a stepwise highway alignment optimization using genetic algorithms

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TRB Paper No. 03-4158

A Stepwise Highway Alignment Optimization

Using Genetic Algorithms

by

Eungcheol Kim, Ph.D.Research Fellow

Department of Highway ResearchThe Korea Transport Institute (KOTI), South Korea

TEL: +82-31-910-3057, FAX: +82-31-910-3235E-MAIL: [email protected]

Manoj K. Jha, Ph.D., P.E.(Corresponding Author)

Assistant ProfessorDepartment of Civil Engineering

Morgan State University5200 Perring ParkwayBaltimore, MD 21251

TEL: 1-443-885-1446, FAX: 1-443-885-8218E-MAIL: [email protected]

and

Bongsoo Son, Ph.D.Associate Professor

Department of Urban Planning and EngineeringYonsei University, South Korea

TEL: +82-2-2123-5891, FAX: +82-2-393-6298E-MAIL: [email protected]

November 2002

Submitted for presentation at the 2003 Annual Meeting of the Transportation Research Boardand for publication in the Transportation Research Record

TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal.


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ABSTRACT

In this paper we propose a stepwise highway alignment optimization approach using

genetic algorithms for improving computational efficiency and quality of solutions. Our previous

work in highway alignment optimization has demonstrated that computational burden is a

significant issue when working with a Geographic Information System (GIS) database requiring

numerous spatial analyses. Furthermore, saving computation time can enhance adoptability of a

model especially when a study area is relatively large or involves many sensitive properties or if

locating complex structures such as intersections, bridges and tunnels is necessary. It is well

acknowledged that in many optimization processes subdividing large problems into smaller

pieces can decrease the computation time and produce a better solution. In this research two

different population sizes are used to develop a stepwise alignment optimization when

employing genetic algorithms in suitably subdivided study areas. An example study shows that

the proposed stepwise optimization gives more efficient results than the existing methods and

also improves quality of solutions.

Key Words: Stepwise optimization, Genetic algorithms, Computational efficiency, Highway

alignment optimization, Geographic information systems, Segmentation



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INTRODUCTION

The highway alignment optimization involves finding the best highway alternative

between a pair of points (1-5). The problem can be stated as follows:

Given two end points in the study area and allowing the existing conditions of the study

area changeable, find the best alignment among alternatives to optimize a specified objective

function, while considering needed structures and satisfying design and operational

requirements.

For more reliable and realistic applications highway alignment optimization processes

should consider many factors, which increase the complexity of the problem. The factors may

include structures, topography, socio-economics, ecology, geology, soil types, land use patterns,

environment and even community concerns. They are considered with different emphasis and

levels of detail at different stages in the alignment selection processes. Traditionally, these

processes have consumed much time and effort of agencies, planners, engineers and residents.

Several models have been developed in response to this need. They can save considerable time

and costs compared to the traditional manual methods using computers and mathematical

formulations (6-8). Recently, a solution approach (1, 4-5) based on genetic algorithms (GAs) for

three-dimensional highway alignment optimization has been developed. The GA advantages to

the highway alignment optimization problem over traditional methods have been extensively

covered in (1-5); therefore, have been skipped here for brevity.

A model integrating geographic information systems (GIS) with such a GA has also been

developed (2). Furthermore, there has been an effort to incorporate structures such as

intersections, tunnels, bridges and interchanges into the optimization process for improving

practical usability of the models (3).



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Although the first objective of the many developed models is to obtain the best alignment

(global optimum or at least near global optimum), computational efficiency of the models is also

of great concern since it largely affects the degree of a models adoptability. The computational

burden especially increases (2) when the number of properties to be analyzed for right-of-way

cost calculation and environmental impact assessment increases.

It is well known that in many optimization processes, subdividing large problems into

suitable pieces can decrease the computation time and produce a better solution. This argument

also applies to this study, since optimizing highway alignments repeatedly involves fine-tuning

search steps during successive search processes.

LITERATURE REVIEW

Theoretically, highway alignment optimization problem involves an infinite number of

alternatives to be evaluated. In previous applications (1-5) the optimization problem was

formulated as a cost minimization problem in which cost functions are non-differentiable, noisy

and implicit. Thus, it is inevitable to use fast and efficient search algorithms to solve such a

problem.

According to Table 1, seven search methods (1-28) are used for alignment optimization

models. Among those, all have some critical defects when applied to the highway alignment

optimization problem except genetic algorithms. Table 2 summarizes these defects.

GENETIC ALGORITHMS AS AN OPTIMAL SEARCH

Genetic Algorithms (GAs) have been proven to be very effective to highway alignment

optimization problems (1-5) since they can effectively search in a continuous search space

without getting trapped in local optima. Goldberg (29) states four important distinctions of GAs



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over other search methods:

(1) GAs work with a coding of the parameter set, not the parameters themselves.

(2) GAs search from a population rather than a single point.

(3) GAs use payoff (objective function) information, not derivatives or other auxiliary

knowledge.

(4) GAs use probabilistic transition rules, not deterministic rules.

In addition it is found that GA is highly efficient means of searching a large solution

space. Some computational details of GA application to optimize three-dimensional highway

alignments (1) relevant to this study is described next.

Data Format for Describing the Region of Interest

A matrix format as shown in Figure 1, is employed to minimize the needed memory and

carry important information for the entire region. The coordinates of the origin (bottom left

corner) are labeled as ),( OO yxO and the dimensions of each cell are Dx and Dy . We further

denote maxx and maxy as the maximal X and Y coordinates of the study region.

Decision Variables (Points of Intersections)

In highway engineering processes, points of intersections ( iP , see Figure 2) are used to

initially locate alignments. Those points are then connected linearly to make tangent sections.

Finally, appropriate curves are fitted to create a smooth and continuous alignment. Genetic

algorithms adopted here exactly follow the above real engineering processes. Therefore, points

of intersections ( iP ) are the decision variables for alignment optimization and a set of points of

intersections describes one specific highway alternative. In Figure 2, iC and iT mean points of

curvature and points of tangency, respectively. For notational convenience, we further denote

SPT == 00 and EPC nn == ++ 11 as the start and end points of the alignment.



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Genetic Encoding of Alignment Alternatives

Each point of intersection is determined by three decision variables, namely the X , Y,

and Zcoordinates (1, 4-5). For an alignment represented by n points of intersections, the

encoded chromosome is composed of n3 genes. Thus, the chromosome is defined as:

[ ] [ ]nnn PPPPPPnnn

zyxzyx ,,,......,,,,,......,,,11131323321

==

(1)

where: = chromosome

i = thethi gene, for all ni 3,.......,1=

iii PPPzyx ,, = the coordinates of the thi point of intersection, for all ni ,.......,1=

Genetic Operators

The genetic operators employed for this study are problem-specific. Each operator is

designed to work on the decoded points of intersections rather than individual genes.

1. Uniform Mutation

Let [ ]nnn 31323321 ,,,......,,, = be the chromosome to be mutated at the encoded

genes of the thk intersection point, where [ ]nrk d ,1= , Then 23 k and 13 k will be replaced by:

[ ], max23 xxr Ock = (2a)

[ ], max13 yyr Ock = (2b)

2. Straight Mutation

Let [ ]nnn 31323321 ,,,......,,, = be the chromosome for mutation. We randomly

generate two independent discrete random numbers i and j , where [ ]1,0 += nri d ,



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[ ]1,0 += nrj d , ji , and ji < . Then the intermediate genes between thethi)3( and thj )23(

will be replaced by:

+=

ijilij

il

2323

1323 )(

, 1,......,1allfor += jil (3a)

+=

ij

ilij

il

1313

1313 )(

, 1,......,1allfor += jil (3b)

+=

ijil

ij

il

33

33 )(

, 1,......,1allfor += jil (3c)

3. Non-Uniform Mutation

Let [ ]nnn 31323321 ,,,......,,, = be the chromosome to be mutated at the encoded

genes of the thk intersection point, where [ ]nrk d ,1= . We first generate two random binary digit

[ ]1,0dr . Then the alleles of 23 k and 13 k in the resulting offspring

[ ]nnnkkk 3132331323321 ,,,...,,,,...,,, = are determined by the following rules:

(1) If the first random digit [ ] 01,0 =d

r , then

),( 232323 Okkk xtf = (4a)

If the first random digit [ ] 11,0 =dr , then

),( 23max2323 += kkk xtf (4b)

where: f is defined

+=

ijil

ij

il

)()(

, 1,......,1allfor += jil

(2) If the second random digit [ ] 01,0 =dr , then

),( 131313 Okkk ytf = (5a)



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If the second random digit [ ] 11,0 =dr , then

),( 13max1313 += kkk ytf (5b)

4. Whole Non-Uniform Mutation

This operator applies the non-uniform operator to each point of intersection of a given

chromosome in a randomly generated sequence to change the entire configuration of the

corresponding horizontal alignment.

5. Simple Crossover

Let two parents [ ])3()13()23(321 ,,,......,,, nininiiiii = and

[ ])3()13()23(321 ,,,......,,, njnjnjjjjj = be crossed after a randomly generated position

k3 , where [ ]nrk d ,1= . Then the resulting offspring are

[ ])3()13()23()13()3(321 ,,...,,,...,,, njnjnjkjkiiiii += (6a)

[ ])3()13()23()13()3(321 ,,...,,,...,,, nininikikjjjjj += (6b)

6. Two-point Crossover

Let [ ])3()13()23(321 ,,,......,,, nininiiiii = and

[ ])3()13()23(321 ,,,......,,, njnjnjjjjj = be the two parents to be crossed between two

randomly generated positions k3 and l3 , where [ ]nrk d ,1= , [ ]nrl d ,1= , lk , and lk< . Then

the resulting offspring are

[ ])3()13()3()13()3(321 ,...,,...,,,...,,, nililjkjkiiiii ++= (7a)

[ ])3()13()3()13()3(321 ,...,,...,,,...,,, njljlikikjjjjj ++= (7b)

7. Arithmetic Crossover

Given two parents [ ])3()13()23(321 ,,,......,,, nininiiiii = and



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[ ])3()13()23(321 ,,,......,,, njnjnjjjjj = , the arithmetic crossover reproduces two offspring

as follows:

jii += )1( (8a)

ijj += )1( (8b)

where [ ]1,0cr=

8. Heuristic Crossover

Let the two parents to be crossed by this operator be denoted by

[ ])3()13()23(321 ,,,......,,, nininiiiii = and

[ ])3()13()23(321 ,,,......,,, njnjnjjjjj = , where we assume )()( jTiT CC (i.e., i is

at least as good as j ). Then the operator generates a single offspring according to the

following rule:

iji += (9)

where [ ]1,0cr=

Further details on genetic encoding and operators can be found in Jong et al. (4), and Jong and

Schonfeld (5).

METHODOLOGY

When obtaining an alignment alternative through an optimization process, the expected

outputs are three-dimensional coordinates of the alignment centerline. To describe highway

alignments (or centerlines of highways), a parametric representation is useful (30-32). In the

proposed method a smooth and continuous alignment is explored in a given solution space.



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Boldface capital letters will be used to denote vectors in space. Let Tuzuyuxu )](),(),([)( =P be

a position vector along the alignment L , where

=1

0

0

)(

)(

dtt

dtt

u

u

P

P

and

222 ))(())(())(()( uzuyuxu ++=P . Basically, P is parameterized by u , which represents

the fraction of arc length traversed to that point. IfL is an alignment connecting

T

SSS zyx ],,[=S andT

EEE zyx ],,[=E , then the position vector )(uP must satisfy SP =)0( ,

and EP =)1( . )(uP must also be continuous and continuously differentiable in the interval

[ ]1,0u .

Alignment Optimization Model Formulation

Model formulation consists of two parts: (1) objective function and (2) constraints. The

objective function is the total cost function having five main components (user cost ( UC ), right-

of-way cost ( RC ), pavement cost ( PC ), earthwork cost ( EC ) and structure cost ( SC )) as shown

in Equation (10).

SEPRUTzyxzyx

CCCCCCnPnPnPPPP

++++=,,,.....,,

11,1

Minimize (10)

subject to nixxxiPO

,.....,1,max = (10a)

niyyyiPO

,.....,1,max = (10b)

where ),( OO yx = the YX , coordinates of the bottom-left corner of the study region (Fig. 1)

),(ii PP

yx = the YX , coordinates of points of intersections, iP

),( maxmax yx = the YX , coordinates of the top-right corner of the study region (Fig. 2)



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Kim et al. 10

where +=u

dttytxuh0

22 ))(())(()(

(4) Vertical Curvature Constraint

As in the case of horizontal curvature constraint, parabolic curves curvature in vertical

alignments should be less than the maximum allowable value, maxV . This constraint can be

expressed as the minimum length of the vertical curve, mL (35-37).

Crest Curve

( )22

22100 od

dm

hh

SAL

+

= , if dm SL > (13)

( )

+=

A

hhSL

od

dm

2

221002 , if dm SL < (14)

where mL = minimum length of vertical curve (ft)

A = algebraic difference in grades (percent), 1 ii gg

dS = sight distance (ft),)(30

67.3)(2

ir

ddd

gf

VViS

++=

dh = height of driver's eye above roadway surface (ft)

oh = height of object above roadway surface (ft)

Sag Curve

d

dm

S

SAL

5.3400

2

+=

, if dm SL > (ord

d

S

SA

5.3400 +>

) (15)



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Kim et al. 12

1.25; average accident cost ($/per accident) 20000; unit cutting cost ($/cub yard) 35; unit filling

cost ($/cub yard) 20; unit transportation cost for moving earth from a borrow pit ($/cub yard) 2;

unit transportation cost for moving earth to a landfill ($/cub yd) 3; analysis period (years) 30;

interest rate (decimal) 0.06; annual average daily traffic 2000; traffic growth rate (decimal)

0.005. In the interest of the page limitations set by TRB it is not possible to give all the details

on how these values are used in the model. Readers should refer to (1-5, 33).

Although the model is designed to automatically select the best crossing type of the new

alignment with the existing road, in this example it is assumed that users specify an intersection

as the crossing type with the existing road. A desktop computer with 1 GHz CPU speed and 261

MB RAM is used to run the program.

Figures 4 shows the optimized solution and other useful information. The figure shows

three main window areas: (1) horizontal alignment, (2) vertical alignment and (3) generation

number and best solution value. The best solution contains two bridges, two tunnels and an

intersection crossing the existing road with approximately 70 degrees.

Table 3 provides general information for the test run. Computation time took 4 minutes

and 50 seconds for 500 generations. Since the existing road initiated an additional module for

intersection evaluation, 4 minutes and 50 seconds are found to be relatively longer when

considering other types of structures. For instance, 3 minutes and 24 seconds took for a grade

separation and 3 minutes and 25 seconds consumed for an interchange. Please note that this is a

relatively simpler example in which saving a few minutes of computing time may not be very

significant; however, for larger problems with heterogeneous land use, especially when the

model is connected to a GIS (2) saving in computing time assumes particular significance.



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Kim et al. 13

Total costs are found to be 21.03 million for approximately 1.5 miles long alignment. It

is also found that user costs account for 33% of the total costs. Structures costs including an

intersection, two bridges and two tunnels are 28% while construction costs only account for 39%.

STEPWISE ALIGNMENT OPTIMIZATION

Now, our concern is to examine if the stepwise approach yields any improvement in

computational efficiency and the quality of solution. In many optimization processes,

subdividing large problems into suitable pieces can decrease the computation time and produce a

better solution. This argument also applies to this study, since modeling intersections and other

structures in alignment optimization repeatedly involves fine-tuning search steps for structures.

Another issue for computational efficiency and search performance is the population size.

Goldberg (29) has shown that the efficiency of a GA in reaching a global optimum instead of

local ones largely depends on the population size.

In our application, the population size for each generation is set proportionally to the

number of decision variables (points of intersections, iP s). For example, if three points of

intersections are used for generating highway alignments, then the population size is set at 30 (=

3 10) while a population of 150 is used for 15 points of intersections.

The artificial study area previously used is chosen for a stepwise alignment optimization

and three scenarios shown in Table 4 are designed to check the search performance and

computational efficiency. Scenarios 1 and 2 are devised for a one-step optimization while

scenario 3 is for a two-step optimization. The results of scenarios 1 and 2 can be used for

assessing the effects of the population size on computational time and the quality of solutions



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Kim et al. 14

while the result of scenario 3 can be compared to the results of both scenario 1 and 2 for

checking how much improvement is found with a two-step optimization.

The crossing type with the existing road (Fig. 3) is again assumed to be intersection, to

preserve comparison basis. The unit excavation cost is assumed to be $100 and the limiting

value beyond which tunnels are considered rather than cuts is assumed to be 20 ft. Since the

optimization processes using GA is stochastic, each program run shows different results.

Therefore, several runs (we call it replications) need to be made to check (1, 4) the variance of

results. Therefore, three replications are run for scenarios 1 and 2. Table 5 shows comparison

between two scenarios and Figures 5 and 6 show the best solutions among three replications

under scenarios 1 and 2.

In scenario 1, two bridges, one tunnel and an intersection are found while scenario 2

shows one bridge, an intersection and no tunnels in the best solutions. Total costs of the best

solutions for each scenario significantly decreased from $22.14 million to $17.29 million ($4.85

million, 21.9% improvement) while computation time for scenario 2 is 4.72 times longer for

scenario 1. These results indicate various tradeoffs between cost and computational time. Please

note that GA does not guarantee a global optimal solution rather it gives a near optimal solution.

Also, for a problem such as ours it is possible to obtain significantly different solution values for

slightly different alignments requiring bridge/tunnel constructions versus cut/fill. Therefore,

caution should be exercised in interpreting the applications of the stepwise approach.

To check computation time and the quality of solutions of the two-step optimization, the

three points of intersections of scenario 1 are obtained after the one-step optimization and used to

subdivide the whole alignment into four segments. Figure 7 shows the resulting segmentation.

Since direct use of the points of intersection for the start or end points of each segment may not



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Kim et al. 16

This study presented a stepwise highway alignment optimization procedure using genetic

algorithms, one of the artificial intelligence (AI) techniques. The stepwise optimization is based

on different population sizes and segmentation of study areas into suitable pieces. The proposed

stepwise approach is implemented in an artificial test example, which indicates that substantial

improvement in computing efficiency can be achieved with the stepwise approach. The

approach also improves the solution (i.e., an economical alignment is obtained) compared to the

traditional one stage approach. More test cases with larger problem size and additional GA

scenarios are needed to be run to investigate the full potential of the stepwise approach.

To subdivide a study area, it is recommended that a one-step optimization be run with a

relatively small number of decision variables ( iP s). Then the relevant iP s location for

subdivision should be selected based on: (1) the possibility for construction of structures and (2)

the precision requirements. The lengths of segments may also differ depending on the precision

requirements and need for savings in computing time.

The method was not implemented on a real map using a GIS. Further research is

necessary to examine how much improvement in computational efficiency and the quality of

solutions can be achieved when the stepwise optimization is adopted for real application.

ACKNOWLEDGEMENTS

The authors wish to thank the four anonymous reviewers whose valuable comments

enhanced the quality of the paper. This research has been partially performed by the Advanced

Highway Research Center funded by the Korea Science and Engineering Foundation affiliated to

the Korea Ministry of Science and Technology.



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Kim et al. 17

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TABLE 2 Defects of the Existing Highway Alignment Optimization Methods

Methods Defects

Calculus of variations Requires differentiable objective functions

Not suitable for discontinuous factors

Tendency to get trapped in local optima

Network optimization Outputs are not smooth

Not for continuous search space

Dynamic programming

Outputs are not smooth

Not suitable for continuous search space

Not applicable for implicit functions

Requires independencies among subproblems

Enumeration Not suitable for continuous search space

Inefficient

Linear programming Not suitable for non-linear cost functions

Only covering limited number of points for gradient

and curvature constraints

Numerical research Tendency to get trapped in local optima

Complex modeling

Difficulty in handling discontinuous cost items



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FIGURE 8 Optimized Solution for Segment 1 under Scenario 3

Bridge



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Kim et al. 37

FIGURE 10 Optimized Solution for Segment 3 under Scenario 3



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a stepwise highway alignment optimization using genetic algorithms

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