On the estimation of trip matrices in the case of missing and uncertain data

The problem of missing data in the base (calibration) year is of major importance when applying the gravity model in transportation planning, because base year data are used for calibration of the model. In this article, this problem is tackled by solving an optimization problem for the prediction year for values of the entropy level and the proportions of trips in a given stratification of the cells of the trip matrix estimated in the base year. The problem of uncertain data in the prediction year is handled by replacing equality constraints for the marginal totals in the prediction year by interval constraints. Computational results are given for trip data from the Linköping area.     

Ridership estimation for short-range transit planning

The objective of this research was to develop a simple transit ridership estimation model system for short-range planning. The main feature of the model system is that it exploits knowledge of transit link volumes which are obtained readily from on-off counts. Extensive use is made of default values for model parameters, taken directly from the transportation literature. The remaining parameters can be derived easily from generally available land-use and socioeconomic data. Expensive household surveys and time-consuming model calibrations are not required. A sequence of simple trip generation, trip distribution and modal split models generate trip-purpose specific transit trip tables, denoted as “trial” trip tables. These trip tables and observed transit link volumes are used in a linear programming model which serves as a correction mechanism. The gain in accuracy is achieved by using the ridership information contained in the transit link volumes. The corrected trip tables may be used in a pivot-point analysis to estimate changes in ridership and revenue. The results of a test application of the model system indicate that it can generate accurate ridership estimates when reliable transit link volumes are available from on-off counts, and when the trial transit trip tables as derived from the first three component models are reasonably accurate.     

A flexible gravity-opportunities model for trip distribution

This paper develops a flexible gravity-opportunities model for trip distribution in which standard forms of the gravity and opportunities models are obtained as special cases of a general opportunities (GO) model. Hence the question of choice between gravity or opportunities approaches is decided empirically and statistically by restrictions on parameters which control the global functional form of the trip distribution mechanism. The test for the gravity model is shown to be equivalent to a test of the IIA axiom where alternatives are destinations.The notational dichotomy between the two approaches is resolved by employing ordered trip matrices and transformations to permit row and column sum constraints to be applied. These constraints, often interpreted in various ways, are treated as normalisation terms and are therefore not strictly part of the form of the model. Doubly constrained, singly constrained and unconstrained versions of both models are developed throughout.A key step in the integration is the specification of an opportunity function which has as arguments destination-attribute variables such population, income or some other measure of opportunities and generalized cost/impedance-type variables relating origin and destination. This device obviates the mutual exclusiveness ordinarily required of these two sets of variables.The opportunity function is incorporated into a general proportionality factor which is defined by the difference in functions of cumulative opportunities; the latter are subjected to a convex combination of direct and inverse Box-Cox transformations. Different values of the parameters controlling these transformations generate contrasting families of models, notably the exponential and logarithmic intervening opportunities models and the gravity model. All models are shown to be embedded in a transformed triangular region over which likelihood function, response surface or simultaneous confidence interval contours may be plotted.These generalised gravity-opportunity concepts are applied to two well-known models: direct demand multimodal travel demand models, and the estimation of the OD matrix from link volumes. The second case is estimated empirically and here it is shown that a significant improvement is obtained over the gravity model, which is rejected, along with the logarithmic intervening opportunity model, in favour of a more general direct opportunities version.     

Simplified transport models based on traffic counts

Having accepted the need for the development of simpler and less cumbersome transport demand models, the paper concentrates on one possible line for simplification: estimation of trip matrices from link volume counts. Traffic counts are particularly attractive as a data basis for modelling because of their availability, low cost and nondisruptive character. It is first established that in normal conditions it may be possible to find more than one trip matrix which, when loaded onto a network, reproduces the observed link volumes. The paper then identifies three approaches to reduce this underspecification problem and produce a unique trip matrix consistent with the counts. The first approach consists of assuming that trip-making behaviour can be explained by a gravity model whose parameters can be calibrated from the traffic counts. Several forms of this gravity model have been put forward and they are discussed in Section 3. The second approach uses mathematical programming techniques associated to equilibrium assignment problems to estimate a trip matrix in congested areas. This method can also be supplemented by a special distribution model developed for small areas. The third approach relies on entropy and information theory considerations to estimate the most likely trip matrix consistent with the observed flows. A particular feature of this group is that they can include prior, perhaps outdated, information about the matrix.These three approaches are then compared and their likely areas for application identified. Problems for further research are discussed and finally an assessment is made of the possible role of these models vis-a-vis recent developments in transport planning.     

Public transport demand estimation by calibrating a trip distribution-mode choice (TDMC) model from passenger counts: A case study in Bandung,Indonesia

A problem always found in developing countries is the lack of information required for short, medium and long term planning purposes due to money and time constraints. This becomes even more valuable for problems which require ‘quick-response’ treatment. A flexible model approach allows monitoring a long term plan in order to check its short term performance at regular intervals using easily-available data. If found necessary, changes to the plan may be evaluated and eventually implemented. For this reason, the approach is deemed appropriate for long term planning and project evaluation even in the case of rapid changes in land-use, socio-economic and population parameters usually occurs in most of developing countries. A key element of the approach is a system to update the forecasting model (in particular its trip distribution and mode choice elements) using low-cost and/or easily-available information. Traffic counts are particularly attractive to be used in developing countries for planning purposes. The estimation of public transport demand, particularly important for planning purposes, is an expensive and time consuming undertaking. The need for a low-cost method to estimate the public transport demand is therefore obvious. The objective of this paper is the development of methods and techniques for modelling the public transport demand using traffic (passenger) count information and other simple zonal-planning data. We will report on a family of aggregate model combined with a family of mode choice logit models which can be calibrated from traffic (passenger) counts and other low-cost data. The model examined was the Gravity (GR) model combined with the Multi-Nominal-Logit (MNL) model. Non-Linear-Least-Squares (NLLS) estimation method was used to calibrate the parameter of the combined model. The combined TDMC model and the calibration method have been implemented into a micro-computer package capable of dealing with the study area consisting of up to 300 zones, 3000 links and 6000 nodes. The approach has been tested using the 1988 Public Transport Data Survey in Bandung (Indonesia). The model was found to provide a reasonably good fit and the calibrated parameter can then be used for forecasting purposes. General conclusion regarding the advantageous and the applicability of the approach to other environments are given.     

A new random utility model with flexible correlation pattern and closed-form covariance expression: The CoRUM

This paper proposes a new random utility model characterised by a cumulative distribution function (cdf) obtained as a finite mixture of different cdfs. This entails that choice probabilities, covariances and elasticities of this model are also a finite mixture of choice probabilities, covariances and elasticities of the mixing models. As a consequence, by mixing nested logit cdfs, a model is generated with closed-form expressions for choice probabilities, covariances and elasticities and with, potentially, a very flexible correlation pattern. Importantly, the closed-form covariance expression opens up interesting application possibilities in some special choice contexts, like route choice, where prior expectations in terms of the covariance matrix can be formulated.     

Path flow estimator for planning applications in small communities

This paper presents an alternative planning framework to model and forecast network traffic for planning applications in small communities, where limited resources debilitate the development and applications of the conventional four-step travel demand forecasting model. The core idea is to use the Path Flow Estimator (PFE) to estimate current and forecast future traffic demand while taking into account of various field and planning data as modeling constraints. Specifically, two versions of PFE are developed: a base year PFE for estimating the current network traffic conditions using field data and planning data, if available, and a future year PFE for predicting future network traffic conditions using forecast planning data and the estimated base year origin–destination trip table as constraints. In the absence of travel survey data, the proposed method uses similar data (traffic counts and land use data) as a four-step model for model development and calibration. Since the Institute of Transportation Engineers (ITE) trip generation rates and Highway Capacity Manual (HCM) are both utilized in the modeling process, the analysis scope and results are consistent with those of common traffic impact studies and other short-range, localized transportation improvement programs. Solution algorithms are also developed to solve the two PFE models and integrated into a GIS-based software called Visual PFE. For proof of concept, two case studies in northern California are performed to demonstrate how the tool can be used in practice. The first case study is a small community of St. Helena, where the city’s planning department has neither an existing travel demand model nor the budget for developing a full four-step model. The second case study is in the city of Eureka, where there is a four-step model developed for the Humboldt County that can be used for comparison. The results show that the proposed approach is applicable for small communities with limited resources.     

Interval estimation in the calibration of certain trip distribution models

This paper argues for interval, rather than point, estimation when calibrating some variants of the trip distribution “gravity” models. Analytic expressions are derived for the approximate asymptotic covariances of least squares and maximum likelihood estimates of the parameters in the impedance function under a variety of conditions. A comparative numerical example, and an application using migration flows, are also presented.     

An analysis of trip length and land use patterns in the Greater Athens Area

The research described in this paper is an attempt to quantify the impact of a certain distribution of land uses upon trip characteristics — notably trip lengths. The idea is to relate trip lengths classified by mode and purpose to the distance of one trip end from the conurbation centre. The latter is defined as the point which represents a reasonable estimate of the place where the economic, administrative, and cultural life of the urban area is centered.By relating trip lengths to the distance of one trip end from the centre, one could obtain a relation which in effect would be a quantitative expression of the relation between transport and land use. The first application of this idea was in London using the 1966 journey to work data, and it gave quite satisfactory results.The area examined in this research is the Greater Athens Area. The method of analysis is similar to that followed in London so the results of the two studies can be compared. Only work trips are considered for four modes: car, bus, train and all modes (total). It is found that in the case of Athens too, when distance of the workplace from the centre is considered, trip lengths change in smoothly varying ways and a series of mathematical curves can be fitted to the data with an acceptable degree of accuracy. These curves are of the Gamma family having a constant spread factor and varying scale factors for each mode considered. When the distance of the residence end of the trip from the centre is considered, the trip length distributions are not very smooth, a clear mathematical curve cannot be fitted, but again a considerable degree of order can be detected. In addition to the above results a discussion is given on their meaning and the possibilities for future research. In fact the results so far are considered to be the first stage of a more extended research programme which will eventually connect trip length distributions to income and other economic or social parameters in an urban area.The author wishes to express his thanks and appreciation for the comments and constructive criticism made on the various drafts of this paper by M.J.M.     

An equivalent optimization problem for combined multiclass distribution,assignment and modal split which obviates symmetry restrictions

This paper presents a model for combined multiclass trip distribution, trip assignment and modal split. Although this model is based on an equivalent optimization problem, it avoids the symmetry restrictions heretofore always associated with such approaches to multiclass trip assignment. This is accomplished by expressing Wardrop's first principle as a set of nonlinear constraints in standard mathematical programming form. An algorithm is proposed, each iteration of which requires solving a nonlinear program with linear constraints.     

An analysis of elevator operation in moderate height buildings—I: A single elevator

Approximate analytic methods are used to describe how the equilibrium trip time of an elevator depends upon the physical characteristics of the elevator, the passenger demand, and possible strategies of operation. The analysis is directed particularly toward elevators in buildings of moderate height (10–15 floors) at traffic levels such that the elevator is seldom idle or fully loaded. Part I describes the model and the general method for estimating means and variances of trip time if a group of floors is served by only one elevator and passenger arrivals define a homogeneous Poisson process.     

Inferences on trip matrices from observations on link volumes: A Bayesian statistical approach

Previous methods for estimating a trip matrix from traffic volume counts have used the principles of maximum entropy and minimum information. These techniques implicitly give as little weight to prior information on the trip matrix as possible. The new method proposed here is based on Bayesian statistical inference and has several advantages over these earlier approaches. It allows complete flexibility in the degree of belief placed on the prior estimate of the trip matrix and also allows for different degrees of belief in diffeent parts of the prior estimate. Furthermore under certain assumptions the method reduces to a simple updating scheme in which observations on the link flows successively modify the trip matrix. At the end of the scheme confidence intervals are available for the estimates of the trip matrix elements.     

Space and time related determinants of public transport use in trip chains

This research aims at gaining a better understanding about time and space related determinants, which are generally acknowledged to be important factors in the choice of transport mode. The effect of trip chaining is taken into account to improve the insight in the relation between the choice of transport mode and time factors. The data source is the first large scale Belgian mobility survey, carried out in 1998–1999, complemented with a newly created database, containing for each trip a calculated public transport trip. This allows comparing for each trip the actual travel time with the calculated travel time by public transport. Using elasticities and regression techniques the relation between travel time components and public transport use is quantified. On trip level, a clear relation is found between waiting and walking time and public transport use. On trip chain level, travel time variables for the whole trip chain such as the maximum and the range in the travel time ratio provide a significant improvement to the explanatory power of the regression model. The results contain parameters for model input and recommendations to public transport companies on information provision, intermodality and supply.     

A model for integrated analysis of household location and travel choices

We develop a model for integrated analysis of household location and travel choices and investigate it from a theoretical point of view.Each household makes a joint choice of location (zone and house type) and a travel pattern that maximizes utility subject to budget and time constraints. Prices for housing are calculated so that demand equals supply in each submarket. The travel pattern consists of a set of expected trip frequencies to different destinations with different modes. The joint time and budget constraints ensure that time and cost sensitivities are consistent throughout the model. Choosing the entire travel pattern at once, as opposed to treating travel decisions as a series of isolated choices, allows the marginal utilities of trips to depend on which other trips are made.When choosing trip frequencies to destinations, households are assumed to prefer variation to an extent varying with the purpose of the trip. The travel pattern will tend to be more evenly distributed across trip ends the less similar destinations and individual preferences are. These heterogeneities of destinations and individual preferences, respectively, are expressed in terms of a set of parameters to be estimated.     

Determining intermediate origin-destination matrices for the analysis of composite mode trips

We consider in this paper the problem of determining intermediate origin-destination matrices for composite mode trips that involve a trip by private car to a parking facility and the continuation of the trip to the destination either by walking or by a transit mode. The intermediate origin-destination matrices relate to each component of the composite mode trip: a matrix from the trip origins to intermediate destinations which are parking lots and a matrix from the parking lots to the final destinations. The approach that we propose to solve this problem is to modify the entropy based trip distribution models to consider inequality constraints related to parking lot capacities. Such models may be easily calibrated by using well known calibration methods or generalization of these methods and may be easily solved by applying a primal feasible direction method of nonlinear programming.     

A note on the calculation and calibration of doubly constrained trip distribution models

The use of growth factor models for trip distribution has given way in the past to the use of more complex synthetic models. Nevertheless growth factor models are still used, for example in modelling external trips, in small area studies, in input-output analysis, and in category analysis. In this article a particular growth factor model, the Furness, is examined. Its application and functional form are described together with the method of iteration used in its operation. The expected information statistic is described and interpreted and it is shown that the Furness model predicts a trip distribution which, when compared with observed trips, has the minimum expected information subject to origin and destination constraints. An equivalent entropy maximising derivation is described and the two methods compared to show how the Furness iteration can be used in gravity models with specified deterrence functions. A trip distribution model explicitly incorporating information from observed trips, is then derived.It is suggested that if consistency is to be maintained between iteration, calibration, and the derivation of gravity models, then expected information should be used as the calibration statistic to measure goodness of fit. The importance of consistency in this respect is often overlooked.Lastly, the limitations of the models are discussed and it is suggested that it may be better to use the Furness iteration rather than any other, since it is more fully understood. In particular its ease of calculation makes it suitable for use in small models computed by hand.     

Traveler delay costs and value of time with trip chains, flexible activity scheduling and information

The delay costs of traffic disruptions and congestion and the value of travel time reliability are typically evaluated using single trip scheduling models, which treat the trip in isolation of previous and subsequent trips and activities. In practice, however, when activity scheduling to some extent is flexible, the impact of delay on one trip will depend on the actual and predicted travel time on itself as well as other trips, which is important to consider for long-lasting disturbances and when assessing the value of travel information. In this paper we extend the single trip approach into a two trips chain and activity scheduling model. Preferences are represented as marginal activity utility functions that take scheduling flexibility into account. We analytically derive trip timing optimality conditions, the value of travel time and schedule adjustments in response to travel time increases. We show how the single trip models are special cases of the present model and can be generalized to a setting with trip chains and flexible scheduling. We investigate numerically how the delay cost depends on the delay duration and its distribution on different trips during the day, the accuracy of delay prediction and travel information, and the scheduling flexibility of work hours. The extension of the model framework to more complex schedules is discussed.     

Discrete time dynamic estimation model for passenger origin/destination matrices on transit networks

This paper addresses the problem of estimating or updating a passenger trip matrix for transit networks from passenger courts. An extension of a model to take into account time information contained in the passenger counts is developed. Several promising optimization formulations of the resulting model are presented and implementation issues are examined.     

A zonal inference model based on observed smart-card transactions for Santiago de Chile

The collection of origin–destination data for a city is an important but often costly task. This way, there is a need to develop more efficient and inexpensive methods of collecting information about citizens’ travel patterns. In this line, this paper presents a generic methodology that allows to infer the origin and destination zones for an observed trip between two public transport stops (i.e., bus stops or metro stations) using socio-economic, land use, and network information. The proposed zonal inference model follows a disaggregated Logit approach including size variables. The model enables the estimation of a zonal origin–destination matrix for a city, if trip information passively collected by a smart-card payment system is available (in form of a stop-to-stop matrix). The methodology is applied to the Santiago de Chile’s morning peak period, with the purpose of serving as input for a public transport planning computational tool. To estimate the model, information was gathered from different sources and processed into a unified framework; data included a survey conducted at public transport stops, land use information, and a stop-to-stop trip matrix. Additionally, a zonal system with 1176 zones was constructed for the city, including the definition of its access links and associated distances. Our results shows that, ceteris paribus, zones with high numbers of housing units have higher probabilities of being the origin of a morning peak trip. Likewise, health facilities, educational, residential, commercial, and offices centres have significant attraction powers during this period. In this sense, our model manages to capture the expected effects of land use on trip generation and attraction. This study has numerous policy implications, as the information obtained can be used to predict the impacts of changes in the public transport network (such as extending routes, relocating their stops, designing new routes or changing the fare structure). Further research is needed to improve the zonal inference formulation and origin–destination matrix estimation, mainly by including better cost measures, and dealing with survey and data limitations.