



Highway 3Existing



NOTE: Improved features are screened.

For the CPM intersection models, certain intersection improve-ments include changing the intersection from a four-leg stop-controlled type to a four-leg signalized type. In this case, appropriateCi, α, β, and γ should be selected for the corresponding items in theobjective function.

LINEAR OPTIMIZATION MODEL TEST CASE

Test Case Network

A network containing three highways ranging from 4 km to 10 kmlong and 13 intersections were used to test the model. Table 2 showsthe characteristics of the existing design and the alternatives forhighway segments. The calculated delays for Highway 1, the onlyhighway with an alternative with a climbing lane, without and withthe climbing lane are 14,413 h/year and 10,995 h/year, respectively.Table 3 shows the characteristics for all improvement combinationsfor intersections as well as the calculated intersection delay. InTables 2 and 3, features improved in the alternatives are grayed out.The ADTs in these tables are averages for the project lifetime.

In the calculations of the delay time for Highway 1 (the onlyhighway with an alternative with a climbing lane) and intersectionsit was assumed that there are 6 h of peak-period traffic with a flowrate of 12% of the ADT and 18 h of off-peak-period traffic with aflow rate of 1.556% of the ADT. At each intersection approach itwas assumed that 10% of the approach traffic turn left and another10% turn right. The signal cycle length was 120 s for all signalizedintersections.


Test Problem Preparation

The codes for developing the highway segment components and itsinput data files are similar to those described by Banihashemi andDimaiuta (3) except that in the objective function the crash costs arecalculated instead of number of crashes. Excel spreadsheets were usedto prepare the intersection components of the test problem. The seg-ment and intersection components were then combined to become thenetwork test problem.

Test Problem Characteristics and Improvement Costs

The test problem characteristics for all three highways and 13 inter-sections are as follows: 135 homogenous segments after matching,3,385 variables, and 535 constraints.

The expected number of crashes, delay time for Highway 1, inter-section delays, and segment and intersection improvement costs areall calculated for the project lifetime of 20 years. Table 4 shows esti-mated unit cost ranges for selected highway segment improvementswith different alternatives for all three highways; features improvedin the alternatives are grayed out.

For intersection improvements the following are the ranges forthe costs:

• Removing skew angle: $600,000 for Intersections 7, 12, and 13;• Adding left-turn lane: $300,000 for Intersections 1 and 2;

$400,000 for Intersections 5 and 6; $600,000 for Intersections 9, 10,and 13; and $100,000 for Intersection 12;

3.00

3.00

3.30

3.60

3.30

3.30

3.65

3.00

3.00

3.60

0.00

0.00

1.80

0.90

1.25

1.25

2.40

1.25

1.25

1.80

Existing

Existing

Redesign (flatten)all curves withR < 230 m

Redesign (flatten)all curves withR < 250 m

Existing

Existing

Existing

Existing

Existing

Existing

Existing

Superelevation deficienciesremoved

Superelevation deficiencies for flattened curvesremoved

Superelevation deficiencies for flattened curvesremoved

Existing


Existing

Existing


Existing

—

—

—

870 m

—

—

—

—

—

—

1

1

1

1

9 to 26

9 to 26

6

4 to 11

4 to 11

4 to 11

3

3

2

1

3

3

3

3

3

3

5,444

5,444

5,444

5,444

7,100

7,100

7,100

9,000

9,000

9,000

• Removing intersection sight distance deficiency: $500,000 forIntersection 7, and $600,000 for Intersection 13.

The traffic control change improvement costs were considerednegligible. Improvement costs were estimated roughly to simulatevariations of improvements in the real world.

Budget Limitation Options and Test Problem Results

There were nine different budget limitation options (except the do-nothing option). The MILP problems for these eight scenarios weresolved by using CPLEX optimization software. The first scenariowas no limitation, for which $15 million was chosen in the mathe-matical model for the total cost. For this scenario, $12,201,510 was

Banihashemi 115

the calculated cost to minimize the network total crash and delaycosts within the improvement options proposed by all alternatives.The eight other budget limitation options were $10 million, $8 mil-lion, $6 million, $5 million, $4 million, $3 million, $2 million, and$1 million.

Table 5 shows the budget limitations, calculated improvementcosts, and selected improvements for highway segments and inter-sections for all scenarios. In Table 5, LW, SW, HC, SE, CL, DD, andRHR stand for improvements related to lane width, shoulder width,horizontal curve, superelevation, climbing lane, driveway density,and roadside hazard rating, respectively. Numbers attached to theseabbreviations stand for the alternative from which each improvementis selected, followed by the percentage of highway length with theselected improvements (e.g., SW2-75 represents Alternative 2 forshoulder width applied to about 75% of the highway length for which

TABLE 3 Intersection Features and Intersection Delay for All Improvement Combinations

Improvement Skew Angle Traffic Left-Turn Right-Turn Sight Distance IntersectionIntersection ADT1 ADT2 Combinations (degree) Control Lane Lane Deficiency Delay (h/year)

1 7,100 2,500 0 (existing) 4.75 Minor stop No No No 3,0391 4.75 Minor stop Yes No No 3,0392 4.75 All stop No No No 14,8523 4.75 All stop Yes No No 9,151


3 7,100 2,500 0 (existing) 9.00 Minor stop No No No 6,0781 9.00 Signalized No No No 98,165

4 7,100 2,500 0 (existing) 3.77 Minor stop No No No 6,0781 3.77 Signalized No No No 72,450



7 7,100 5,000 0 (existing) 43.14 Minor stop No No Yes 67,6981 43.14 Minor stop No No No 67,6982 0.00 Minor stop No No Yes 67,6983 0.00 Minor stop No No No 67,698

8 9,000 2,500 0 (existing) 23.25 Minor stop No No No 5,4381 23.25 All stop No No No 66,412


10 7,975 2,500 0 (existing) 13.25 Signalized No No No 81,6071 13.25 Signalized Yes No No 83,030

11 6,950 2,500 0 (existing) 17.50 Minor stop No No No 2,9351 17.50 All stop No No No 8,994

12 6,950 2,500 0 (existing) 7.00 Minor stop No No No 5,8701 7.00 Minor stop Yes No No 5,8702 0.00 Minor stop No No No 5,8703 0.00 Minor stop Yes No No 5,870

13 6,950 2,500 0 (existing) 8.00 Minor stop No No Yes 5,8701 8.00 Minor stop No No No 5,8702 8.00 Minor stop Yes No Yes 5,8703 0.00 Minor stop Yes No Yes 5,8704 8.00 Minor stop Yes No No 5,870

NOTE: Improved features are screened.

this improvement is considered). Int 1 through Int 13 stand for Inter-sections 1 through 13. Also, SA, LTL, and ISD stand for improve-ments related to skew angle, left-turn lane, and intersection sightdistance, respectively. The results show that as the improvementcost grows from zero to around $12 million, the total of crash anddelay cost is decreased from $174 million to $141 million. This is atotal cost saving of $33 million. The increase in this savings is notlinear.

The results for benefit/cost (B/C) analysis are reflected in Figures 1and 2. In this analysis the total decrease in crash and delay costs isconsidered as benefit and the total of improvement costs is consid-ered as cost. Figure 1 shows these ratios for different improvementcosts. Figure 2 shows such ratios for the increments of the crash anddelay cost savings to the increments of the improvement cost. Thesetwo B/C ratios could be used to justify the level of safety improvementsuggested for the highway network.

OTHER CRASH PREDICTION MODELS

This optimization model was developed on the basis of the crashprediction models with productive AMFs. For any other crash pre-diction model to estimate the expected number of crashes for dif-ferent segments of highway and for different combinations ofalternatives, a similar optimization model could be developed. Thekey assumption in so doing is that the expected number of crashesfor each homogenous segment is independent from that for theothers.

INTERPRETATION OF TEST CASE RESULTS

Result summaries such as those shown for the case study in Table 5could help decision makers identify which set of improvements wouldminimize crash and delay costs for a given improvement cost. It isnotable in the results for intersection improvements that no traffic


control improvement is chosen in the solutions. The reason was thateven though these improvement options were assumed to have nocost, the delay costs added by these options were more than the safetycost savings.

Graphical presentations such as Figures 1 and 2 would help deci-sion makers determine the B/C ratios for different safety improve-ments, both ratio of total benefits to total costs and the ratio of theincrements of benefits to the increments of costs. In the case study, theB/C ratio of the totals is very high for a budget range of $0 to $2 mil-lion and stays above 2 for all improvements. The B/C ratio of theincrements is very high for a budget range of $0 to $2 million (i.e., asmall investment reaps significant safety and operational benefits).The ratio stays above 1 until the improvement cost is a little bit above$8 million and then falls below 1. The whole analysis may justifyspending up to around $6 million for improvements when the B/C ratioof the last increments is above 2. By this time, the improvement optionshave already gained close to 88% of the crash and delay cost savings(88% of $33 million). This factor may make the results acceptable evenwith a high error margin in the crash prediction model.

CONCLUSION

An MILP model for minimizing the crash and delay costs for crashprediction models with productive AMFs was presented. The modelwas used to identify the best combination of improvements for differ-ent sections of a highway network from a predefined set of improve-ment alternatives, given cost constraints. Expected crash and delaycosts were the measures for this evaluation. The IHSDM CPM wasused as an example of a crash prediction model with productiveAMFs. A network of three rural highways and 13 intersections wasselected as a case study, and unit costs associated with highway andintersection improvements were estimated. The optimization modelwas built and solved with the CPLEX optimization solver for ninebudget constraints. The solution time for the linear programming testproblem was negligible (less than 5 s). The size of the MILP prob-

TABLE 4 Unit Cost Ranges for Segment Improvements for Different Alternatives

Shoulder Horizontal Climbing Driveway Roadside HazardAlternative Lane Width Width Curves Superelevation Lane Density Rating

Highway 1 0 0 0 0 0 0 0Existing

Highway 1 0 0 0 15 0 0 0Alternative 1

Highway 1 100–150 50–100 150–250 15 0 0 30Alternative 2

Highway 1 200–300 50–100 200–350 15 500 0 60Alternative 3



Highway 2 120–200 20–80 0 0 0 80–250 0Alternative 2



Highway 3 100–300 20–100 0 0 0 0 0Alternative 2

NOTE: Improved features are screened; costs are given in dollars per meter.

Banihashemi 117

TABLE 5 Improvement Costs, Crash and Delay Costs, and Selected Improvements

Improvement CostImprovement Descriptions and Percentages

(budget limit) Crash and Delay Cost Highway Segments Intersections

12,201,510 (no limit)

$9,992,641 ($10,000,000)

$7,988,325 ($8,000,000)

$5,982,823 ($6,000,000)

$4,998,954 ($5,000,000)

$3,998,954 ($4,000,000)

$2,999,227 ($3,000,000)

$1,999,418 ($2,000,000)

$999,451 ($1,000,000)

$0 ($0)

Int 1: LTL; Int 2: LTL; Int 5: LTL;Int 6: LTL; Int 7: SA, ISD; Int 9: LTL; Int 10: LTL; Int 12: SA, LTL; Int 13: SA, LTL, ISD

Int 1: LTL; Int 2: LTL; Int 5: LTL;Int 6: LTL; Int 7: SA, ISD; Int 9: LTL;Int 12: LTL; Int 13: SA, LTL, ISD

Int 1: LTL; Int 2: LTL; Int 5: LTL;Int 6: LTL; Int 7: SA, ISD; Int 12: LTL; Int 13: LTL

Int 7: SA, ISD; Int 12: LTL; Int 13: LTL

Int 7: SA, ISD; Int 12: LTL; Int 13: LTL

Int 12: LTL; Int 13: LTL

Int 12: LTL; Int 13: LTL

IInt 12: LTL

Int 12: LTL

No improvement

$140,805,988

$141,348,942

$142,594,267

$144,834,436

$146,805,888

$149,467,886

$152,363,771

$155,763,241

$160,478,348

$173,874,383

Hwy 1: LW3-100, SW2-100, HC3-100, SE1-100, CL3-100, RHR3-100

Hwy 2: LW2-100, SW2-100, SE1-100, DD2-100Hwy 3: LW2-100, SW2-100, SE1-100

Hwy 1: LW2-70, SW2-100, HC3-100, SE1-100, CL3-100, RHR3-100

Hwy 2: LW2-100, SW2-100, SE1-100, DD2-100 Hwy 3: LW2-100, SW2-100, SE1-100

Hwy 1: SW2-100, HC2-25, HC3-75, SE1-100,CL3-100, RHR3-100

Hwy 2: LW2-25, SW2-100, SE1-100, DD2-100Hwy 3: LW2-100, SW2-100, SE1-100

Hwy 1: SW2-100, HC2-25, HC3-75, SE1-100,CL3-100, RHR3-100

Hwy 2: SW2-100, SE1-100, DD2-100 Hwy 3: LW2-90, SW2-100, SE1-100

Hwy 1: SW2-100, HC2-50, SE1-100, CL3-90,RHR3-100

Hwy 2: SW2-100, SE1-100, DD2-100Hwy 3: LW2-70, SW2-75, SE1-100

Hwy 1: SW2-100, HC2-25, SE1-100, CL3-90,RHR3-100

Hwy 2: SW2-100, SE1-100, DD2-100Hwy 3: LW2-70, SW2-75, SE1-100

Hwy 1: SW2-100, HC2-50, SE1-100, CL3-30, RHR3-100


Hwy 1: SW2-90, HC2-50, SE1-100, CL3-20,RHR3-100


Hwy 1: SW2-70, HC2-50, SE1-100, RHR3-45 Hwy 2: SW2-45, SE1-100, DD2-55 Hwy 3: SW2-5, SE1-100

No improvement

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0 2 4 6 8 10 12 14

Improvement Cost ($Millions)

B/C

Rat

io f

or

To

tal I

mp

rove

men

t C

ost

FIGURE 1 B/C ratio for total improvements.

lem and the solution time suggest that the optimization problem fora real-world network would be quite manageable.

In the case study, a crash and delay cost of $174 million is expectedfor the do-nothing option for the next 20 years. For a $1 millioninvestment, the crash and delay cost is expected to decrease to $161million, a reduction of $13 million. From $2 million up to about $8million in investments, the incremental decrease in crash and delaycost for each $1 million would stay above $1 million (a B/C ratio ofthe incremental benefits to the incremental costs above 1). Thediminishing rate of return shown in Figures 1 and 2 would be impor-tant for decision makers to note when they determine where andhow to allocate safety and operational improvement funding at thenetwork level.

On the network level, the predicted crash and delay cost savingversus improvement costs gives decision makers a basis for select-ing the most safety and operationally cost-effective projects. Theresults also would assist decision makers in maximizing networksafety and operational benefits, given a fixed amount of fundingavailable for improvements.

The presented optimization model is developed for crash predic-tion models without an EB procedure. These models usually couldbe used with an EB procedure, in which the available crash historydata would affect the predicted number of crashes. In a separatepaper (9), the author offers an expansion of this optimization modelfor highway segments to cases with an EB procedure but withoutconsideration of delays.

If crash data for all highways and intersections of the network areavailable and all of the improvements are of the type that make theuse of an EB procedure appropriate, this optimization process wouldbetter be used with the EB method. However, if crash data are avail-able for only part of the network, or if some of the improvements donot allow the use of the EB procedure, the author recommends thatthe optimization model for the whole network be developed withoutthe EB procedure.

The optimization model presented has a deterministic characteris-tic. However, the crash prediction models are inherently stochastic.A reasonable future direction of work for this problem is to use astochastic linear programming approach instead of the current deter-ministic approach. In the stochastic linear programming approach,each input datum would be a probability distribution rather than adeterministic number.


ACKNOWLEDGMENTS

The author acknowledges Raymond Krammes, of the FHWA Officeof Safety Research and Development, and Michael Dimaiuta, of theLENDIS Corporation and the FHWA Geometric Design Laboratory,for their support of this effort.

REFERENCES

1. Harwood, D. W., F. M. Council, E. Hauer, W. E. Hughes, and A. Vogt. Pre-diction of the Expected Safety Performance of Rural Two-Lane Highways.Report FHWA-RD-99-207. FHWA, U.S. Department of Transportation,Dec. 2000.

2. Interactive Highway Safety Design Model (IHSDM). http://www.tfhrc.gov/safety/ihsdm/ihsdm.htm. Accessed March 12, 2007.

3. Banihashemi, M., and M. Dimaiuta. Maximizing Safety ImprovementBenefits in Crash Prediction Models with Accident Modification Fac-tors. In Transportation Research Record: Journal of the Transporta-tion Research Board, No. 1908, Transportation Research Board of theNational Academies, Washington, D.C., 2005, pp. 9–18.

4. Harwood, D. W., E. R. Kohlman Rabbani, and K. R. Richard. SystemwideOptimization of Safety Improvements for Resurfacing, Restoration, orRehabilitation Projects. In Transportation Research Record: Journal ofthe Transportation Research Board, No. 1840, Transportation ResearchBoard of the National Academies, Washington, D.C., 2003, pp. 148–157.

5. White Paper for Module 3—Economic Appraisal and Priority Ranking.FHWA, U.S. Department of Transportation. http://www.safetyanalyst.org/docs.htm. Accessed March 12, 2007.

6. Highway Capacity Manual. TRB, National Research Council, Washington,D.C., 2000.

7. Vogt, A., and J. G. Bared. Accident Models for Two-Lane Rural Roads:Segments and Intersections. Report FHWA-RD-98-133. FHWA, U.S.Department of Transportation, Oct. 1998, p. 149.

8. Vogt, A., and J. G. Bared. Accident Models for Two-Lane Rural Seg-ments and Intersections. In Transportation Research Record 1635, TRB,National Research Council, Washington, D.C., 1998, pp. 18–29.

9. Banihashemi, M. EB Analysis in the Micro Optimization of the Improve-ment Benefits of Highway Segments for Models with Accident Modi-fication Factors (AMFs). Presented at 86th Annual Meeting of theTransportation Research Board, Washington, D.C., 2007.

The research reported here was partly conducted at the FHWA Turner–FairbankHighway Research Center as work related to development of the IHSDM. How-ever, the methods, findings, and conclusions are solely those of the author anddo not necessarily reflect the point of view of FHWA.

The Safety Data, Analysis, and Evaluation Committee sponsored publication ofthis paper.

B/C

Rat

io f

or

Incr

emen

ts o

fIm

pro

vem

ent

Co

sts

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0 2 4 6 8 10 12 14

Improvement Cost ($Millions)

FIGURE 2 B/C ratio for increments of improvement costs.

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