- Email: [email protected]
Stochastic User Equilibrium Assignment with Explicit Path Enumeration: Comparison of Models and Algorithms
Copyright © IFAC Transportation Systems Chania, Greece, 1997
STOCHASTIC USER EQUILIBRIUM ASSIGNMENT WITH EXPLICIT PATH ENUMERATION: COMPARISON OF MODELS AND ALGORITHMS
Ennio Cascetta*, Francesco Russoo, Antonino Vitetta°
* Dept. Transportation Engineering - Univ. of Naples Federico 11 -
Via Claudio 21 - 80125 Napoli - Italy - Tel. +3981 7683351 - E. Mail [email protected] o Dept. of Engineering and Mathematics - Univ. of Reggio Calabria - Via E. Cuzzocrea, 48 89128 Reggio Calabria - Italy - Tel. +39-965-875209 - E. mail [email protected]
Abstract: In this paper a preliminary analysis of alternative models for "feasible"path generation and choice is presented. In particular a k-shortest path multi-criteria model for path enumeration is explored and different choice models (Logit, recently proposed C-Logit and Probit) are tested by comparing SUE assignment link flows with counts on an urban road network. Flows are also compared for more traditional DUE and SUE Probit implicit path enumeration models. The results obtained show that a limited number (4-7) of paths generated with rather "simple" criteria give satisfactory results, SUE with explicit path enumeration is largely comparable with, and in some cases superior to, traditional implicit SUE and DUE models. Explicit path enumeration allow also the specification of more sophisticated non additive attributes in the utility function of route choice models. From the computational point of view the explicit path C-Logit and Probit SUE algorithms are from three to twenty times superior to the implicit Probit SUE assignment.
Keywords: behaviour, network, road traffic
somehow restricted by the computing powers available until recently. On the other hand explicit path enumeration allows a greater modelling flexibility insomuch it is possible to model both choice set generation and choice among alternatives, including path-related attributes. Also some numerical problems, e.g. negative link costs for Monte-Carlo Probit algorithms, can be easily solved in such a context.
1. INTRODUCTION
A very significant body of literature dealing with assignment models to urban transport networks, both for road and transit systems, has been produced over the years. The problem of path choice modelling, which is the kernel of any assignment model, has received relatively little attention. As a matter of fact all assignment models and algorithms are based on route choice models (deterministic, Logit and Probit) coupled with implicit path enumeration and simplified ''utility'' specification. This choice was
Limited exceptions can be found for intercity networks where some route choice models are imbedded in assignment models based on explicit
1031
path enumeration; Ben Akiva et al. (1984), Cascetta et al. (1996).
•
Section 2 describes the different approaches and models which have been proposed to simulate the generation of the feasible route set and path choice; section 3 a Stochastic User Equilibrium assignment model and the MSA algorithm are formalised; section 4 describes the main results of the empirical work carried out by comparing different choice set generation criteria and explicit path based on SUE models on the basis of flows measured and modelled. Finally some conclusions are drawn up in section 5.
•
the "exhaustive" approach that considers all loopless paths connecting OlD centroids as belonging to the choice set; the "selective" approach that consider only a subset of topologically feasible paths as perceived choice alternatives, following some heuristic rules that can be calibrated by comparison with the set of path chosen by a sample of user.
In this paper the second approach will be followed and assignment results will be compared with those obtained by the traditional exhaustive models.
2.2 Route choice models
2. PATH CHOICE MODELLING Virtually all route choice models are based on random or deterministic utility theory. Specification of a model requires the definition of the attributes included in the systematic utility fimction and of the probability distribution fimction of random residual, if any. The perceived utility of path k belonging to the set of possible paths Irs can be formally expressed as:
Two main, interconnected problems arise in the definition and modelling of path choice behaviour: choice set definition and choice model. A short state of the art describing how these two aspects have been dealt with in the literature and setting the stage for the empirical work is reported in the following.
2.1 Choice-Set generation
"if k E In
The principal hypotheses for the specification of the choice set definition model are relative to the level of users' knowledge of the network and the criteria considered for the generation of the paths perceived as feasible or available alternatives.
(1)
where VkD is the average, or systematic, utility of path k, and EkD the random residual for user n. In path choice models the average perceived utility is usually expressed as a linear combination of attributes:
The models proposed in literature (De La Barra et al., 1993; Laurent 1993 and 1994; Ben Akiva et al., 1984; Russo and Vitetta, 1995), use for the generation of the choice set a certain number of criteria (time, distance, cost, number of signals, high capacity path, hierarchical level, use of highway, scenic quality, minimum congestion, presence of commercial activity, quality of pavement, etc.).
(2) The attributes influencing the path choice of user n are usually assumed as level of service attributes with a negative sign (e. g. travel time, monetary cost, distance, etc.). Implicit path enumeration models used in an assignment context usually identify the sistematic utility with minus the generalised transportation cost:
The ''feasible" paths choice set, Irs, connecting the OlD pairs (r,s) should be defined according to a behavioural model giving the probability that each feasible path to belongs to the set of alternatives perceived by a generic user. In Ben Akiva et al. (1984) one such model is proposed for the generation of the choice set on intercity network.
"if k E In
The definition, specification and calibration of a disaggregate behavioural explicit model for the generation of a set of feasible paths, requires a database containing the paths considered by a sample of users and in practice has not been yet carried out. The models proposed in literature and commonly used are essentially aggregated; they could be classified as belonging to two approaches:
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(3)
and in general it is assumed that the average perceived cost Ck is purely additive, i. e. it is obtainable as the sum of costs c; of the links belonging to the path: T
cADD=A c
(4)
with: C ADD vector of additive path cost; A link path incidence matrix; c vector of link cost In some models the sistematic utility of a path could include attributes relative to the path itself and not
obtainable as sum of link costs independent from the path itself (not additive cost, cNAD~; a specification with not additive costs generally requires the explicit enumeration of the paths.
specification for the systematic utility of a Multinomial Logit model (Cascetta, 1995; Cascetta et al., 1996). C-Logit tackles the problem arising from the IIA property of the Logit model while maintaining an easy analytical specification. The CLogit model has the following form:
Deterministic utility models simply assume that random residuals £k are zero and path choice probabilities can be expressed only indirectly as:
VheI,.
Ct -CF,,) Plc. rs = Lhexp( - Cl C -CF) exp( -Cl
h
(5)
(8)
where the CFk term, named Commonality Factor, can be specified in different ways. The specification which insofar has given the best numerical and empirical results (Cascetta et al., 1996) is the following:
due to the possibility of different minimum cost paths. Random utility path choice models usually adopted have a Logit or Probit specification depending on the hypotheses made up on the joint distribution of the residuals Ek. The Multinomial Logit model:
where Cbk represents the cost of the links shared by the paths h and k. The CFk attribute is an inverse measure of the degree of independence of a path, in fact it is equal to zero if all the links of path k do not belong to other paths, the larger is CFk the more paths share the links of path k. In conclusion the C-Logit model (8), reduces the probability of choosing paths with heavy overlaps and it increases that of "independent" paths.
(6)
originates from the assumption that El< are i.i.d. Gumble variates of zero mean and parameter a. . The hypothesis of independent residuals and the resulting property of independence of the irrelevant alternatives (IlA), is not theoretically acceptable when alternative paths shares many links (Florian and Fox, 1976; Sheffi, 1985; Daganzo and Sheffi, 1977). From that it results that the Logit models should be used in congestion with explicit path enumeration excluding strongly overlapping paths even though this does not solve the problem completely.
The C-Logit model has an interesting behavioural interpretation, as a joint model of perception (availability) of the alternatives and of choice among perceived alternatives, Cascetta et al. (1996).
3. USER EQUILmRIUM ASSIGNMENT The equilibrium state of a network can be defined as the state in wich demand, flows and costs are mutually consistent. In formal terms, the equilibrium could be seen as solution of a fixed point problem (Daganzo 1989, Cascetta, 1990; Cantarella, 1996):
Alternatively, if the residuals are assumed to be jointly distributed as a Multivariate Normal of zero mean the Probit path choice model is obtained. The usual specification of the variances and covariances for the Probit route choice model, proposed by Daganzo and Sheffi (1983), is: var (£t) = () C" COV(£h £;) = ()Chk
h
(10) f* = A P[C(f*)] d where: r is the vector of link flows; A is the link path incidence matrix; P is matrix of path choice probabilities ordered for each OlD pair.; C is the vector of path costs; d is the vector of OlD demand flows. In stochastic user equilibrium (SUE) models path choice probabilities, i.e. elements of P matrix, are obtained through a probabilistic path choice model, generally a random utility model.
(7)
where Cbk is the part of the (additive) cost relative to
links shared by paths h and k, and a is a calibration parameter. The use of Probit model is limited by the impossibility to obtain explicit fimctional form of choice probabilities; this limit is commonly overcome in path choice modelling through the use of a MonteCarlo algorithm. This algorithm coupled with implicit path enumeration may cause other problems with the possibility of generating negative link costs (Bifulco and Adamo, 1995; Van Vuren, 1994).
The problem of computating of SUE flows on large size networks is usually solved by the algorithm of the Method of Successive Averages (MSA) proposed by Powell and Sheffi (1982). This algorithm guarantees the convergence (Blum, 1954) to the
Recently has been proposed a path choice model, named C-Logit, that is based on a different
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optimum solution of equilibrium under certain hypotheses on the path choice model and on link cost functions, in particular the symmetry of cost function lacobian. The basic iteration of the MSA algorithm is the following: repeat k=k+1 ft.L =
f
le
Ap[C(f
lc J - )]
d
k -1 fle-J =i1 fieSNL +-k-
until convergence where convergence should be tested as (Cantarella, 1996): fie == f~L
Deterministic User Equilibrium (DUE) on the other hand is based on the deterministic route choice model (5) and can be formulated as a variation inequality problem and an equivalent optimization problem, under the assumption of a symmetric lacobian of cost functions.
4. THE EMPffiICAL ANALYSES Different empirical tests were carried out with the pw-pose of calibrating and comparing path enumeration models as well as assignment models based on different route choice assumptions and explicit path enumeration. Other analyses were conducted in order to compare these models with more conventional implicit path enumeration equilibrium assignment models both in terms of "goodness-of-fit" and computational requirements. The empirical work has been carried out with reference to the urban road network of Salerno (Italy). The city was subdivided into 53 homogeneous internal zones and 9 external ones. The network is thus composed by 62 centroids, 466 real nodes and 1127 links. The network is significantly congested. The average flow capacity ratio is 0.4 and 5% of links leave flow/capacity ratio larger than 0.9. The link travel time flow function has been considered depending on the flow/capacity ratio through a strictly monotone increasing separable function. The OlD demand for the a.m. peak hour was derived from a sample of users interviewed both at home and at cordon sections. Resulting OlD demand matrix was "corrected" for level and spatial structure using traffic counts on cordons.
MSE=
Li{ti -
In
2 /
N
RMSE = --) MSE / (L/;· / N)
(11)
(12)
where: N is the number of links where the flow has been measured; f is the vector of measured flow and f is the vector of modelled flows. The main results are reported in the following. Further details on the test network can be founded in Cascetta et al. (1993). 4.1 Path enumeration models
In the first phase different "selection" criteria were tested in order to find the subset of most efficient ones. The criteria initially taken into account were the following: 1) free flow travel time to (minimisation); 2) congested (DUE assignment) travel time le (minimisation); 3) motorway use (maximisation of motorway links on the path as computed by considering to on motorway links and 20 to on the other links); 4) hierarchical network (path with primary links on the central part computed as to on links wider than 6 m, 5 to on links of width between 3 m and 6 m and 50 to on the other links); 5) minimum number of junction. Their relative performances were assessed by comparing MSE statistics of link flows obtained by assigning the OlD demand with a C-Logit SUE model to the first k shortest paths generated by each iteration. For each considered criterion the first 40 paths between each OlD pair were generated through a k shortest paths algorithm with heap management of the nodes waiting list to analyse. This algorithm allows to obtain the first k trees for each origin with computational time equal to k times the one for the shortest path tree from the same root. For each generated choice set (5 choice sets each of which is composed of 40 paths) only the paths which differ from each other by at least a fraction A. were considered The overlapping factor between two generic paths h and k generated according to the generic criterion has been calculated as:
(13) For each considered criterion the choice set with paths different from each other by a fraction A. variable from 0.0 to 0.25 with step 0.05 was generated. It has been verified that for each set generated there is a significant improvement of the RMSE indicator for values of A. increasing up to the value 0.1. The RMSE indicator does not show
All the analyses were carried out by comparing flows measured on 69 sections with flows predicted by assignment models on the same links. The comparison between measured flows and flows obtained from models was carried out by using the following indicators:
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on the best path sets (Le. the first three sets of Tab. 1) were analysed.
significant improvements for values of A. greater than that value. Furthermore the best results with monocriterion sets were obtained with the following criteria: 1) free flow travel time to; 2) congested (DUE assignment) travel time tc; 3) motorway use (maximisation of motorway links on the path). Finally for each OlD pair a choice set composed of paths generated with the three criteria specified above and differing from each other by at least 10% (I..> 0.1) was selected.
Tab. 1 - RMSE and MSE for the best choice using a SUE C-Logit model Criterion to tc m all
MSE RMSE ~veic2/h:l
For each criterion the possibility to use sets with a different numerousness and composition was verified. A mono-criterion choice set with a numerousness which varying from 1 to 8 and some multi-criteria choice sets with a numerousness varying from 2 to 18 (from 1 to 6 paths for each criterion) were genereted.
Explicit SNL and SUE assignment models have been used with Probit, Logit and C-Logit with overlapping factor (9) and different variance. The value of 'Y is equal to 0.5. The values used for a and
a=c/c..
The best results (Fig. 1) in terms of RMSE were obtained by using a multi-criteria choice set with a global number path between 6 to 8, and composed of 1 or 2 paths generated with the criterion of free-flow travel time, between 3 and 4 paths generated with the criterion of congested travel time and between 2 and 4 paths generated with the criterion of maximum motorway. The sets giving the best results are reported in Tab. 1. RMSE and MSE values refer to C-Logit SUE model have been obtained using for the assignment the standard deviation mean ratio (Cv) equal to 0.2. One relevant result is that an increase in the total number of paths does not necessarily imply an improvement in the assignment model performances. This could be interpreted as an indication that only a limited number of paths are considered as choice alternative by users.
Tab. 2 - RMSE for the best choice set for explicit assignment Criterion
60000
40000
\
Cv
SNLLogit SNL C-Logit SNLProbit SUELogit SUEC-Logit SUE Probit SNLLogit SNLC-Logit SNLProbit SUE Logit SUE C-Logit SUE Probit
0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2
m all
RMSE
RMSE
i
\
Number of paths 1 1 1 4 4 4 4 2 3 7 8 9 0.474 0.476 0.476 0.472 0.473 0.471 0.468 0.472 0.473 0.274 0.276 0.276 0.267 0.267 0.266 0.263 0.264 0.264 0.424 0.430 0.434 0.420 0.423 0.424 0.414 0.421 0.424 0.284 0.285 0.286 0.276 0.276 0.275 0.268 0.270 0.270
From the analysis of the results it can be concluded that: • the results obtained with SUE models are significantly better compared to those obtained with SNL models; • the C-Logit model reproduces observed flows better than the Logit model and is very close to the results obtained with the Probit models; in particular it is for Cv=O.2 slightly worse and equivalent for Cv=O.I. • the best values are obtained with Cv equal to 0.1; • the algorithms based on the Logit and C-Logit models give equilibrium flows depending only on
20000
o
Assignment
to tc
I
1\
(15)
c..
120000
80000
(14)
where is the minimumum congested travel time on the network for each OlD pair (r,s). The main results are reported in Tab. 2.
MSE (veic'1Ir)
100000
Number of paths 1 2 2 2 4 4 4 4 4 4 4 4 2 3 2 3 7 10 8 9 8 9 0.276 0.276 0.275 0.279 0.279 0.279 34257 33884 33817 34532 34427 34404
U~
23456789
Fig. 1. Best value ofMSE indicator for multi-criteria choice set 4.2 Comparison of assignment models The second phase of the analysis was focused on the comparison of the performances of different assignment models. At first assignment models based
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•
the stop test because for these models the probability of choice could be expressed in a closed form; the Probit models give estimated SUE flows that depends on the MSA stop test and on the sequence of random numbers used to generate the flows through a Montecarlo extraction. Some results obtained in this work and from other researchers, show in fact that the SNL Probit flows calculated with more than 50 iterations have a Cv around 0.17 (Fig. 2).
paths algorithm with heap management of the waiting nodes results equal to:
time for paths=30 time for minimum trees. It resulted that a set of this dimension allows the generation of a filtered set, for almost all the OlD pairs (99% of the cases), composed of 4 paths for each criterion differing at least 10%. The operation of filtering in terms of computational time is equal to Y2 to that used to generate the paths(Russo and Vitetta, 1997). The total time to generate and filter the choice set results at most equal to:
time for filtered paths =45 times for minimum trees. average cv for all links
Furthermore for Probit assignment with explicit paths enumeration Monte Carlo extraction of link costs have to be generated in each iteration with an increase in computation time of 113 time for minimum trees in each iteration inside the SNL.
O:: tF=~~____~__~~::::::= 0.15
+1 _ _ _ _ _ _ _ _ _ _ _ _- _
0.1
t-- - - - - - - - - - - -
0.05
t------------2
6 10 14 18 22 26 30 34 38 42 46 SO AON itelation with extracted costs inside SNL with equilibrium cost
Fig. 2. Average Cv for all links on the network versus AON iterations with extracted costs inside SNL with equilibrium costs
On the other hand Probit assignment with implicit enumeration can be carried out following two algorithmic approaches. The first approach (internal SNL) at each MSA iteration an SNL is carried out with NSNL iterations. Applying this algorithm to the study case on average 16 external iterations of SNL have to be done and in each iteration 50 internal iterations of AON with the "extracted" costs have to be performed. The total number of trees needed to each origin results equal to: 50· 16 = 800. The time spent to generate the paths results equal to:
time for paths =800 time for minimum trees.
RMSE values obtained for implicit path enumeration models are reported in Tab. 3. The Deterministic User Equilibrium flows can be seen as a particular case of the SUE model with Cv of random terms equal to zero. The results show that DUE flows are outperformed by C-Logit and Probit SUE flows. Implicit Probit SUE is superior for Cv=O.l and comparable for Cv=O.2 to both explicit C-Logit and Probit SUE models.
The other algorithm approach (powell and Sheffy, 1982) for SUE Probit assignment with implicit path enumeration uses a single Monte Carlo extraction of link costs at each MSA iteration. The convergence test requires each N iterations a full SNL Probit assignment with NSNL iterations. It resulted that the most effective N and NSNL values are 10 and 50 respectively. Applying this algorithm to the study case on average 25 iterations (and in some events more than 30) have to be done. The time spent to generate the paths results equal to:
Tab. 3 RMSE for the best choice set for different implicit assignment models
o RMSE
0.280
0.1 0.257
0.2 0.269
time for paths = 123 time for minimum tree (and in some events more than 177 time for minimum tree).
0.3
0.284
4.3 Comparison of computational requirements
The situation changes if we consider an algorithm for Deterministic User Equilibrium (DUE). For the solution of DUE a Frank and Wolfe algorithm is used. In the algorithm on average 15 iterations (in each iteration we have to perform one AON and one linear optimization) have to be done. If we consider that one linear optimization in term of computation time is equivalent to generating the minimum trees for all the network, the total computation time result equal to:
Finally it is possible to compare the computation times needed for implicit and explicit path enumeration models in the study case.
In the explicit enumeration models initially all the paths forming the choice set for each OlD pairs need to be generated and afterward a model of equilibrium assignment needs to be applied Fixed an origin, the necessary computation time in order to' generate the first 30 trees (10 for each criterion), using a k shortest
time for DUE=30 time for minimum trees.
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Thus, for the generation of the paths, the computation time for the implicit enumeration algorithms with Probit choice model is from 3 to 20 times larger than the computation time for the explicit path algorithm. The explicit path enumeration algorithm generates paths once for all which can be stored in the RAM memory of the computer (this consideration is less significant with increasing storage spaces). On the other hand algorithms with implicit enumeration generate paths repeatedly but require smaller memory spaces. We can conclude that the explicit SUE Probit and CLogit models are computationally more efficient than SUE Probit models with implicit enumeration and that the C-Logit outperforms the explicit path enumeration Probit. 5. CONCLUSIONS The preliminary results reported in this paper show the potential of SUE assignment models with explicit path enumeration on medium sized networks. It seems that a limited number of paths generated with few criteria are able to reproduce observed flows satisfactorily. Furthermore the possibility of enumerating paths allows more sophisticated attributes for route choice models and is computationally more efficient, even though it requires longer storage memory. Finally the C-Logit model has performances very close to the Probit, but it has a better accuracy and computational efficiency due to the possibility of expressing route choice probabilities in a closed form.
REFERENCES Ben-Akiva, M., M. 1. Bergman, AJ. Daly. and R. Ramaswamy. (1984). Modelling inter urban route choice behaviour. In: Ninth International Symposium on Transportation and Traffic Theory. VNU Science Press, 299-330. Bifulco G. N . and V. Adamo (1995): SUE assignment model and MSA algorithm: some condictions on travels and effectiviness. In: Proceedings of the IV EURO working grup on Urban Traffic and Transportatio. Barcel/ona. Blum 1. (1954). "Multidimensional stochastic approximation methods. In: Ann. Math. Stat. 22. Cantarella, G.E. (1996). A General Fixed Point Approach to Multi-mode Multi-user Equilibrium Assignment with Elastic Demand. Internal Report
Department of Transport and Engineering. University of Napoli. Transportation Science. to be published. Cascetta, E. (1990). Metodi quantitativi per la pianijicazione dei sistemi di trasporto. CEDAM, Padova Italy. Cascetta, E. (1995). A modified Logit model for route choice. Internal Report Department of Transport Engineering, Napoli, Italy. Cascetta, E., A Nuzzolo, V. Velardi (1993).A system of mathematical models for the evaluation of integrated traffic planning and control policies. In Preprint of Workshop on integration problems in urban transportation planning. Capri. Cascetta E., A Nuzzolo, F. Russo and A Vitetta (1996). A modified logit route choice model overcoming path overlapping problems. specification and some calibration results for interurban networks. Proceedings of IS17T conference, Lyon, France, 1996, In "Transportation and Traffic Theory", J. B. Lesort editor, Pergamon. Daganzo, C.F. and Y. Sheffi (1977). On Stochastic Models of Traffic Assignment. In: Transportation Science, vol 11 (3), 253-274. De La Barra, T., B. Perez and J. Anez (1993). Multidimentional path search and assignment. Proceedings of the 21 PTRC Summer Meeting ( PTRCed.). Florian, M, and B. Fox (1976). On the Probabilistic Origin of Dial's Multipath Traffic Assignment Model. In: Transportation Research, Vol. 10(5), 339-341. Laurent, F. (1993). Cost versus Time Equilibrium over a Network. Internal report INRETS, France. Laurent, F. (1994). Modele d'affectation routiere statique. Internal report INRETS, France. Powell W.B.and Y. Sheffi (1982). The convergence of equilibrium algorithms with predetermined step sizes. In: Transportation Science, Vo1.16, 45-55. Russo, F. and A Vitetta (1997). Algoritmi per la ricerca di insiemi di percorsi. Internal report, Department of Engeenering and Mathematics. University ofReggio Calabria, Italy. Russo, F. and A Vitetta (1995). Network and assignment models for the italian national transportation systems. In: Proceeding of WCTR95, Sidney, Australia Sheffi, Y. (1985). Urban transportation networks. Prentice Hall, Englewood Cliff. Van Vuren, T. (1994). The truble with SUE stochastic assignment options in practice. In Proceeding of 22nd European Transport Forum, Seminar Hpp.( PTRC ed.), 41-52.
1037
STOCHASTIC USER EQUILIBRIUM ASSIGNMENT WITH EXPLICIT PATH ENUMERATION: COMPARISON OF MODELS AND ALGORITHMS
Ennio Cascetta*, Francesco Russoo, Antonino Vitetta°
* Dept. Transportation Engineering - Univ. of Naples Federico 11 -
Via Claudio 21 - 80125 Napoli - Italy - Tel. +3981 7683351 - E. Mail [email protected] o Dept. of Engineering and Mathematics - Univ. of Reggio Calabria - Via E. Cuzzocrea, 48 89128 Reggio Calabria - Italy - Tel. +39-965-875209 - E. mail [email protected]
Abstract: In this paper a preliminary analysis of alternative models for "feasible"path generation and choice is presented. In particular a k-shortest path multi-criteria model for path enumeration is explored and different choice models (Logit, recently proposed C-Logit and Probit) are tested by comparing SUE assignment link flows with counts on an urban road network. Flows are also compared for more traditional DUE and SUE Probit implicit path enumeration models. The results obtained show that a limited number (4-7) of paths generated with rather "simple" criteria give satisfactory results, SUE with explicit path enumeration is largely comparable with, and in some cases superior to, traditional implicit SUE and DUE models. Explicit path enumeration allow also the specification of more sophisticated non additive attributes in the utility function of route choice models. From the computational point of view the explicit path C-Logit and Probit SUE algorithms are from three to twenty times superior to the implicit Probit SUE assignment.
Keywords: behaviour, network, road traffic
somehow restricted by the computing powers available until recently. On the other hand explicit path enumeration allows a greater modelling flexibility insomuch it is possible to model both choice set generation and choice among alternatives, including path-related attributes. Also some numerical problems, e.g. negative link costs for Monte-Carlo Probit algorithms, can be easily solved in such a context.
1. INTRODUCTION
A very significant body of literature dealing with assignment models to urban transport networks, both for road and transit systems, has been produced over the years. The problem of path choice modelling, which is the kernel of any assignment model, has received relatively little attention. As a matter of fact all assignment models and algorithms are based on route choice models (deterministic, Logit and Probit) coupled with implicit path enumeration and simplified ''utility'' specification. This choice was
Limited exceptions can be found for intercity networks where some route choice models are imbedded in assignment models based on explicit
1031
path enumeration; Ben Akiva et al. (1984), Cascetta et al. (1996).
•
Section 2 describes the different approaches and models which have been proposed to simulate the generation of the feasible route set and path choice; section 3 a Stochastic User Equilibrium assignment model and the MSA algorithm are formalised; section 4 describes the main results of the empirical work carried out by comparing different choice set generation criteria and explicit path based on SUE models on the basis of flows measured and modelled. Finally some conclusions are drawn up in section 5.
•
the "exhaustive" approach that considers all loopless paths connecting OlD centroids as belonging to the choice set; the "selective" approach that consider only a subset of topologically feasible paths as perceived choice alternatives, following some heuristic rules that can be calibrated by comparison with the set of path chosen by a sample of user.
In this paper the second approach will be followed and assignment results will be compared with those obtained by the traditional exhaustive models.
2.2 Route choice models
2. PATH CHOICE MODELLING Virtually all route choice models are based on random or deterministic utility theory. Specification of a model requires the definition of the attributes included in the systematic utility fimction and of the probability distribution fimction of random residual, if any. The perceived utility of path k belonging to the set of possible paths Irs can be formally expressed as:
Two main, interconnected problems arise in the definition and modelling of path choice behaviour: choice set definition and choice model. A short state of the art describing how these two aspects have been dealt with in the literature and setting the stage for the empirical work is reported in the following.
2.1 Choice-Set generation
"if k E In
The principal hypotheses for the specification of the choice set definition model are relative to the level of users' knowledge of the network and the criteria considered for the generation of the paths perceived as feasible or available alternatives.
(1)
where VkD is the average, or systematic, utility of path k, and EkD the random residual for user n. In path choice models the average perceived utility is usually expressed as a linear combination of attributes:
The models proposed in literature (De La Barra et al., 1993; Laurent 1993 and 1994; Ben Akiva et al., 1984; Russo and Vitetta, 1995), use for the generation of the choice set a certain number of criteria (time, distance, cost, number of signals, high capacity path, hierarchical level, use of highway, scenic quality, minimum congestion, presence of commercial activity, quality of pavement, etc.).
(2) The attributes influencing the path choice of user n are usually assumed as level of service attributes with a negative sign (e. g. travel time, monetary cost, distance, etc.). Implicit path enumeration models used in an assignment context usually identify the sistematic utility with minus the generalised transportation cost:
The ''feasible" paths choice set, Irs, connecting the OlD pairs (r,s) should be defined according to a behavioural model giving the probability that each feasible path to belongs to the set of alternatives perceived by a generic user. In Ben Akiva et al. (1984) one such model is proposed for the generation of the choice set on intercity network.
"if k E In
The definition, specification and calibration of a disaggregate behavioural explicit model for the generation of a set of feasible paths, requires a database containing the paths considered by a sample of users and in practice has not been yet carried out. The models proposed in literature and commonly used are essentially aggregated; they could be classified as belonging to two approaches:
1032
(3)
and in general it is assumed that the average perceived cost Ck is purely additive, i. e. it is obtainable as the sum of costs c; of the links belonging to the path: T
cADD=A c
(4)
with: C ADD vector of additive path cost; A link path incidence matrix; c vector of link cost In some models the sistematic utility of a path could include attributes relative to the path itself and not
obtainable as sum of link costs independent from the path itself (not additive cost, cNAD~; a specification with not additive costs generally requires the explicit enumeration of the paths.
specification for the systematic utility of a Multinomial Logit model (Cascetta, 1995; Cascetta et al., 1996). C-Logit tackles the problem arising from the IIA property of the Logit model while maintaining an easy analytical specification. The CLogit model has the following form:
Deterministic utility models simply assume that random residuals £k are zero and path choice probabilities can be expressed only indirectly as:
VheI,.
Ct -CF,,) Plc. rs = Lhexp( - Cl C -CF) exp( -Cl
h
(5)
(8)
where the CFk term, named Commonality Factor, can be specified in different ways. The specification which insofar has given the best numerical and empirical results (Cascetta et al., 1996) is the following:
due to the possibility of different minimum cost paths. Random utility path choice models usually adopted have a Logit or Probit specification depending on the hypotheses made up on the joint distribution of the residuals Ek. The Multinomial Logit model:
where Cbk represents the cost of the links shared by the paths h and k. The CFk attribute is an inverse measure of the degree of independence of a path, in fact it is equal to zero if all the links of path k do not belong to other paths, the larger is CFk the more paths share the links of path k. In conclusion the C-Logit model (8), reduces the probability of choosing paths with heavy overlaps and it increases that of "independent" paths.
(6)
originates from the assumption that El< are i.i.d. Gumble variates of zero mean and parameter a. . The hypothesis of independent residuals and the resulting property of independence of the irrelevant alternatives (IlA), is not theoretically acceptable when alternative paths shares many links (Florian and Fox, 1976; Sheffi, 1985; Daganzo and Sheffi, 1977). From that it results that the Logit models should be used in congestion with explicit path enumeration excluding strongly overlapping paths even though this does not solve the problem completely.
The C-Logit model has an interesting behavioural interpretation, as a joint model of perception (availability) of the alternatives and of choice among perceived alternatives, Cascetta et al. (1996).
3. USER EQUILmRIUM ASSIGNMENT The equilibrium state of a network can be defined as the state in wich demand, flows and costs are mutually consistent. In formal terms, the equilibrium could be seen as solution of a fixed point problem (Daganzo 1989, Cascetta, 1990; Cantarella, 1996):
Alternatively, if the residuals are assumed to be jointly distributed as a Multivariate Normal of zero mean the Probit path choice model is obtained. The usual specification of the variances and covariances for the Probit route choice model, proposed by Daganzo and Sheffi (1983), is: var (£t) = () C" COV(£h £;) = ()Chk
h
(10) f* = A P[C(f*)] d where: r is the vector of link flows; A is the link path incidence matrix; P is matrix of path choice probabilities ordered for each OlD pair.; C is the vector of path costs; d is the vector of OlD demand flows. In stochastic user equilibrium (SUE) models path choice probabilities, i.e. elements of P matrix, are obtained through a probabilistic path choice model, generally a random utility model.
(7)
where Cbk is the part of the (additive) cost relative to
links shared by paths h and k, and a is a calibration parameter. The use of Probit model is limited by the impossibility to obtain explicit fimctional form of choice probabilities; this limit is commonly overcome in path choice modelling through the use of a MonteCarlo algorithm. This algorithm coupled with implicit path enumeration may cause other problems with the possibility of generating negative link costs (Bifulco and Adamo, 1995; Van Vuren, 1994).
The problem of computating of SUE flows on large size networks is usually solved by the algorithm of the Method of Successive Averages (MSA) proposed by Powell and Sheffi (1982). This algorithm guarantees the convergence (Blum, 1954) to the
Recently has been proposed a path choice model, named C-Logit, that is based on a different
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optimum solution of equilibrium under certain hypotheses on the path choice model and on link cost functions, in particular the symmetry of cost function lacobian. The basic iteration of the MSA algorithm is the following: repeat k=k+1 ft.L =
f
le
Ap[C(f
lc J - )]
d
k -1 fle-J =i1 fieSNL +-k-
until convergence where convergence should be tested as (Cantarella, 1996): fie == f~L
Deterministic User Equilibrium (DUE) on the other hand is based on the deterministic route choice model (5) and can be formulated as a variation inequality problem and an equivalent optimization problem, under the assumption of a symmetric lacobian of cost functions.
4. THE EMPffiICAL ANALYSES Different empirical tests were carried out with the pw-pose of calibrating and comparing path enumeration models as well as assignment models based on different route choice assumptions and explicit path enumeration. Other analyses were conducted in order to compare these models with more conventional implicit path enumeration equilibrium assignment models both in terms of "goodness-of-fit" and computational requirements. The empirical work has been carried out with reference to the urban road network of Salerno (Italy). The city was subdivided into 53 homogeneous internal zones and 9 external ones. The network is thus composed by 62 centroids, 466 real nodes and 1127 links. The network is significantly congested. The average flow capacity ratio is 0.4 and 5% of links leave flow/capacity ratio larger than 0.9. The link travel time flow function has been considered depending on the flow/capacity ratio through a strictly monotone increasing separable function. The OlD demand for the a.m. peak hour was derived from a sample of users interviewed both at home and at cordon sections. Resulting OlD demand matrix was "corrected" for level and spatial structure using traffic counts on cordons.
MSE=
Li{ti -
In
2 /
N
RMSE = --) MSE / (L/;· / N)
(11)
(12)
where: N is the number of links where the flow has been measured; f is the vector of measured flow and f is the vector of modelled flows. The main results are reported in the following. Further details on the test network can be founded in Cascetta et al. (1993). 4.1 Path enumeration models
In the first phase different "selection" criteria were tested in order to find the subset of most efficient ones. The criteria initially taken into account were the following: 1) free flow travel time to (minimisation); 2) congested (DUE assignment) travel time le (minimisation); 3) motorway use (maximisation of motorway links on the path as computed by considering to on motorway links and 20 to on the other links); 4) hierarchical network (path with primary links on the central part computed as to on links wider than 6 m, 5 to on links of width between 3 m and 6 m and 50 to on the other links); 5) minimum number of junction. Their relative performances were assessed by comparing MSE statistics of link flows obtained by assigning the OlD demand with a C-Logit SUE model to the first k shortest paths generated by each iteration. For each considered criterion the first 40 paths between each OlD pair were generated through a k shortest paths algorithm with heap management of the nodes waiting list to analyse. This algorithm allows to obtain the first k trees for each origin with computational time equal to k times the one for the shortest path tree from the same root. For each generated choice set (5 choice sets each of which is composed of 40 paths) only the paths which differ from each other by at least a fraction A. were considered The overlapping factor between two generic paths h and k generated according to the generic criterion has been calculated as:
(13) For each considered criterion the choice set with paths different from each other by a fraction A. variable from 0.0 to 0.25 with step 0.05 was generated. It has been verified that for each set generated there is a significant improvement of the RMSE indicator for values of A. increasing up to the value 0.1. The RMSE indicator does not show
All the analyses were carried out by comparing flows measured on 69 sections with flows predicted by assignment models on the same links. The comparison between measured flows and flows obtained from models was carried out by using the following indicators:
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on the best path sets (Le. the first three sets of Tab. 1) were analysed.
significant improvements for values of A. greater than that value. Furthermore the best results with monocriterion sets were obtained with the following criteria: 1) free flow travel time to; 2) congested (DUE assignment) travel time tc; 3) motorway use (maximisation of motorway links on the path). Finally for each OlD pair a choice set composed of paths generated with the three criteria specified above and differing from each other by at least 10% (I..> 0.1) was selected.
Tab. 1 - RMSE and MSE for the best choice using a SUE C-Logit model Criterion to tc m all
MSE RMSE ~veic2/h:l
For each criterion the possibility to use sets with a different numerousness and composition was verified. A mono-criterion choice set with a numerousness which varying from 1 to 8 and some multi-criteria choice sets with a numerousness varying from 2 to 18 (from 1 to 6 paths for each criterion) were genereted.
Explicit SNL and SUE assignment models have been used with Probit, Logit and C-Logit with overlapping factor (9) and different variance. The value of 'Y is equal to 0.5. The values used for a and
a=c/c..
The best results (Fig. 1) in terms of RMSE were obtained by using a multi-criteria choice set with a global number path between 6 to 8, and composed of 1 or 2 paths generated with the criterion of free-flow travel time, between 3 and 4 paths generated with the criterion of congested travel time and between 2 and 4 paths generated with the criterion of maximum motorway. The sets giving the best results are reported in Tab. 1. RMSE and MSE values refer to C-Logit SUE model have been obtained using for the assignment the standard deviation mean ratio (Cv) equal to 0.2. One relevant result is that an increase in the total number of paths does not necessarily imply an improvement in the assignment model performances. This could be interpreted as an indication that only a limited number of paths are considered as choice alternative by users.
Tab. 2 - RMSE for the best choice set for explicit assignment Criterion
60000
40000
\
Cv
SNLLogit SNL C-Logit SNLProbit SUELogit SUEC-Logit SUE Probit SNLLogit SNLC-Logit SNLProbit SUE Logit SUE C-Logit SUE Probit
0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2
m all
RMSE
RMSE
i
\
Number of paths 1 1 1 4 4 4 4 2 3 7 8 9 0.474 0.476 0.476 0.472 0.473 0.471 0.468 0.472 0.473 0.274 0.276 0.276 0.267 0.267 0.266 0.263 0.264 0.264 0.424 0.430 0.434 0.420 0.423 0.424 0.414 0.421 0.424 0.284 0.285 0.286 0.276 0.276 0.275 0.268 0.270 0.270
From the analysis of the results it can be concluded that: • the results obtained with SUE models are significantly better compared to those obtained with SNL models; • the C-Logit model reproduces observed flows better than the Logit model and is very close to the results obtained with the Probit models; in particular it is for Cv=O.2 slightly worse and equivalent for Cv=O.I. • the best values are obtained with Cv equal to 0.1; • the algorithms based on the Logit and C-Logit models give equilibrium flows depending only on
20000
o
Assignment
to tc
I
1\
(15)
c..
120000
80000
(14)
where is the minimumum congested travel time on the network for each OlD pair (r,s). The main results are reported in Tab. 2.
MSE (veic'1Ir)
100000
Number of paths 1 2 2 2 4 4 4 4 4 4 4 4 2 3 2 3 7 10 8 9 8 9 0.276 0.276 0.275 0.279 0.279 0.279 34257 33884 33817 34532 34427 34404
U~
23456789
Fig. 1. Best value ofMSE indicator for multi-criteria choice set 4.2 Comparison of assignment models The second phase of the analysis was focused on the comparison of the performances of different assignment models. At first assignment models based
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•
the stop test because for these models the probability of choice could be expressed in a closed form; the Probit models give estimated SUE flows that depends on the MSA stop test and on the sequence of random numbers used to generate the flows through a Montecarlo extraction. Some results obtained in this work and from other researchers, show in fact that the SNL Probit flows calculated with more than 50 iterations have a Cv around 0.17 (Fig. 2).
paths algorithm with heap management of the waiting nodes results equal to:
time for paths=30 time for minimum trees. It resulted that a set of this dimension allows the generation of a filtered set, for almost all the OlD pairs (99% of the cases), composed of 4 paths for each criterion differing at least 10%. The operation of filtering in terms of computational time is equal to Y2 to that used to generate the paths(Russo and Vitetta, 1997). The total time to generate and filter the choice set results at most equal to:
time for filtered paths =45 times for minimum trees. average cv for all links
Furthermore for Probit assignment with explicit paths enumeration Monte Carlo extraction of link costs have to be generated in each iteration with an increase in computation time of 113 time for minimum trees in each iteration inside the SNL.
O:: tF=~~____~__~~::::::= 0.15
+1 _ _ _ _ _ _ _ _ _ _ _ _- _
0.1
t-- - - - - - - - - - - -
0.05
t------------2
6 10 14 18 22 26 30 34 38 42 46 SO AON itelation with extracted costs inside SNL with equilibrium cost
Fig. 2. Average Cv for all links on the network versus AON iterations with extracted costs inside SNL with equilibrium costs
On the other hand Probit assignment with implicit enumeration can be carried out following two algorithmic approaches. The first approach (internal SNL) at each MSA iteration an SNL is carried out with NSNL iterations. Applying this algorithm to the study case on average 16 external iterations of SNL have to be done and in each iteration 50 internal iterations of AON with the "extracted" costs have to be performed. The total number of trees needed to each origin results equal to: 50· 16 = 800. The time spent to generate the paths results equal to:
time for paths =800 time for minimum trees.
RMSE values obtained for implicit path enumeration models are reported in Tab. 3. The Deterministic User Equilibrium flows can be seen as a particular case of the SUE model with Cv of random terms equal to zero. The results show that DUE flows are outperformed by C-Logit and Probit SUE flows. Implicit Probit SUE is superior for Cv=O.l and comparable for Cv=O.2 to both explicit C-Logit and Probit SUE models.
The other algorithm approach (powell and Sheffy, 1982) for SUE Probit assignment with implicit path enumeration uses a single Monte Carlo extraction of link costs at each MSA iteration. The convergence test requires each N iterations a full SNL Probit assignment with NSNL iterations. It resulted that the most effective N and NSNL values are 10 and 50 respectively. Applying this algorithm to the study case on average 25 iterations (and in some events more than 30) have to be done. The time spent to generate the paths results equal to:
Tab. 3 RMSE for the best choice set for different implicit assignment models
o RMSE
0.280
0.1 0.257
0.2 0.269
time for paths = 123 time for minimum tree (and in some events more than 177 time for minimum tree).
0.3
0.284
4.3 Comparison of computational requirements
The situation changes if we consider an algorithm for Deterministic User Equilibrium (DUE). For the solution of DUE a Frank and Wolfe algorithm is used. In the algorithm on average 15 iterations (in each iteration we have to perform one AON and one linear optimization) have to be done. If we consider that one linear optimization in term of computation time is equivalent to generating the minimum trees for all the network, the total computation time result equal to:
Finally it is possible to compare the computation times needed for implicit and explicit path enumeration models in the study case.
In the explicit enumeration models initially all the paths forming the choice set for each OlD pairs need to be generated and afterward a model of equilibrium assignment needs to be applied Fixed an origin, the necessary computation time in order to' generate the first 30 trees (10 for each criterion), using a k shortest
time for DUE=30 time for minimum trees.
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Thus, for the generation of the paths, the computation time for the implicit enumeration algorithms with Probit choice model is from 3 to 20 times larger than the computation time for the explicit path algorithm. The explicit path enumeration algorithm generates paths once for all which can be stored in the RAM memory of the computer (this consideration is less significant with increasing storage spaces). On the other hand algorithms with implicit enumeration generate paths repeatedly but require smaller memory spaces. We can conclude that the explicit SUE Probit and CLogit models are computationally more efficient than SUE Probit models with implicit enumeration and that the C-Logit outperforms the explicit path enumeration Probit. 5. CONCLUSIONS The preliminary results reported in this paper show the potential of SUE assignment models with explicit path enumeration on medium sized networks. It seems that a limited number of paths generated with few criteria are able to reproduce observed flows satisfactorily. Furthermore the possibility of enumerating paths allows more sophisticated attributes for route choice models and is computationally more efficient, even though it requires longer storage memory. Finally the C-Logit model has performances very close to the Probit, but it has a better accuracy and computational efficiency due to the possibility of expressing route choice probabilities in a closed form.
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Department of Transport and Engineering. University of Napoli. Transportation Science. to be published. Cascetta, E. (1990). Metodi quantitativi per la pianijicazione dei sistemi di trasporto. CEDAM, Padova Italy. Cascetta, E. (1995). A modified Logit model for route choice. Internal Report Department of Transport Engineering, Napoli, Italy. Cascetta, E., A Nuzzolo, V. Velardi (1993).A system of mathematical models for the evaluation of integrated traffic planning and control policies. In Preprint of Workshop on integration problems in urban transportation planning. Capri. Cascetta E., A Nuzzolo, F. Russo and A Vitetta (1996). A modified logit route choice model overcoming path overlapping problems. specification and some calibration results for interurban networks. Proceedings of IS17T conference, Lyon, France, 1996, In "Transportation and Traffic Theory", J. B. Lesort editor, Pergamon. Daganzo, C.F. and Y. Sheffi (1977). On Stochastic Models of Traffic Assignment. In: Transportation Science, vol 11 (3), 253-274. De La Barra, T., B. Perez and J. Anez (1993). Multidimentional path search and assignment. Proceedings of the 21 PTRC Summer Meeting ( PTRCed.). Florian, M, and B. Fox (1976). On the Probabilistic Origin of Dial's Multipath Traffic Assignment Model. In: Transportation Research, Vol. 10(5), 339-341. Laurent, F. (1993). Cost versus Time Equilibrium over a Network. Internal report INRETS, France. Laurent, F. (1994). Modele d'affectation routiere statique. Internal report INRETS, France. Powell W.B.and Y. Sheffi (1982). The convergence of equilibrium algorithms with predetermined step sizes. In: Transportation Science, Vo1.16, 45-55. Russo, F. and A Vitetta (1997). Algoritmi per la ricerca di insiemi di percorsi. Internal report, Department of Engeenering and Mathematics. University ofReggio Calabria, Italy. Russo, F. and A Vitetta (1995). Network and assignment models for the italian national transportation systems. In: Proceeding of WCTR95, Sidney, Australia Sheffi, Y. (1985). Urban transportation networks. Prentice Hall, Englewood Cliff. Van Vuren, T. (1994). The truble with SUE stochastic assignment options in practice. In Proceeding of 22nd European Transport Forum, Seminar Hpp.( PTRC ed.), 41-52.
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