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19

Transportation Research Record: Journal of the Transportation Research Board,

No. 1985, Transportation Research Board of the National Academies, Washington,

D.C., 2006, pp. 1928.

An algorithm to solve explicitly the path enumeration problem is pro-

posed. This algorithm is based on the branch-and-bound technique

and belongs to the class of deterministic methods along with existing

approaches that combine heuristic or randomization procedures with

shortest-path search. The branch-and-bound algorithm is formulated,

and a methodology is designed for the application of deterministic ap-

proaches to a real case study. Path sets generated with different methods

are compared for behavioral consistency, namely, the ability to reproduce

actual routes chosen by individuals driving habitually from home to

work. Choice set compositions for modeling purposes are determined for

the consistency of the path generation process with the observed behav-

ior. Further, model estimates and performance for different route choice

specifications are examined for both path set compositions. Results sug-

gest that the proposed branch-and-bound algorithm generates realistic

and heterogeneous routes, reproduces better the observed behavior of the

interviewed drivers, and produces a good choice set for route choice

model estimation and performance comparison.

Route choice behavior modeling generally considers separately the

individuation of available alternative routes and the calculation of the

probability of choosing a certain route from the generated choice set.

Enumeration of alternatives is not a straightforward problem

since the actual size of real networks creates problems in the choice

set definition. The generated set should exclude unrealistic pathsthat no traveler would ever consider and highly similar paths that no

traveler would ever differentiate between and include relevant and

heterogeneous routes that different travelers would choose.

Several approaches found in the literature to generate routes are

based on variations to the shortest-path search. Definition of objective

functions to be minimized, formulation of heuristic rules, and imple-

mentation of randomization techniques are combined with K-shortest-

path algorithms in order to create a path set. The analysis of relevance

and heterogeneity of paths generated with methods relying on the

shortest-path search suggests a different approach to the problem.

An algorithm based on the branch-and-bound technique is pro-

posed in this study, which intends to explicitly generate realistic and

heterogeneous routes while limiting computational costs. This proce-

dure constructs a connection tree between the origin and destination

of a trip by processing sequences of links according to a branching

rule that accounts for logical constraints formulated to increase route

likelihood and heterogeneity. Each sequence of links connecting ori-

gin and destination and satisfying all the constraints enters the choice

set as a feasible solution to the path enumeration problem.

The application of the branch-and-bound algorithm to a real case

study is illustrated here. The procedure is compared with different

deterministic techniques with respect to the ability to reproduce actual

route choices of individuals driving habitually from home to work in

an urban network. A choice set for modeling purposes resulting from

the application of the method is constructed by considering path set

consistency with the observed behavior and is used for comparing

estimates and performance of different model specifications.

DETERMINISTIC APPROACHESFOR PATH GENERATION

The exhaustive approach to path generation assumes unrealistically

that all physical routes connecting origin and destination of a trip are

considered. Selective approaches account for deterministic and prob-

abilistic procedures, depending on the methods used to generate the

paths. The proposed branch-and-bound algorithm enters the class

of deterministic approaches reviewed in this study. A review of the

probabilistic methods for choice set generation may be found else-

where (1). Prashker and Bekhor review the route choice models

estimated for comparing the different generated choice sets (2).

The most straightforward path generation approach searches for

the first K-shortest paths that minimize the generalized path costs.

Shortest-path algorithms, for example, that of Dijkstra (3), assumeimplicit awareness of all the link attributes. Van der Zijpp and

Fiorenzo-Catalano (4) present a constrained method to generate

routes by finding directly feasible K-shortest paths and exploiting

a wide class of constraints.

Ben-Akiva et al. (5) propose an approach for generating possible

paths by labeling each route according to a criterion for which the

path is optimum. This approach assumes that travelers may have dif-

ferent objective functions. Each criterion corresponds to a different

preferred route, and each route can be labeled according to a differ-

ent objective function. Dial (6) generalizes the labeling method by

constructing a set of efficient paths in which being efficient means

minimizing a linear combination of label costs.

Azevedo et al. (7) define an approach in which all the shortest-path

links are removed from the network to find the next best path. Themain problem with the link elimination approach is related to net-

work disconnection, since removing centroid connectors and major

junctions does not guarantee the existence of more paths between the

origin and the destination. A variant to this approach obviates the

problem by eliminating individual links or a combination of links

from the shortest path.

De la Barra et al. (8) illustrate the link penalty approach, in which

impedances of the shortest-path links are increased in order to cal-

culate the next best path. The process continues until no more new

paths are produced. Park and Rilett (9) modify this approach by not

increasing the impedance on links within a certain distance from the

Applying Branch-and-Bound Techniqueto Route Choice Set Generation

Carlo Giacomo Prato and Shlomo Bekhor

Transportation Research Institute, TechnionIsrael Institute of Technology, Haifa

32000, Israel.

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origin or the destination. Scott et al. (10) optimize the program for

determining the penalizing factor for impedances on shortest-path

links in order to generate a next best path that overlaps with the

shortest path by no more than a given number of links.

Simulation methods assume that travelers erroneously perceive

link attributes, and therefore extraction of random draws from a dis-

tribution that might represent drivers perceptions appears to be

suitable. Sheffi and Powell (11) apply the Monte Carlo technique

with multinomial probit to the traffic assignment problem. Since theassignment algorithm does not present the network loading steps,

the application presents analogies with path generation procedures.

Fiorenzo-Catalano and Van der Zijpp (12) implement a version of the

Monte Carlo technique by gradually increasing the variance of the

random components in the model in order to keep the frequency with

which new paths are found at a constant rate. Bekhor et al. (13) ver-

ify the suitability of simulation methods to produce paths similar to

those observed for a real case study in Boston.

The aspect common to the foregoing approaches is the shortest-path

search. The method proposed in this study provides an alternative to

the shortest-path-based methods by applying the branch-and-bound

technique to solve the path enumeration problem.

In transportation-related problems, Friedrich et al. (14) apply a

branch-and-bound assignment procedure for transit networks byusing a timetable-based search algorithm. Hoogendoorn-Lanser (15)

adapts the same procedure for choice set generation in the analysis

of multimodal transport networks. In these transit applications, the

method exploits predefined route sections. In the following section,

a different approach is designed for a road network.

BRANCH-AND-BOUND ALGORITHM

FOR PATH GENERATION

The branch-and-bound algorithm enumerates paths by generating a

tree of routes connecting an origin node o to a destination node d.

The preprocessing phase of the algorithm creates arrays containing

the network elements to be processed:

An array lists the generic linksLi,j by tracing initial node i, finalnodej, predecessors ofi, successors ofj, lengthD(Li,j), travel time

T(Li,j), and straight linear distanceDj,dfrom the final nodej of the link

to the destination node d; Defining a path segment Po,kas the sequence of links connecting

the origin node o to the initial node kof the next link processed, an

array records path segments Po,kby registering lengthD(Po,k), travel

time T(Po,k), number of left turns LT(Po,k), and straight linear distance

Dk,dfrom the final node kof the path segment to the destination node

d; and

An array catalogs the set Ck of path segments Po,k arriving atnode k.

The processing phase of the algorithm represents the centroid node

of the origin zone as the root of the tree and the centroid connector

to the origin node o as the only outgoing branch.

Starting from the origin node o, given a path segment Po,x from the

current tree level, all possible successors of nodex are considered. Let

L*x,y be the currently processed link starting at nodex and terminating

at nodey. Let P*o,y be the processed path segment between origin node

o and nodey, formed by adding the linkL*x,y to the path segment Po,xarriving at nodex. Let Cy be the set of all the connections to nodey.

L*x,y is inserted into the tree as an appendix of the connection set C*x if

and only if all the following conditions hold:

20 Transportation Research Record 1985

Directional constraint. Consider the straight linear distanceD*y,dfrom the final nodey of the processed linkL*x,y to the destination d

and the straight linear distanceD*x,d from the arriving nodex of the

processed connection set C*x to the destination d. The processed link

enters the connection set Cy if

where D is a distance factor larger than 1. This directional con-straint excludes from consideration links that take the driver sig-

nificantly farther from the destination and closer to the origin. The

distance factor introduces a tolerance because it considers drivers

that move slightly farther from the destination to reach a faster

road, for example, a highway.

Temporal constraint. Consider the travel time T(P*o,y) of theprocessed path segment P*o,y and the minimum travel time Tmin(Po,y)

among the path segments belonging to the connection set Cy. The

connection set Cy includes the processed path segment if

where T is a time factor larger than 1. This temporal constraintrejects path segments that travelers would consider unrealistic sincetheir travel time is excessively high for connecting the origin node o

to the arrival nodey.

Loop constraint. Consider each linkL*i,j belonging to the set of

links {L*o,(o+1),L*(o+1),(o+2), . . . ,L*(o+n),y} that form the processed path seg-

ment P*o,y. Define the functionsNs(Li,j) andNe(Li,j) that map the initial

and the final node of each linkLi,j. Consider a subsequence of links

{L*r,(r+1), L*(r+1),(r+2), . . . , L*(r+m),s} that compose a path subsegment P*r,sconnecting two generic nodes rand s on the processed path segment

P*o,y. Consider the length D(L*r,s) of each link belonging to the path

subsegment P*r,s and the minimum distance D[Ns(L*r,(r+1)),Ne(L*(r+m),s)]

between nodes rand s. The path segment P*o,y is excluded from the

connection set Cy if there exists at least one path subsegment P*r,sfor which

where L is a detour factor larger than 1. This loop constraint dis-cards path segments that travelers would not consider because they

constitute a detour larger than an acceptable value.

Similarity constraint. Considering the same definitions as forthe previous constraint, the path segment P*o,y is not taken into con-

sideration in the connection set Cy if there exists at least one path

subsegment P*r,s for which

where o is an overlap factor smaller than 1. This similarity con-straint removes highly overlapping path segments that travelers

would not consider as separate alternatives. The sequence of links

{L*r,(r+1),L*(r+1),(r+2), . . . , L*(r+m),s} is actually a detour and the overlap

factor is the minimum required detour length in order to individuate

a different path segment.

D L D N L N Lr sP

s r r e r m s

r s

,*

,*

,*

,*

,( ) > ( ) ( ) +( ) +( )1

( ) ( )( )

+( ) +( ) 1 ss r s o yP P* ,* ,* ( )( ) 3

T P T Po y T o y,*

min ,( )( ) ( ) 2

D Dy d D x d,*

,

* ( ) 1

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Movement constraint. Consider the number of left turns LT(P*o,y)in the processed path segment P*o,y. The connection set Cy includes the

processed path segment if

where LT is the maximum number of left turns for each route. Espe-cially in urban networks where traffic light regulation does not reserve

green time for left turns, these movements markedly increase traveltime by adding long waiting times at junctions before a left turn. This

movement constraint removes unrealistic path segments, causing

delay in terms of travel time and apprehension in drivers approaching

the junction.

The algorithm processes a tree level completely before path seg-

ments of the next level are considered. Two queues store unprocessed

segments of the current and the next level, and the level is completed

when the queues are empty. The algorithm completes the connection

search when all the tree levels are processed and nodey corresponds

to destination node dfor all the branches.

Figure 1 outlines the structure of the generic connection tree. The

tree width depends on the network structure and may be wider than the

usual shortest-path tree. According to operational research theory,the speed of the branch-and-bound algorithm depends linearly on

the width of the connection tree but exponentially on its depth. Con-

sequently the proposed technique behaves conveniently from the

computational perspective.

APPLICATION OF PATH

GENERATION ALGORITHMS

For modeling purposes, choice sets are considered consistent with the

observed behavior if they contain the actual chosen route among paths

produced with generation techniques. For this reason, different deter-

LTLTPo y,

* ( )( ) 5

Prato and Bekhor 21

ministic approaches to path enumeration are tested with respect to the

ability to reproduce the observed behavior of individuals driving

habitually from home to work in an urban network.

Actual route choices were collected among faculty and staff mem-

bers of Turin Polytechnic, in Italy, who participated voluntarily in a

Web-based survey. Prato et al. (16) describe the details of the ques-

tionnaire, which in the first part collected information about spatial

abilities and driving attitudes of the respondents and in the second

part recorded the chosen route and possible alternatives to reach theirworkplace. The application of path generation algorithms exploits the

network of the city of Turin, built on the basis of the urban traffic plan

designed by the municipality in 2001 and composed of 419 nodes

and 1,427 links (17).

Different path generation methods are evaluated with respect to the

coverage of the collected routes. The coverage is the percentage of

observations for which an algorithm generates a route that satisfies a

threshold for the overlap measure:

whereI() is the coverage function, equal to 1 when its argument is

true and zero when its argument is false; Onris the overlap percentage;and is a threshold for the overlap measure.

The overlap measure evaluates the consistency of a path enumer-

ation method with respect to the observed behavior by considering

the length of the links shared between generated and collected routes:

whereLnr is the overlapping length between the path generated by

algorithm rand the observed path for driver n, andLn is the length

of the observed path for driver n.

OL

Lnr

nr

n

= ( )7

max ( )r

nr

n

N

I O ( )=

1

6

tree width

node d

8

7

2

1

3

4

5

6

no e

node d

node d

node d

node d

node d

node d

node d

node d

node d

node o

level 1 level 2 level 3 level 4 level 5

tree depth

FIGURE 1 Structure of generic connection tree.

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The ideal algorithm would reproduce perfectly the observed

behavior by replicating link by link all the routes collected in the sur-

vey and would result in 100% coverage for a 100% overlap thresh-

old. The actual techniques partially reproduce the observed behavior,

and an index measures the behavioral consistency of path generation

methods with respect to the ideal algorithm by accounting for total

overlap over all the observations:

where CIris the consistency index of algorithm r, Onr,max is the max-

imum overlap measure obtained with the paths generated by algo-

rithm rfor the observed choice of each driver n, and Omax is the 100%

overlap over all theNobservations for the ideal algorithm.

Five different deterministic algorithms are applied with the

objective of maximizing the coverage function:

The branch-and-bound algorithm is implemented by definingthe parameters for the constraints of the branching rule: the dis-

tance factor is 1.10, the time factor is 1.50, the loop factor is 1.20,

the overlap factor is 0.80, and the maximum number of left turns

is 4. The first factor introduces tolerance with respect to links that

take the driver farther from the destination. The following three

factors guarantee heterogeneity among paths, since routes that are

too similar and overcircuitous are excluded from the generated

path set.

The labeling approach is applied by calculating the shortestpath with respect to route attributes such as distance, free-flow time,

travel time, and delay, which measure the level of congestion by

evaluating the difference between travel times in congested and in

free-flow conditions.

The link elimination approach is modified from the original for-

mulation described by Azevedo et al. (7) by repeating for 10 itera-tions the following three-step method: (a) computation of the shortest

path by considering the travel time, (b) elimination of a link belong-

ing to the current shortest path, and (c) computation of the next-

shortest path. Shortest-path links are eliminated if they take the driver

farther from the destination or compel the driver to turn from a high

hierarchical road to a low hierarchical road.

The link penalty approach is adapted from the original methodproposed by De la Barra et al. (8) by replicating for 15 iterations

the following three-step procedure: (a) computation of the shortest

path by considering the travel time, (b) penalization of the links

belonging to the current shortest path with a factor equal to 5% of

the travel time, and (c) computation of the next-shortest path. The

penalizing factor is a compromise between a low value for which

the same path is identified repeatedly and a high value for whichlonger paths are generated before routes that are more similar to the

shortest path.

Two simulation approaches are implemented by computing theshortest path for each draw of impedances of the links belonging to

the network. The two approaches exploit the same procedure to draw

impedances from a truncated normal distribution characterized from

the following parameters: (a) mean equal to the travel time, (b) vari-

ance equal to a percentage of the mean, (c) left truncation limit equal

to the free-flow time, and (d) right truncation limit equal to the travel

time calculated for a minimum speed assumed equal to 10 km/h. The

first simulation approach sets the variance equal to 20% of the mean

CIr

nr

n

N

O

N O= =

,maxmax

( )1 8


and extracts 25 draws. The second simulation approach defines the

variance equal to the mean and extracts 35 draws.

For all algorithms based on the shortest-path search, the number

of iterations or draws is determined by the asymptotically decreasing

ability of each technique to generate unique routes with an increas-

ing number of repetitions, and the postprocessing step consists of the

elimination of duplicated paths.

The generated path sets are compared with respect to the observed

routes, and the resulting coverage measures the goodness of fit of each

method. Considering the number of reproduced routes and the differ-

ent nature of each technique and looking for consistency of path sets

with the observed behavior, two different choice sets are built con-

sidering the same reproduced observations: the first contains all the

paths generated by the approaches relying on the shortest-path search,

and the second consists of all the paths generated by the branch-and-

bound algorithm. Six route choice models are estimated and compared

within the different choice sets: multinomial logit (MNL), C-logit,

path-size logit (PSL), generalized-nested logit (GNL), cross-nested

logit (CNL), and link-nested logit (LNL).

RESULTS OF PATH GENERATION ALGORITHMS

The application of the described methodology allows comparison

of coverage results from path sets generated with different methods

and model performance within different choice sets.

Coverage Measures of Path

Generation Algorithms

The database of the Web-based survey responses contains 276 ob-

servations. Incomplete observations, presenting either missing val-

ues in the attitudinal section or any incorrectly coded route in the

path-recording section, are excluded from consideration.Of the 236 observations entering the clean data set 90% contain

information not only about the chosen route but also about the alter-

native routes considered for reaching the workplace. A total of 236

actual chosen routes and 339 possible alternatives, covering 182 dif-

ferent origindestination pairs, constitute the comparison term for

measuring the coverage of the path generation algorithms described

in the methodological section.

Table 1 presents coverage results according to different overlap

thresholds varying from complete replication to the reproduction of

70% of the collected routes.

Each of the single labels performs weakly, which suggests that

the actual behavior of habitual drivers does not usually correspond to

the shortest-path selection. The analysis of the combined effect of the

four labels shows that only 40% of the chosen routes are replicated,

whereas almost 45% are reproduced with 80% overlap threshold.

The link elimination approach duplicates almost 60% and covers

almost 70% of the observed routes with an 80% overlap threshold.

It also outperforms the link penalty and simulation techniques, with

a small variance for the probability distribution. The simulation

method with large variance surpasses the thresholds of 60% of the

replicated routes and 70% of at least 80% overlapping routes.

The branch-and-bound algorithm largely outperforms each single-

generation algorithm by replicating over 90% and reproducing over

96% of the observed routes. The branch-and-bound algorithm also

slightly outperforms the path set resulting from the combination of all

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Prato and Bekhor 23

null overlap, mainly among the alternatives rather than among the

chosen paths and mainly for shortest-path-based techniques rather

than for the branch-and-bound algorithm.

Behavioral consistency constitutes the most desirable property of

the generation techniques, but providing guidelines for selecting the

preferred method should consider the trade-off with computational

performance. Table 2 shows the computational costs for the tech-

niques applied to this case study and allows examination of the

trade-off between implementation time and consistency.For each origindestination pair, minimizing each label requires

time for shortest-path calculation. The link elimination and link pe-

nalty approaches demand longer times for rewriting the network by

removing one link or updating costs at each iteration. The simulation

approaches require additional time for drawing random travel times

before the shortest path is calculated. The branch-and-bound method

exploits partial path subsegments calculated for different origin

destination pairs, with consequent reduction of computational costs

while additional origindestination pairs are processed.

The labeling approach shows excessive inconsistency with respect

to the observed behavior and provides limited increase in coverage

with additional labels, and consequently further implementation of the

method appears inadvisable. The link penalization, link elimination,

and simulation techniques present asymptotical behavior with respectto the ability to produce unique routes and consequently additional

iterations give reduced gain in coverage.

The branch-and-bound technique gives excellent results in terms

of coverage, computational costs similar to those of techniques

largely outperformed, and shows a good trade-off between behav-

ioral consistency and implementation time. For example, with time

comparable with that of the simulation method with small variance,

the 40% coverage increase of replicated routes indicates that the

proposed algorithm performs excellently.

Choice Set Composition for Model Estimation

Reproduction of the actual chosen route indicates the consistency ofeach algorithm with respect to the observed behavior of each respon-

dent. Inconsistent observations for which path enumeration methods

fail to reproduce the actual choice are excluded from choice sets pre-

pared for model estimation.

To visualize the consistency of the applied algorithms regarding the

observed behavior, Figure 2 represents the distribution of coverage

over the cumulative percentage of observations.

The ideal algorithm would replicate all the observed routes and

draw a horizontal line at the 100% overlap threshold. The area below

TABLE 1 Coverage Resu lts of Appli ed Alg or ith ms

Overlap Threshold

Algorithm 100% 90% 80% 70%

Chosen routes

Labeling approach 26.69 26.69 30.93 34.32(least length)

Labeling approach 26.27 26.27 27.54 29.24(least free flow time)

Labeling approach 17.80 17.80 18.22 22.46(least travel time)

Labeling approach 21.19 21.19 22.88 23.73(least delay)

Link elimination 58.47 58.47 69.92 81.78approach

Link penalty approach 53.81 53.81 62.29 68.22

Simulation with small 49.15 49.15 54.24 59.32variance

Simulation with large 61.44 61.86 71.19 81.36variance

Branch-and-bound 91.10 91.53 96.61 97.88algorithm

Alternative routesLabeling approach 13.27 13.57 22.42 26.25

(least length)

Labeling approach 21.53 21.53 22.42 30.68(least free flow time)

Labeling approach 15.93 15.93 16.22 19.17(least travel time)

Labeling approach 17.40 17.40 17.99 20.35(least delay)

Link elimination 54.87 54.87 66.37 76.11approach

Link penalty approach 43.95 43.95 51.62 61.65

Simulation with small 42.77 42.77 48.08 61.06variance

Simulation with large 55.16 55.16 64.90 76.11variance

Branch-and-bound 82.60 82.60 89.38 95.58algorithm

TABLE 2 Compu tatio na l Performance of App lied Algor ithms

Algorithm Computational Time Number of Unique Routes Consistency Index

Labeling approach (length) 1.5 h 182 53.54

Labeling approach (free flow time) 1.5 h 182 49.36

Labeling approach (travel time) 1.5 h 182 43.34

Labeling approach (delay) 1.5 h 182 44.46

Link elimination approach 36 h 958 87.16

Link penalty approach 54 h 1164 81.29

Simulation with small variance 42 h 1097 75.49

Simulation with large variance 58 h 3305 88.12

Branch-and-bound method 40 h 2038 97.91

NOTE: Computations performed using Borland Pascal and Microsoft Excel XP on an Intel Pentium IV 3.06 GHz with 512 MbRAM running Windows XP Home.

the other methods since the coverage of the merged path set duplicates

87% of the routes and covers the 95% with an 80% overlap threshold.

Results show the tendency of all algorithms to perform better

with respect to the actual choices than to the possible alternatives

considered by the respondents. More than 700 generated routes are

completely inconsistent with the observed behavior by presenting

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each line representing the distribution of the coverage measures the

consistency of each algorithm with respect to the rectangular area

that the ideal algorithm would individuate. The consistency of the

applied methods varies from 67.2% for the labeling approach to

97.9% for the branch-and-bound method.

With the objective of constructing choice sets for model estimation,

the 80% overlap threshold individuates the observations to be in-

cluded for each algorithm. With the objective of increasing the relia-

bility of choice set comparison by considering the same observations

and a high number of observations, two choice sets are constructed.

The first choice set merges the path sets generated with the algo-

rithms based on the shortest-path search, since each single algorithm

reproduces a limited number of observations. The second choice set

corresponds to the path set obtained with the branch-and-bound tech-

nique. Both choice sets account for the same 223 observations and

results consistent with the same observed behavior.

Figure 3 illustrates differences between the two choice sets in

terms of number of alternatives and number of links for each obser-

vation. For the choice set obtained by merging paths generated with

different algorithms, the median size counts 32 routes, more than

one quarter of the observations contain at least 40 paths, and the

maximum number of alternatives reaches 55. For the choice set

generated with the branch-and-bound technique the median size is

17 routes, only 6% of the observations consist of 40 paths or more,

and the maximum number of alternatives is 44.

Both choice sets include a high number of alternatives and pre-

sumably contain routes that drivers would not consider. Considering

the number of links for each observation with respect to the merged

path set, the branch-and-bound path set presents a higher ratio be-

tween links and routes. This finding indicates that the paths share

fewer links and are most likely to be more heterogeneous.


Route Choice Model Estimation

Data sets for model estimation account for

Level-of-service variables, such as distance, free-flow time, andtravel time;

Landmark dummy variables, equal to 1 if the route crosses thelandmark and zero otherwise; and

Behavioral variables, measured at the individual level byapplying factor analysis to the behavioral indicators (16).

In contrast to the MNL model, different model specifications

account for similarity among alternatives and require the estima-

tion of additional parameters. The C-logit and PSL models main-

tain the logit structure by including a correction term within the

deterministic part of the utility function. The following formula-

tions for commonality factor and path size are applied for C-logit

and PSL estimation (18, 19):

where

CFk = commonality factor of route k,Lk = length of route k,Ll = length of each route l in choice set Cn,

Lkl = common length between routes kand l, and0 = 1.

CFkkl

k ll C

k l

k kl

t

L

L L

L L

L Ln

= +

0 1lnkkl

( )9

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80

cumulative percentage of the observations

overlapthreshold

100

branch & bound simulation with large variance simulation with small variance

link elimination link penalty labeling approach

FIGURE 2 Distribution of coverage over 236 observations.

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where

PSk = path size of route k,k = set of links of route k,La = length of linka,al = linkpath incidence dummy (equal to 1 if route l uses link

a, and zero otherwise), and

= positive parameter.

Generalized nested structures relate the model coefficients to the

network topology for adaptation to route choice. The GNL, CNL,

and LNL models consider the following functional relationship for

the inclusion coefficients (20):

where

mk = inclusion coefficients (with 0 mk 1),Lm = length of linkm,Lk = length of route k, andmk = linkpath incidence dummy (equal to 1 if route kuses link

m and zero otherwise).

For GNL estimation, nesting coefficients are considered unique

for each nest and are expressed with the following parameterized

formulation (21):

mkm

k

mk

L

L= ( )11

PSkq

k k

t

al

l C

a

L

L L

Ln

k

=

1 10

( )

Prato and Bekhor 25

where m are the nesting coefficients (with 0 m 1) and is aparameter to be estimated.

CNL is a particular case of the GNL model, in which all the links

share a common nesting coefficient m to be estimated. LNL is a

particular case of the CNL model, in which the common nesting

coefficient m approaches zero and is not estimated (22).

Tables 3 and 4 illustrate the best estimates for route choice models

considering both choice sets.

The same interpretation of the results is possible for both choice sets.

Parameter estimates suggest that choices are influenced by experience

since habit and familiarity negatively influence the utility whereas

landmarks positively affect drivers behavior. The same conclusions

about the goodness of fit across models are reachable for both choice

sets. The LNL model largely outperforms all other models, and PSL

and C-Logit models also produce very good results. GNL and CNL

models tend to collapse to MNL and present worse results than MNL.

Statistical comparison across data sets is not available, for the

impossibility of measuring the covariance across data sets that are

correlated since they also include similar routes. The better likeli-

hood ratio index obtained for the branch-and-bound choice set is

justified by the lower number of alternatives in the branch-and-

bound choice set, which leads to higher probabilities of choosing the

observed choice and consequently to higher likelihood values.

Since the choice sets include the same observations and the

same conclusions are drawn in terms of result interpretation and

model comparison, the analysis of the choice sets suggests that the

m

ml

l C

ml

l C

n

n

=

1 12( )

0

5

10

15

20

25

30

35

40

45

50

55

numberofuniqueroute

s

shortest pathbased methods branch-&-bound algorithm

0 20 40 60 80

cumulative percentage of the observations

100

FIGURE 3 Distribution of number of alternatives in choice sets.

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TABLE3

ModelEstimationwithMergedChoiceSet

MNL

C-Logit

PSL(=

9)

GNL

CNL

LNL

Variable

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Distance

0.882

3.58

1.025

4.36

1.028

4.41

0.80

3

3.84

1.215

2.91

0.929

2.97

Traveltime

0.422

6.29

0.412

6.44

0.416

6.46

0.30

5

3.72

0.572

3.38

0.234

2.83

Sabotinosquaredummy

2.692

6.10

2.544

5.85

2.156

5.17

2.15

4

4.53

3.837

3.53

2.357

4.37

Adrianosquaredummy

1.113

2.51

1.113

2.46

1.148

2.58

0.81

6

2.35

1.418

1.99

1.214

2.12

Sommeillerbridgedummy

4.083

9.06

4.040

8.69

3.659

8.47

3.21

1

5.18

5.640

3.83

3.684

6.64

Dantebridgedummy

3.256

4.97

3.335

5.06

3.054

4.88

2.68

9

4.16

4.509

3.31

3.279

3.91

Rivolisquaredummy

1.061

2.89

1.449

3.92

1.214

3.45

0.84

3

2.72

1.490

2.32

0.968

1.92

Berninisquaredummy

0.648

1.34

0.379

0.81

0.545

1.14

0.52

8

1.44

0.805

1.07

0.653

0.90

Acajasquaredummy

0.527

1.31

0.195

0.50

0.309

0.79

0.27

9

0.92

0.467

0.77

0.432

0.76

Habit

0.482

2.73

0.514

2.97

0.530

3.14

0.26

4

2.33

0.996

1.88

0.478

2.94

Spatialability

0.225

1.23

0.239

1.34

0.241

1.36

0.07

7

0.90

0.222

0.61

0.082

0.50

Familiarity

0.154

0.97

0.163

1.09

0.199

1.34

0.17

6

2.75

0.310

0.93

0.649

4.46

Commonalityfactor

1.197

4.45

Lnofpathsize

4.245

5.89

Commonnestingcoefficient

0.986

3.49

Gammauniquenestingco

efficient

4.24

5

1.35

Loglikelihoodatestimates

592.30

58

3.46

575.45

597.19

596.49

442.58

Likelihoodratioindex

0.175

0.188

0.199

0.169

0.170

0.384

N=2

23;loglikelihoodforall

coefficientsatzerois718.35.

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TABLE4

ModelEstimationwithBranch-and-BoundChoiceSet

MNL

C-Logit

PSL(=

9)

GNL

CNL

LNL

Variable

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Estimate

t-Stat.

Distance

1.544

6.29

1.271

5.13

1.033

4.04

1.617

5.52

1.548

4.24

1

.299

4.46

Traveltime

0.355

5.29

0.335

5.13

0.332

5.18

0.302

4.06

0.347

3.63

0

.160

2.07

Sabotinosquaredummy

1.930

4.33

1.746

3.99

1.465

3.34

1.775

3.66

1.854

3.23

1

.632

3.17

Adrianosquaredummy

1.138

2.56

1.169

2.69

1.159

2.71

1.070

2.30

1.165

2.16

1

.112

1.82

Sommeillerbridgedummy

3.089

8.00

2.903

7.47

2.720

7.08

2.748

5.76

2.979

4.20

2

.478

4.54

Dantebridgedummy

1.846

3.27

1.936

3.43

1.903

3.41

1.630

2.83

1.592

2.44

2

.049

2.49

Rivolisquaredummy

0.345

1.13

0.745

2.57

0.555

1.99

0.141

0.43

0.103

0.23

0

.338

0.82

Berninisquaredummy

1.006

2.02

0.619

1.31

0.686

1.48

1.458

2.59

1.434

2.05

0

.883

1.34

Acajasquaredummy

0.605

1.45

0.250

0.64

0.288

0.75

0.830

1.86

0.811

1.54

0

.500

0.91

Habit

0.319

2.34

0.343

2.53

0.338

2.51

0.294

2.12

0.445

1.76

0

.296

2.10

Spatialability

0.270

1.86

0.272

1.87

0.297

2.05

0.219

1.50

0.147

0.59

0

.212

1.28

Familiarity

0.209

1.76

0.227

1.95

0.309

2.61

0.114

0.94

0.165

0.77

0

.352

2.88

Commonalityfactor

1.218

2.88

Lnofpathsize

3.455

4.33

Commonnestingcoefficien

t

0.941

4.63

Gammauniquenestingco

efficient

0.091

0.06

Loglikelihoodatestimates

466.27

46

2.28

457.19

470.79

463.22

364.89

Likelihoodratioindex

0.223

0.230

0.238

0.215

0.228

0.392

N=2

23;loglikelihoodforallcoefficientsatzerois600.07.

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branch-and-bound algorithm performs better from the behavioral

perspective by reproducing the actual chosen routes while including

more heterogeneous alternatives in smaller choice sets.

SUMMARY AND CONCLUSIONS

A path enumeration algorithm based on the branch-and-bound tech-

nique is proposed. This method is applicable to any urban networksince its implementation requires existing resources and computa-

tional speed depends more on the tree depth than on the tree width.

Implementation in a real case study shows the applicability of the

method and evaluates the performance of the algorithm.

Labeling, link elimination, link penalty, and simulation approaches

and branch-and-bound techniques are applied to an urban network. A

comparison with respect to actual routes chosen by individuals driv-

ing habitually from home to work shows that the proposed algorithm

is significantly better from the perspective of behavioral efficiency

with respect to the ideal algorithm. In parallel, the designed technique

shows a good trade-off between computational costs and efficiency

with respect to the methods that demonstrate a small performance

increase with increasing implementation time.

Construction of different choice sets characterized by similar behav-ioral efficiencyone consisting of a path set resulting from the com-

bination of methods relying on the shortest-path search and the other

resulting from the application of the branch-and-bound algorithm

enables evaluation of results and comparison of model estimation. The

two choice sets produce estimates that are qualitatively comparable

and suggest similar conclusions in terms of model comparison.

Results from model estimation suggest the need for further inves-

tigation into generalized nested structures, since GNL and CNL mod-

els tend to collapse to the MNL model and the LNL model largely

outperforms all other route choice models. Another area for fur-

ther investigation is the route choice mechanism, since the influence

of habit and landmarks on the utility suggests that distance and travel

time are not the only elements considered in choosing a route.

The parameters for defining the constraints in the bounding rulewere arbitrarily defined in this study on the basis of common sense.

Further research is needed to verify the sensitivity of the branch-

and-bound algorithm to the constraints parameters and to measure

the effectiveness of the proposed method with respect to different

data sets.

ACKNOWLEDGMENTS

The authors are grateful to the anonymous reviewers, who provided

many insightful comments and corrections to improve this paper.

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The Traveler Behavior and Values Committee sponsored publication of this paper.


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